1

PROBING NANOSCALE STRUCTURES –

THE SANS TOOLBOX

Boualem Hammouda

National Institute of Standards and Technology

Center for Neutron Research

Gaithersburg, MD 20899-6102

hammouda@nist.gov

http://www.ncnr.nist.gov/staff/hammouda/the_SANS_toolbox.pdf

BEAM BEAM

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METERS

Small-Angle Neutron Scattering

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NANOMETERS

Polymers Complex Fluids Biology Materials Science
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LIST OF CHAPTERS

Preliminaries

Outline

Preface Page

Chapter 1: Introduction 6

Part A. Neutron Sources and Neutron Flux

Chapter 2: The Neutron Probe 10

Chapter 3: Neutron Sources 12

Chapter 4: Cold Neutron Moderators 25

Chapter 5: Neutron Flux on Sample 30

Part B. Neutron Scattering

Chapter 6: Introduction to Neutron Scattering 39

Chapter 7: Neutron Scattering Theory 44

Chapter 8: Elastic and Quasielastic/Inelastic Neutron Scattering 55

Chapter 9: Coherent and Incoherent Neutron Scattering 62

Part C. SANS Technique and Instrumentation

Chapter 10: The SANS Technique 86

Chapter 11: The SANS Instrument 94

Chapter 12: Velocity Selectors and Time-of-Flight Measurements 107

Chapter 13: Neutron Area Detectors 123

Chapter 14: Sample Environments 137

Part D. SANS Resolution and Smearing

Chapter 15: The SANS Instrumental Resolution 148

Chapter 16: Neutron Focusing Lenses 166

Chapter 17: Gravity Correcting Prisms 179

Chapter 18: Neutron Beam Current 188

Chapter 19: The Smearing Effect 192

Part E. SANS Data Corrections and Data Reduction

Chapter 20: SANS Data Corrections 205

Chapter 21: SANS Data Reduction 213

Part F. Simple SANS Data Interpretation

Chapter 22: Standard Plots 221

Chapter 23: Empirical Models 238

Chapter 24: Representative SANS Data 250

Chapter 25: SANS Data from Oriented Samples 256

Part G: SANS Data Modeling

Chapter 26: Radius of Gyration Calculations 265

Chapter 27: Single-Particle Form Factors 277
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Chapter 28: Form Factors for Polymer Systems 291

Chapter 29: Effect of Polydispersity 304

Chapter 30: Scattering from Dilute Polydisperse Systems 311

Chapter 31: Structure Factors for Polymer Systems 316

Chapter 32: Structure Factors for Particulate Systems 330

Chapter 33: Scattering from Fractal Systems 341

Chapter 34: The Multicomponent Random Phase Approximation 349

Part H. SANS from Polymers

Chapter 35: Introduction to Polymers 364

Chapter 36: Polymer Contrast Factors 368

Chapter 37: SANS from Polymer Solutions 374

Chapter 38: SANS from Polymer Blends 384

Chapter 39: SANS from Block Copolymers 394

Chapter 40: SANS from Ternary Polymer Blends 403

Chapter 41: SANS from Polymers Literature Review 411

Part I. SANS from Complex Fluids

Chapter 42: Phase Diagrams for Micellar Systems 429

Chapter 43: SANS from Crystalline Lamellae 445

Chapter 44: SANS from a Pluronic 463

Chapter 45: SANS from Ionic Micelles 478

Chapter 46: SANS from Complex Fluids Literature Review 487

Part J. SANS in Biology

Chapter 47: Elements of Biology 501

Chapter 48: SANS from Phospholipid Bilayers Under Pressure 522

Chapter 49: SANS from DNA 527

Chapter 50: SANS from a Protein Complex 541

Chapter 51: SANS in Biology Literature Review 554

Part K. Other SANS Topics

Chapter 52: SANS from Polymer Blends Under Pressure 565

Chapter 53: SANS Under Shear 581

Chapter 54: Solvation in Mixed Solvents 594

Chapter 55: Clustering in Macromolecular media 602

Chapter 56: SANS from Polymeric Materials 613

Chapter 57: Neutron Scattering with Spin Polarization 624

Chapter 58: Other SANS Topics Literature Review 631

Part L. Even Lower SANS Scales

Chapter 59: SANS Resolution with Slit Geometry 644

Chapter 60: The VSANS Technique 650

Chapter 61: The USANS Instrument 668

Part M. Final Issues
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Chapter 62: Gallery of SANS Data Images 697

Chapter 63: Brief History and Future Prospect 689

Part N. Appendices

Appendix 1: Useful Mathematical Expressions 697

Appendix 2: Elements of Quantum Mechanics 708

Part O. Indexes

List of Symbols and Notation 713
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PREFACE

This book grew out of my twenty-five years as a Small-Angle Neutron Scattering practitioner 

mostly at the National Institute of Standards and Technology. I helped build, maintain, 

improve and schedule the 30 m SANS instruments. I also acted as local contact for a 

multitude of user experiments and strived to keep a healthy research program of my own 

using the SANS technique. 

Many notes were accumulated over the years relating to topics as varied as instrumentation, 

experimental work and theoretical calculations. These topics were stimulated by questions 

from users, by lecturing needs or just by personal curiosity and research interests. 

This “SANS Toolbox” has been put together in a tutorial format with a broad intended 

audience. It is meant to be for a wide variety of users of the SANS technique as well as for 

hardcore practitioners such as instrument scientists. 

This work is dedicated to my colleagues and collaborators, to my dear children and to my 

sweetheart wife Fatima. 

“When you reach the heart of maturity, you find beauty in everything”. 

Quote from Khalil Gibran. 

Boualem Hammouda

Gaithersburg, Maryland

June 2016
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Chapter 1 - INTRODUCTION

Nanometer scale structures include sizes from the near atomic (nanometer) scale to the near 

optical (micrometer) scale. This includes most structures of interest to science for the past 

100 years, i.e., since the advent of non-optical probes such as diffraction methods and 

electron microscopy. Before this period, the optical microscope was the main tool for 

observation. 

Diffraction methods include neutron scattering which has found wide use in the 

characterization of materials. Partial deuteration has made neutron scattering unique. Use of 

deuterated molecules in a non-deuterated environment is comparable to the staining method 

used in electron microscopy and helps enhance the contrast of particular structural features.

Small-angle neutron scattering (SANS) is a well-established characterization method for 

microstructure investigations in various materials. It can probe inhomogeneities in the 

nanometer scale. Since the construction of the first SANS instrument over 40 years ago, this 

technique has experienced a steady growth. SANS instruments are either reactor-based using 

monochromated neutron beams or time-of-flight instruments at pulsed neutron sources. 

SANS has had major impact in many fields of research including polymer science, complex 

fluids, biology, and materials science. This technique has actually become a "routine" 

analytic characterization method used even by non-experts. 

This book is intended to help SANS users acquire (or brush up on) basic knowledge on the 

technique and its applications. Readers need not be experts in the various subjects covered 

here. Basic knowledge in areas like nuclear physics, basic chemistry, statistical mechanics 

and mathematics is of course helpful. The covered topics are organized into broad categories 

(parts) which are divided into chapters. Each chapter contains a number of related topics 

included as sections. Helpful questions (and answers) are included at the end of each chapter. 

The outlines of the various parts and the section titles are color coded; blue has been chosen 

for essential knowledge sections. Readers would benefit by first focusing on these sections. 

After a brief review of basic neutron properties, the various methods of neutron production 

and various neutron sources are introduced first along with discussion of neutron flux. The 

major neutron sources are listed along with their overall characteristics. Production of cold 

neutrons (essential for SANS applications) is discussed along with description of cold 

neutron remoderators. Basic elements of neutron scattering follow. These include advantages 

and disadvantages of the technique, scattering lengths and cross sections, coherent/incoherent 

scattering contributions, and example calculations. This is followed by discussion of 

elastic/inelastic and coherent/incoherent neutron scattering. Elements of quantum mechanics 

are used to derive the scattering cross section. 

The SANS technique is described next. SANS instrumentation is examined in no great detail 

focusing on the major components and pointing out differences between reactor-based and 

spallation source-based instruments. Neutron velocity selectors and area detectors are 

included here along with their calibration and discussion of their performance. SANS 
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resolution and the various elements of instrumental smearing are described next. These 

include contributions from the instrument focusing geometry, wavelength spread and 

detector resolution as well as the effect of gravity on neutron trajectories. Instrumental 

resolution is also discussed when refractive optics (neutron lenses or prisms) are included. 

Description of the various elements of SANS data correction and data reduction are included 

next. The main SANS data interpretation methods include standard plots, the use of empirical 

models and nonlinear least-squares fits to realistic models. Representative SANS data are 

presented. Elements of SANS data modeling include calculations of the radius of gyration, of 

the single-particle form factor and of inter-particle structure factors. The effect of 

polydispersity is also discussed. Since "most SANS spectra look alike", SANS is a heavily 

model-dependent method. The major theories used to interpret SANS data are discussed 

including the Random Phase Approximation (RPA) for polymer systems and the OrnsteinZernike (OZ) equation for scattering from particles. 

The major SANS research topics are covered in turn in a series of chapters. These various 

“parts” include: Polymers, Complex Fluids, Biology, and Other Topics that includes 

Materials Science. In each chapter, typical topics borrowed from the research efforts of this 

author are described at the tutorial level. The part on “SANS from Polymers” includes 

polymer solutions, polymer blends and copolymers. The Random Phase Approximation 

approach is described in detail and applied to realistic homogeneous polymer mixtures. The 

thermodynamics of phase separation are described for multi-component homogeneous 

polymer mixtures. The part on “SANS from Complex Fluids” includes a discussion of the 

phase diagram for micellar systems and contains chapters on ionic and nonionic “selfassembling systems”. The main scattering features include single-particle and inter-particle 

contributions. Material balance equations help in the understanding of some details of the 

probed structures. The part on “SANS in Biology” introduces elements of biology then 

covers representative basic topics such as a phospholipid membranes, the helix-to-coil 

transition in DNA and the structure of a protein complex. 

The “Other SANS Topics” part is covered next. These include the effect of pressure or shear 

on nanoscale structures, solvation in mixed solvents, and molecular orientation of polymeric 

materials. SANS measurements involving in-situ pressure or in-situ shear have been the 

focus of research for many years. The effects of pressure on phase separation and miscibility 

are discussed. In-situ shear allows investigations of the rheology and structure 

simultaneously. 

Chapters covering more topics borrowed from research in this author was involved have been 

included. 

Two other small-angle neutron scattering techniques are discussed in no-great detail in the 

part on “Even Lower SANS Scales”. These are the Ultra small-angle (USANS) range 

probing structures as large as 20 microns and the merging VSANS technique (V is for very 

small-angle) which bridges the two probing ranges. 
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A gallery of interesting SANS data images is included. These images have been collected by 

this author over several years. They are included here in order to show the full richness of the 

SANS technique and for their esthetic value. Some brief concluding topics are covered along 

with two appendices; one on “Useful Mathematical Expressions” and the other on “Elements 

of Quantum Mechanics”. These appendices gather material used throughout.

This document is meant to be used in a pdf (not print) format so that it could be searched for 

subject or author keywords. For this reason, no indexes have been included at the end of the 

book. It is also meant to be used in an environmentally friendly way which helps minimize 

printing. 
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Part A – NEUTRON SOURCES AND NEUTRON FLUX

Chapter 2. The Neutron Probe

2.1 What are Neutrons?

2.2 Why Use Neutrons?

Chapter 3. Neutron Sources

3.1 Introduction

3.2 Nuclear Fission Reactions

3.3 Nuclear Reactors

3.4 The NIST Thermal Neutron Instruments

3.5 The NIST Guide Hall

3.6 The HFIR Guide Hall

3.7 Spallation Sources

3.8 Some Other Neutron Sources

References

Questions

Answers

Chapter 4. Cold Neutron Moderators

4.1 Cold Neutron Source

4.2 Cold Neutron Spectrum

References

Questions

Answers

Chapter 5. Neutron Flux on Sample

5.1 The Cold Neutron Source Spectrum

5.2 Neutron Flux on Sample

5.3 Case of Specific Configurations

5.4 Measured Flux on Sample

5.5 Neutron Beam Monitor Count Rate

References

Questions

Answers
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Chapter 2 - THE NEUTRON PROBE

1. WHAT ARE NEUTRONS?

The neutron was discovered by Chadwick in 1932. It has zero charge, a mass of 1.0087 

atomic mass unit, a spin of 1/2 and a magnetic moment of -1.9132 nuclear magnetons. It has 

a lifetime of 894 seconds and decays into a proton, an electron and an antineutrino. Its 

interactions with matter are confined to the short-range nuclear and magnetic interactions. 

Since its interaction probability is small, the neutron usually penetrates well through matter 

making it a unique probe for investigating bulk condensed matter. Since the neutron can be 

reflected by some surfaces when incident at glancing angles, it can also be used as a surface 

probe. Neutrons are scattered by nuclei in samples or by the magnetic moments associated 

with unpaired electron spins (dipoles) in magnetic samples. The nuclear scattering potential 

is short range so that most neutron scattering can be described by "s wave" scattering (zero 

orbital angular momentum) and the scattering cross section can be described by the first Born 

approximation. Higher order term in the Born expansion series are required for neutron 

reflection from surfaces. Reflection involves the refraction (not diffraction) limit. 

Some useful properties follow:

Mass: m = 1.675*10-24 gm

Magnetic Moment: μn = 6.031*10-12 eV/gauss

Energy: 

[Å ]

81.787 E[meV]

2 2





Wavelength:  [Å] = 3955/v [m/sec] 

Velocity: v = 1 m/msec (at =4 Å)

Useful relationship: 

mv  h . 

Thermal neutrons correspond to 25 meV energies and 1.8 Å wavelength. 

2. WHY USE NEUTRONS?

Neutrons are both a bulk and a surface probe for investigating both structures and dynamics. 

Some of the advantages of neutrons as a probe for condensed matter follow. 

-- Neutrons interact through short-range nuclear interactions. They are very penetrating and 

do not heat up (i.e., destroy) samples. Neutrons are good probes for investigating structures 

in condensed matter. 

-- Neutron wavelengths are comparable to atomic sizes and inter-distance spacing. Neutron 

energies are comparable to normal mode energies in materials (for example phonons, 

diffusive modes). Neutrons are good probes to investigate the dynamics of solid state and 

liquid materials.
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-- Neutrons interactions with hydrogen and deuterium are widely different making the 

deuterium labeling method an advantage. 

Someone once stated that “neutrons never lie!”. Trust what they’re telling you. 

QUESTIONS

1. The neutron decays into what particles? How about the proton? Does it decay?

2. Why are neutrons a good probe to investigate condensed matter?

3. Can neutrons get reflected from surfaces at large angles like light does?

4. Define the electronvolt (eV) in terms of the SI energy unit, the joule (J). 

ANSWERS

1. The neutron decays into an electron, a proton and an anti-neutrino. The proton is stable. Its 

decay has not been observed. 

2. Neutrons are a good probe to investigate condensed matter because it is very penetrating 

(due to its charge neutrality) and to its just-right typical wavelengths and kinetic energies. 

3. Neutrons can be reflected from surfaces only at low glancing angles. They cannot be 

reflected at large angles from surfaces. 

4. The electrostatic energy is the product of the charge by the applied voltage. The 

electronvolt is the energy of 1 electron in a potential of 1 volt. The charge of 1 electron is 

1.602*10-19 coulomb. Therefore, 1eV = 1.602*10-19 J. 
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Chapter 3 - NEUTRON SOURCES

1. INTRODUCTION

Since the early days of neutron scattering, there has been an insatiable demand for higher 

neutron fluxes. Neutron sources are based on various processes that liberate excess neutrons 

in neutron rich nuclei such as Be, W, U, Ta or Pb. Presently, the highest fluxes available are 

around a few *1015 n/cm2sec. Even though various neutron sources exist, only a few are 

actually useful for scattering purposes. These are:

-- continuous reactors

-- spallation sources

-- some other neutron sources.

Only minor improvements in flux increase of continuous reactors are expected because of the 

saturation of the technology (i.e., limit of heat removal rate and operating safety 

considerations). Pulsed sources are expected to go to higher fluxes (non-continuous operation 

allows for a better heat removal rate). 

Continuous reactors operate in a continuous neutron generation mode whereas spallation 

sources function in a pulsed (or time-of-flight) mode. 

monochromation

source

collimation scattering detection

time

intensity 

at source

intensity 

at detector

time

Continuous Reactors

Measure some of the neutrons all of the time

single wavelength
13

Figure 1: The two main types of neutron sources: continuous reactors and pulsed sources. 

Schematic representations of SANS instruments are shown. 

2. NUCLEAR FISSION REACTIONS

Some heavy nuclides undergo fission reaction into lighter ones (called fission products) upon 

absorption of a neutron (Duderstadt-Hamilton, 1974; Lamarsh, 1977). Known fissile nuclides 

are U-233, U-235, Pu-239 and Pu-241, but the most used ones are U-235 and Pu-239. Each 

fission event releases huge energies (200 MeV) in the form of kinetic energy of the fission 

fragments, gamma rays and several fast neutrons. Fission fragments are heavy and remain 

inside the fuel elements therefore producing the major source of heat while energetic 

gammas and fast neutrons penetrate most everything and are carefully shielded against. 

Gamma rays and fast neutrons are a nuisance to neutron scatterers and are not allowed to 

reach the detectors as much as possible. After being slowed down by the moderator material 

(usually light or heavy water) neutrons are used to sustain the fission reaction as well as in 

beam tubes for low energy (thermal and cold) neutron scattering. 

chopper

source

collimation

scattering

detection

time

intensity 

at source

intensity 

at detector

time

Pulsed Sources

t

Measure all of the neutrons some of the time

wavelength range

time-of-flight
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Figure 2: Typical fission chain reaction. 

3. NUCLEAR REACTORS

Nuclear reactors are based on the fission reaction of U-235 (mainly) to yield 2-3 

neutrons/fission at 2 MeV kinetic energies. Moderators (D2O, H2O) are used to slow down 

the neutrons to thermal (0.025 eV) energies. Reflectors (D2O, Be, graphite) are used to 

maintain the core critical by reflecting neutrons back into the core. Electrical power 

producing reactors use wide core sizes and low fuel enrichment (2-5 % U-235), while 

research reactors use compact cores and highly enriched fuel (over 90 % U-235) in order to 

achieve high neutron fluxes. Regulatory agencies encourage the use of intermediate 

enrichment (20-50 %) fuel in order to avoid proliferation of weapon-grade material. Note that 

the relative abundance of U-235 in natural uranium is 0.7 %. 

Nuclear research reactors have benefited from technological advances in power producing 

reactors as well as in nuclear submarines (compact cores operating with highly enriched fuel 

and foolproof safety control systems). The most popular of the present generation of reactors, 

the pressurized water reactor (PWR), operates at high pressure (70 to 150 bars) in order to 

achieve high operating temperatures while maintaining water in its liquid phase. 

Neutrons that are produced by fission (2 MeV) can either slow down to epithermal then 

thermal energies, be absorbed by radiative capture, or leak out of the system. The slowing 

down process is maintained through collisions with low Z material (mostly water is used both 

as moderator and coolant) while neutron leakage is minimized by surrounding the core by a 

reflector (also low Z material) blanket. Most of the fission neutrons appear instantaneously 

Fission Chain Reaction

fission 

fragment

incident

neutron

2 to 3 fission

neutrons

radiative 

capture

gamma

neutrons used for 

neutron scattering

fissile nucleus

(U-235, Pu-239)
15

(within 10-14 sec of the fission event); these are called prompt neutrons. However, less than 1 

% of the neutrons appear with an appreciable delay time from the subsequent decay of 

radioactive fission products. Although the delayed neutrons are a very small fraction of the 

neutron inventory, these are vital to the operation of nuclear reactors and to the effective 

control of the nuclear chain reaction by "slowing" the transient kinetics. Without them, a 

nuclear reactor would respond so quickly that it could not be controlled.

A short list of research reactors in the USA used for neutron scattering follows: 

-- HFIR-Oak Ridge National Laboratory (100 MW), a horizontal cold source has recently 

been installed. 

-- NIST-The National Institute of Standards and Technology (20 MW), contains third 

generation cold neutron source. 

-- MURR-University of Missouri Research Reactor (10 MW), does not contain a cold 

neutron source. 

These reactors were built during the1960's but have undergone various upgrades. 

There is one major research reactor in Canada:

-- CRNL-Chalk River, Canada (135 MW).

A short list of research reactors in Europe follows: 

-- ILL-Grenoble, France (57 MW), 

-- NERF-Petten, Netherland (45 MW), 

-- FRM-II Munich, Germany (20 MW),

-- KFKI-Budapest, Hungary (15 MW), 

-- LLB-Saclay, France (14 MW), 

-- HMI-Berlin, Germany (10 MW), 

-- Riso-Roskilde, Denmark (10 MW), 

-- VVR-M Leningrad, Russia (10 MW). 

A short list of research reactors in Asia follows:

-- DRHUVA-Bombay, India (100 MW), 

-- CIAE-Beijing, China (60 MW), 

-- NLHEP-Tsukuba, Japan (50 MW), 

-- Bhabha ARC-Bombay, India (40 MW), 

-- HFANAR, KAERI, Hanaro, Korea (30 MW) 

-- JRR3-Tokai Mura, Japan (20 MW), 

-- CMRR-Mianyang, China (20 MW), 

One reactor exists in Oceania. It is the Bragg Institute, ANSTO, Australia (20 MW). 
16

Most of these facilities either have or are planning to add a cold source in order to enhance 

their cold neutron capability and therefore allow effective use of SANS instruments.

4. THE NIST THERMAL NEUTRON INSTRUMENTS

The NIST Center for Neutron Research (CNR) facility has a split-core geometry whereby 

thermal neutron beam tubes do not look at the fuel elements directly. This helps minimize 

epithermal neutrons and gamma radiation in the beams. There is a host of thermal neutron 

instruments located in the confinement building. These comprise triple axis instruments for 

inelastic neutron scattering, a powder diffractometer, a single crystal instrument also used for 

texture studies, a neutron radiography station, and a Bonse-Hart USANS instrument. 

Location of the cold neutron source is optimized. It is located at the peak flux position within 

the reflector region. A set of neutron guides transport cold neutrons to a guide hall.

Figure 3: Schematics of the NIST confinement building showing the thermal neutron 

scattering instruments and the cold neutron source along with the beginning of the cold 

neutron guides leading to the current guide hall. The USANS instrument is located on a 

thermal neutron beam tube. 

NIST Thermal Instruments

USANS

cold 

neutron 

source

NG1

NG2 NG3

NG4

NG5

NG6

NG7

0 5 m
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5. THE NIST GUIDE HALLS

The NIST CNR old and new guide halls contain a set of seven plus three guides looking at 

the cold source. Cold neutron instruments include three SANS instruments, three 

reflectometers, a time-of-flight instrument, a cold triple axis, a backscattering spectrometer, a 

neutron spin-echo spectrometer and other fundamental physics stations (interferometry, 

measurement of the neutron half-life, etc). 

The guides in the old guide hall are straight (with no curvature) and looking at the cold 

source directly. Guide dimensions are 12 cm*5 cm for some and 15 cm*6 cm for others. The 

guides’ inner surfaces are coated with either natural Ni or Ni-58 on the sides and with either 

Ni-58 or supermirror coating on the top and bottom. The critical angle for natural Ni is 0.1 

o

/Å, that for Ni-58 is 0.115 o/Å and that for supermirror coating is 0.3 o/Å. This critical angle 

for total reflection increases with neutron wavelength as c= c where c = 

b / 

is given 

in terms of the atomic number density  and scattering length b of the reflecting material. 

Neutron guides are anchored onto a thick concrete base in order to decouple them from the 

rest of the guide hall. Neutron guides are encased in jackets that are evacuated or filled with 

helium. Neutron losses in neutron guides are estimated to be around 1 % per meter. 

Filters are used to remove epithermal neutrons and gamma radiation from the neutron guides. 

Crystal filters include beryllium for neutrons and bismuth for gamma rays. They are kept at 

liquid nitrogen temperature. Optical filters are also used to steer the neutron beam out of the 

direct line-of-sight from the cold source and with minimum losses. Optical filters are 

characterized by high transmission gains over crystal filters for long wavelength neutrons. 

The old guide hall contains seven straight guides and the new guide hall contains 3 curved 

guides that are split (top and bottom parts) in order to accommodate more end-guide 

positions. 

Note that curved guides are used to get out of the direct line of sight from the neutron source 

and therefore completely avoid the use of filters. Curved guides however transmit neutrons 

above a cutoff wavelength that depends on the guide curvature and width. A curved guide of 

width W and radius of curvature R has a characteristic angle c = 

2W/R

. This is the 

minimum angle that the guide subtends (in the horizontal plane) in order to get out of the 

direct line-of-sight. This curved guide has a cutoff wavelength c = c/c below which no 

neutrons are transmitted. 
18

Figure 4: Schematics of the NIST old and new guide halls. Note the two 30 m SANS 

instruments on the NGB30 guide (new guide hall) and NG7 guide (old guide hall), the 10 m

SANS instrument and the 40 m VSANS instrument. 
19
20

Figure 5: Photograph of the NIST CNR old and new guide halls. The confinement building 

wall is at the rear end of the picture. The red color scattering vessel of the NG7 30m SANS 

instrument is seen to the left in the old guide hall. The purple color instrument in the new 

guide hall is the NGB30 SANS instrument. 

6. THE HFIR GUIDE HALL

The High Flux Isotope Reactor (HFIR) located at Oak Ridge National Lab has built two 

SANS instruments and a horizontal cold source. These are 35 m and 30 m long respectively 

and both use 1 m*1 m size area detectors. 

Figure 6: Schematic representation of the HFIR guide hall with the two 30 m SANS 

instruments. The CG2 SANS instrument is slightly longer. 

7. SPALLATION SOURCES

Beams of high kinetic energy (typically 70 MeV) hydrogen ions are produced (by linear 

accelerator) and injected into a synchrotron ring to reach much higher energies (500-800 

MeV) and then steered to hit a high Z (neutron rich) target (W-183 or U-238) and produce 

about 10-30 neutrons/proton with energies about 1 MeV. These neutrons are then 

moderated, reflected, contained, etc., as in the case of nuclear reactor. Most spallation 

sources operate in a pulsed mode. The spallation process produces relatively few gamma rays 

CG2 SANS

CG3 BioSANS

The HFIR Guide Hall
21

but the spectrum is rich in high energy neutrons. Typical fast neutron fluxes are 1015-1016

n/sec with a 50 MeV energy deposition/neutron produced. Booster targets (enriched in U235) give even higher neutron fluxes.

Figure 7: Spallation nuclear reaction.

Spallation sources in the USA:

-- WNR/PSR LANSCE (Los Alamos): 800 MeV protons, W target, 100 μA (12 Hz), pulse 

width = 0.27 μsec, flux = 1.5*10

16 n/sec, operating since 1986.

-- SNS (Oak Ridge National Lab): 1.3 GeV, Hg target, 2 mA (60 Hz), pulse width = 0.945 

μsec, operation started in 2006. 

Spallation sources elsewhere in the world:

-- ISIS (Rutherford, UK): 800 MeV protons, U target, 200 μA (50 Hz), pulse width = 0.27 

μsec, flux = 4*1016 n/sec, operating since 1984.

-- KENS (Tsukuba, Japan): 500 MeV protons, U target, 100 μA (12 Hz), pulse width = 0.07 

μsec, flux = 3*1014 n/sec, operating since 1980.

-- SINQ, Paul Scherrer Institut (PSI), Switzerland, 590 MeV protons, Pb target, 1.8 mA, flux 

= 5*1014 n/sec, operating since 2002. 

-- ESS, European Spallation Source in Lund, Sweden, 2 GeV protons, is under construction. 

-- CSNS, Chinese Spallation Neutron Source in Dongguan is under construction. 

Spallation Nuclear Reaction

incident

H ion

10 to 30 

neutrons 

emitted

neutrons used for 

neutron scattering

high Z nucleus

(W-183, U-238)
22

Figure 8: Schematic of the LANSCE (LANL) instruments hall. Note that the SANS (LQD) 

instrument on the right hand side is no longer operated.

8. SOME OTHER NEUTRON SOURCES

“Pulsed reactors” include a moving element of fuel (or reflector material) which moves 

periodically causing regular variation of the reactivity. A fast rising burst of neutrons occurs 

when the reactivity exceeds prompt critical. One such reactor exists at:

-- IBR-II (Dubna, Russia), with mean power of 2 MW, pulse width of 50 μsec, repetition rate 

of 5 Hz. Neutron in pulse fluxes are of order of 5*1015 n/cm2sec.

Stripping (p,n) nuclear reactions can be used to produce neutrons. The following reaction:

p + 9Be  n + 9B

LQD

Los Alamos Neutron Scattering 

Center
23

is used to produce pulsed neutrons at the following facility:

-- The Low Energy Neutron Source at the University of Indiana with pulse width between 5 

sec and 1 msec. 

REFERENCES

J.J. Duderstadt and L.J. Hamilton, "Nuclear Reactor Analysis", J. Wiley and Sons, Inc., 

(1976).

L.R. Lamarsh, "Introduction to Nuclear Engineering", Addison Wesley Pub. Co., (1977).

International Atomic Energy Commission web site (http://www.iaea.org). 

QUESTIONS

1. When was the first research reactor built?

2. Name a few applications of nuclear research reactors besides neutron scattering.

3. Why can’t neutron sources be designed for much higher fluxes? 

4. What is the origin of delayed neutrons?

5. Are there nuclear reactors that use non-enriched uranium?

6. Name the research reactor and the spallation source closest to your home institution.

7. Instruments at pulsed sources use a range of wavelengths whereas reactor-based 

instruments use single wavelength. How could the same scattering information be obtained 

from these two different types of instruments?

8. Why are most SANS instruments installed in neutron guide halls?

9. What is a dosimeter?

ANSWERS

1. The first nuclear reaction was performed by Enrico Fermi and his team in a sports facility 

close to the University of Chicago stadium in 1942. This is the first nuclear reactor built in 

the US called CP1 for Chicago Pile 1. A series of reactors were built at Oak Ridge, Los 

Alamos, Brookhaven, and Argonne National Labs and were referred to as CP2 to CP5. The 

first university-based research reactor was built in 1955 at Penn State University. The second 

one was built in 1957 at the University of Michigan.

2. There are many practical applications of nuclear research reactors besides neutron 

scattering. A few are mentioned here: neutron activation analysis, radioisotopes production, 

neutron radiography, transmutation doping of silicon, coloration of gemstones, etc. 

3. Neutron sources cannot deliver much higher fluxes because they are at their limit of heat 

removal rate from the core (cooling rate). 

4. Delayed neutrons are emitted from the decay of fission fragments. Their half-lives range 

from seconds to minutes. 
24

5. The Canadian CANDU design uses U-238 (natural uranium). 

6. There are two main research reactors in the US, one at the NIST Center for Neutron 

Research and one at the Oak Ridge High Flux Reactor. 

7. Reactor-based neutron scattering instruments use some of the neutrons all of the time 

while spallation source-based instruments (time-of-flight) use all of the neutrons some of the 

time. They both measure scattered neutrons intensity with increasing scattering variable Q. 

8. SANS instruments are located mostly in guide halls because they are long (30 m). 

Moreover guide halls are characterized by low neutron and gamma background. 

9. A dosimeter is a special type of detector to monitor radiation levels and doses. It is worn 

by experimenters. 
25

Chapter 4 - COLD NEUTRON MODERATORS

1. COLD NEUTRON SOURCE

"Cold" (slow) neutrons are often needed for better spatial resolution in scattering applications 

(long wavelength scattering). Atoms with low Z (such as H or D) are good moderators 

making them ideal as cold source material. Cold neutrons are generated in a neutron 

remoderator also called "cold source" using either hydrogen or deuterium in the liquid form, 

supercooled gas form, or solid form (methane or ice). The Maxwellian neutron spectral 

distribution (peaking at 1.8 Å for thermal neutrons) is shifted to lower energies by neutron 

slowing down (through inelastic scattering) processes. The mean free path (average distance 

between collisions) of neutrons in hydrogen (0.43 cm) is smaller than in deuterium (2.52 

cm). 

Liquid cold sources (hydrogen or deuterium) operate at low temperature (between 20 K and 

30 K) and 2 bar pressure (Russell-West, 1990). Note that hydrogens boils at 21 K. Vacuum 

and helium jackets isolate the remoderating liquid from the surrounding. Supercritical gas 

cold sources (hydrogen or deuterium) operate at 40 K and 15 bars of pressure (one phase 

system); thicker walls are necessary for the containment of the higher gas pressure. Solid 

methane at 50 K and solid ice at 35 K have been used as cold source material. Radiation 

damage in solid state cold sources produces stored (so called "Wigner") energy due to 

ionization. In order to avoid sudden release of this energy (explosion!), a recombination of 

radiolysis products is induced in the cold source material by warming it up on a regular basis 

(once every couple of days). 

Use of a cold source yields high gains (one to two orders of magnitude) at high wavelengths. 

The Cold Neutron Source

NG0

NG1

NG2

NG3

NG4

NG5

NG6

NG7

core
26

Figure 1: The NIST liquid hydrogen cold source and neutron guide system.

Figure 2: Schematic view of the liquid hydrogen cold source with optimized re-entrant 

geometry. 

2. COLD NEUTRON SPECTRUM

Neutrons are produced by fission with energies around 2 MeV, then they slow down to form 

a Maxwellian spectrum distribution which is peaked around the moderator temperature kBT 

(in energy units).

The neutron flux 

(E)

is the number of neutrons emitted in all directions per second and per 

unit energy at neutron kinetic energy E. 

E exp(-E/k T)

(k T)

(E) 2 B

B

0   . (1)

Its integral is the neutron current (total number of neutrons produced by the cold source per 

second):





  

0

dE (E) . (2)

Liquid hydrogen 

moderator

Hydrogen vapor

Heavy water

coolant
27

Neutron conservation is expressed as 

| (E)dE || ()d |

. The neutron kinetic energy E can 

be expressed in terms of the wavelength  as 

2

2

1

2m

h

E





















. Using 

3

2

( 2)

2m

h

d

dE

























, ()

can be expressed as:



























    2

2

T

5

4

T

0

exp

2

( ) . (3)

The variable 

2mk T

h

B

2

2

T

has been defined for simplicity in notation and h is Planck's 

constant. 

()

is the neutron current per unit wavelength. Its units are n/s.Å. The angular 

spectral neutron distribution simply referred to as neutron flux (or current density) is given 

by 

2

L0 4

( )



 

at a distance L0 from the cold source. Its units are n/cm2.s.sr.Å. Note that the 

steradian (symbol sr) is the unit of solid angle. 

For high neutron wavelength  ()decreases as 1/

5

. A cold source effectively shifts the 

Maxwellian peak to higher wavelengths therefore increasing the population of cold neutrons 

and yielding better small-angle neutron scattering resolution. For elastic scattering, this 

means the ability to resolve larger structures (close to micron size). 

The spectral neutron distribution of the NIST Center for Neutron Research cold source is 

plotted (Williams-Rowe, 2002) for the hydrogen cold source supplying the two guide halls 

and the deuterium cold source (so-called peewee) supplying one guide tube. Note that 

hydrogen has a stronger “1/v”; this is what makes the deuterium cold source spectral neutron 

distribution higher than that for hydrogen. 
28

109

1010

1011

1012

1013

0 5 10 15 20

NCNR Hydrogen Cold Sources

Proposed NCNR Cold Source

Current NCNR Cold Source

Spectral Neutron Distributions 

(n/cm

2.s.ster.Å)

Neutron Wavelength (Å)

Figure 3: Spectral neutron distributions for the current and the proposed NIST Center for 

Neutron Research cold sources. The current one supplies neutrons to the old guide hall and to 

the new guide hall addition. The proposed smaller and brighter cold source (referred as 

“peewee”) will supply cold neutrons to one instrument inside the confinement building. 

REFERENCES

G.J. Russell, and C.D. West, “International Workshop on Cold Neutron Sources”, Los 

Alamos National Lab, March 5-8 (1990). 

R. E. Williams and J. M. Rowe, “Developments in Neutron Beam Devices and an Advanced 

Cold Source for the NIST Research Reactor”, Physica B 311, 117-122 (2002).

QUESTIONS

1. What are the main types of cold neutron sources?

2. What is the primary safety issue associated with solid cold sources?

3. What is the boiling temperature of hydrogen?

4. What is the spectral distribution of cold neutrons?
29

5. Why are cold neutrons necessary for the SANS technique?

6. What is the definition of the steradian?

ANSWERS

1. Cold sources are of the liquid, gas or solid types. Most of them use eith liquid hydrogen or 

deuterium to slow down neutrons to cold energies. 

2. Solid state cold sources (either solid methane or solid heavy ice) store Wigner energy that 

needs to be released by annealing the cold source. If not annealed, the solid cold source could 

explode. 

3. Liquid hydrogen boils at 21 K.

4. Cold neutrons follow a Maxwellian spectral distribution with a tail varying like 1/

5 where 

 is the neutron wavelength. 

5. Cold neutrons are characterized by long wavelengths  which yield lower scattering 

variables 











 







2

sin 4Q

(at fixed scattering angle ). Lower Q values correspond to longer 

d-spacing in the probed structures. 

6. The steradian (symbol sr) is the unit of solid angle. The sr is equal to the square of the 

radian (symbol rad) which is an angular unit. Note that an angle of 3.14159 rad corresponds 

to180 o. 
30

Chapter 5 - NEUTRON FLUX ON SAMPLE

Flux on sample is an important factor in characterizing the performance of a neutron 

scattering instrument. It depends on many factors as discussed here. 

1. THE COLD NEUTRON SOURCE SPECTRUM

The liquid hydrogen neutron cold source is characterized by the following angular spectrum 

distribution (neutrons/cm2.s.Å.ster):



































 

2

2

T

5

4

T

2

0

0

2

0

exp

2

4 L 4 L

( )

(1)

It is also referred to as the “Maxwellian” distribution.  is the neutron wavelength and T is a 

cold source constant defined as

T h / 2mk BT . T can be expressed as:

T = A /

Te

. (2)

The constant 

A  30.9Å K

, Te is the cold source effective temperature Te = 32 K. Note that 

the cold source real temperature is the condensation temperature of hydrogen (around 20 K). 

Therefore T = 5.5 Å is a good estimate in our case. The cold neutron wavelength distribution 

is therefore peaked around 3.5 Å and falls off with a 1/

5

tail. The normalization factor 

0

is 

determined through flux measurements. 

2. NEUTRON FLUX ON SAMPLE

The neutron current on sample (neutrons/s) can be estimated for a typical SANS instrument 

configuration as:

2

1

2

2

0

1

2

1 2

L

A

L

A

16

( )

4 4

( ) 



 











  



 is the wavelength spread,  is the solid angle subtending the source aperture defined by 

the area A1 and  is the solid angle subtending the sample aperture defined by the area A2, 

L0 and L1 are the cold source-to-source aperture and source aperture-to-sample aperture 

distances respectively. 
31

Figure 1: Typical pre-sample SANS collimation geometry. This figure is not to scale. 

Vertical scale is of order of centimeters while horizontal scale is of order of meters.

This quantity can be expressed as:

2

1

2

2

0

1

2

2

T

5

4

T

2

1 2 0

L

A

L

A

exp

4 4 8

( ) 









































  

(4)

with T = 5.5 Å. In order to make the neutron flux expression match the measured flux at the 

NG3 SANS instrument the following factor is chosen:

1.65*10 n / cm .s

8 L

12 2

2

0

2

0







. (5)

The estimated flux (or current density) on sample (n/cm2.s) is given by: 







































































 

 

  

2

1

1

2

2

T

4

4

T

2

0

2

1 2 0

2 L

A

exp

A 4 4 8 L

( )

( )

(6)



















































  

2

1

1

4 2

15

L

30.25 A

exp

1.507 *10 ( )

circular

source 

aperture circular

sample

aperture

A1 A2

L0 L1

1

2

cold 

source 
32

Consider a typical neutron wavelength and wavelength spread:

Neutron wavelength:  = 6 Å. 

Wavelength spread:  = 0.15. 

So that:

















 

2

1

10 1

L

A

(6 Å) 7.53*10

n/cm2.s (7)

This expression is used in the following section. 

3. CASE OF SPECIFIC CONFIGURATIONS

Consider two instrument configurations both using:

Neutron wavelength:  = 6 Å. 

Wavelength spread:  = 0.15.

The first configuration corresponds to high flux on sample:

Source aperture radius: R1 = 2.5 cm.

Area of source aperture: A1 =  2.52 = 19.63 cm2. 

Source-to-sample distance: L1 = 3.82 m. 

So that 

7

(6Å) 1.01*10

n/cm2.s for the high flux configuration.

The second configuration corresponds to low flux on sample:

Source aperture radius: R1 = 1.9 cm.

Area of source aperture: A1 =  1.92 = 11.34 cm2. 

Source-to-sample distance: L1 = 16.22 m. 

So that 

5

(6Å)  3.24*10

n/cm2.s for the low flux configuration.

4. MEASURED FLUX ON SAMPLE

The two previously considered cases correspond to two specific configurations on the NG3 

30 m-SANS instrument at NIST. Flux on sample measurements were made for these two 

configurations described above and for a range of wavelengths. These results are plotted 

here. 
33

1000

104

105

106

107

108

109

4 6 8 10 12 14 16 18 20

NG3 Flux 

R1 = 2.5 cm, L

1

 = 382 cm (measured) 

R1 = 2.5 cm, L

1

 = 382 cm (estimated) 

R1 = 1.9 cm, L

1

 = 1622 cm (measured) 

R1 = 1.9 cm, L

1

 = 1622 cm (estimated) 

Neutron Flux on Sample (n/cm

2.s)

Neutron Wavelength (Å)

Figure 2: Measured neutron flux on sample with varying wavelength for the high flux 

configuration (R1 = 2.5 cm, L1 = 3.82 m) and the low flux configuration (R1 = 1.9 cm, L1 = 

16.22 m). Estimates values are also plotted. 

Note that the neutron current on sample (n/s) is obtained by multiplying the neutron flux by 

the area of the sample aperture A2 (= R2

2

). In our notation, that quantity is given 

by

A2 ()  ()

. Note that  and  are not per unit wavelength, but are calculated at 

wavelength . 

Considering a sample aperture of radius R2 = 0.635 cm, the following neutron currents can be 

estimated:

(6Å)

= 1.28*107n/s for the high flux configuration.

(6Å)

= 4.10*105n/s for the low flux configuration.

These are reasonably high numbers for a SANS instrument (Cook et al, 2005). 
34

5. NEUTRON BEAM MONITOR COUNT RATE

The neutron beam monitor count rate is measured on a regular basis for increasing 

wavelength. Measurements shown here were taken on the NG3 30 m SANS instrument at the 

NIST CNR before the optical filter was installed. The beam monitor is a low-efficiency 

fission counter and is placed just after the velocity selector. It detects neutrons through their 

absorption in a thin U-235 plate. The absorption cross section varies like “1/v” (v being the 

neutron velocity). It is proportional to the neutron wavelength , i.e., a() = c where c is a 

constant. 

The measured monitor count rate m() is compared to the following empirical expression:



































 

2

3

7

7.37

exp

2.25*10

m( ) . (8)

The multiplicative constant depends on the fission counter used. Note the characteristic -

dependence. The tail drops out like 1/

3

. Recall that the cold source spectrum drops out like 

1/

5

. Use of a velocity selector (with constant  changes the tail of the transmitted 

spectrum to 1/

4

. Therefore, the tail of the corrected monitor count rate varies like m()/a() 

~ 1/

4 where a() is the neutron absorption cross section. The wavelength dependence of the 

monitor count rate/wavelength and the neutron current density are the same. It is not clear as 

to why the constants in the exponential are different. 
35

0

1000

2000

3000

4000

4 6 8 10 12 14 16 18 20

 Neutron Beam Monitor 

measured 

predicted 

Monitor Count Rate/Wavelength 

Neutron Wavelength (Å)

Figure 3: Variation of the neutron beam monitor count rate divided by the neutron 

wavelength with increasing wavelength. 

REFERENCE

J.C. Cook, C.J. Glinka, and I.G. Schroder, ”Performance of the vertical optical filter for the 

NG-3 30 m SANS instrument at the National Institute of Standards and Technology's Center 

for Neutron Research”, Review of Scientific Instruments, 76, no. 2, 25108-1-8, (2005). 

QUESTIONS

1. What is the neutron current? 

2. What is the neutron flux (or current density) at the sample? 

3. What is the highest neutron flux on sample for 6 Å neutrons at the NG3 SANS instrument? 

4. How do neutron fluxes compare with x-ray fluxes?

5. Is the neutron current crossing the sample aperture the same as the detector count rate?

ANSWERS
36

1. The neutron current is the number of neutrons per second. 

2. The neutron flux at the sample is expressed in n/cm2.s. It is independent of sample area.

3. The highest neutron flux on sample for 6 Å neutrons at the NG3 SANS instrument is 

around 107n/cm2.s. It is obtained for a high-Q high flux configuration. 

4. Neutron fluxes are orders of magnitude lower than x-ray fluxes. Even fluxes for a rotating 

anode x-ray source are higher than the highest neutron source fluxes. 

5. The neutron current crossing the sample aperture is not the same as the detector count rate 

because of loss due to attenuation in the scattering flight path, due to neutrons that are 

scattered outside of the detector solid angle and due to the detector absorption cross section 

and non-perfect detector efficiency. 
37

Part B – NEUTRON SCATTERING

Chapter 6. Introduction to Neutron Scattering

6.1 Characteristics of Neutron Scattering

6.2 Types of Neutron Scattering

6.3 Diffractometer Types

References

Questions

Answers

Chapter 7. Neutron Scattering Theory

7.1 Solution of the Schrodinger Equation

7.2 Scattering Cross Sections

7.3 The Bra-Ket Notation

7.4 Simple Model for Neutron Scattering Lengths

7.5 Measurements of Neutron Scattering Lengths

References

Questions

Answers

Chapter 8. Elastic and Quasielastic-Inelastic Neutron Scattering

8.1 Definitions

8.2 Scattering Sizes and Energy Ranges

8.3 Diffraction and Refraction

8.4 The Master Formula of Neutron Scattering

8.5 The Various Structure Factors

References

Questions

Answers

Chapter 9. Coherent and Incoherent Neutron Scattering 

9.1 Coherent and Incoherent Cross Sections

9.2 Spin Incoherence

9.3 Coherent Scattering Cross Section

9.4 Incoherent Scattering Cross Section

9.5 Total Scattering Cross Section

9.6 Scattering Length Density

9.7 Contrast Factors

9.8 Macroscopic Scattering Cross Sections

9.9 Summary for H2O and D2O

9.10 General Case

9.11 Tabulated Scattering Lengths and Cross Sections

9.12 Neutron Transmission

9.13 Measured Macroscopic Cross Section for Water

9.14 Cross Sections For H2O/D2O Mixtures
38

References

Questions

Answers
39

Chapter 6 - INTRODUCTION TO NEUTRON SCATTERING

Neutron scattering is the technique of choice for condensed matter investigations in general 

because thermal/cold neutrons are a non-invasive probe; they do not change the investigated 

sample since they do not deposit energy into it. 

1. CHARACTERISTICS OF NEUTRON SCATTERING

A few advantages of neutron scattering are included here. 

-- Neutron scattering lengths vary "wildly" with atomic number and are independent of 

momentum transfer Q. This is used to advantage in deuterium labeling using the fact that the 

scattering lengths for hydrogen and deuterium are widely different (bH= -3.739 *10-13cm 

and bD= 6.671 *10-13 cm respectively). The negative sign in front of bH means that the 

scattered neutrons wavefunction is out of phase with respect to the incident neutrons 

wavefunction. 

-- Neutrons interact through nuclear interactions. X-rays interact with matter through 

electromagnetic interactions with the electron cloud of atoms. Electron beams interact 

through electrostatic interactions. Light interacts with matter through the polarizability and is 

sensitive to fluctuations in the index of refraction. For this, neutrons have high penetration 

(low absorption) for most elements making neutron scattering a bulk probe. Sample 

environments can be designed with high Z material windows (aluminum, quartz, sapphire, 

etc) with little loss. 

-- In neutron scattering, scattering nuclei are point particles whereas in x-ray scattering, 

atoms have sizes comparable to the wavelength of the probing radiation. In the very wide 

angle (diffraction) range, x-ray scattering contains scattering from the electron cloud, 

whereas neutron scattering does not. In the SANS range, this is not the case. 

-- Neutrons have the right momentum transfer and right energy transfer for investigations of 

both structures and dynamics in condensed matter. 

-- A wide range of wavelengths can be achieved by the use of cold sources. Probed size range 

covers from the near Angstrom sizes to the near micron sizes. One can reach even lower Q's 

using a double crystal monochromator (so called Bonse-Hart) USANS instrument. 

-- Since neutron detection is through nuclear reactions (rather than direct ionization for 

example) the detection signal-to-noise ratio is high (almost 1 MeV energy released as kinetic 

energy of reaction products).
40

Figure 1: Neutrons are scattered from nuclei while x-rays are scattered from electrons. 

Scattering lengths for a few elements are compared. Negative neutron scattering lengths are 

represented by dark circles.

A few disadvantages of neutron scattering follow. 

-- Neutron sources are very expensive to build and to maintain. It costs millions of US 

dollars annually to operate a nuclear research reactor and it costs that much in electrical bills 

alone to run a spallation neutron source. High cost (billions of dollars) was a major factor in 

the cancellation of the Advanced Neutron Source project in the mid 1990s. 

-- Neutron sources are characterized by relatively low fluxes compared to x-ray sources 

(synchrotrons) and have limited use in investigations of rapid time dependent processes.

-- Relatively large amounts of samples are needed: typically 1 mm-thickness and 1 cm 

diameter samples are needed for SANS measurements. This is a difficulty when using 

expensive deuterated samples or precious (hard to make) biology specimens.

X-rays interact with the electron cloud

C O Si Cl Ti

U

H-1

D-2

C O Si

Cl-37

Ti-46

U

Ti-50

Ti-49

Ti-48

Ti-47

Neutrons interact with the nuclei

Nuclei Seen by X-Rays

H

Nuclei Seen by Neutrons

Cl-35

Ti
41

2. TYPES OF NEUTRON SCATTERING

There are four main types of neutron scattering. 

(1) The simplest type consists in a measurement of the sample transmission. This 

measurement requires a monochromatic beam (or the time-of-flight method), some 

collimation and a simple neutron detector (end-window counter). Transmission 

measurements contain information about the sample content and the relative fractions of the 

various elements. For example, the relative ratio of carbon to hydrogen in crude oils (the socalled cracking ratio) could be measured accurately.

(2) Elastic neutron scattering consists in measuring the scattered intensity with varying 

scattering angle. This is a way of resolving the scattering variable Q = (4) sin(/2) where 

 is the neutron wavelength and  is the scattering angle. This is performed by either stepscanning or using a position-sensitive detector. The main types of elastic scattering 

instruments are diffractometers (either for single-crystal, powder diffraction or for diffuse 

scattering from amorphous materials), reflectometers and SANS instruments. Diffractometers 

probe the high Q range (Q > 0.5 Å-1) whereas reflectometers and SANS instrument cover the 

low-Q range (Q < 0.5 Å-1). They all investigate sample structures either in crystalline of 

amorphous systems. 

(3) Quasielastic/inelastic neutron scattering consists in monochromation, collimation, 

scattering from a sample, analysis of the neutron energies then detection. The extra step uses 

a crystal analyzer (or the time-of flight method) in order to resolve the energy transfer during 

scattering. In this case both 

Q ks ki

  

  and E = Es – Ei are resolved. Quasielastic scattering 

corresponds to energy transfers around zero, whereas inelastic scattering corresponds to finite 

energy transfers. The main types of quasielastic/inelastic spectrometers are the triple axis, the 

time-of-flight, and the backscattering spectrometers. These instruments cover the eV to 

meV energy range. They investigate sample dynamics and structure. Inelastic instruments are 

used to investigate phonon, optic and other types of normal modes. Quasielastic instruments 

are used to investigate diffusive modes mostly. 

(4) The spin-echo instrument is another type of quasielastic spectrometer. It is singled out 

here because it measures correlations in the time (not energy) domain. It uses polarized 

neutrons that are made to precess in the pre-sample flight path, get quasielastically scattered 

from the sample, then are made to precess again but in the other direction in the post-sample 

flight path. A neutron spin analyzer keeps track of the number of spin precessions. The 

difference in the number of spin precessions before and after the sample is proportional to the 

neutron velocity change during scattering and therefore to the energy transfer. Scanned Q 

ranges are between 0.01 Å-1and 0.5 Å-1and probed times are in the nanoseconds range. This 

instrument is useful for investigating diffusive motions in soft materials. 
42

Figure 2: Schematic representation of the four types of neutron scattering methods. 

3. DIFFRACTOMETER TYPES

The main types of diffractometers include (1) single-crystal and powder diffractometers, (2) 

diffuse and liquid scattering instruments, (3) small-angle neutron scattering instruments and 

(4) reflectometers. All of these diffractometers correspond to “double axis” diffraction, i.e., 

they are schematically represented by a monochromator (first axis) and diffraction from the 

sample at an angle  (second axis). Types (1) and (2) probe the high Q scale with Q > 0.1 Å-1

(i.e., small d-spacings d < 60 Å). The third and fourth type probe the lower Q scale 0.4 Å-1 > 

Q > 0.001 Å-1(i.e., 16 Å < d < 6000 Å). The measurement window for SANS instruments 

and reflectometers covers from the near atomic sizes (near Å) to the near optical sizes (near 

m). Type (1) measures purely crystalline samples whereas the other types are used mostly 

for amorphous systems. SANS however can measure both amorphous and crystalline 

samples. Types (1), (2) and (3) measure bulk samples whereas type (4) (reflectometers) 

measure surface structures only. Similar discussions can be found elsewhere (Price-Skold, 

1986). 

monochromation

sample

detector

TRANSMISSION 

MEASUREMENT

DIFFRACTOMETER

monochromation

sample

detection

or VS

QUASIELASTIC/INELASTIC 

SCATTERING

monochromator

sample

analyzerdetector

or TOF method

NEUTRON SPIN ECHO

polarizer

sample

spin analyzer

detector

flipper

or VS

or VS

monochromation
43

REFERENCES

D.L Price and K. Skold, "Introduction to Neutron Scattering" Methods of Experimental 

Physics 23A, 1 (1986)

"NIST Cold Neutron Research Facility and Instruments", a series of articles covering the 

entire issue, National Institute of Standards and Technology Journal of Research, 98, Issue 

No 1 (1993).

QUESTIONS

1. Name a couple of advantages of neutron scattering. 

2. Neutrons interact with what part of the atom?

3. Name a couple of disadvantages of neutron scattering. 

4. Name the four types of neutron scattering instruments. 

5. What type is the SANS instrument?

ANSWERS

1. Neutrons are very penetrating, they do not heat up or destroy the sample, deuterium 

labeling is unique. They have the right wavelengths (Angstroms) and kinetic energies (eV 

to meV) to probe structures and dynamics of materials. 

2. Neutrons interact with the nuclei.

3. Neutron sources are characterized by low flux compared to x-ray sources. Relatively large 

amounts of sample (gram amounts) are required for neutron scattering measurements. 

4. The four types of neutron scattering instruments are: transmission, elastic, 

quasielastic/inelastic and neutron spin echo. 

5. The SANS instrument is a “diffractometer” for diffuse elastic neutron scattering. 
44

Chapter 7 - NEUTRON SCATTERING THEORY

Elements of neutron scattering theory are described here. The scattering amplitude, scattering 

lengths and cross sections are introduced and discussed. 

1. SOLUTION OF THE SCHRODINGER EQUATION

Neutron scattering theory involves quantum mechanics tools such as the solution of the 

Schrodinger equation even though the scattering problem is not a quantum mechanical 

problem (no bound states are involved). A simple solution of the Schrodinger equation 

involving perturbation theory is presented here. This is to so-called Born Approximation

method. 

Figure 1: Incident plane wave and scattered spherical wave.

The Schrodinger equation is expressed as follows:

Hii  Eii

(1)

H  Es

H  Hi  V.

H is the full Hamiltonian operator, Hi is the incident neutron kinetic energy operator and V is 

the neutron-nucleus interaction potential. Ei and Es are the eigenvalue energies for the 

incident neutron and for the scattered neutron. i and  are the eigenfunctions for the 

incident (non-interacting) neutron and for the interacting neutron-nucleus pair.

r

r' 



r r'

 


45

2

2 2

i

2m

-

2m

p

H   



(2)

 







p i

is the momentum operator.

2m

k

E

2

i

2

i



 .

Ei is the incident neutron kinetic energy and ki is its incident wavenumber. i is the solution 

of the homogeneous differential equation:

( k ) (r) 0

2m

(H E ) (r) i

2

i

2

2

i  i i      

  

. (3)

The solution is an incident plane wave 

(r) exp(ik .r)

i i





 

using vector notation. The full 

differential equation is written as:

( k ) (r) V(r) (r)

2m

2

s

2

2

   

       . (4)

Its solution is an integral equation of the form:



 













    dr'G(r r')V(r') (r')

2

m

(r) (r)

i 2

    



 

(5)

Here 

G(r r')

 

 is a Green’s function satisfying the following differential equation:

( k )G(r) (r)

2m

(H E )G(r)

2

s

2

2

s

   

      

(6)

ks is the scattered neutron wavenumber. Its solution is a spherical outgoing wave of the form:

r

exp(ik r)

G(r)

s





. (7)

In order to verify this result, the following relations valid in spherical coordinates are used:

2

2

r

1

r

1

r

r r

1

 















 













(8)

(r)

r

1

r

r

r r

1

r

1 2

2

2

  











































 











 .
46

Therefore:

 



















    V(r') (r')

| r r'|

exp(ik | r r'|) dr'

2

m

(r) (r)

s

i 2

 

 

 





 

(9)

Vector 

r'



is within the sample and 

r



is far from the sample so that r >> r’ and therefore one 

can approximate 

r

r.r'

| r r'| r

 

 

   . 

exp( ik .r')

r

exp(ik r)

r

ik r.r'

exp

r

exp(ik r)

G(| r r'|) s

s s s





 

 

  









 

  . (10)

Here, the scattered neutron wavevector 

s k



has been defined as 

k k r r

s s





 . 

The general solution of the Schrodinger equation involves an integral equation that can be 

solved iteratively through the expansion:



   













    dr'G(r r')V(r') (r')

2

m

(r) (r)

i 2 i

    



 

 

    













dr'G(r r')V(r') dr"G(r r") (r") ...

2

m

i

2

2

       



(11)

Keeping only the first integral term corresponds to the first Born approximation which can be 

presented in the form:



 













    dr'G(r r')V(r') (r')

2

m

(r) (r)

i 2 i

    



 

(12)



 













   dr'exp( ik .r')V(r')exp(ik .r')

2

m

r

exp(ik r)

(r) exp(ik .r)

2 s i

s

i



   





 

f( )

r

exp(ik r)

(r) exp(ik .r)

s   i  



 

.

The scattering amplitude f() has been defined as:

 













  dr'exp( iQ.r')V(r')

2

m

f( )

2

 

 



(13)

Q ks ki

  

  is the scattering vector. f() is the Fourier transform of the interaction potential 

V(r’). f() has been assumed to be independent of the azimuthal angle. 
47

The first Born approximation applies to thermal/cold neutrons neutron scattering 

corresponding to "s wave" scattering (i.e., corresponding to a zero orbital angular quantum 

number). This includes all of neutron scattering except for neutron reflectivity whereby 

higher order terms in the Born expansion have to be included. Neutron reflectometry 

involves refraction (not diffraction). 

Q



characterizes the probed length scale and its magnitude is given for elastic scattering in 

terms of the neutron wavelength  and scattering angle  as Q = (4 sin(2). For small 

angles (SANS), it is simply approximated by Q = 2. Since Q is the Fourier variable (in 

reciprocal space) conjugate to scatterer positions (in direct space), investigating low-Q 

probes large length scales in direct space and vice versa. 

In summary, the solution of the Schrodinger equation is an incident plane wave plus a 

scattered spherical wave multiplied by the scattering amplitude. 

2. SCATTERING CROSS SECTIONS

The microscopic differential scattering cross section is defined here. It represents the fraction 

of neutrons scattered into solid angle d with a scattering angle . 

Figure 2: Representation of neutrons scattered with angle  inside a solid angle d. 

Consider incident neutrons of wavenumber ki and scattered neutrons of wavenumber ks. The 

incident neutron flux also called current density (neutrons/cm2.s) is given by:

  i i i

*

i

*

i

2m

i

J     

   

. (14)

Here * represents the complex conjugate and 

exp(ik .r)

i i





 

is the incident plane wave. 

Performing the simple operation 

  iki

 

, one obtains 

Ji ki m









. Similarly for the 

scattered neutron flux:

   

   

* *

2m

i

Js

. (15)



i k



s k



 

 

dS r d

2

r





d
48

Where  =  - i = 

f( )

r

exp(ik r)

s



. Here also, performing the differentiations, one obtains: 

2

2

s

s

r

f( )

m

k

J







 

. Note that the current densities 

i

J



and 

s

J



have units of velocity (speed). In 

order to obtain the standard units for a current density (neutrons/cm2.s), one has to divide by 

the volume formed by a unit area and the distance travelled by the neutrons per second. 

The differential neutron scattering cross section is defined as:

  



   | f( ) | d

k

k

J

J r d

d ( )

2

i

s

i

2

s

s

(16)

This is the ratio of the neutron flux scattered in d over the incident neutron flux. Within the 

first Born approximation (also called the Fermi Golden Rule):

2

i

s s

| f( ) |

k

k

d

d ( )

 



 

(17)

2

2

i

s s

dr'exp( iQ.r')V(r')

2

m

k

k

d

d ( )



 

















   





.

This cross section contains information about what inhomogeneities are scattering and how 

they are distributed in the sample. The microscopic scattering cross section is its integral over 

solid angles: 



 















  d

d

d s

s

. Cross sections are given in barn units (1 barn = 10-24cm

2

). 

Given the (atomic) number density N/V (number of scattering nuclei/cm3) in a material, a 

macroscopic cross section is also defined as: s = (N/V) s (units of cm-1). SANS data are 

often presented on an "absolute" macroscopic cross section scale independent of instrumental 

conditions and of sample volume. It is given by ds/d = (N/V) ds/d.

3. THE BRA-KET NOTATION

The <bra|ket> approach is useful for simplifying notation. Consider the following definitions:

r | k exp(ik .r)

i i



 



 

(18)

r | (r)

 

   

 r | V  V(r)

r | G | r') G(r r')

   

  

r | r') (r r')

   

    .
49

Define the following closure relations:



| r' dr' r'|1

  



| k  dk  k |1

  

. (19)

The integrations are over all direct 

r



or reciprocal 

k



space. The scattering amplitude is 

expressed as:



 













  dr'exp( iQ.r')V(r')

2

m

f( )

2

 

 



(20)



 













 dr'exp( ik .r')exp(ik .r')V(r')

2

m

2 s i

 

  



.

Using the <bra|ket> notation, f() can be also manipulated to the form:

  













 

2 s ki k | V |

2

m

f( )

 



. (21)

The scattering cross section is therefore given in terms of the transition probability 

 ks| V | ki 

 

as:

2

s 2 i

i

s s V | k

2

m

k |

k

k

d

d ( )

 













 



  





. (22)

This result ignores the effect of spin interactions and therefore does not apply to scattering 

from magnetic systems.

4. SIMPLE MODEL FOR NEUTRON SCATTERING LENGTHS

A simple argument is used here in order to appreciate the origin of the scattering length 

(Squires, 1978). Consider a neutron of thermal/cold incident energy Ei being elastically 

scattered from a nucleus displaying an attractive square well potential -Vo (note that Vo >> 

Ei). Recall the Schrodinger equation for this simplest potential. 

V(r) (r) E (r)

2m s

2

2

  











  



. (23)
50

The Schrodinger equation can be solved in 2 regions (inside and outside of the well region).

Figure 2: Neutron scattering from the quantum well of a nucleus.

Outside of the well region (i.e., for r > R) where V(r) = 0, the solution has the form:

r

exp(ik r) b

k r

sin(k r) (r)

s

i

Out i    (s-wave scattering). (24)

Here b is the scattering length and for elastic scattering ks = ki = 

2mEi

/



. Note that in this 

case, the scattering amplitude is simply f() = -b. Note also that the incident plane wave has 

been averaged over orientation: 

k r

sin(k r)

d exp(ik r )

2

1

i

i

1

1

  i  



. 

Inside of the well (r < R) where V(r) = -V0the solution is of the form:

 

qr

sin(qr) (r) A

In   with 

q  2m(Ei  V0)/  . (25)

Ei

Energies

r

incident

neutron

scattered

neutron

V0

R

Out In Out

0

Ei ~ 0.025 eV

V0 ~ MeV
51

Note that this wavefunction 

(r)

In 

represents a randomly oriented plane wave 

















   



1

1

d exp(iqr )

2

1

qr

sin(qr)

. The boundary conditions (continuity of the wavefunction and 

its derivative) are applied at the surface (r = R): 

 (r R) (r R)

In Out     

(26)

 [ψ ](r R)

dr

d

[ψ ](r R)

dr

d In Out

   .

Here 

kir  ksr  2mEir  1

(nuclear interactions are short ranged) and therefore 

Out

~ 

1 - b/r. Finally:

 

 

R

b

1

qR

sin(qR) A   (27) 

2 2 R

b

qR

sin(qR)

A

qR

cos(qR) Aq   .

 

In another form:

cos(qR)

1

A 

(28)

qR

tan(qR) 1

R

b

  .

The solution of this transcendental equation:

 

qR

tan(qR) 1

R

b

  (29)

gives a first order estimate of the scattering length b as a function of the radius of the 

spherical nucleus R and the depth of the potential well V0. 
52

-10

-5

0

5

1 0

0 1 2 3 4 5

b/R vs qR

b/R = 1-tan(qR)/qR

qR

H

D

Figure 3: Solution of the Schrodinger equation subject to the boundary conditions. 

Due to the steep variation of the solution to the above transcendental equation, adding only 

one nucleon (for example, going from H to D) gives a very large (seemingly random) 

variation in b. The scattering length can be negative like for H-1, Li-7, Ti-48, Ni-62, etc. The 

H and D nuclei have been added to the figure knowing their scattering lengths (bH = -0.374 

fm and bD = 6.671 fm) and assuming RH = 1 fm and RD = 2 fm. The Fermi (1 fm = 10-13 cm) 

is a convenient unit for scattering lengths. The neutron-nucleus interaction potential can be 

estimated for the case of H as V0 = 30 MeV. These are huge energies compared to the 

thermal neutron kinetic energy of 25 meV. 

The scattering length itself can be complex if absorption is non negligible: b = bR – ibI. 

Neutron absorption is small for most organic materials. It has been neglected completely in 

the simple model discussed above. 

Since no nucleus is completely free, bound scattering lengths should be used instead: bbound = 

bfree (A + 1)/A, where A is the atomic number. Free and bound scattering lengths are 

substantially different only for low mass elements such as hydrogen. 

5. MEASUREMENTS OF NEUTRON SCATTERING LENGTHS
53

Note that the index of refraction n is related to the material atomic density  (atoms/cm3), the 

neutron scattering length b, and the neutron wavelength  as:

2

2π

ρb

n  1   (30)

The scattering length b can be measured by measuring the index of refraction n using optical 

methods. Note that most materials have an index of refraction less than one for neutrons and 

greater than one for light. 

Neutron interferometry methods are another way of measuring scattering lengths. 

REFERENCES

G.L. Squires, “Introduction to the Theory of Thermal Neutron Scattering” Dover 

Publications (1978).

QUESTIONS

1. What is the neutron scattering length of an element?

2. What is the scattering cross section of an element? How does it relate to the scattering 

length?

3. What is the differential scattering cross section?

4. What is the strength of typical neutron-nucleus interaction potentials? What is a typical 

neutron kinetic energy? 

5. Write down the Schrodinger equation. 

6. What is the first Born approximation? What type of neutron scattering is not well modeled 

by the first Born approximation? 

7. What is a simple description of the solution of the Schrodinger equation in terms of 

waves?

ANSWERS

1. The neutron scattering length of an element represents the apparent “size” of this element 

during scattering.

2. The scattering cross section of an element is the apparent area that it offers during 

scattering. The scattering cross section  is related to the scattering length b as  = 4b

2

. 

3. The differential scattering cross section is the cross section per unit solid angle d/d. 

4. Typical neutron-nucleus interaction potentials are of order MeV. Typical neutron kinetic 

energies are of order meV (thermal neutron energy is 25 meV). 

5. The Schrodinger equation is [

2m

2

 



2

+ V(r)] (r) = E (r) where the first term is the 

kinetic energy, the second term is the potential energy, V(r) is the neutron-nucleus interaction 
54

potential, E is the so-called system energy and (r) is the so-called eigenfunction. This 

equation can also be written as H = E where H is the system Hamiltonian. 

6. The first Born approximation corresponds to keeping only the first term in the expansion 

solution of the Schrodinger equation. The first Born approximation does not model 

reflectivity well. 

7. The solution of the Schrodinger equation corresponds to an incident plane wave and a 

scattered spherical wave. 
55

Chapter 8 - ELASTIC AND QUASIELASTIC/

INELASTIC NEUTRON SCATTERING

Structures are investigated using elastic scattering instruments whereas dynamics are probed 

using quasielastic/inelastic scattering instruments. 

1. DEFINITIONS

Defining the momentum and energy for the incident neutron as (

i k



, Ei) and for the scattered 

neutron as (

s k



, Es), the momentum transfer (scattering vector) is 

Q ks ki

  

  and the energy 

transfer is E = Es-Ei during the scattering event. Elastic scattering occurs when there is no 

energy transfer E = 0 (zero peak position and peak width). Inelastic scattering occurs when 

there is a transfer of both momentum and energy. Qasielastic scattering is a form of inelastic 

scattering where the energy transfer peak is located around E = 0 (zero peak position but with 

a finite peak width). In practice, the peak width is always limited by the instrumental energy 

resolution. 

Figure 1: Schematic representation of the momentum and energy initial state (

i k



, Ei) and 

final state (

s k



, Es). 

2. SCATTERING SIZES AND ENERGY RANGES

The various elastic and quasielastic/inelastic neutron scattering instruments have specific 

window ranges in the (Q,E) space. 

 incident neutrons

momentum 

i k



, energy Ei

 scattered neutrons

momentum 

s k



, energy Es

i k



, Ei

s k



, Es Q ks ki

  

  momentum transfer

E = Es-Ei energy transfer 
56

Figure 2: Schematic representation of the various elastic and quasielastic/inelastic neutron 

scattering instrument windows in (Q,E) space. 

3. DIFFRACTION AND REFRACTION

Most neutron scattering methods operate in the “diffraction” mode. They involve single 

scattering and avoid multiple scattering events. Neutron reflectometry on the other hand 

operates in the “refraction” mode. It involves a large number of incremental scattering events 

that tend to steer the incident neutron beam until it is completely reflected. Describing 

reflection therefore involves a completely different theoretical basis than all other (single) 

scattering methods. The focus here will be on these methods that do not involve reflection. 

Within the first order perturbation theory, the so-called “master formula of neutron 

scattering” is derived next. 

4. THE MASTER FORMULA OF NEUTRON SCATTERING

The single-scattering theory is based on the first Born approximation (the so-called Fermi 

Golden rule) describing s-wave scattering (Schiff, 1955; Bee, 1990). This corresponds to 

spin echo

backscattering

triple axis

time-of-flight

SANS

Reflectometry

Diffraction

0.01 0.1 1 10

10-7

10-9

10-11

10-13

Q (Å-1)

Times (s) E (eV)

Lengths (Å)

100 10

10-2

10-0

10

+2

10+4

1

ELASTIC

INELASTIC

QUASIELASTIC
57

most forms of neutron scattering except for neutron reflectometry which requires higher 

order terms in the Born expansion. Defining an initial state for the neutron-nucleus system as 

|i> = |

i k



ni> where 

i k



is the incident neutron momentum and ni is the initial nuclear state and 

a scattered state as |s> = |

s k



ns>, the double differential neutron scattering cross section can 

expressed as:

V(Q) | i | (E )

2

m

| s|

k

k

dEdΩ

d σ

s i

2

2

i

s

2

    E  E













 



(1)

P | k n | V(Q) | k n | (E )

k

k

2

m

s i

n ,n

2

n s s i i

i

s

2

2

i s

i

      E  E















 



.

Here m is the neutron mass, and Pniis the probability of finding a scattering nucleus in initial 

state |ni>. Eniand Ensare the energy states of the nucleus before and after scattering and V is 

the interaction potential. Note that due to the conservation of energy Ei – Es = Ei – Es ~ E 

where Ei and Es are the incident and scattering neutron energies and E is the transferred 

energy. Averaging over initial states and summing up over final states has also been 

performed. 

Since neutron-nucleus interactions are short ranged, the following Fermi pseudo-potential is 

used. 

b (r r )

m

2

V(r)

j

N

j

j

2

  

  















 

 . (2)

b exp( iQ.r )

m

2

V(Q)

j

N

j

j

2

  

 















 



Here bj is the scattering length for nucleus j and N is the number of scattering nuclei in the 

sample. The following closure relation is introduced:

| r  dr  r |1 

  

. (3)

The <bra|ket> notation is used as follows:

r | k exp(ik .r)

i i



 



 

(4)

k | r exp( ik .r)

s s









  

The transition matrix element is calculated as:
58

         













 s i i

j

s 2

V(Q) | ki b j k | r (r r )dr r | k

2

m

k |

      





(5)

  

j

j j b exp( iQ.r )





Here 

Q ks ki

  

  and a property of the Dirac Delta function have been used. Moreover a 

special representation of the Delta function is used to express the following term as:











   





   



  

i(E )t

dt exp

2

(E )

s i

s i

E E

E E . (6)

Finally the Heisenberg operator helps represent time dependence as follows:























 









 iHt

r (0)exp

iHt

r

j

(t) exp j

. (7)

Here H is the scattering system Hamiltonian.

H | ni  Ei| ni  , H | ns  Ei

| ns . (8)

Putting all terms together, the cross section is expressed as follows:

        











 



 



j l i

j,l

s j l

n ,n

n

i

s

2

n | b b exp iQ.r (0) exp iQ.r (t) | n

iEt P dt exp

k

k

dEd

d

i s

i



 



S(Q,E)

k

k

i

s

 . (9)

This is the most general neutron scattering cross section within the first order perturbation 

theory. The dynamic structure factor S(Q,E) has been defined in terms of the scattering 

density n(Q,t) as follows:

   



N

j 1

j j n(Q,t) b exp iQ.r (t)





(10)

    









  







n( Q,0)n(Q,t)

iEt dt exp2

S(Q,E)

 

. (11)

The averaging notation 

  s i 

ns, ni

ni

P n | ... | n

has also been simplified to <...>. The 

summations are over scattering nuclei. Note that at this level the scattering lengths are still 

included in n(Q,t) and in S(Q,E). 
59

Note that the differential cross section 

dΩ

d(Q)

used in elastic scattering is related to the 

double differential cross section 

dEdΩ

d σ(Q,E) 2

used in quasielastic/inelastic scattering through 

an integral over energy transfers. 

 







dEd

d (Q,E)

dE

dΩ

d (Q) 2

. (12)

There are many definitions for S(Q,E) in the literature. 

5. THE VARIOUS STRUCTURE FACTORS

Many textbooks discuss the various structure factors (Bacon, 1962; Marshall-Lovesey, 

1971). The Fourier transform of S(Q,E) is in the time domain. 

S(Q,E)

iEt S(Q,t) dEexp 











 



 

(13)

S(Q,t) is the time-dependent density-density correlation function also called time-dependent 

structure factor. 

S(Q,E) is measured by most quasielastic/inelastic neutron scattering spectrometers such as 

the triple axis, the backscattering and the time-of-flight instruments. S(Q,t) is measured by 

the neutron spin echo instrument. 

The initial value S(Q,t = 0) is the so-called static scattering factor S(Q). S(Q) is what 

diffractometers and SANS instruments measure. Note that S(Q) is also expressed as:



S(Q)  S(Q,t  0)  S(Q,E)dE. (14)

Elastic scattering does not really mean with energy transfers E = 0 (zero peak and zero 

width); it rather means integrated over all energy transfers (summing up over all energy 

modes). 

S

(Q) is the density-density correlation function. 

S(Q)  n(Q)n(Q) 

(15)

Another form of the density-density correlation function S(Q) is related to the pair 

correlation function 

g(r)



through the space Fourier transform:
60

S(Q) 1 N dr exp(iQ.r)g(r) 1

 





. (16)

Here 

N  N V

is the particle number density. Note also that the scattering lengths have still 

not been separated out. These will be averaged for each scattering unit to form the contrast 

factor which will be multiplying S(Q). 

REFERENCES

M. Bee, “Quasielastic Neutron Scattering”, Adam Hilger (1990). 

L.I. Schiff, “Quantum Mechanics”, McGraw Hill (1955). 

G.E. Bacon, “Neutron Diffraction”, Oxford, Clarendon Press (1962).

W. Marshall and S.W. Lovesey, "Theory of Thermal Neutron Scattering", Clarendon Press, 

Oxford (1971).

QUESTIONS

1. What is the difference between quasielastic and inelastic scattering?

2. Define the terms in the following expression: Q = 

k k 2k k cos( )

i s

2

s

2

i    . 

3. What is “s-wave scattering”? What does it correspond to?

4. Can reflectometry data be described by the first Born approximation?

5. What is the Fermi pseudo-potential?

6. What is the differential cross section? How about the double differential cross section? 

7. Write down the double differential cross section (the Master formula) for neutron 

scattering. 

ANSWERS

1. Quasielastic scattering is characterized by energy transfer peaks centered at zero energy 

(with finite widths). Inelastic scattering is characterized by energy transfer peaks centered at 

finite energy (ev to meV). 

2. ki and ks are the incident and scattered neutron momentums and  is the scattering angle. 

3. s-wave scattering corresponds to a zero angular orbital momentum (l = 0). It corresponds 

to single (not multiple) scattering. 

4. Reflectometry involves refraction (not single diffraction). It cannot be described by first 

Born approximation. Higher order terms of the perturbation theory would have to be 

accounted for. 

5. The Fermi pseudo-potential describes the short range neutron-nucleus interactions. It is 

formed of a series of Dirac Delta functions. 
61

6. The differential cross section is d/d. The double differential cross section is d2/ddE. 

7. The double differential cross section is written as: 

dEdΩ

d σ

2

= 

i

s

k

k

S(Q,E). Here ks and ki are 

the scattering and incident neutron momentums and S(Q,E) is the dynamic structure factor. 
62

Chapter 9 - COHERENT AND INCOHERENT NEUTRON SCATTERING

Neutron scattering is characterized by coherent and incoherent contributions to scattering. 

Coherent scattering depends on Q and is therefore the part that contains information about 

scattering structures, whereas SANS incoherent scattering is featureless (Q independent) and 

contains information about the material scattering density only. Here only elastic scattering is 

considered. 

1. COHERENT AND INCOHERENT CROSS SECTIONS

The coherent and incoherent parts of the elastic scattering cross section are separated. 

Consider a set of N nuclei with scattering lengths biin the sample. The scattering cross 

section is given by:



   













  



  2

2

2

2

| dr exp( iQ.r)V(r) |

2

m

| f( ) |

d

d ( ) 

 



. (1)

Here f() is the scattering amplitude, 

Q ks ki

  

  is the scattering vector and V(r) is the 

Fermi pseudopotential describing neutron-nucleus interactions:

  















 





N

i 1

i i

2

b (r r )

m

2

V(r)

  

. (2)

Here 

i

r



is the position and bithe scattering length of nucleus i. Therefore, the differential 

scattering cross section is the sum of the various scattering phases from all of the nuclei in 

the sample properly weighed by their scattering lengths:

  

 

   



 

N

i 1

N

j 1

i j i j b b exp iQ.(r r )

d

d ( )  



(3)

where <...> represents an “ensemble” average (i.e., average over scatterers’ positions and 

orientations).

Consider an average over a set consisting of a number m of nuclei:

{...} = 



m

i 1

...

m

1

 (4)

This average could be over different atoms, over different isotopes or different atoms in a 

small molecule. 
63

Define average and fluctuating parts for the scattering lengths bi={b}+ biand positions 

i R S i

r 

  

 



as well as the following:

Ri



: position of the center-of-mass of molecule 

Si



: relative position of scatterer i inside molecule 

m: number of nuclei per molecule

M: number of molecules in the sample (Note that N = mM).

Figure 1: Parametrization for two scattering molecules.

The various terms of the scattering cross section can be separated as:

           



 



N

i,j

i j i j

{b} b {b} b exp iQ.(r r

d

d ( )  



(5)

               



 

  

N

i,j

N

i,j

N

i,j

ij i j ij i ij

2

{b} exp iQ.r b b exp iQ.r 2{b} b exp iQ.r

d

d ( ) 

  

where 

ij i j

r r r

  

  . If 

ij r



is approximated by 

R



which is equivalent to Si<< R(all nuclei 

of one molecule are located very close to each other) the term:

b expiQ.r  b expiQ.R  0

N

i

N

i

 i ij i   











     

  

(6)

Si



R



Sj



R



ij r



R



i

r



j

r


64

can be neglected. This is due to the definition of the center-of-mass 

{ b } b 0

N

i

 i   i  .

The term 

     

N

i,j

i j ij b b exp iQ.r





contributes only when i=j. When i



j, 

b b expiQ.r  b b expiQ.R  0

N

j

j

N

i,j

N

i

i j ij i   















 











         

  

.

Therefore, the scattering cross section can be written simply as the sum of two contributions:

     



 

N

i,j

2

i

N

i,j

ij

2

{b} exp iQ.r b

d

d ( ) 



(7)

= 

dΩ coh

dσ(θ)













+ 

dΩ inc

dσ(θ)













.

The last term is the incoherent cross section for the whole sample:

2 2 2

inc

{ b } {b } {b}

dΩ

dσ(θ)

    











. (8)

Note that these are cross sections per atom. Cross sections can also be defined per molecule 

instead as m

dΩ

dσ(θ)

where m is the number of atoms per molecule. 

Incoherent scattering has two contributions: one from spin incoherence (different atoms) and 

the other from isotopic incoherence (different isotopes). Disorder scattering (different atoms 

in a small molecule) could be considered as coherent (not incoherent) scattering even though 

it is Q-independent in the SANS range due to the smallness of the molecules. Note that in xray scattering, there is no equivalent for spin-incoherence and that contributions from 

disorder scattering are coherent. Spin-incoherence, isotopic incoherence and disorder 

scattering will be described in the next sections. 

2. SPIN INCOHERENCE

Nuclei with nonzero spin contribute to spin incoherence since neutron and nuclear spins 

could be either parallel or antiparallel during the scattering process. The neutron is a Fermion 

with spin 1/2 which couples to the nuclear spin I to give: 
65

-- 2I + 2 states (for which the scattering length is noted b+) corresponding to parallel neutron 

and nuclear spins, or

-- 2I states (for which the scattering length is noted b-) corresponding to antiparallel neutron 

and nuclear spins.

There are 2(2I + 1) total states with the following weighing factors: 

2I 1

I 1

2(2I 1)

(2I 2) W















and 

2I 1

I

2(2I 1)

2I W







  . (9)

Note that W+ + W-=1. In the case of the hydrogen nucleus (I = 1/2), W+ corresponds to a 

triplet state and W- corresponds to a singlet state. 

Figure 2: Representation of the neutron spin 

1 2

with its up and down values and the nuclear 

spin 

I



. 

The averages over spin states are calculated for coherent and spin-incoherent scatterings 

using:

2I 1

(I 1)b Ib {b} W b W b



 

  

 

   

(10)

2I 1

(I 1)b Ib {b } W b W b

2 2

2 2 2



 

  

 

   

.

Either the two scattering lengths b+ and b- or the other two scattering lengths bc and bi could 

be tabulated for each (isotope) element. Most often, it is the bc and bi scattering lengths that 

are tabulated. Tables are based on measurements made using thermal neutrons.

  W b  Wb b {b} c

(11)

b {b } {b} W W (b b )

2 2

i       

.

spin operators

2(I-1/2)+1=2I

I + 1/2 2(I+1/2)+1=2I+2

eigenvalues

number of

degenerate 

states

I - 1/2

2

1

I






66

For the scattering from hydrogen (pure H-1) nuclei:

bc = -3.7406 fm, bi = 25.274 fm, c = 1.7583 barn, i = 80.27 barn.

The corresponding microscopic cross sections are obtained as c = 4 bc

2

and i = 4 bi

2

. 

The b+ and b- scattering lengths can be calculated from bc and bi as follows:

c i c bi

I 1

I

b b

W

W b b



   







(12)

c i c bi

I

I 1

b b

W

W b b



   







.

For hydrogen (I=1/2), b+ = 10.851 fm and b- = -47.517 fm. A spin-dependent scattering 

length is expressed in terms of the neutron and nuclear spin operators 

s



and 

I



as:

c bi

I(I 1)

2s.I

b





 

. (13)

In order to separate out the coherent and incoherent components, three options present 

themselves. (1) Use of polarized neutrons along with polarization analysis. Polarization 

analysis is not easy to achieve due to the large divergence angles in the SANS scattered 

beam. (2) Aligning the nuclear spins along with using polarized neutrons. Aligning nuclear 

spins is extremely difficult and has not matured to be a viable alternative. Note that the field 

of magnetism involves aligning electron (not nuclear) spins. (3) Using deuterium labeling. 

Deuterium labeling is used routinely to minimize the incoherent signal from hydrogen atoms 

(deuterium has a low incoherent component). 

3. COHERENT SCATTERING CROSS SECTION

The coherent scattering cross section is given by the integral of the differential cross section 

over solid angle:

 





  d

d

d c

c

.

Consider the case of a generic molecule AmBn made out of m atoms of element A and n 

atoms of element B. Define the number fractions fA = m/(m+n) and fB = n/(m+n) obeying 

fA+fB=1. The coherent scattering cross section per atom for molecule AmBn is given by:

2

A A B B

2

c m n  (A B )  4{b}  4(f b  f b ) . (14)
67

Consider the case of water (H2O) with bH = -3.739 fm, bO = 5.803 fm, fH = 2/3 and fO = 1/3 

yielding c(H2O) = 4(-2*3.739/3 + 5.803/3)2fm2 = 0.039 barn. Note that bH = -3.739 fm for 

hydrogen with natural abundance of H-1, H-2 (D), and H-3 (T), whereas bH = -3.7406 fm for 

pure H-1. Recall that 1 barn = 10-24 cm2and 1 fm = 10-13 cm. This is the cross section per 

atom. The cross section per H2O molecule is 3*0.039 = 0.117 barn.

Similarly, for heavy water (D2O) with bD = 6.671 fm, one obtains c(D2O) = 5.12 barn. The 

cross section per D2O molecule is 3*5.12 = 15.35 barn.

4. INCOHERENT SCATTERING AND DISORDER SCATTERING CROSS 

SECTIONS

Every element in the periodic table has a spin incoherence scattering cross section except if 

the nuclear spin is zero. Mixing isotopes contributes isotopic incoherence as well. Isotopic 

incoherence and disorder scattering are lumped into “composition incoherence” since the 

same formalism is used to estimate them. 

Consider the AmBn molecule again.

The spin incoherence cross section per atom for the AmBn mixture is given by:

 (A B ) f (A) f (B) i m n spin

 Ai  Bi

. (15)

The isotopic incoherence and “disorder” scattering cross sections per atom involve the 

following difference:

 (A B ) 4 ({b } {b} )

2 2

i m n comp

    . (16)

The two averages are {b} = fAbA + fBbB and {b2} = fAbA

2 + fBbB2

. Therefore:

 (A B ) 4 [f b f b (f b f b ) ]

2

A A B B

2

B B

2

i m n comp  A A   

(17)

2

A B A B

 4f f (b  b ) .

Putting both contributions together, the incoherent cross section per atom is obtained as:

2

i m n A i B i A B A B  (A B )  f  (A)  f  (B)  4f f (b  b ) . (18)

Note that the incoherent cross section per molecule is obtained by multiplying by the number 

of atoms per molecule (m+n).

Consider the case of water (H2O):

2

i 2 H i O i H O H O  (H O)  f  (H)  f  (O)  4f f (b  b ) . (19)
68

Using i(H) = 80.27 barn, i(O) = 0 barn, bH = -3.739 fm, bO = 5.803 fm, fH = 2/3 and fO = 

1/3, the following result is obtained i(H2O) = 53.5 + 2.54 = 56.04 barn. This is the cross 

section per atom. The cross section per H2O molecule is 3*56.04 = 168.12 barn.

Similarly, for heavy water (D2O). Using i(D) = 2.05 barn and bD = 6.671 fm, the following 

result is obtained i(D2O) = 1.37 + 0.021 = 1.39 barn. This is the cross section per atom. The 

cross section per D2O molecule is 3*1.39 = 4.17 barn. 

Note that in both cases the spin incoherence cross section (first term in units of barn) 

dominates over the composition incoherence cross section (second term in units of fm2).

5. TOTAL SCATTERING CROSS SECTION

The total scattering cross section is the sum of the coherent and incoherent contributions. 

(A B ) (A B ) (A B ) s m nc m n i m n

(20)

4 {b } f (A) f (B) A i B i

2

     

4 [f b f b ] f (A) f (B) A i B i

2

B B

2

  A A      .

The first term {b2} contains both coherent and composition incoherence contributions and 

the other two terms contain the spin incoherence contribution. 

For the cases of H2O and D2O, one obtains s(H2O) = 56.08 barn and s(D2O) = 6.51 barn 

respectively. These are the cross sections per atom. The cross sections per molecule are 

3*56.08 = 168.24 barn and 3*6.51 = 19.53 barn. 

6. SCATTERING LENGTH DENSITY

What is needed to calculate neutron contrast factors is the scattering length density (not the 

scattering length). The scattering length density is defined as the ratio of the scattering length 

per molecule and the molecular volume. Assuming an AmBn molecule, the scattering length 

density is given by:

  

v

mb nb

v

f b f b m n

v

b A A B B A B

AmBn

AmBn





 

 











  . (21)

Here mbA + nbB is the scattering length per molecule and v is the volume of molecule AmBn

comprising m atoms A and n atoms B. 
69

The molecular volume v is given in terms of the density d and molar mass m for molecule 

AmBn and Avogradro’s number Nav (Nav = 6*1023 /mol) as:

N d

m

v

av

 . (22)

For H2O the density is d = 1 g/cm3, the molar mass is m = 18 g/mol so that the molecular 

volume is v = 3*10-23 cm3and (b/v)H2

O = -5.58*109

cm-2 = -5.58*10-7 Å

-2

. For D2O, d = 1.11 

g/cm3, m = 20 g/mol so that v = 3*10-23 cm3and (b/v)D2

O = 6.38*10-10 cm-2 = 6.38*10-6 Å

-2

. 

7. CONTRAST FACTORS

The scattering intensity is proportional to the contrast factor. Consider the simple case of 

scattering inhomogeneities consisting of A molecules in a background of B molecules (think 

polymers, proteins or micelles in solution). The scattering length densities are (bA/vA) and 

(bB/vB). The contrast factor often referred to as 

2

is defined as:

 

2

B

B

A

2 A

A B

2

v

b

v

b

















       . 

Scattering length densities can be changed through the deuteration process. Adjusting the 

relative amount of deuterated to non-deuterated solvent is called a contrast variation series 

and yields the zero contrast condition whereby the scattering length density of the mixed 

solvent matches that of the scattering inhomogeneities. 

8. MACROSCOPIC SCATTERING CROSS SECTIONS

Keeping the same notation for molecule AmBn, the macroscopic scattering cross section  is 

the product of the microscopic cross section per molecule (m+n) times the number of 

molecules per unit volume N/V. N is the total number of molecules and V is the total sample 

volume. For a sample comprising pure AmBn substance, V/N = v is the molecular volume. 

This applies to coherent c, incoherent i or total scattering s cross sections.

(m n)

V

N

c c 











 

(23)

(m n)

V

N

i i 











 

(m n)

V

N

s s 











  .
70

The number density of molecules is given in terms of the density d and molar mass m for 

molecule AmBn and Avogradro’s number Nav as:

m

N d

V

N av  











. (24)

For H2O, the macroscopic scattering cross sections per molecule are

c(H2O) = 

23

24

3*10

0.117 *10





= 3.9*10-3cm-1

i(H2O) = 

23

24

3*10

168.12*10





= 5.6 cm-1

s(H2O) = 

23

24

3*10

168.24*10





= 5.6 cm-1.

For D2O, the macroscopic scattering cross sections per molecule are

c(D2O) = 

23

24

3*10

15.35*10





= 0.512 cm-1

i(D2O) = 

23

24

3*10

4.17 *10





= 0.139 cm-1

s(D2O) = 

23

24

3*10

19.53*10





= 0.651 cm-1.

The results for H2O and D2O are summarized next. 

9. SUMMARY FOR H2O AND D2O

For the case of H2O

Coherent cross section per atom c(H2O) = 0.039 barn. 

Coherent cross section per H2O molecule is 3*c(H2O) = 0.117 barn.

Incoherent cross section per atom i(H2O) = 53.51 + 2.54 = 56.04 barn

Incoherent cross section per molecule is 3*i(H2O) = 168.12 barn. 

Total scattering cross section per atom s(H2O) = 56.08 barn

Total scattering cross section per molecule is 3*s(H2O) = 168.24 barn

Molecular volume v = 3*10-23 cm3

Scattering length density (b/v)H2

O = -5.58*10-7 Å

-2

Macroscopic coherent cross section c(H2O) = 3.9*10-3cm-1
71

Macroscopic incoherent cross section i(H2O) = 5.604 cm-1

Macroscopic total cross section s(H2O) = 5.608 cm-1

Table 1: Scattering lengths and cross sections (per atom) for water. In order to obtain cross 

sections per molecule, one has to multiply by the number of atoms per molecule (i.e., by 3). 

Mixing 

Fraction

b

Fermi

c

Barn

i-spin

Barn

i-comp

Barn

i

Barn

s

Barn

Hydrogen H 2/3 -3.739 ------ 80.27 ------ ------ ------

Oxygen O 1/3 5.803 ------ 0 ------ ------ ------

Water H2O 1 -0.558 0.039 53.51 2.54 56.04 56.08

For the case of D2O

Coherent cross section per atom c(D2O) = 5.12 barn. 

Coherent cross section per D2O molecule is 3*c(D2O) = 15.35 barn.

Incoherent cross section per atom i(D2O) = 1.37 + 0.021 = 1.39 barn

Incoherent cross section per molecule is 3*i(D2O) = 4.17 barn. 

Total scattering cross section per atom s(D2O) = 6.51 barn

Total scattering cross section per molecule is 3*s(D2O) = 19.53 barn

Molecular volume v = 3*10-23 cm3

Scattering length density (b/v)D2

O = 6.38*10-6 Å

-2

Macroscopic coherent cross section c(D2O) = 0.512 cm-1

Macroscopic incoherent cross section i(D2O) = 0.139 cm-1

Macroscopic total cross section s(D2O) = 0.651 cm-1

Table 2: Scattering lengths and cross sections (per atom) for heavy water. In order to obtain 

cross sections per molecule, one has to multiply by the number of atoms per molecule (i.e., 

by 3). 

Mixing 

Fraction

b

Fermi

c

Barn

i-spin

Barn

i-comp

Barn

i

Barn

s

Barn

Deuterium D 2/3 6.671 ------ 2.05 ------ ------ ------

Oxygen O 1/3 5.803 ------ 0 ------ ------ ------

Heavy

Water

D2O 1 6.382 5.118 1.367 0.0210 1.387 6.505

Note that natural hydrogen contains 99.985 % of the H-1 isotope and 0.015 % of the D (or H2) isotope. 

Two quantities are relevant to SANS measurements: the scattering length density that enters 

in the contrast factor and the macroscopic incoherent scattering cross section which appears 
72

as a constant (Q-independent) background. These two quantities are summarized for H2O and 

D2O. 

Table 3: Calculated scattering length densities and macroscopic scattering cross sections (per 

molecule) for water and heavy water. 

v

cm3

b/v

Å

-2

c

cm-1

i

cm-1

s

cm-1

Water H2O 3*10-23-5.583*10-70.00392 5.608 5.612

Heavy 

Water

D2O 3*10-23 6.382*10-60.512 0.139 0.651

The measured and calculated total cross section (T = s + a) for H2O and D2O are included 

in a table for thermal neutrons (wavelength of  = 1.8 Å). Note that the absorption cross 

sections are small. Cross sections were obtained from the Evaluated Nuclear Data File 

(ENDF) online database. This is the so-called “Barn Book”. 

Table 4: Measured and calculated macroscopic cross sections for H2O and D2O for thermal 

neutrons 

T Measured T Calculated

H2O 3.7 cm-15.612 cm-1

D2O 0.49 cm-10.651 cm-1

The calculated and measured values are different for a number of reasons including the 

neglect of inelastic scattering effects. 

10. GENERAL CASE

Consider the general case of a molecule AnABnBCnC... containing nA atoms of element A, etc. 

The total number of atoms per molecule is n = nA+nB+nC... and the number fractions are fA = 

nA/n, etc. 

The scattering length density is given by:

v

n b n b n b ...

v

b A A  B B  C C

 











(25)

Here v is the molecular volume and bA, bB, bC... are the tabulated scattering lengths. 

The macroscopic scattering cross section is given by:

     i i spin i comp V

N

   











 

(26)
73

N is the number of molecules in the sample of volume V. The spin incoherence and 

composition incoherence microscopic cross sections per molecule are given by:

  nf (A) f (B) f (C)... i spin

 Ai  Bi  Ci

(27)

   

2 2

i comp

  n {b }{b}

(28)

     

2

A A B B C C

2

C C

2

B B

2

i comp A A   n f b  f b  f b ... f b  f b  f b ...   nf f (b b ) f f (b b ) f f (b b ) ...

2

B C B C

2

A C A C

2 i comp

 A B A  B    

The relation fA + fB + fC...= 1 has been used. For pure substances, the molecules number 

density is (N/V) = 1/v where v is the molecular volume given by:

N d

m

v

av



(29)

m and d are the molecular mass and mass density and Nav is Avogadro’s number. 

11. TABULATED SCATTERING LENGTHS AND CROSS SECTIONS 

The coherent and incoherent thermal neutron scattering lengths and cross sections are 

tabulated here for a few elements (Koester, 1991; Sears, 1992). That table also contains the 

absorption cross section a for thermal neutrons (i.e., with a wavelength of  = 1.8 Å or a 

kinetic energy of 0.025 eV). Neutron absorption is small for most elements except for boron, 

cadmium and gadolinium (that are used as neutron absorbers). The absorption cross section is 

related to the imaginary part ba of the scattering length b-iba. Only neutron absorbing 

materials have an imaginary part. The absorption cross section is expressed as

k

4 {b }a

a



  . (30)

k is the incident neutrons wavenumber 

k  mv 

where 



is Planck’s constant, m is the 

neutron mass and v is the incident neutron speed. {ba} is the composition averaged 

absorption length. The absorption cross section varies like ~1/v where v is the neutron speed. 

The macroscopic absorption cross section is given by a = (N/V)a = a/Vsp where (N/V) is 

the number density and Vsp is the specific volume of the material.

Table 5: Coherent and incoherent thermal neutron scattering lengths (bcand bi) and cross 

sections (cand i) as well as absorption cross section (abs) for some nuclei. Note that 1 

fermi=10-13 cm and 1 barn=10-24cm

2

.
74

Element b

Fermi

c

Barn

i

Barn

s

Barn

a

Barn

H -3.739 1.757 80.30 82.057 0.333

D 6.671 5.592 2.05 7.642 0.000

C 6.646 5.550 0.001 5.551 0.003

N 9.36 11.01 0.50 11.51 1.90

0 5.803 4.232 0.000 4.232 0.000

F 5.654 4.232 0.001 4.233 0.000

Na 3.63 1.66 1.62 3.28 0.530

Si 4.149 2.163 0.004 2.167 0.171

P 5.13 3.307 0.005 3.312 0.172

S 2.847 1.017 0.007 1.024 0.53

Cl 9.577 11.526 5.3 16.826 33.5

The coherent scattering length for a mixture involves the mean and the incoherent scattering 

length involves the standard deviation. If one considers a fictitious sample comprising a few 

low-Z elements in equal amounts, then the deviation from the incoherent scattering length 

would be the deviation from the average of the scattering lengths plotted vs Z. 

3

4

5

6

7

8

9

10

0 5 10 15 20 25

plot_b_vs_z_2

scattering lengths 

coherent scattering length b

c Scattering Lengths (fermi)

Atomic Number Z

D C

N

O F

Na

average = b

coherent

b

incoherent

Figure 3: Variation of the scattering length for a few low-Z elements. 
75

12. NEUTRON TRANSMISSION

Neutrons incident on a sample have three choices: they either cross it without interaction, get 

scattered or get absorbed. Neutron absorption is small for most materials. The neutron 

transmission of a flat slab sample (appropriate for SANS measurements) of thickness d and 

total macroscopic cross section:

t s  a c  i  a

(31)

is given by:

T exp( d)  t

. (32)

Here d is the sample thickness. The transmission is measured as the ratio of the direct beam 

intensity with and without the sample. Transmission gets lower for longer neutron 

wavelengths. 

The total cross section can be estimated in terms of the sample transmission T and sample 

thickness d as: 

t  ln(T)/ d

. Note that the incoherent component is the dominant form of 

scattering from samples containing hydrogen in the flat high-Q region. 

13. MEASURED MACROSCOPIC CROSS SECTION FOR WATER

Water scatters mostly incoherently and is characterized by a flat (Q-independent) SANS 

signal. Water is used as a secondary absolute intensity standard since its macroscopic 

scattering cross section is well known. The SANS measured cross section corresponds to the 

macroscopic cross section T/4. The wavelength-dependent microscopic cross section was 

obtained from the Evaluated Nuclear Data File (ENDF) online database (the modern version 

of the so-called Barn Book).
76

3

4

5

6

7

8

9

0 2 4 6 8 10 12

 H2O 

Measured 

Macroscopic Cross Section 

T (cm-1

)

Neutron Wavelength (Å)

Y = M0 + M1*x + ... M8*x8 + M9*x9

M0 1.2865

M1 1.5188

M2 -0.13603

M3 0.0052822

R 0.99943

Figure 5: Variation of the macroscopic cross section T for H2O. Note that the cross section 

measured by SANS corresponds to T/4. 

14. CROSS SECTIONS FOR H2O/D2O MIXTURES

Often H2O/D2O mixtures are used to vary the contrast factor. Ignoring H/D exchange, the 

variation of the incoherent and coherent microscopic scattering cross sections i and c are 

summarized. The following scattering lengths and cross sections are used:

bH 3.739 fm

(33)

bO  5.803fm bD

 6.671fm
77  (H) 80.27 barn i spin  

 (O) 0 barn i spin  

 (D) 2.05 barn i spin   .

The incoherent cross section is dominated by the spin-incoherent part. Defining the D2O 

number fraction (relative number of molecules) as D2

O in an H2O/D2O mixture, the spinincoherent cross section is given by:

      i 2 2 spin D O i 2 spin D O i 2 spin (H O/ D O) (1 ) (H O) (D O)

2 2

       

(34)

 (H O) 160.54 barn i 2 spin  

 (D O) 4.10 barn i 2 spin  

 (H O/ D O) 160.54*(1 D O

) 4.10*

D O barn i 2 2 spin 2 2

      .

There are two levels of composition-incoherence; one for disorder within the (H2O or D2O) 

molecules and one for disorder in the H2O/D2O mixture. Contribution to the disorder within 

the molecules is given by:

  (b b ) 7.63 barn

3

2

(H O) 4

2 i 2 comp

  H  O 

(35)

  (b b ) 0.063 barn

3

2

(D O) 4

2 i 2 comp

  D  O  .

This is the cross section “per molecule”. Contribution to the composition incoherence for the 

H2O/D2O mixture is given by:

          

2

i 2 2 comp D2O D2O bD2O bH2O

(H O/ D O) 4 (1 )

    D O i 2 comp D O i 2 comp

(D O) (1- ) (H O)

2 2

    (36)

b b  4b b  4.33 barn 2

H D

2

D2O  H2O    .

Gathering the terms, one obtains:

 (H O/ D O) (1 D O

) D O 54.47 D O 0.063 (1- D O) 7.63 barn i 2 2 comp 2 2 2 2        

(37)

The total incoherent cross section is the sum of the two (spin- and composition-incoherent) 

contributions. It is equal to:

 i

(H2O/ D2O) (1 D O) D O 54.47 D O 4.16 (1- D O)168.17 barn

2 2 2 2

         .

The coherent scattering cross section for the mixture is given by:
78  

2

c D2O bH2O D2ObH2O   4*3 (1  )  

(38)

c 0.117 (1 D O

) 15.35 D O 1.342 D O(1 D O) barn

2 2 2 2

          .

The scattering length density for the H2O/D2O mixture is given by:

D2O H2O D2O D2O   (1  )   

(39)

7 2

H O 5.6*10 Å

2

 

  

6 2

D O 6.4*10 Å

2

 

  .

The incoherent and coherent cross sections and the scattering length density are plotted using 

different scales. 

0

50

100

150

200

-1 10-6

0

1 10-6

2 10-6

3 10-6

4 10-6

5 10-6

6 10-6

7 10-6

0 0.2 0.4 0.6 0.8 1

H2O/D2O Mixtures without H/D Exchange

total incoherent

composition incoherent

coherent

spin incoherent

scattering length density

Cross Sections (Barn) 

 Scattering Length Density (Å-2

)

 D2O Fraction 

Figure 6: Incoherent and coherent microscopic scattering cross sections for H2O/D2O 

mixtures without D/H exchange. 

Now assume that there is full H/D exchange to the point that the H2O/D2O/HDO mixture is 

randomized. The relative fraction of D2O molecules in the mixture is D2

O

2

. This is the 

probability of picking out a pair of D atoms from the randomized H/D mixture. The relative 
79

fraction of H2O molecules is (1-D2

O)

2

and the relative fraction of HOD and DOH molecules 

is 2D2

O(1-D2O). 

With full H/D exchange, the spin-incoherent cross section is the same before and after 

exchange.

      i spin D O i 2 spin D O i 2 spin (D O) (1 ) (H O)

2 2

       

        i 2 spin D O D O i spin

2

i 2 spin D O

2

i spin D O

(D O) (1 ) (H O) 2 (1 ) (DOH)

2 2 2 2

            

(40)

 (DOH) 82.32 barn i spin  

  4.10* 160.54*(1 ) 164.64 D O

(1 D O) barn 2

D O

2

i spin D2O 2 2 2

          .

Here also, there are two levels of disorder; one within the molecules and one for the 

molecular mixture. The composition-incoherent cross section becomes:

         

2

D O H O

2

D O

2

i 2 2 comp D2O 2 2 2

(H O/ D O) 4 (1 ) b b

       

2

D O D O D O HOD

2

D O 2 (1 ) b b

2 2 2 2

        

2

D O D O H O HOD

2

(1 D O) 2 (1 ) b b

2 2 2 2

   i 2comp D O D O icomp 2

i 2 comp D O

2

D O

(D O) (1- ) (H O) 2 (1- ) (HOD)

2 2 2 2

        

(41)

Note that:

   

2

D H

2

bD O bH O 4 b b

2 2

   (42)

   

2

D H

2

bD O bHOD b b

2

  

   

2

D H

2

bH O bHOD b b

2

  

  (b b ) (b b ) (b b )  8.38 barn

3

1

(DOH) 4

2

D H

2

H O

2 i comp

  D  O      . 

After manipulations, 

          

2

i 2 2 comp D O D O 2 bD bH

(H O/ D O) 4 (1 )

2 2

          

2

H O

2

D O

2

D O

2

D O b b

3

2

b b (1 )

3

2

2 2

     



















       

2

H O

2

H O

2

D O D O D O b b

3

1

b b

3

1

b b

3

1

2 (1 )

2 2

(43)
80         

2

i 2 2 comp D2O D2O 0.063 D2O

(H O/ D O) 27.23(1 )

7.63(1 ) 16.77 D O(1- D O) barn 2

D2O 2 2

     .

The total incoherent cross section is the sum of the two (spin- and composition-incoherent) 

contributions. It is equal to:

 (H O/ D O) 4.16* 168.17*(1 ) 181.41 D O

(1 D O) barn 2

D O

2

i 2 2 D2O 2 2 2

         

(44)

Including full H/D exchange, the coherent scattering cross section becomes:

 

2

H O D O D O HOD

2

H O D O

2

c 4 *3 (1 D O

) b b 2(1 ) b

2 2 2 2 2 2

           . (45)

This expression can be shown to reduce to the one obtained without D/H exchange:

 

2

c D2O bH2O D2ObH2O   4*3 (1  )  

(46)

c 0.117 (1 D O

) 15.35 D O 1.342 D O(1 D O) barn

2 2 2 2

          .

The scattering length density for the H2O/D2O mixture becomes:

H O D O D O HDO

2

H O D O

2

D O

(1 ) 2 (1 )

2 2 2 2 2 2

            

(47)

6 2

HDO 2.92*10 Å

 

  .

Here also, the incoherent and coherent cross sections and the scattering length density are 

plotted for the case of full H/D exchange. 
81

0

50

100

150

200

-1 10-6

0

1 10-6

2 10-6

3 10-6

4 10-6

5 10-6

6 10-6

7 10-6

0 0.2 0.4 0.6 0.8 1

H2O/D2O Mixtures with H/D Exchange

total incoherent 

coherent

composition incoherent

spin incoherent

scattering length density

Cross Sections (Barn) 

 Scattering Length Density (Å-2) 

 D2O Fraction 

Figure 7: Incoherent and coherent microscopic scattering cross sections for H2O/D2O 

mixtures with full D/H exchange. 

Note that just like the incoherent scattering cross section, the isothermal compressibility 

contribution is also Q-independent but is small. 

REFERENCES

L. Koester, H. Rauch, and E. Seymann, "Neutron Scattering Lengths: a Survey of 

Experimental Data and Methods", Atomic Data and Nuclear Data Tables 49, 65 (1991)

V.F. Sears, “Neutron Scattering Lengths and Cross Sections”, Neutron News 3, 26-37 

(1992). 

The Barn Book, Evaluated Nuclear Data File online database is found at the following web 

address http://www.nndc.bnl.gov/exfor7/endf00.htm. 

A scattering length density calculator can be found online at: 

http://www.ncnr.nist.gov/resources/sldcalc.html
82

QUESTIONS

1. Neutrons are scattered by what part of the atom?

2. Are higher fluxes achieved in research reactors (neutron sources) or in synchrotron x-ray 

sources?

3. Is deuteration always needed for neutron scattering?

4. What is the origin of the name for neutron cross sections (barn)?

5. Work out the relative composition of an H2O/D2O mixture that would have zero average 

coherent cross section (so called semi-transparent mixture). 

6. Comparing the coherent scattering cross sections for a deuterated polymer in hydrogenated 

solvent and a hydrogenated polymer in deuterated solvent, which one has the highest signalto-noise ratio for dilute solutions?

7. Why does carbon have a negligible incoherent scattering cross section?

8. What is the meaning of a negative scattering length?

9. Work out the scattering contrast for a mixture of your choice (or of your own research 

interest). 

10. Calculate the incoherent microscopic scattering cross section per molecule for H2O. 

11. Define the neutron transmission for a SANS sample. Does it depend on neutron 

wavelength? 

ANSWERS

1. Neutrons are scattered by the nuclei.

2. Synchrotron x-ray sources have much higher fluxes than neutron sources. 

3. Deuteration is not always needed for neutron scattering. Many systems are characterized 

by “natural contrasts”. Deuteration is however necessary to enhance the contrast of specific 

structures. 

4. The word barn was first used by Fermi in 1942 when initial measurements came up with 

estimates for the size of neutron-nuclear cross sections. These estimates were larger than 

expected (as large as a barn!). 1 barn = 10-24 cm2. 1 fm2 = 10-26 cm2 was expected. 

5. The scattering length density for H2O (density = 1 g/cm3) is -5.6*10-7 Å

-2

. The scattering 

length density for D2O (density = 1.11 g/cm3) is 6.38*10-6 Å

-2

. The H2O/D2O mixture that 

would give zero scattering length density corresponds to 92 % H2O and 8 % D2O. 

6. The contrast factor is the same for the two systems: deuterated polymer in hydrogenated 

solvent and hydrogenated polymer in a deuterated solvent (this is the so-called Babinet 

principle). However, for dilute polymer solutions the level of incoherent scattering 

background is lower when the solvent is deuterated. The signal to noise (contrast 

factor/incoherent background level) is higher when deuterated solvent and hydrogenated 

polymer is used. 

7. Carbon has a negligible incoherent scattering cross section because it has zero spin and 

therefore no spin-incoherence. 

8. A negative scattering length (such as in the case of H) means that the phase of the 

eigenfunction (solution of the Schrodinger equation) is shifted by 180 oduring the scattering 

process. 
83

9. Consider the polymer solution comprising poly(ethylene oxide) or hPEO in d-water or 

D2O. The following calculates the scattering lengths b, scattering length densities b/v and 

then the contrast factor. The specific volume v is needed in each case. 

hPEO: C2H4O, bhPEO = 4.139*10-13cm, vhPEO = 38.94 cm3/mol

d-water: D2O, bD2O = 19.14*10-13 cm, vD2O = 18 cm3/mol

Contrast Factor: (bhPEO/VhPEO – bD2

O/vD2O)

2 Nav = 5.498*10-3 mol/cm4

, Navis 

Avogadro's number.

10. Use the standard expression for the cross section per atom incoh(H2O) = 4fHfO(bH-bO)

2 + 

fHincoh(H) + fOincoh(O) where bH = -3.739 fm and bO = 5.803 fm are the coherent scattering 

lengths for H and O, incoh(H) = 80.27 barn and incoh(O) = 0 barn are the spin-incoherent 

scattering cross sections and fH = 2/3 and fO = 1/3 are the relative number fractions. Note that 

1 fm = 10-13 cm and 1 barn = 10-24 cm2. The result for the cross section per atom is 

incoh(H2O) = 56 barn. The cross section per molecule is 3incoh(H2O) = 168 barn. 

11. The SANS transmission from a flat sample of thickness d and total macroscopic 

scattering cross section t is given by T = exp(-td). Transmission decreases with increasing 

neutron wavelength. 
84

Part C – SANS TECHNIQUE AND INSTRUMENTATION

Chapter 10. The SANS Technique

10.1 Reciprocal Space

10.2 Comparing SANS to Other Techniques

10.3 The SANS Technique

10.4 The Measured Macroscopic Scattering Cross Section

10.5 Neutron Contrast Conditions

10.6 The Phase Problem

References

Questions

Answers

Chapter 11. The SANS Instrument

11.1 Continuous SANS Instrument Components

11.2 Time-of-Flight SANS Instrument Components

11.3 Sample Environments

11.4 SANS Measurements

11.5 SANS Instruments in the World

References

Questions

Answers

Chapter 12. Velocity Selectors and Time-of-Flight

12.1 Velocity Selectors Characteristics

12.2 Typical Velocity Selector

12.3 Velocity Selector Calibration by Time-Of-Flight

12.4 Other Wavelength Calibration Methods

12.5 Discussion

References

Questions

Answers

Chapter 13. Neutron Area Detectors

13.1 Neutron Detector Basics

13.2 Neutron Area Detector Specifics

13.3 Neutron Measurements

13.4 Other Neutron Detectors

References

Questions

Answers

Chapter 14. Sample Environments

14.1 Standard Sample Cells

14.2 Heating and Cooling Blocks

14.3 The Pressure Cell
85

14.4 The Couette Shear Cell

14.5 The Plate/Plate Shear Cell

14.6 Other Sample Environments

References

Questions

Answers
86

Chapter 10 - THE SANS TECHNIQUE

1. RECIPROCAL SPACE

Small-Angle Neutron Scattering (SANS) is a technique of choice for the characterization of 

structures in the nanoscale size range (Hammouda, 1995). This covers structures from the 

near Angstrom sizes to the near micrometer sizes. How small are the small angles? They are 

typically from 0.2 oto 20 oand cover two orders of magnitude in two steps. A low-Q 

configuration covers the first order of magnitude (0.2 oto 2 o) and a high-Q configuration 

covers the second one (2 oto 20 o). The scattering variable is defined as Q = (4/) sin(/2) 

where  is the neutron wavelength and  is the scattering angle. Within the small-angle 

approximation, Q simplifies to Q = 2/. The SANS scattering variable Q range is typically 

from 0.001 Å-1to 0.45 Å-1. This corresponds to d-spacings from 6,300 Å down to 14 Å. 

Scattering measurements are performed in the Fourier (also called reciprocal) space, not real 

space like microscopy. For this, scattering data have to be either inverted back to real space 

or fitted to models describing structures in reciprocal space. Scattering methods measure 

correlation functions. These are not the Fourier transform of the density of inhomogeneities 

within the sample. They are the density-density correlation functions instead. It should be 

noted that because of this, the “phase” information is completely lost. It is not possible to 

reconstruct a complete image of the sample structure by scattering from one sample. Trying 

to recover phase information is complicated and involves measuring a series of samples with 

identical structures but different contrasts. 

2. COMPARING SANS TO OTHER TECHNIQUES

The advantage of SANS over other small-angle scattering methods (such as small-angle xray or light scattering) is the deuteration method. This consists in using deuterium labeled 

components in the sample in order to enhance their contrast. This is reminiscent of contrast 

variation in microscopy whereby the level of light incident upon a sample is varied using a 

diaphragm. SANS can measure either naturally occurring contrasts or artificial contrasts 

introduced through deuteration. Labeling is difficult to achieve with x-rays (SAXS) since this 

involves heavy atom labels that change the sample drastically. SANS can measure density 

fluctuations and composition (or concentration) fluctuations. SAXS can measure only density 

fluctuation. The deuteration method allows this bonus. 

SANS is disadvantaged over SAXS by the intrinsically low flux of neutron sources (nuclear 

reactors or spallation sources using cold source moderators) compared to the orders of 

magnitude higher fluxes for x-ray sources (rotating anode or synchrotrons). Neutron 

scattering in general is sensitive to fluctuations in the density of nuclei in the sample. X-ray 

scattering is sensitive to inhomogeneities in electron densities whereas light scattering is 

sensitive to fluctuations in polarizability (refraction index). 
87

Microscopy has the advantage that data are acquired in direct (real) space whereas scattering 

methods (such as SANS) measure in reciprocal space. Electron microscopy (EM) and SANS 

are complementary methods. EM is applied on very thin samples only, it cannot measure 

samples at different concentrations and temperatures directly, and the observed images are a 

2D projection. SANS can do all these things but cannot produce an image in real space. 

SANS data analysis is performed at many levels. The initial level consists of “follow the 

trends” type of approach using standard plot methods. The next level uses nonlinear least 

squares fits to realistic models. The final trend makes use of sophisticated ab-initio or “shape 

reconstruction” methods in order to obtain insight into the structure and morphology within 

the investigated sample. Oftentimes, it takes independent information obtained from other 

methods of characterization to obtain a thorough understanding of SANS data because “most 

SANS data look alike”. SANS is not known for abundance in scattering peaks (unlike singlecrystal diffraction, Nuclear Magnetic Resonance, Infra-Red spectroscopy, etc) but enough 

features (i.e., “clues”) are available. Available models describe scattering from compact 

shape objects in dilute or concentrated systems as well as “non-particulate” scattering such as 

in the case of gel-like or porous media. SANS has been used for single-phase as well as 

multi-phase systems. Phase transitions have been investigated as well as the thermodynamics 

of demixing. 

Figure 1: Various classes of samples and morphologies investigated by SANS. 

3. THE SANS TECHNIQUE

Polymers

Spherical

Morphology

Cylindrical 

Morphology

Lamellar

Morphology

Micelles Biological

Systems

Bicontinuous 

Structure
88

SANS involves the basic four steps used in all scattering techniques: monochromation, 

collimation, scattering and detection. Monochromation is performed mostly using a velocity 

selector. Collimation is preformed through the use of two apertures (a source aperture and a 

sample aperture) placed far (meters) apart. Scattering is performed from either liquid or solid 

samples. Detection is performed using a neutron area detector inside an evacuated scattering 

vessel. The large collimation and scattering distances make SANS instruments very large 

(typically 30 m long) compared to other scattering instruments. 

Figure 2: This figure represents the schematics of the SANS technique. It is not to scale with 

vertical sizes in centimeters whereas horizontal distances are in meters. 

The SANS technique has been an effective characterization method in many area of research 

including Polymers, Complex Fluids, Biology, and Materials Science. Other areas such as 

magnetism also benefited from SANS. SANS instruments have been essential components 

for any neutron scattering facility for almost three decades. They provide the main 

justification for growth and are highly oversubscribed. New sample environments have given 

new momentum to the technique. These include in-situ shear cells, flow cells and rheometers, 

pressure cells, electromagnets and superconducting magnets, vapor pressure cells, humidity 

cells, in-situ reaction cells, etc. New advances in electronics, data handling methods and 

computers have made SANS a sophisticated “user friendly” characterization method for the 

non-experts and for “routine” characterization as well as cutting edge research. 

4. THE MEASURED MACROSCOPIC SCATTERING CROSS SECTION

Incident Beam

Area Detector

Scattered 

Beam Sample Q

Source

Aperture

Sample

Aperture

Monochromatic

Neutron Beam

Monochromation Collimation Scattering Detection
89

Consider a simple scattering system consisting of globular (think spherical) inhomogeneities 

in a matrix (think solvent). If this system is assumed to be incompressible, the SANS 

coherent macroscopic scattering cross section (scattering intensity in an absolute scale) can 

be modeled as:

V P(Q)S (Q)

V

N

d

d (Q)

I

2 2

P

c

 

















. (1)

(N/V) is the number density of particles, VP is the particle volume, 

2

is the contrast factor, 

P(Q) is the single particle form factor and SI(Q) is the inter-particle structure factor. Note that 

P(Q) and SI(Q) are normalized as follows: P(Q0) = 1, P(Q) = 0 and SI(Q) = 1. 

SI(Q) has a peak corresponding to the average particle inter-distance (the so-called 

coordination shell) in the case of “concentrated” system where the particle inter-distance is of 

the same order as the particle size. The inter-distance is much larger than the particle size for 

“dilute” system. 

The incoherent scattering cross section di/d = i/4 is a constant (Q-independent) 

background to be added to the coherent scattering level. Its contribution is mostly from 

hydrogen scattering in the sample. 

5. NEUTRON CONTRAST CONDITIONS

Consider a scattering system made of spheres in a solvent background. The following figures 

consider four types of contrast conditions: (1) finite contrast, (2) zero contrast for two 

component systems, (3) multiple contrasts and (4) the scattering length density match 

condition for three component systems. The scattering length density match condition 

corresponds to zero contrast for the blue spheres. 
90

Figure 3: Various contrast conditions. 

6. THE PHASE PROBLEM

The so-called “phase problem” affects all scattering methods because measurements are 

performed in reciprocal (Fourier) space. In order to explain the issue, let us consider the 

simple case of a scattering medium (think solvent) of scattering length density g (think 

“grey” color), and two set of structures, one comprised of “white” spheres of scattering 

length density w and one comprised of “black” spheres of scattering length density b. 

Assume that the white and black spheres are identical except for their scattering length 

densities (i.e., “color” as appearing to neutrons) that are opposite. Also assume that the white 

spheres are hydrogenated (w < g) and the black spheres are deuterated (b > g). 

Microscopy is sensitive to the following differences w-g <0 and b-g >0 whereas 

scattering methods are sensitive to the following “contrast factors” (w-g)

2 > 0 and (b-g)2

>0. Both are positive and therefore appear the same. In order to defeat the phase problem, a 

second sample is necessary whereby the scattering length density of the solvent matches that 

Finite contrast Zero contrast

Multiple contrasts Contrast match
91

of the black spheres for example (g = b). In this case the black spheres will be invisible and 

the white spheres will be distinct. 

Figure 4: Microscopy sees the white spheres and the black spheres as distinct. Scattering with 

one sample sees the black spheres and the white spheres as similar. A second sample (where 

the scattering length density of the solvent matches that of the black spheres) shows the white 

spheres. 

This is an oversimplified view of the more complex phase problem. The central aspect of the 

phase problem comes from the square nature in the form factor P(Q) = F2(Q). Consider the 

case of scattering from a sphere of radius R for which F(QR) = 3j1(QR)/QR where j1(QR) is 

the spherical Bessel function given by j1(QR) = sin(QR)/(QR)2-cos(QR)/QR. In order to 

obtain the variation of F(QR) from P(QR), one needs more information in order to 

reconstruct the negative values of F(QR). Here also, scattering from one sample does not 

suffice. 

Microscopy Scattering -

Same Sample

Scattering -

Second Sample
92

-0.2

0

0.2

0.4

0.6

0.8

1

0.1 1 10 100

 Sphere 

F(QR)

|F(QR)|

|F(QR)|2 Scattering Factors 

QR 

Figure 5: Comparison of the three scattering factors for a sphere F(QR, |F(QR)| and P(QR) = 

|F(QR)|2. 

The phase problem is resolved for x-ray single crystal diffraction by including heavy atoms 

in the structure and in neutron reflectometry and SANS by preparing samples with the same 

structures but different deuteration schemes. 

REFERENCE

B. Hammouda, “SANS from Polymers Tutorial”, NIST Center for Neutron Research Report 

(1995)

QUESTIONS

1. What is reciprocal space?

2. What is the phase problem? How to go around it?

3. What are the four basic steps involved in the concept of the SANS instrument?

4. What is the range of scattering angles used in SANS?

5. What are typical sample environments for in-situ SANS measurements?
93

6. What are the major SANS research areas?

7. Why are SANS instruments bigger than SAXS instruments? 

ANSWERS

1. Reciprocal space is the Fourier transform space. 

2. The phase problem is due to the fact the scattering contrast factor involves the square of 

the difference in the scattering length densities so that differences that are opposite in sign 

show the same contrast. The phase problem is resolved by preparing more than one sample 

with different deuteration strategies. 

3. The four basic steps are monochromation, collimation, scattering and detection. 

4. SANS uses scattering angles between 0.2 oand 20 oin two steps. 

5. In-situ SANS environments include: shear cells/rheometers, pressure cells, 

electromagnets/superconducting magnets, humidity cells, etc. 

6. SANS research areas include: polymers, complex fluids, biology, materials science, 

magnetism, etc.

7. SANS instruments are bigger than SAXS instruments because of the inherently lower flux 

neutron sources. Neutron current on sample is increased for SANS instruments by making 

larger samples. Larger samples imply longer SANS instruments in order to cover the same Q 

range. 
94

Chapter 11 - THE SANS INSTRUMENT

The first SANS instruments utilizing long flight paths, long wavelength neutrons from a 

reactor cold source and position sensitive detectors were developed over 40 years ago. Smallangle neutron scattering instruments should really be called low-Q instruments. Q is the 

scattering variable which is expressed in terms of the neutron wavelength  and low 

scattering angle  as Q = 2/. Low Q can be realized either through the use of small angles 

or long wavelengths (or both). In order to obtain small angles, good collimation and good 

resolution area detectors are needed. Good collimation is achieved through the use of long 

neutron flight paths before and after the sample. SANS instruments on continuous neutron 

sources use velocity selectors to select a slice of the (often cold) neutron spectrum while 

time-of-flight SANS instruments use a wide slice of the spectral distribution with careful 

timing between the source chopper and the detector to separate out the various wavelength 

frames. In this last case (TOF instruments) the maximum length of an instrument is 

determined by the pulse frequency so as to avoid frame overlap (whereby the fastest neutrons 

of one pulse would catch up with the slowest neutrons of the previous pulse).

1. CONTINUOUS SANS INSTRUMENT COMPONENTS

A brief description of the main components of reactor-based SANS instruments follows. This 

description covers the main features found on the NG3 30 m SANS instrument at the NIST 

Center for Neutron Research (Hammouda et al, 1993; Glinka et al, 1998). 

-- Cold neutrons are transported through total internal reflection at glancing angles inside 

neutron guides. These transmit neutrons from the cold source to the entrance of scattering 

instruments with little loss (1 % per meter). Neutron guides are coated with natural Ni or Ni58 which has a wider critical angle for reflection. This critical angle increases linearly with 

neutron wavelength thereby allowing more cold neutrons to reach the SANS instrument. 

Note that supermirrors (characterized by even higher critical angles) are not used due to the 

tight collimation requirement of SANS instruments; neutrons that experience too many 

reflections never make it through the tight SANS collimation. 

-- Beam filters (for example Be for neutrons and Bi for gammas) are used to clean up the 

beam from unwanted epithermal neutrons and gamma rays. Gammas are stopped by high-Z 

materials such as Bi. Be transmits neutrons with wavelengths > 4 Å. Note that if a curved 

guide is used, no crystal filter is needed because there is no direct line-of-sight from the 

reactor source (no gammas in the beam). Curved guides transmit only wavelengths above a 

cutoff value (no epithermal neutrons in the beam). Typical crystal filter thickness is between 

15 cm and 20 cm. For better efficiency, filters are cooled down to liquid nitrogen temperature 

(77 K = -196 oC). 

-- Optical filters are devices that stir a neutron beam away from the direct line-of-sight and 

replace crystal filters. They consist of tapered neutron guides that transmit only neutrons that 
95

are reflected. They have the advantage of gains in flux over crystal filters at long 

wavelengths. 

Figure 1: Schematic side view representation of an optical filter used on the NG3 SANS 

instrument at the NIST CNR facility. The reflecting guide surfaces are made out of Ni and 

Ni-58. Since there was no room horizontally, the neutron beam is steered vertically. This 

figure is not to scale. 

-- A velocity selector yields a monochromatic beam (with wavelengths  between 4 Å and 20 

Å and wavelength spreads  between 10 % and 30 %). Some SANS instruments that 

need sharp wavelength resolution use crystal monochromators (with wide mosaic spreads to 

give  < 10 %) instead. Since  is constant, the neutron spectrum transmitted by the 

velocity selector falls off as 1/

4

(instead of the 1/

5

coming from the moderator produced 

Maxwellian distribution).

15 cm

6 cm

Velocity

Selector

Optical Filter

Pre-sample

Collimation

Horizontal Distance from Cold Source

0 m 10 m 20 m 30 m 40 m
96

Figure 2: Schematic representation of a multi-disk velocity selector. A white neutron 

spectrum is incident from the left and a monochromated beam is transmitted toward the right. 

Changing the rotation speed controls the neutron wavelength. Tilting the selector horizontally 

changes the wavelength spread. Magnetic coupling is used to drive the selector rotation to 

high rotation speed. Temperature and vibration sensors insure reliable operation. 

-- The collimation usually consists of a set of circular (source and sample) apertures that 

converge onto the detector. An evacuated pre-sample flight path contains the beam 

collimation system. Typical adjustable flight path distances are from 1 m to 20 m depending 

on resolution and intensity considerations. Inside the pre-sample flight path, more neutron 

guides (with reflecting inner surfaces) are included in parallel with the collimation system for 

easy insertion into the beam. This allows a useful way to adjust the desired flux on sample 

along with the desired instrumental resolution by varying the effective source-to-sample 

distance. A normal configuration consists of a certain number of guides inserted into the 

neutron beam, followed by a source aperture right after, then a series of empty beam 

positions up to the sample aperture located inside the sample chamber. 
97

Figure 3: Cross section of a pre-sample flight path box showing an aperture, a neutron guide 

and an empty beam position that can be inserted in the beam using an actuator and a movable 

translation table. 

-- A sample chamber usually contains a translation frame that can hold many samples 

(measured in sequence). Heating and cooling of samples (-10 oC to 200 oC) as well as other 

sample environments (cryostats, electromagnets, ovens, shearing devices, etc) are often 

accommodated. The oversized sample environments are mounted on a 22” diameter Huber 

sample table instead. This sample table can be rotated around a vertical axis and translated in 

and out of the neutron beam. This translation is useful for moving the in-situ rheometer (for 

example) from the radial position to the tangential position. All of these motions are 

computer controlled. 

lead shield

linear actuator

neutron guide

aperture

empty beam position

movable table
98

Figure 4: Schematic drawing of the sample area showing the sample chamber on the right 

and a sample table for oversized environments on the left. Two gate valves are used to isolate 

the evacuated areas (pre-sample and post-sample flight path) when pumping down or 

evacuating the sample chamber. This softens the shock on the brittle neutron windows during 

such rapid pressure surges. 

-- A set of three main neutron windows are used: (1) One at the entrance to the pre-sample 

flight path. This window is before the source aperture and can be made out of quartz. (2) A 

second window just before the sample. This window is between the two defining apertures 

and could therefore produce unwanted diffuse scattering. It is often made out of sapphire 

(more expensive than quartz but with better neutron transmission). Sapphire is transparent 

thereby allowing a laser beam (parallel to the neutron beam) through for rapid sample holder 

alignment. The laser itself is installed on one of the collimation boxes and produces a beam 

that gets reflected (90 o) by a silicon mirror placed at 45 ofrom the beam axis. The silicon 

wafer reflects the laser beam but is transparent to neutrons. (3) A wide silicon window is 

used at the entrance to the scattering vessel (just after the sample). Silicon has the best 

neutron transmission and is the best window material when optical transparency is not 

required. These windows are between 3 mm and 6 mm thick. 

-- Precise alignment of sample blocks with respect to the sample aperture is performed using 

a “neutron camera”. A double exposure picture is taken with and without the sample 

aperture. A neutron camera is a regular flat camera outfitted with a scintillation plate (using 

material such as Li-6). 

pre-sample collimation

silicon window

sample aperture

Huber sample table

post-sample 

collimation

sample chamber

sapphire window
99

-- The post sample flight path is usually an evacuated cylindrical tube (to avoid scattering 

from nitrogen in air) that permits the translation of an area detector along rails in order to 

change the sample-to-detector distance. The vacuum level is kept at less than 100 mT. In 

order to evacuate such a large volume, a large capacity vacuum pump and a roots blower are 

used. 

-- The area detector is often a gas detector with 0.5 cm to 1 cm resolution and typically 

128*128 cells. The detection electronics chain starts with preamplifiers on the back of the 

detector and comprises amplifiers, coincidence and timing units, plus encoding modules and 

a means of histogramming the data and mapping them onto computer memory. In order to 

avoid extensive use of vacuum feedthroughs, high count rate area detector design 

incorporates most electronics modules (amplification, coincidence, encoding, etc) inside an 

electronics chamber located on the back of the detector. In this design, flexible hoses are, 

however, needed to ventilate the electronics and to carry the high voltage and powering 

cables in and the encoded signal out.

Figure 5: Schematic representation of a neutron area detector. 

-- Detector protection is performed in two ways: (1) through an analog monitoring of the 

total count rate and (2) through software monitoring of each detector cell count rate. If either 

the total detector count rate or a preset number of cells overflow, the data acquisition system 

pauses, the next attenuator is moved in and data acquisition is restarted. Typical presets are 

50,000 cps for the total detector count rate and 100 cps for a maximum of 10 cells. 

-- A set of beam stops is used to prevent the unattenuated main beam from reaching the 

detector and therefore damaging it due to overexposure. Use of glass seeded with Li-6 as 

neutron absorber avoids the gamma-ray background obtained with Cd, B or Gd containing 

3He

 + 

CF4

4He

detection 

chamber

membrane

dome electronics

chamber
100

materials. For easy alignment, motion of the beam stops should be independent of that of the 

area detector.

-- Between the velocity selector and the pre-sample collimation system, a low-efficiency 

fission chamber detector is used to monitor the neutron beam during data acquisition.

-- Just before the pre-sample collimation flight path a set of calibrated attenuators are used to 

attenuate the neutron beam. This system consists of a slab of plexiglass milled stepwise so as 

to provide attenuators of varying thickness. The insertion of this attenuator set is computer 

controlled. For example, if the area detector count rate is above a preset ceiling, the thinnest 

attenuator is moved into the beam by the data acquisition software. If this does not attenuate 

the beam enough, the next attenuator in thickness is moved in, etc. Another option for an 

attenuator system would be to use neutron absorbing material (such as cadmium) plates with 

holes milled into them. The density of holes would determine the attenuation factor. 

-- Gamma radiation produced by neutron capture in various neutron absorbing materials (Cd, 

Gd, B) is stopped using high-Z shielding materials (Fe, Pb, concrete). Shields surround the 

velocity selector (especially the front disk that produces most of the dose) and beam defining 

apertures. The scattering vessel is also shielded in order to minimize background radiation 

from reaching the detector.

-- The pre-sample and post-sample flight paths are often made out of non magnetic metals 

(like aluminum or non-magnetic steel) in order to allow the use of polarized neutrons. 

-- A neutron polarizer consists of a Fe/Si coating on 1 mm thick silicon plates aligned to form 

a V inside a copper-coated neutron guide. This polarizing cavity is 1.2 m long and polarizes a 

4*5 cm2neutron beam for a wavelength between 5 Å and 15 Å. Immediately following the 

polarizing cavity is a flat coil  spin flipper for reversing the direction of polarization. 

Permanent magnets maintain a 500 gauss vertical field to magnetize the supermirror coating 

and a 50 gauss field from the supermirror to the sample area to maintain neutron polarization. 

-- In order to avoid diffuse scattering from the beam defining apertures, these are tapered (5 o

taper angle is enough) and made out of material like boron nitrite or lithiated glass with the 

smaller inner edge made out of cadmium. This keeps the beam sharp and any emitted gamma 

radiation to a minimum. 

-- Data acquisition is computer controlled within menu-driven screen management 

environments and on-line imaging of the data is usually available. Encoded 2D data are 

received from the area detector electronics, binned into histogramming memories, then 

regularly displayed and saved. The data acquisition software interface also controls the 

various peripheral functions such as controlling the various motors, the sample 

heating/cooling protocols, and handshaking with the various other stand-alone sample 

environments (shear cells or rheometers, pressure cells, etc). 
101

-- There are many figures of merit used to judge the performance of SANS instruments. 

These include: instrumental resolution, minimum scattering variable (Qmin), flux on sample, 

dynamic Q range and background level. 

Figure 6: Schematics of a 30m SANS instrument at NIST.

velocity 

selector collimation

sample

area

area 

detector

beam

shutter

15 m 15 m

BEAM BEAM

 ON OFF
102

Figure 7: Photograph of the NG3 30 m SANS instrument. The picture was taken from the 

bridge walkway just before the velocity selector shield. 

Table 1: 30 m NIST-SANS Instruments Characteristics.

________________________________________________________________________

Source: neutron guide (NG3), 6 * 6 cm2

Monochromator: mechanical velocity selector with variable speed and pitch

Wavelength Range: variable from 5 Å to 20 Å

Wavelength Resol.: 10 to 30 % for  (FWHM)

Source-to-Sample Dist.: 3.5 to 15 m in 1.5 m steps via insertion of neutron guides

Sample-to-Detector Dist.: 1.3 to 13.2 m continuously variable for NG3

Collimation: circular pinhole collimation

Sample Size: 0.5 to 2.5 cm diameter

Q-range: 0.001 to 0.6 Å

-1

Size Regime: 10 to 6000 Å

Detector: 64 * 64 cm2He-3 position-sensitive ORDELA type

proportional counter (0.508 * 0.508 cm2spatial resolution)

________________________________________________________________________

Table 2: Short list of ancillary equipment used on SANS.

________________________________________________________________________

Ancillary Equipment: - Automatic multi-specimen sample changer with 

temperature control from -10 to 200 oC.

- Electromagnet (0 to 9 Tesla).

- Couette flow shearing cell, plate/plate shear cell,

in-situ rheometers.

- Cryostats and closed cycle refrigerators (1 K to 300 K).

- Oven for in-situ use (300 K to 1800 K).

- Pressure cell (0 to 1*108Pa, 25 oC to 160 oC).

________________________________________________________________________

2. TIME-OF-FLIGHT SANS INSTRUMENT COMPONENTS

In order to avoid frame overlap, time-of-flight SANS instruments tend to be shorter at typical 

pulsed sources. TOF SANS instruments comprise some of the main features described above 

(collimation, sample chamber, flight paths, area detector, etc) as well as some specific 

features described here:

-- A source chopper is used to define the starting neutron pulse.
103

-- The area detector is synchronized to the source chopper so that a number of wavelength 

frames (for example 128) are recorded for each pulse. No monochromator is necessary with 

the time-of-flight method.

-- A supermirror bender can be used to remove short wavelengths and let the instrument get 

out of the direct line of sight from the source. Note that curved guides have a cutoff 

wavelength below which neutrons are not transmitted. This bender replaces the crystal filter.

-- High wavelengths (say above 14 Å) have to be eliminated in order to avoid frame overlap. 

This can be done by gating the detector or through the use of frame overlap mirrors.

Reflecting mirrors are set at a slight angle (1 o) from the beam direction so as to reflect only 

long wavelength neutrons (note that the reflection critical angle varies linearly with 

wavelength). 

-- Prompt gamma rays emitted during the spallation reaction are eliminated by paralyzing the 

detection system for the first microsecond after each pulse. 

Because of the wide wavelength range used in time-of-flight instruments, materials that 

display a Bragg cutoff (such as sapphire windows) cannot be used. Data reduction becomes 

more complex with time-of-light instruments since most corrections (transmission, monitor 

normalization, detector efficiency, linearity, uniformity, etc) become wavelength dependent. 

Time-of-flight instruments have the advantage, on the other hand, of measuring a wide Q 

range at once. Also the large number of wavelength frames can be kept separated therefore 

yielding very high wavelength resolution ( < 1 %) which is useful for highly ordered 

scattering structures (characterized by sharp peaks). 

3. SAMPLE ENVIRONMENTS

Typical sample thickness for SANS measurements is of order of 1 mm for hydrogenated 

samples and 2 mm for deuterated samples. Liquid samples (polymer solutions, 

microemulsions) are often contained in quartz or demountable cells into which syringes can 

be inserted. Solid polymer samples are usually melt-pressed above their softening 

temperature, then confined in special cells between quartz windows. 

Flexibility of design for some instruments allows the use of typical size samples under 

temperature control or bulky sample environments. Temperature is easily varied between 

ambient temperature and 200 oC using heating cartridges or between -10 oC and room 

temperature using a circulating bath. Other sample environment equipment such as lowtemperature cryostats (4 to 350 K) and electromagnets (1-10 Teslas) are sometime made 

available to users. Various shear cells (Couette, plate-and-plate, in-situ rheometers, etc) help 

probe "soft" materials at the molecular level in order to better understand their rheology. 

Pressure cells are also finding wide use for investigations of compressibility effects on the 

thermodynamics of phase separation as well as on structure and morphology.
104

4. SANS MEASUREMENTS

SANS measurements using cold neutrons take from a few minutes to a few hours depending 

on the scattering sample, the neutron source and the instrument configuration used. The 

process starts by sample preparation, loading into cells and measurement of the sample 

thickness. 

A reasonable instrument configuration is chosen at first by setting a low wavelength (5 to 6 

Å) and varying the sample-to-detector distance so as to optimize the desired Q-range. If the 

maximum available sample-to-detector distance of that instrument is reached, wavelength is 

then increased. Choice of the source-to-sample distance, wavelength spread, and aperture 

sizes are dictated by the desired instrumental resolution (sharp scattering features require 

good resolution) and flux on sample. Scattered intensity is controlled by many factors that 

have to be optimized. Transmission measurements are required as well. In order to avoid 

complicated multiple scattering corrections, sample transmissions are kept high (> 60 %). 

Many experiments require a wide Q range covering two orders of magnitude (from Q = 0.003 

Å

-1

to Q = 0.3 Å

-1

). This range is obtained over two instrument configurations. In order to 

improve counting statistics, a third configuration is often used. The use of focusing lenses 

lowers the minimum Q down to slightly below Q = 0.001 Å

-1

. 

A complete set of data involves measurements from the sample, from an incoherent (usually 

nondeuterated) scatterer that yields a flat (Q-independent) signal, from the empty cell and 

blocked beam and from a calibrated (absolute standard) sample. The beam flux measurement 

method (measurement of the direct beam transmission) can be used to replace the absolute 

standard measurement. 

SANS data are corrected, rescaled to give a macroscopic cross section (units of cm-1) then 

averaged (circularly for isotropic scattering or sector-wise for anisotropic scattering). 

Reduced data are finally plotted using standard linear plots (Guinier, Zimm, Kratky, etc) in 

order to extract qualitative trends for sample characteristics (radius of gyration, correlation 

length, persistence length, etc) or fitted to models for more detailed data analysis. 

5. SANS INSTRUMENTS IN THE WORLD

Since the first SANS instrument went into operation at the Institut Laue Langevin (Grenoble, 

France) in the mid-1970s, many more SANS instruments have been built. Every neutron 

scattering facility has at least one such instrument. The SANS technique has managed to 

keep a steady growth and a high level of user subscription. A web site keeps a SANS 

instruments directory in the world (http://www.ill.fr/lss/SANS_WD/sansdir.html). 
105

REFERENCES

B. Hammouda, S. Krueger and C. Glinka, "Small Angle Neutron Scattering at the National 

Institute of Standards and Technology", NIST Journal of Research 98, 31-46 (1993).

C.J. Glinka, J.G. Barker, B. Hammouda, S. Krueger, J. Moyer and W. Orts, “The 30m SANS 

Instruments at NIST”, J. Appl. Cryst. 31, 430-445 (1998). 

K. Ibel, “World Directory of SANS Instruments”, available online at the address 

http://www.ill.fr/lss/SANS_WD/sansdir.html

QUESTIONS

1. Why are small-angle neutron scattering instruments bigger than small-angle x-ray 

scattering instruments?

2. Why are crystal monochromators not used (instead of velocity selectors) in SANS 

instruments?

3. Could one perform SANS measurements without using an area detector?

4. What is the useful range of cold neutron wavelengths?

5. When is it necessary to use wide wavelength spread ?

6. Find out how a velocity selector works?

7. How does a He-3 area detector work?

8. What is the cost of building a SANS instrument?

9. Name some materials used for neutron windows.

10. Do cold neutrons destroy samples?

11. Why are time-of-flight SANS instruments short?

ANSWERS

1. Neutron fluxes are lower than x-ray fluxes. SANS samples are made bigger than SAXS 

samples in order to enhance the neutron current on sample. Bigger samples require larger 

flight paths in order to cover the same Q range. 

2. Crystal monochromators are characterized by narrower wavelength spreads than velocity 

selectors and therefore lower neutron currents. Moreover, the use of a crystal monochromator 

would require the pivoting of the entire SANS instrument around the monochromator axis in 

order to change the neutron wavelength because they operate in reflection geometry. 

Velocity selectors operate in transmission geometry. 

3. SANS measurements can be performed using an end-window or a 1D position-sensitive 

detector. Count rate would however be prohibitively low. 

4. Cold neutron wavelengths range from 4 Å to 20 Å. The range used is effectively from 5 Å 

to 12 Å. Longer wavelengths are characterized by low fluxes. 

5. SANS uses wide wavelength spread in order to increase the neutron current on sample. 
106

6. Velocity selectors rotate at a specific speed for every neutron wavelength. The helical 

selector slot lets neutrons of the right speed through. Those that are either too slow or too fast 

are absorbed and never get transmitted. 

7. He-3 absorbs a neutron to give two charged particles: a proton (H-1) and a triton (H-3). 

These two charges create an electron detection cloud that drifts towards the anode (at high 

voltage) and therefore get sensed by the cathodes. 

8. A SANS instrument costs as much as its level of sophistication. A deluxe model costs over 

$ 1 million. 

9. Neutron windows have to be very transparent to neutrons. Silicon is the most transparent 

but is opaque to visible light. Sapphire is less transparent to neutrons, very transparent to 

visible light but rather expensive. Quartz is like sapphire but less expensive. In practice, 

sapphire is used for windows before the sample. They can transmit neutrons as well as let a 

laser beam through for sample alignment purposes. After the sample, silicon windows are 

used. 

10. Cold neutrons do not destroy most samples. Unlike x-rays they do not heat them up. 

Samples containing elements that can be activated by neutrons (such as Fe for example) have 

to be handled differently. Most SANS samples (polymers, complex fluids and biology) 

contain organic molecules that do not get activated (C, H, D, O, N, etc). 

11. Time-of flight SANS instruments are short in order to avoid frame overlap (when the fast 

neutrons of one pulse catch up with the slow neutrons of the previous pulse). 
107

Chapter 12 - VELOCITY SELECTORS AND

TIME-OF-FLIGHT MEASUREMENTS

Characteristics of velocity selectors and time-of-flight measurement of the neutron 

wavelength distribution are described. Monochromation is a necessary step for SANS 

instruments that do not use the time-of-flight method. Continuous SANS instruments use 

velocity selectors instead of crystal monochromators. The basic concept for velocity selectors 

is to allow neutrons to travel in a rotating helical path. Neutrons that are either too fast or too 

slow get absorbed. Only neutrons with the right velocities are transmitted thereby 

transforming a white incident neutron spectral distribution into a monochromated distribution 

with mean wavelength  and wavelength spread (FWHM) . Velocity selectors are either of 

the solid drum type (with helical slot) or of the multidisk type. 

Figure 1: Schematics of a velocity selector explicitly showing three absorbing sectors and 

two transmitting windows between them. 

1. VELOCITY SELECTORS CHARACTERISTICS

Velocity selectors rotate at constant frequency  which is varied to change the transmitted 

neutron wavelength . A typical selector has an overall length L (length of the rotating 

“drum”) and a radius R (between the selector rotation axis and the neutron window). The 

helical path is characterized by a pitch angle . This is the angle by which the selector rotates 

while neutrons cross its length L. 

Equating the time it takes for neutrons to travel that distance L to the time it takes the selector 

to rotate the angle  gives a relationship between the neutron wavelength  and the rotation 

speed . 

FRONT

VIEW

SIDE

VIEW absorbing

region transmitting

region




108

Lmω

αh   . (1)

Here h is Planck’s constant, and m is the neutron mass. This relationship is expressed in more 

convenient units as:

ω[rpm]L[mm]

α[deg] [Å] 6.59*105

  . (2)

Here, the selector rotation frequency  is given in rpm (rotations per minute). 

The selector transmission is proportional to the uncovered area of the input face. Two angles 

are defined.  is the angle subtending the transmitting window and  is the angle 

corresponding to the absorbing region between two transmitting windows. The selector 

transmission is expressed as the following ratio:

β δ

β

T



 . (3)

The wavelength spread  is defined as the FWHM of the selector output distribution. The 

selector resolution (relative wavelength spread) is simply expressed as the ratio of the two 

relevant angles. 







λ

Δλ

. (4)

Note that  is independent of  so that the incoming 

-5 wavelength distribution from the 

neutron source becomes 



















4

1

after the selector. Since  is constant, this becomes a 

1/

4

distribution. The transmitted wavelength distribution is of the triangular form with 

slightly rounded angles (as will be described later). 

To decrease the wavelength resolution, one can either (1) decrease the transmitting window 

angle , or (2) increase the pitch angle  at the design stage. The first possibility is limited by 

the accompanying loss in selector neutron transmission. The second possibility comes with 

an increase in rotation speed in order to reach the same wavelength. In order to keep the same 

rotation speed, both the pitch angle  and the selector length L could be varied 

proportionately. 

In order to change the wavelength spread of a selector, the selector axis is tilted (in the 

horizontal plane) by an angle  relative to the beam axis thereby modifying the effective 

pitch angle as: 
109

R

νL

eff    . (5)

Tilting modifies both the wavelength spread  and the neutron wavelength . 

Figure 2: Schematic top view of a velocity selector showing the horizontal tilt angle . 

2. TYPICAL VELOCITY SELECTOR

The following parameters correspond to a velocity selector used on a 30 m SANS instrument 

at the NIST Center for Neutron Research (Hammouda, 1992). It is a multi-disk unit of 

Hungarian type design. 

Selector length L = 42 cm.

Radius to the middle of the window R = 17.5 cm.

Helix rotation angle  = 17o.

Open window aperture angle  = 2.25o.

Absorbing region between windows  = 0.75o.

Number of disks n = 22.

These design parameters yield the following predicted selector characteristics.

Neutron transmission T = 0.75.

Wavelength Å] = 2.669*104/[rpm].

Relative wavelength spread  = 0.132 for a tilt angle of  = 0. 

Effect of tilt angle  on the helix angle eff = 17o + 2.4 .

Therefore 

17 2.4 ν

2.25

o

o



 















.

These are theoretical numbers predicted based on design parameters. Measured 

characteristics using the time-of-flight method are described in the following section. 

tilt angle  incident white 

neutron beam

transmitted 

monochromatic 

beam
110

3. VELOCITY SELECTOR CALIBRATION BY TIME-OF-FLIGHT

The time-of-flight (TOF) method consists in chopping the neutron beam (using a rotating 

chopper) and gating a neutron detector with the time-zero chopper pulse. All neutrons cross 

the chopper at the same time despite their spread in velocities. As neutrons travel beyond the 

chopper, they spread out with faster neutrons arriving to the detector first. The neutron pulse 

is sharp at the chopper level and becomes spread out at the detector level. 

Here the time-of-flight method is used to characterize the wavelength distribution coming out 

of the velocity selector described earlier. The setup consists of a neutron disk-chopper 

followed by two pencil detectors installed close to the SANS instrument sample area. The 

pencil detectors have a diameter of 1.27 cm. The second pencil detector is used for 

redundancy and in order to obtain an exact measurement of the SANS sample-to-detector 

distance. The two pencil detectors are located 0.5 m apart and the distance between the 

second pencil detector and the area detector is 3 m. The chopper has a vertical neutron slit 1 

mm wide and 1.27 cm tall. A fixed slit aperture of the same size is located just ahead of the 

chopper in order to define the neutron beam. The chopper rotation is synchronized with the

neutron detectors data acquisition system through the time-zero pulse. 

Figure 3: Schematic representation of the time-of-flight setup comprising a chopper, two 

pencil detectors and the area detector. The fixed slit aperture is located just before the 

chopper. 

chopper

pencil 

detector 1

pencil 

detector 2

fixed slit area detector

aperture
111

A multi-channel scaler electronic unit was used to record the neutron pulses from the neutron 

detectors using the gated signal from the chopper. A typical spectrum corresponding to a 

wavelength around 6 Å, a source-to-chopper distance of 14.27 m, a detector 1-to-detector 2 

distance of 0.5 m and a pencil detector 2-to-area detector distance of 3 m is shown below. 

250

200

150

100

50

0

0 2 4 6 8 10 12 14 16

x103

Figure 4: Neutron spectrum obtained by time-of-flight. The first two peaks were recorded by 

the two pencil detectors and the third peak was recorded by the SANS area detector. The 

horizontal axis is in time channel numbers (0.5 sec/channel) and the vertical axis is in 

neutron counts. The chopper frequency was set to 113 Hz. 

Fits of the various peaks to Gaussian shapes were performed in order to obtain peak positions 

and standard deviations. Peak positions yielded flight times (and therefore wavelengths) and 

standard deviations yielded wavelength spreads. 

Wavelength Measurement

Knowing the distances between any two detectors, the neutron wavelength is proportional to 

the measured flight time between them and inversely proportional to the inter-distance 

between them.
112

v[m/sec]

3.956*10 [Å]

3

 

(6)

v[m/sec] = inter-distance[m]/flight time[sec]. 

The velocity selector rotation speed was varied and the neutron wavelength was measured in 

each case. Using different detector pairs gave the same result. 

6

8

10

12

14

16

0.0002 0.00025 0.0003 0.00035 0.0004 0.00045 0.0005 0.00055 0.0006

TOFF

using detectors 2 and 3

using detectors 1 and 2

Wavelength (Å)

Inverse Rotation Frequency [1/rpm]

Y = M0 + M1*X

M0 0.074128

M1 26910

R 0.99996

Figure 5: Variation of the measured wavelength with inverse velocity selector rotation 

frequency. 

A linear fit to the [Å] with 1/[rpm] gives the following measured relationship:

[Å] = 0.0741 + 2.691*104/[rpm]. (7)

The measured slope of 2.691*104agrees with the predicted one of 2.669*104reported earlier. 

Variation of the neutron wavelength with inverse rotation frequency is plotted for three tilt 

angles. 
113

5

10

15

20

25

0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014

Time of Flight Calibration

NG3 SANS Instrument

-3 deg tilt 

0 deg tilt 

+3 deg tilt 

Neutron Wavelength (Å) 

 1/Frequency 

0 deg tilt -3 deg tilt +3 deg tilt

Figure 6: Variation of the measured wavelength with inverse frequency for three tilt angles. 

Wavelength Spread Measurement

Gaussian fits to the measured spectra gave average flight times t1, t2 and t3 and standard 

deviations 1, 2 and 3 for the peaks corresponding to detectors 1, 2 and 3 (recall that 1 and 

2 are pencil detectors and 3 is the area detector). The relative wavelength spread obtained 

from detectors 1 and 3 is obtained as:

3 1

3 1

3 1

2

1

2

3

1,3 L L

L L

(t t )

2.355







  

 















. (8)

Subtracting 1

2

insures that smearing contributions from the chopper’s finite size slit (1 mm 

wide) and from the pencil detector’s finite width (1.27 cm diameter) are removed. The factor 

2.355 = 2(2ln2)1/2 is used to convert the standard deviation of the Gaussian shaped 

distribution  into a full-width at half maximum (FWHM);  = 2(2ln2)1/2. 
114

The last term 

3 1

3 1

L L

L L





is obtained through the following argument. The variance of the 

pulse time at a distance L3 from the chopper is given by:

(L ) L (0)

2

t

2 2

3 3

2

t    . (9)

Here 

(0) t

is the standard deviation at the chopper position and  is the standard deviation 

of the time-of-flight distribution. Applying this relation to two positions L1 and L3, one 

obtains the following relation:

3 1

3 1

3 1

2

1

2

3

L L

L L

(t t ) 





  

 















 













 

. (10)

In practice L1 << L3 so that the last term (square root ratio) becomes unity.

Varying the velocity selector tilt angle decreases the relative wavelength spread. 

0.1

0.15

0.2

0.25

0.3

-4 -3 -2 -1 0 1 2 3 4

Using detectors 1 and 3

Relative Wavelength Spread

Tilt Angle (degrees)

Figure 7: Variation of the measured relative wavelength spread with increasing selector tilt 

angle. 
115

The measured relative wavelength spreads () corresponding to the three measured tilt 

angles (-3

o

, 0oand +3o) are 0.269, 0.141 and 0.113. 

The measured inverse of the relative wavelength spread gives the following linear variation 

with the tilt angle .

   















6.55 0.85 . (11)

This variation is far from the predicted value of

   















7.55 1.07 . (12)

The wavelength spread is a very sensitive measurement to make. This is due to many factors: 

the assumption of Gaussian shape (for fitting purposes), smearing due to the defining slit’s 

finite width, smearing due to the detectors finite detection depth, etc... For example, the 

pencil detectors are 1.27 cm in diameter and the area detector has a detection depth of 2.54 

cm. Measured wavelength spreads are expected to be larger than predicted ones. 

Wavelength Distribution Profile

60

40

20

0

-20

10 11 12 13

x103

Figure 8: Wavelength distribution peak measured using the area detector with a wavelength 

around 6 A, a source-to-chopper distance of 14.27 m (corresponding to 1 pre-sample 

collimation guide inserted), and a pencil detector 2-to-area detector distance of 3 m. This 

distribution is characterized by a triangular shape. 
116

150

100

50

0

-50

10 11 12 13

x103

Figure 9: Wavelength distribution peak measured using the area detector with a wavelength 

around 6 A, a source-to-chopper distance of 3.38 m (corresponding to 8 pre-sample 

collimation guides inserted), and a pencil detector 2-to-area detector distance of 3 m. This 

distribution is characterized by a rounded Gaussian shape. 

The source-to-chopper distance is varied by inserting neutron guides into the SANS 

instrument’s pre-sample flight path. The monochromated neutron beam is therefore reflected 

(by the guides surface) before reaching the chopper. The first case shown corresponds to 1 

guide inserted (source-to-chopper distance of 14.27 m) whereas the second case shown 

corresponds to 8 guides inserted (source-to-chopper distance of 3.42 m). The fist case is 

characterized by a triangular wavelength distribution whereas the second case is 

characterized by a rounded up Gaussian distribution (due to time-of-flight smearing). 

Inserted neutron guides introduce smearing to the wavelength distribution because they 

smear the flight time distribution. 

Assuming a triangular wavelength distribution, the second moment is expressed as:





























  



     

2

2 2

6

1

1 . (13)

Assuming a Gaussian wavelength distribution, this quantity is expressed as:





























  



     



2

2 2

1 . (14)
117

Here 



is the variance of the Gaussian distribution defined as 

2 = <2> - <>2

. Recall that 

the FWHM of a Gaussian distribution defined as  is given by  = 2(2ln2)1/2 = 2.355. 

In this case:





























  



     

2

2

2 2

2.355

1

1 . (15)

In order to simplify the notation, (/<>) is often represented by (). 

Discussion

Wavelength calibration can be performed by either measuring a sample with a well-defined 

SANS peak or by the time-of flight method. The measured neutron wavelength agrees with 

prediction whereas it is hard to precisely predict the wavelength spread. Many time-smearing 

(or pulse-broadening) factors contribute to the uncertainty in wavelength spread. For 

example, increasing the chopper frequency decreases this time smearing. Time-of-flight 

calibration measurements are better performed with a high chopper frequency even if peaks 

corresponding to consecutive time frames overlap. It is easier to unravel what peak 

corresponds to what time frame than introduce a systematic uncertainty due to changing 

chopper frequency. 

The finite depth of the detector volume in area detectors introduces more pulse broadening. 

Moreover, increasing the neutron wavelength decreases the sample-to-detector distance 

measurably (by as much as 1.27 cm equivalent to the active depth up to the anode plane in 

the area detector) because slower neutrons are stopped closer to the entrance side of the 

detection volume. This is due to the “1/v”-dependence of the neutron absorption crosssection in He-3. 

Inserting neutron guides between the velocity selector and the sample (done to reduce the 

SANS source-to-sample distance) rounds off the edges of the neutron spectral distribution 

from a triangular shape to a Gaussian shape. This increases the wavelength spread. 

Such “second order effect” corrections could include (1) slight variation of the wavelength 

spread with wavelength and with number of guides in the incident pre-sample flight path 

collimation, and (2) slight variation of the sample-to-detector distance with wavelength.

Some of the issues discussed here are essential in understanding the resolution of time-offlight (TOF) SANS instruments located at pulsed neutron sources. 

The Graphite Bragg Diffraction Edge

In order to independently check the wavelength calibration, it is nice to use other methods. 

The Bragg edge method is reliable. The Bragg law  = 2d sin(/2) relates the neutron 
118

wavelength , the d-spacing of a crystal d, and the scattering angle . The Bragg edge occurs 

when the incident neutrons are parallel to the crystal lattice planes. This is obtained when 

 = 0oor 180o. There is a drop in diffraction intensity at that condition (neutrons are 

transmitted through rather than diffracted). Note that the scattering angle is defined as  in 

SANS terminology (not as 2 as done in some diffraction books). 

Polycrystalline graphite is characterized by a Bragg diffraction edge at 6.708 Å. Placing a 

graphite block in the neutron beam along with the time-of-flight setup gives an independent 

check of the wavelength calibration. A pencil detector and the area detector are used in the 

time-of-flight setup. The spectrum shown in the figure corresponds to a neutron wavelength 

around 6.7 Å, a source-to-chopper distance of 5.42 m, and a pencil detector-to-area detector 

distance of 6 m. Here only one pencil detector and the area detector are used. A chopper 

frequency of 70 Hz and a dwell time of 1 sec were used. The first sharp peak corresponds to 

the pencil detector located just after the chopper. The broad peak corresponds to the area 

detector and shows the graphite edge. The second narrow peak corresponds to the next 

neutron pulse on the pencil detector. 

160

140

120

100

80

60

40

20

0 Counts

16x1030 2 4 6 8 10 12 14

Channel Number - 1

Figure 10: Time-of flight spectrum using a pencil detector and the area detector and placing a 

(4 cm thick) polycrystalline graphite block just after the pencil detector. Graphite is 

characterized by a sharp Bragg diffraction edge at 6.708 Å (located by the arrow). 

4. OTHER WAVELENGTH CALIBRATION METHODS

There are other methods to calibrate the neutron wavelength based mostly on scattering 

samples that are characterized by Bragg peaks in the SANS range. Here Silver Behenate and 

Kangaroo tail tendon are discussed briefly. 
119

Silver Behenate

Silver Behenate is characterized by a sharp Bragg ring with a d-spacing of 58.38 Å. It is 

useful for a “quick” wavelength check. It cannot, however, be used to determine the 

wavelength spread. 

0.4

0.6

0.8

1

1.2

1.4

1.6

0.05 0.1 0.15 0.2 0.25 0.3 0.35

Silver Behenate 

measured

Scattering Intensity (cm-1

)

Scattering Variable Q (Å-1)

Figure 11: SANS spectrum from Silver Behenate showing a sharp first peak at Q = 0.1076 Å1

(d-spacing of 58.38 Å). 

Kangaroo Tail Tendon

Kangaroo tail tendon is characterized by a regular periodic structure along the fiber with a dspacing of 667 Å. SANS scattering from Kangaroo tail tendon in D2O is anisotropic. Sector 

averaging along the Bragg spots shows many order reflections. The first peak is strong, the 

second reflection is extinct and the third peak is well defined even with typical SANS 

smearing ( = 0.13). 
120

0

1 104

2 104

3 104

4 104

5 104

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Kangaroo Tail Tendon

measured

Scattered Intensity

Q (Å-1) 

Figure 12: SANS spectrum from Kangaroo tail tendon characterized by a first sharp peak at 

Q = 0.00942 Å-1corresponding to a d-spacing of 667 Å. The third order peak is also strong. 

5. DISCUSSION

Monochromation is an essential step for SANS as well as other diffraction methods. 

Instruments located at pulsed neutron sources use the time-of-flight method. Continuous 

beam instruments use either velocity selectors or crystal monochromators. Velocity selectors 

are preferred because monochromation occurs without change in the incident beam direction. 

When using crystal monochromators, the entire SANS instrument has to be rotated 

horizontally around the crystal monochromator axis in order to change neutron wavelength. 

This is a prohibiting factor in guide halls where experimental space between close together 

guides is at a premium. An advantage of crystal monochromators is the tight wavelength 

spread due to the typically small mosaic spread of crystals. That spread can be widened by 

using superlayers of slightly misaligned crystals. Velocity selectors commonly cover a wide 

wavelength spread (from  = 10 % to 30 %). Crystal monochromations cover the lower 

scale (from  = 0.1 % to 5 %). The use of two velocity selectors in parallel (for either low 

or wide ) would be a nice option for both low-resolution and high-resolution SANS 

measurements. 
121

REFERENCE

L. Rosta, “Multi-Disk Neutron Velocity Selectors”, Physica B156-157, 615-618 (1989). 

B. Hammouda, “Multidisk Neutron Velocity Selectors”, Nuclear Instruments and Methods in 

Physics Research A321, 275-283 (1992). 

QUESTIONS

1. How does a velocity selector work?

2. How does a crystal monochromator work?

3. What is the main characteristic of a SANS instrument that uses a crystal monochromator?

4. Does the velocity selector transmission vary with neutron wavelength? 

5. Does the predicted relative wavelength spread vary with neutron wavelength? 

6. What is the wavelength dependence of the neutron spectrum before and after the velocity 

selector?

7. What is the purpose of tilting a velocity selector to an angle with respect to the neutron 

beam? What is the range of effective tilt angles? 

8. How would you measure the transmission of a velocity selector?

9. What are the main pieces of equipment necessary to perform time-of-flight measurements?

10. What samples are characterized by sharp peaks in the SANS range and could be used for 

cold neutron wavelength calibration?

11. The standard deviation of a Gaussian distribution  can be related to its full-width at half 

maximum (FWHM) by FWHM = 2(2ln2)1/2. Derive this factor. 

12. Velocity selectors transform the neutron wavelength distribution from Maxwellian tail to 

a triangular distribution. Why is that?

13. Assuming a triangular wavelength distribution outputted by a velocity selector calculate 

the variance 

2 = <2> - <>2

. 

14. What causes a Bragg diffraction edge? Bragg diffraction edges occur at what 

wavelengths for graphite and for beryllium? 

15. Find out possible suppliers of velocity selectors. What are the essential characteristics to 

provide to a potential supplier. 

16. Find out possible suppliers of neutron choppers, pencil detectors and multi-channel 

scalers. 

ANSWERS

1. A velocity selector works by letting through only neutrons of the right speed. 

2. A crystal monochromator uses the Bragg law of diffraction. It works by scattering 

neutrons of a certain wavelength into a specific scattering angle. 

3. A SANS instrument that uses a crystal monochromator has to pivot around the vertical 

monochromator axis in order to change the neutron wavelength. 

4. The predicted velocity selector transmission does not vary with wavelength. 
122

5. The predicted relative wavelength spread does not vary with wavelength. 

6. The tail of the Maxwellian neutron spectrum from the cold source varies like 1/

5 whereas 

after the velocity selector the spectrum varies like 1/

4 where  is the neutron wavelength. 

7. Velocity selectors are tilted horizontally in order to vary the wavelength spread. Tilt angles 

vary between -3 oto +3 o. 

8. The transmission of a velocity selector could be measured similarly to the transmission of 

any SANS sample, by using a second selector operating at the same wavelength at the sample 

location. Transmission is the ratio of the detector counts with the selector in over that with 

the selector out (i.e., removed). 

9. Time-of-flight measurements can be performed using a chopper, two detectors positioned 

a known distance apart and a multi-channel scaler gated with the time zero from the chopper. 

10. Examples of samples that are characterized by sharp peaks in the SANS range include: 

Silver Behenate, phase separated copolymers, fibers like collagen from a Kangaroo tail 

tendon, highly packed silica or latex particles. 

11. Consider a Gaussian function P() = (1/2

2

)

1/2 exp(-2

/2

2

) where  is the standard 

deviation. Setting P() = 1⁄2, two solutions can be found for  = 

 2ln(2)

. This yields a 

band FWHM =  = 2

2ln(2)

 = 2.355. 

12. The output of a velocity selector is a triangular wavelength distribution because of the 

geometry of neutron trajectories through the selector windows. 

13. Consider an isosceles triangular distribution of FWHM  (and base 2) and centered 

at a wavelength 0. The left side of the triangle is given by F() = (-0)/ + 1. The right 

side of the triangle is given by F() = (-+0)/ + 1. The variance 

2 = <2>-<>2

involves 

the following integrations 

2 = <2>-<>2 = 



















   

   













  

  



 



 

0

0

0

0

d 1 d 1

2 0 2 0

= 

 

2

6

1

 .

14. A Bragg diffraction edge occurs when the incident neutrons are parallel to the crystal 

lattice planes and the crystal is probed edgewise. Bragg diffraction edges for graphite and 

beryllium occur at neutron wavelengths of 6.708 Å and 4.05 Å respectively. 

15. Possible suppers of velocity selectors are the KFKI Hungarian and the Dornier German 

companies. 

16. There are many suppliers of choppers and a wide range in prices. For multi-channel 

scaler suppliers, the name Ortec comes to mind. Neutron pencil detectors are sold by Reuter 

Stokes and by Lehnard Neutron Detector (LND). Both are companies based in the USA.
123

Chapter 13 - NEUTRON AREA DETECTORS

1. NEUTRON DETECTOR BASICS

Two-dimensional area detectors are essential components for SANS instruments. The 

position sensitive detection is achieved in one of two ways. (1) Delay line detectors sense the 

position of the detection event through the time delay at both ends of each cathode. Only two 

winding cathodes wires (one for X and one for Y positioning) are used. (2) The other (and 

most used) detection scheme uses the coincidence method whereby only X and Y events that 

arrive in time coincidence are counted. This last method uses 128 wires for X and 128 wires 

for Y cathodes and can handle higher count rates. 

Most neutron area detectors use 3He as the detection gas that undergoes the following nuclear 

reaction:

1

n0 + 3He2  1H1 + 3H1 + 765 keV (1)

The reaction products consist of two charged particles: a proton (1H1) and a triton (3H1) 

released in opposite directions with a combined kinetic energy of 765 keV. This kinetic 

energy is dissipated by ionization of the proportional counting gas (CF4 mostly). Since the 

incident neutron kinetic energy is very small (1/40 eV for thermal neutrons), thermal neutron 

detectors cannot measure neutron energies; they can only detect neutron positions. The 

released charged particles are attracted by the anode plane high voltage and liberate 

electrons. These are accelerated towards the anodes and therefore create a detection cloud 

through secondary ionization (charge multiplication). The two cathode planes (for detection 

in X and Y) are located on both sides of the anode and are kept at a low bias voltage in order 

to increase detection behind the cathodes. The detection cloud which is created close to the 

anode induces a charge on the closest cathodes (through capacitive coupling) which can be 

sensed by the charge sensitive preamplifiers. An X-Y coincidence pair is then selected and 

processed as real event. 

The two main suppliers of neutron area detectors are CERCA (Grenoble, France) and 

ORDELA (Oak Ridge, Tennessee, USA). Both types of area detectors use the coincidence 

method. The NIST Center for Neutron Research SANS group has experience with both 

detector types. A third type of area detectors uses the charge division method which is similar 

to the delay line method but involves measuring the produced charges on both sides of each 

wire. This type is not discussed. 

2. NEUTRON AREA DETECTOR SPECIFICS

The 3He/CF4 gas mixture (60%/40%) is kept at high pressure (2.4 Bars = 243 kPa) inside the 

detection chamber. An additional detector “dome” chamber filled with neutron transparent 

4He is added in front of the detection chamber in order to equilibrate pressure on both sides 

of the entrance window. This helps avoid the use of thick detector entrance windows that 
124

would attenuate the scattered neutrons beam. The detector localization gap (distance between 

the two cathodes) is 1.5 cm and the total detection gap is 2.5 cm. 

In the CERCA detector, both anode and cathode wires are made out of a CuBe alloy. Each 

cathode consists of a band of nine narrowly spaced stretched wires; the bands themselves are 

spaced 1 cm apart (center-to-center). The ORDELA detector uses one wire per cathode. 

The active detection area of typical neutron area detectors is 64 cm*64 cm with a spatial 

resolution of either 1 cm*1 cm for the CERCA detector or 0.5 cm*0.5 cm for the ORDELA 

unit. The detector efficiency is high (around 75 %) for typical neutron wavelengths (around 

= 6 Å). Count rates of order 5*104counts per second over the whole detector are achieved. 

Figure 1: Schematic representation of a neutron area detector. This figure is not to scale. The 

detection chamber is 2.5 cm wide and 64 cm high. 

Since the detector operates inside an evacuated chamber, and in order to avoid using a large 

number of vacuum feed-throughs for the cathode signals, all of the signal processing is 

performed using electronics that are mounted on the back of the detector. The detection 

electronics chain comprises amplification of the analog signals for each cathode wire, 

monitoring of X-Y coincidences and encoding to produce a digital signal which is sent out to 

the data acquisition system. 

3He

 + 

CF4

4He

detection 

chamber

membrane

dome

electronics

chamber

X cathodes Y cathodes

Anodes
125

3. NEUTRON MEASUREMENTS

Measurements of the performance of area detectors have been conducted on both CERCA 

and ORDELA type detectors. Results for one or the other type are described in each of the 

following sections. All measurements were made using a monochromatic neutron beam. 

Pulse Height Spectrum

The pulse height spectrum is measured using a multi-channel analyzer (MCA). A figure 

shows the pulse height spectrum of the anode plane measured on a CERCA detector at a high 

voltage of 2550 V. A narrow neutron peak with a resolution of about 16 % (FWHM divided 

by the average peak position) is observed. This main peak corresponds to the 765 keV energy 

released as kinetic energy during neutron capture by 3He. That energy is split into 191 keV 

for the triton 3H and 574 keV for the proton 1H. When the detection reaction occurs close to 

the detector wall, one of the products (either the proton or the triton) ends up absorbed in the 

wall while the other one deposits its kinetic energy in the stop gas. This “wall effect” is 

manifested by two more peaks and the long plateau region. The triton peak can be observed 

at 191 keV but the proton peak has merged with the main neutron peak and cannot be 

resolved. The low pulse height noise is due mainly to low energy electrons that are knocked 

off by gamma rays that are absorbed in the detector walls. The main neutron peak at 765 keV 

corresponds to both proton and triton being absorbed in the detection gas. 

0

2000

4000

6000

8000

1 104

1.2 104

0 200 400 600 800 1000

Pulse Height Spectrum 

Measured 

Number of Counts

Pulse Height (Channel Numbers)

765 keV 

191 keV 

Wall Effect 

Neutron Peak 

Gamma Rays,

Noise, etc... 
126

Figure 2: The anode pulse height spectrum for a CERCA detector showing a sharp neutron 

detection peak and low noise. The horizontal scale is in arbitrary MCA channel numbers and 

represents the pulse heights (measured in mV to represent the absorbed energies in keV). 

Gas Multiplication Factor

Using the variation of the anode pulse height for increasing detector high voltage yields the 

gas multiplication curve and the gas multiplication factor which represents the number of 

charges produced by the detection of one neutron. A figure shows measurements made on the 

CERCA detector. In order to express this variation in an absolute scale, an electronics pulse 

equivalent to the absorption of one neutron (i.e., the creation of a charge of 0.0035 pC) is 

injected into the anode plane. Measuring the amplified output of this signal on the cathodes 

and comparing it to that output during “normal” detector operation yields a gas multiplication 

factor of 117 at a high voltage setting of 2700 V. 

10

100

1.8 2 2.2 2.4 2.6 2.8

detector

measured

Pulse Height

High Voltage (kV)

Figure 3: Variation of the gas multiplication factor with high voltage.
127

Amplifier Gain and Threshold

The detection electronics chain comprises a preamplifier then an amplifier for each cathode. 

The preamplifier plays the role of impedance matching mostly. The amplifier gain and lower 

level discrimination (LLD) settings must produce “healthy” amplified analog signals around

1 V in height. 

Gain adjustments on the cathode amplifiers are made using a square wave signal (20 mV 

amplitude) injected into the anode plane and equalizing the output signals from the various 

cathodes. Final adjustments are made for the normal operation condition using a uniform 

scatterer such as (1 mm thick) plexiglass or water characterized by mostly incoherent (Qindependent) scattering. 

Setting of the LLD also called “threshold” is described here for the CERCA detector. At the 

chosen high voltage setting of 2700 V, the LLD value is estimated by measuring the total 

detection count rate on the cathodes for increasing values of the LLD as shown in a figure. At 

low LLD settings, the electronics system is paralyzed by the processing of low amplitude 

noise, while at high LLD values, the count rate decreases due to the loss of neutrons 

detection events; this gives a reliable operating LLD around 275 mV. 

500

1000

1500

2000

2500

3000

3500

4000

4500

100 200 300 400 500 600 700

detector

measurement and fit

Detector Counts

Cathode Amplifier Threshold (mV)

Figure 4: Setting of the cathode amplifier LLD (threshold) level. Optimal level is around the 

peak. 
128

The Detector Proportional Region

Neutron detectors are “proportional” counters because the total amount of charge created 

remains proportional to the amount of charge liberated in the original neutron detection 

event. The neutron detector proportional counting region is mapped out by monitoring 

increases in detector count rate for increasing high voltage. A convenient operating high 

voltage is chosen in the proportional region and well below the “plateau” region as shown in 

a figure for a CERCA detector. 

0

1000

2000

3000

4000

5000

2.5 2.6 2.7 2.8 2.9

Detector 

measurement

 and fit

Detector Counts

High Voltage (kV)

plateau region

 proportional region

Figure 5: Determination of a convenient operating high voltage in the proportional region. 

Spatial Resolution

Position sensitive detectors are characterized by their spatial resolution. The spatial 

resolution for a CERCA detector is determined as follows. Using a narrow (1 mm*2.54 cm) 

vertical slit to define a neutron beam, a scan of the detector response along the X cathodes is 

made by recording the count rates of individual cathodes when the detector is moved 

stepwise perpendicular to the neutron beam. Counts for two adjacent cathodes are shown in a 

figure. The detector spatial resolution is confirmed to be 1 cm and the counting efficiency is 
129

seen to remain reasonably constant within each detection band. This is seen by summing up 

counts for the two adjacent cathodes. 

-1 104

0

1 104

2 104

3 104

4 104

5 104

6 104

7 104

0 4 8 12 16 20

detector

channel 67

channel 66

sum 

Detector Counts

Detector Displacement (mm)

Figure 6: Determination of the detector spatial resolution. 

Detector Efficiency

The detector absolute efficiency is measured using another (pencil) detector of known 

efficiency. The high gas pressure in the pencil detector gives it a very high efficiency at all 

wavelengths making it nearly “black”. The detector efficiency was measured for an 

ORDELA detector and shown here. The He-3 neutron absorption cross section increases with 

wavelength (“1/v” absorber). This combined with various losses gives the observed variation. 
130

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.02

0.04

0.06

0.08

0.1

0.12

0.14

4 6 8 10 1 2 1 4 16 18 20

 Detector Efficiency Measurement 

detect efficiency

detect efficiency/wavelength

Detector Efficiency

Detector Efficiency/Wavelength

Neutron Wavelength (Å) 

Figure 7: Variation of the detector efficiency (left axis) with increasing wavelength. 

Performing the “1/v” absorber correction involves dividing by the neutron wavelength (right 

axis). 

Estimation of Dead Time

Dead time is inherent in most detection systems. Defining the “true” count rate as NT, the 

“measured” count rate as NM and the detection rate time constant as , the following 

argument is made. The fraction of total time for which the detector is dead is NT, and the 

rate at which true events are lost is NMNT. That rate of loss is also given by NT-NM so that:

NT-NM = NMNT (2)

This assumes “nonparalyzable” systems whereby the detection system does not get paralyzed 

by detected events. It keeps counting during signal processing. The true count rate is 

therefore estimated as:

1 N τ

N

N

M

M

T



 . (3)
131

Consider two measurements made with two different source apertures. These correspond to 

NT1 and NT2 and NM1 and NM2. The ratio RT = NT1/NT2 can be expressed in terms of the ratio 

RM = NM1/NM2 as follows.

1 N τ

1 N τ

R R

M1

M2

T M





 . (4)

Or:

RM = RT + NM1(1-RT). (5)

Plotting RM vs NM1 yields a linear behavior with intercept RT and slope m = (1-RT). The 

dead time  can therefore be obtained from  = m/(1-RT). 

In order to implement this procedure, the following measurement method is followed for an 

ORDELA detector. Two beam defining (sample) apertures of 1.27 cm and 2.27 cm diameters 

are used in turn. The neutron current crossing each of them is measured for different 

attenuation conditions. Different thickness plexiglass pieces are used to attenuate the neutron 

beam. The neutron currents are measured as count rates on the detector. An isotropic 

scatterer (thick piece of plexiglass) is used to “diffuse” the neutron beam therefore 

broadening the neutron spot on the detector. Plotting the ratio of the count rates for the two 

apertures with increasing count rate (for the 1.27 cm aperture) yields a linear behavior as 

shown in a figure. The intercept RT is of course close to the ratio of aperture areas RT = 

(1.27/2.27)2 = 0.313 and the slope is around m = 3.535*10-6sec giving an estimated dead 

time of  = m/(1-RT) = 5.16 sec. This is the dead time for the entire detection system 

comprising the detector, the detection electronics chain and data acquisition system. 
132

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0 5000 1 1041.5 1042 1042.5 104

detector

measured

Ratio of Count Rates RM

1.27 cm Diameter Aperture Count Rate NM1

slope m

 intercept RT

Figure 7: The dead time is estimated as  = m/(1-RT) where m is the slope and RT is the 

intercept on the linear part of the measured curve. 

Using the estimated dead time of  = 5.16 sec, detector losses can be estimated when count 

rates are increased. The relative (percent) loss factor is given by:

1 N τ

N

N

N N

L

T

T

T

T M











(6)
133

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 104 1 105 1.5 105 2 105 2.5 105 3 105

 Detector 

Loss Fraction

Relative Loss Factor L 

Count Rate NT

Figure 8: Percent loss factor as a function of count rate for a dead time of 5.16 sec. 

Table 1 gives estimates of dead time losses for increasing count rate. 

Table 1: Estimation of dead time losses. 

Count Rate NT

(counts per second)

Percent Loss Factor

L = (NT-NM)/NT

10,000 cps 4.85 %

30,000 cps 13.3 %

60,000 cps 23.4 %

90,000 cps 31.5 %
134

Detector Reliability

Using an intense localized neutron beam (> 10,000 cps), close inspection of the full 2D 

detector image shows whether miscoding “ghost” features are observed. These appear as 

faint spot “shadows” of the main neutron spot. Such shadows have been observed for both 

types of detector systems but are less severe in the ORDELA system. They are however not a 

problem since neutron area detectors usually operate in a less harsh neutron current 

condition. 

4. OTHER NEUTRON DETECTORS

There are many types of other neutron detectors. Old type boron (BF3) neutron detectors are 

hardly ever used anymore due to safety considerations (the BF3 gas is highly toxic). They 

have been replaced by He-3 detectors. Neutron scintillators use a conversion plate made of a 

neutron absorbing material (mostly Gd2O3) that emits gammas upon neutron absorption. The 

gamma rays are then detected as any other photons would be through the use of 

photomultipliers. Neutron scintillators are very sensitive to gamma ray background. 

Fission chambers are used as neutron beam monitors. They use a thin plate of fissile material 

(mostly 235U) that releases two highly energetic fission fragments upon fission reaction with 

a total kinetic energy of 2 MeV. Fission chambers have very low efficiency (of order 10-4) 

but large signal to noise ratios due to the high degree of ionization generated in the gas.

Note that the absorption cross section in neutron detectors varies inversely with neutron 

speed (1/v absorber) or linearly with neutron wavelength a(). Assuming a flat detection 

volume of thickness d and an atomic density  (number of absorbing atoms per cm3), the 

detector efficiency is estimated as 1-T where T is the transmission through the detection 

volume and is given as T = exp[-.a().d]. 

Table 2: Comparing a few characteristics for three types of neutron detectors. The B-10 and 

the He-3 types are proportional counters. The Li-6 type is a scintillator. 

Detector Type B-10(n,)Li-7 He-3(n,p)T-3 Li-6(n,)T-3

Energy of Reaction 2.79 MeV 0.76 MeV 4.78 MeV

Charged Particles 

Energies

 = 1.77 MeV

Li = 1.01 MeV

p = 0.57 MeV

T = 0.19 MeV

T = 2.73 MeV

 = 2.05 MeV

Particles Range  = 3 mm

Li = 2 mm

p = 30 mm

T = 6 mm

T = 0.04 mm

 = 0.007 mm

Emitted Gammas 0.48 MeV None None

Typical Thickness 5 mm 20 mm 2 mm

Atomic Density 0.053*1020 cm-30.81*1020 cm-3173*1020 cm-3

Absorption Cross 10,653 Barn 14,825 Barn 2,614 Barn
135

Section at 5 Å

Efficiency at 5 Å 3 % 80 % 100 %

REFERENCES

G. Knoll, “Radiation Detection and Measurement”, John Wiley & Sons Inc. (1979). 

P. Convert and J.B. Forsyth, “Position Sensitive Detection of Thermal Neutrons”, Academic 

Press (1983). 

L.H. Schwartz and J.B. Cohen, “Diffraction from Materials”. Springer Verlag (1987).

G Marx, B. Hammouda, N. Thillosen, J. Jacobe, J.G. Barker and C. Glinka, “An ILL_Type 

Area Detector for SANS”, private communication (1993).

QUESTIONS

1. What are the two main gases used to detect neutrons? Which one is the most used 

nowadays? 

2. Why is 3He referred to as a “1/v” absorber?

3. What is the pulse height spectrum of a detector? How is it measured? 

4. What is the “wall effect” feature in a pulse height spectrum? What is the “neutron peak”?

5. How does the coincidence method of detection work?

6. What is a fission chamber? How does it operate? What is it used for?

7. What are the two typical sizes of neutron area detectors used on SANS instruments? What 

is the typical detector spatial resolution?

8. Name four measurement tests for characterizing neutron area detectors?

9. How is the proportional detection region test performed? How is the multiplication gain 

factor test performed?

10. What is the gas multiplication factor?

11. How is the detector and electronics dead time test performed?

12. How is the spatial resolution test performed for neutron area detectors?

13. How to choose an operational threshold setting for an amplifier?

14. What is the percent loss for a non-paralysable detector system with 5.16 sec dead time 

and 10,000 cps neutron current?

15. What are the two major suppliers of neutron area detectors for SANS instruments?

16. Find out a possible supplier of multi-channel analyzers (MCAs)?

ANSWERS

1. Neutron detectors use either BF4 or 3He. BF4 is no longer used because if is highly toxic.
136

2. 3He is referred to as a “1/v” absorber because its absorption cross section varies like 1/v (v 

being the neutron velocity or speed). This absorption cross section increases with neutron 

wavelength. 

3. The pulse height spectrum of a detector is the distribution of electronics signal amplitudes 

outputted by the detector electronics. It is measured using a multi-channel analyzer (MCA). 

4. The wall effect represents nuclear reaction products (either proton or triton) hitting the 

detector wall. The neutron peak corresponds to both reaction products being entirely 

absorbed in the gas (no wall effect). 

5. The coincidence method registers a real detected event when an X and a Y cathode signals 

arrive in coincidence (i.e., within a specified time window). 

6. A fission chamber is a very low efficiency neutron detector. It uses fissionable material 

(

235U mostly) to detect neutrons. An energy of 2 MeV is released as kinetic energy for the 

fission fragments. Fission counters are used as neutron beam monitors. 

7. Neutron area detectors used on SANS instruments are either 64 cm*64 cm or 1 m*1 m in 

area. The spatial resolution is either 1 cm* 1 cm or 0.5 cm*0.5 cm. 

8. The various tests performed to characterize neutron area detectors are: pulse height 

spectrum, multiplication factor, amplifier gain and threshold settings, gas proportional 

region, spatial resolution, detector efficiency, detector and electronics dead time. 

9. The gas proportional region is determined by increasing the HV and recording the number 

of detector counts (see Figure 5).

10. The gas multiplication factor represents the number of electrons released from the 

absorption of one neutron.

11. The dead time is measured using two different apertures and varying the count rate each 

time by inserting attenuators in the beam. The dead time is given by  = m/(1-RT) where m is 

the slope and RT the intercept of the ratio of counts (for the two apertures) vs count rate. 

12. The spatial resolution test is performed by stepping the area detector laterally 

(perpendicular) to a neutron beam defined through a thin vertical slit. 

13. The threshold (also called lower level discriminator or LLD) level for an amplifier is 

chosen as that setting that gives the maximum number of neutron counts. 

14. Eq (6) gives the formula and Table 1 gives the answer of L = 4.85 % loss for a dead time 

of 5.16 sec and 10,000 cps neutron current. 

15. The two major suppliers of area detectors for SANS instruments are CERCA (Grenoble, 

France) and ORDELA (Oak Ridge, Tennessee, USA).

16. The company Canberra is a possible supplier of MCAs. A Google search with “multi 

channel analyzer suppliers” comes up with dozens of other possible suppliers. 
137

Chapter 14 - SAMPLE ENVIRONMENTS

A number of sample environments are available for SANS measurements. These consist in 

standard cells and cell holders that can be heated (up to 300 oC) using heating cartridges or 

cooled (down to 0 oC) using circulating baths. Between 0 oC and 10 oC (depending on the 

ambient dew point), the sample chamber must be evacuated and/or filled with inert gas 

(nitrogen or helium) in order to avoid condensation on cooling blocks. A host of special 

sample environments are also available at the NIST CNR. These include in-situ pressure 

cells, in-situ (Couette or plate/plate) shear cells as well as rheometers, electromagnets (up to 

2 Teslas), a superconducting magnet (9 Teslas), low temperature closed cycle helium 

refrigerators for temperatures down to 5 K and even lower (below 1 K), and a furnace for 

temperatures up to 450 oC. Only a few highly-used pieces of equipment are described here. 

1. STANDARD SAMPLE CELLS 

SANS measurements involve a variety of different cells. The first type is the standard offthe-shelf “banjo cell” owing to its characteristic shape. This type is used for photon scattering 

as well and has quartz windows (transparent to both neutrons and light). Their diameter is 2 

cm and are characterized by a sample gap thickness of either 1 mm or 2 mm corresponding to 

a sample volume of 0.3 ml or 0.6 ml respectively. This type of cell is appropriate for liquid 

samples that can be handled using a syringe. 

The second type of cell used for SANS has grown out of successive iterations at the NIST 

CNR. It is of the demountable type with titanium body and quartz windows. An inner spacer 

ring of thickness either 1 mm or 2 mm is part of the cell body and sets the sample gap. The 

sample thickness is uniform between the two quartz windows. Sealing is performed through 

back-up o-rings and tightening retainers on each side. This type of cell can handle liquids, 

gels, wafers and powders. Gel and powder samples are loaded from one side after tightening 

the retainer piece on the other side. Slightly larger volumes than for banjo cells are required. 
138

Figure 1: Sample holder cells used for SANS at the NIST CNR. 

2. HEATING AND COOLING BLOCKS

A 7-position heating block made out of aluminum is used for temperature control. It uses two 

pieces of bakelite at the base to thermally decouple the main heated block from the other 

pieces in the sample chamber. This heating block controls temperature between ambient and 

300 oC with a precision of less than 1 oC. The actual sample temperature lags slightly behind 

the block temperature. A resistance temperature detector (RTD) is used to monitor the block 

temperature. 

quartz window

backup o-ring

tightening

 retainer

DEMOUNTABLE CELL

filling hole

4.32 cm

3.49 cm

2.16 cm

BANJO CELL

2 cm
139

Figure 2: A 7-position heating block using heating cartridges and an RTD to monitor the 

temperature. 

A similar (10-position) block uses flow of coolant instead (50 % water and 50 % antifreeze) 

to cool samples down to 0 oC. The heating range for that block is limited to 90 oC. Another 

heating/cooling block uses circulating silicon oil to reach an even wider temperature range 

(up to 150 oC). 

3. THE PRESSURE CELL

Two generations of pressure cells for in-situ SANS measurements were designed to handle 

polymer samples. The polymer wafer is first melt pressed inside a metal ring to set a uniform 

thickness (of 1 mm). This produces a homogeneous clear sample. The right amount of 

sample (0.3 ml) is used in order to fill the available volume. This wafer is then transferred to 

the middle of a confining o-ring. This “encapsulated” o-ring (flexible rubber for the inside 

and Teflon coating for the outside) transmits pressure from the pressurizing fluid (water in 

this case) to the sample. The sample is also confined between two sapphire windows with a 1 

mm gap between them. The cell body is made out of Iconel metal (75 % nickel, 15 % 

chromium) which is good for its high corrosion resistance and tensile strength at high 

temperatures. The pressure cell is surrounded by a heating jacket using heating cartridges for 

temperatures from ambient to 160 oC. Another (cooling) jacket uses a circulating fluid to 

reach down to 10 oC. The second generation pressure cell can handle up to 3 kbar pressures. 

The pressurizing system consists of a pressure pump, two remotely controlled valves, high 

pressure tubing, two gauges, etc. The pump and the two valves are computer controlled and 

use a feedback signal from the digital gauge. The pressuring system and the main SANS data 

acquisition system follow a handshake protocol through a two bit process (“acknowledge” 

and “release” lines). When using a liquid sample, a separator is inserted between the pump 

heating cartridge

resistance 

temperature 

detector

thermal decoupling piece

38 cm

7.62 cm

5.08 cm
140

and the pressure cell. This consists of a cylindrical tube with a piston inside to separate out 

the pressurizing fluid from the liquid sample. 

Figure 3: The first generation pressure cell for in-situ pressure measurements. 

Retaining Lens

Sapphire Window

Pressurizing Fluid 

valve

Sample

Confining O-ring

Seal Backup

7.62 cm

9.525 cm
141

Figure 4: Schematic representation of the pressure cell system including the computer 

controlled pump and valves, the pressure transducer, the main gauge, and the various high 

pressure tubing. Note that the pressurizing fluid bottle is standing vertical (out of the page). 

4. THE COUETTE SHEAR CELL

The Couette shear cell used for in-situ SANS measurements is a simple device for aligning 

samples. It consists of a stator which is lowered into a rotor cup to shear liquid samples 

within the 0.5 mm gap. Note that the neutron beam crosses the gap twice. The stator and the 

rotor are made out of quartz. The rotor base is made out of a material called Invar which has 

a thermal expansion coefficient similar to quartz. A set of x-y translation Verniers allows the 

precise alignment of the stator with respect to the rotor. The rotor cup takes about 12 ml. 

When the stator is lowered, the fluid sample level rises (in the gap between the stator and 

rotor) until it covers the neutron beam level. 

The shear cell is used in one of two main geometries: (1) the radial mode whereby the 1.27 

cm diameter neutron beam goes through the middle of the cell and (2) the tangential mode 

whereby a vertical slit (1.27 cm*0.3 mm) defines the neutron beam incident tangential to the 

cell. 

This shear cell has been used for easily flowing liquids as well as for highly viscous fluids. 

Temperature control is performed using a circulating fluid. Coolant circulates inside the 

stator without getting in the neutron beam. Cell temperature can be controlled from 10 oC up 

Computer Controlled 

Pressure Pump

End-of-Travel

Limit Switches

Motor

Pressurizing 

Fluid (Water)

Bottle

Digital Pressure 

Transducer and 

Rupture Disk

Valves 1 and 2

Analog 

Pressure 

Gauge

Tubing to 

Pressure 

Cell
142

to 90 oC. The shear cell is used in either the steady shear mode or the reciprocating shear (or 

jiggle) mode. Shear rates up to 5,000 Hz for are possible. 

Figure 5: Schematic representation of the Couette shear cell setup with the stator in the raised 

position. 

Stator

Rotor

Electric

Motor

Lowering

Knob

Lowerable 

Arm

x-y Translation

Verniers

Hole for

Neutron 

Beam

Cooling 

Ports 75 cm

75 cm
143

Figure 6: Schematics of the rotor and stator for the in-situ Couette shear cell. The neutron 

beam is perpendicular to the plane of this drawing. 

5. THE PLATE/PLATE SHEAR CELL

A plate/plate shear cell is available at the NIST CNR for in-situ SANS measurements. This 

device was designed for investigations of oriented block copolymers. It consists of two arms; 

one fixed and one moving. The sample is melt pressed into a special (1 mm thick) holding 

cell which is mounted between the two arms. The fixed arm holding the sample is tightened 

in order to squeeze the sample between the two arms. The translation screw transforms the 

rotational motion (from the motor) into a translational motion (up and down). Two limit 

switches limit the travel range and therefore the strain. A strain of 100 % is obtained for a 1 

mm travel. 

Invar Top

O-ring

Fused Silica

Stator Piece

Nylon Vapor

Seal

Fused Silica

Rotor Cup

Circulating Fluid Port

Radial Beam 

Position

Tangential Slit

Position

Fluid Sample

75 cm

6.35 cm
144

Figure 7: Schematics of the in-situ plate/plate shear cell. 

6. OTHER SAMPLE ENVIRONMENTS

Other sample environments are available for in-situ SANS measurements at the NCNR. 

These include a couple pf rheometers for in-situ rheology. The shear cells described above 

can orient samples but cannot measure torque. The rheometers are standard equipment that 

were modified to allow a neutron beam to be incident on the sample and in-situ SANS 

measurements. The main modification consisted in raising the sample cup from its standard 

location (inside a temperature trough) to a higher (more accessible) location in the neutron 

path. Temperature control is performed through controlled air circulation. 

Other pieces of equipment include electromagnets (up to 2 Teslas) and a superconducting 

magnet (up to 9 Teslas). A humidity chamber and a vapor cell allow sample humidity and 

vapor control. Other cells are available. 

Translation Screw

Heating

Cartridges

Limit

Switches

75 cm

23 cm

Fixed Arm

Moving Arm

Electric Motor

Tightening Screw

Sample

Neutron

Beam
145

REFERENCES

G. Straty, “The NIST SANS in-situ Couette Shear Cell”, Internal Report (1993)

B. Hammouda and S. Kline, “The NIST SANS in-situ Pressure Cell”, Internal Reports (1995, 

1998).

QUESTIONS

1. Why use quartz windows for sample cells?

2. What is the maximum reachable temperature for the heating block? What is the 

temperature range for the cooling block?

3. What are the units for ambient pressure?

4. Why doesn’t water boil above 100 oC inside the pressure cell?

5. What is the characteristic of most SANS data with in-situ shear?

ANSWERS

1. Quartz windows are fairly transparent to neutrons and to light; quartz is less expensive 

than sapphire. 

2. The heating block can reach up to 300 oC. The cooling block uses 50 % water and 50 % 

ethylene glycol (antifreeze) and can reach from 0 oC to 90 oC. The range between 0 oC and 

10 oC is above the dew point (where water condensation occurs on windows). This range 

should be used only in inert (either nitrogen or helium) atmosphere to avoid water 

condensation. 

3. Ambient pressure corresponds to 14.7 psi = 1 atm = 1 Bar. This is equivalent to 760 mm of 

mercury or 760 torr and converts to 101,325 Pa in SI units.

4. Water does not boil above 100 oC inside the pressure cell because a positive pressure (at

least 100 psi) is always maintained. 

5. Most SANS data with in-situ shear are characterized by anisotropic scattering with 

oriented contour maps. 
146

Part D – SANS RESOLUTION AND DATA SMEARING

Chapter 15. The SANS Instrumental Resolution

15.1 The Resolution Function

15.2 Variance of the Q Resolution

15.3 SANS Resolution Variance

15.4 Minimum Q

15.5 Measured SANS Resolution

15.6 Discussion

References

Questions

Answers

Chapter 16. Neutron Focusing Lenses

16.1 Focusing Lenses’ Basic Equations

16.2 Resolution with Focusing Lenses

16.3 Minimum Q with Focusing Lenses

16.4 Measured SANS Resolution

16.5 Lens Transmission

References

Questions

Answers

Chapter 17. Gravity Correcting Prisms

17.1 Neutron Trajectory

17.2 The Prism Deflection Angle

17.3 Contribution to the Q Resolution

17.4 Contribution to Qmin

17.5 Measurements with Gravity Correcting Prisms

17.6 Prisms Transmission

17.7 Discussion

References

Questions

Answers

Chapter 18. Neutron Beam Current

18.1 Variation of Beam Current with Spot Size

18.2 Variation of Beam Current with Qmin

References

Questions

Answers

Chapter 19. The Smearing Effect

19.1 The Resolution Function

19.2 The Resolution Correction
147

19.3 Iso-Intensity Contour Maps with Gravity Effect

19.4 Numerical Application

19.5 Smearing For Hard Spheres

19.6 SANS From Silica Particles

References

Questions

Answers
148

Chapter 15 - THE SANS INSTRUMENTAL RESOLUTION

Instrumental smearing affects SANS data. In order to analyze smeared SANS data, either desmearing of the data or smearing of the fitting model function is required. The second 

approach is more common because it is a direct method. Smearing corrections use the 

instrumental resolution function.

1. THE RESOLUTION FUNCTION

Instrumental smearing is represented by the following 1D convolution smearing integral 

(suitable for radially averaged data):

 

 

 













 

0

1D

smeared d

d (Q Q')

dQ'P (Q')

d

d (Q)

. (1)

Here Q is the scattering variable, 

d(Q) d

is the scattering cross section and the 1D 

resolution function is defined as a Gaussian function:





































2

Q'

2

1/ 2

2

Q'

1D

2σ

Q'

exp

2πσ

1

P (Q') . (2)

The Q standard deviation Q is a measure of the neutron beam spot size on the detector (Q = 

0). It is also a measure of the instrumental part of the width of scattering peaks from samples 

(Q 



0). Q is related to the spatial standard deviation (i.e., standard deviation of the neutron 

beam spot at the detector) r by Q = (2L2r, where L2 is the sample-to-detector distance. 

2. VARIANCE OF THE Q RESOLUTION

Scattering measurements are made in reciprocal (Fourier transform) space where the 

magnitude of the scattering vector is given by:











 







2

sin 4Q . (3)

Here  is the neutron wavelength and  is the scattering angle. At small angles, Q is 

approximated by: Q = 2. 

In order to express Q, differentiate Q on both sides:

 





 





 d

2

d

2

dQ 2

. (4)
149

Take the square:

     

 

  





   















  















 d d

2

d 2

2

d

2

dQ 3

2

2 2

2

2

2

2

2

. (5)

Then perform the statistical averages:

   

    





    





    















  d d

2

(d ) 2

2

(d )

2

(dQ)

3

2

2 2

4

2

2

2

2

(6)

Note that 

 dd  d  d 

because the scattering angle  and the wavelength 

 distributions are uncorrelated. Moreover, 

 d  (   )        0

. This 

cancels out the last term. 

Define the different variances:

Q

2 = <(dQ)2>=<Q2>-<Q>2

, 



2 = <(d)2>=<2>-<>2

, 



2 = <(d)2>=<2>-<>2

(7)

The SANS resolution variance has two contributions:

Q

2 = [Q2

]geo + [Q

2

]wav = 

2

2

λ

4π



2 + Q2

2

2

λ

λ

σ

(8)

These correspond to the “geometry” part (first term) and to the “wavelength spread” part 

(second term) of the Q resolution variance. 

3. SANS RESOLUTION VARIANCE

The main parts of the resolution variance Q

2

are derived for a SANS instrument with 

circular apertures (Mildner-Carpenter, 1984; Mildner et al, 2005). 

Geometry Contribution to the Q Resolution

Consider the geometry contribution to the Q resolution variance:

2

2

2

2 r

L



 

(9)
150

L2 is the sample-to-detector distance. The variance for the radially averaged data corresponds 

to 1D. The 1D case of x

2

(in the horizontal x direction) is considered first. 

Figure 1: Typical SANS geometry with circular source and sample apertures and 2D area 

detector. This figure is not to scale. The horizontal scale is in meters whereas the vertical 

scale is in centimeters. Aperture sizes have been drawn out of scale compared to the size of 

the area detector. 

Consider a uniform neutron distribution within the source and sample apertures. The 

horizontal contribution can be written:

3

2

2

2

2

1

1 2

1

2

2

1

2 2

x x x

L

L L

x

L

L

     













 

   















  . (10)

L1 is the source-to-sample distance, L2 is the sample-to-detector distance, <x2>1 is the 

averaging over the source aperture, <x2>2 is the averaging over the sample aperture and 

<x2>3 is the averaging over a detector cell. R1 and R2 define the source and sample aperture 

radii respectively. In order to see the origin of the (L2/L1) scaling factor, consider the case 

where R2 = 0. Then the spot at the detector would be similar to the source aperture size scaled 

by (L2/L1). Similarly, in order to see the origin of the (L1+L2)/L1 scaling factor, consider the 

case of R1 = 0. The spot would be similar to the sample aperture size scaled by (L1+L2)/L1. 

circular

source 

aperture circular

sample

aperture

R1

R2

L1 L2

2D area 

detector

x

y
151

Figure 2: Geometry of the circular source aperture.

The various averages can be readily calculated:

4

R

rdr d

r dr cos ( )d

rdrd

r cos ( )rdrd

x

2

1

R

0

2π

0

R

0

2π

0

3 2

2π

0

R

0

2π

0

2

R

0

2

1

2

1

1

1

1





 





 

  

 

 

 

 

. (11)

Similarly <x2>2 = 

4

R

2

2

. Averaging over the square (or rectangular) detector cell of sides x3

and y3 follows.

2

3

2

3

Δ x /2

Δ x /2

Δ x /2

Δ x /2

2

3

2

2

x

3

1

12

x

dx

dxx

x

3

3

3

3











 











  





. (12)

Therefore:

12

x

4

R

L

L L

4

R

L

L

2

3

2

2

2

1

1 2

2

1

2

1

2 2

x



 













 

 















  . (13)

Similarly for the vertical part (assuming no effect of gravity on the neutron trajectory):

x

y

r cos()



R1 r
152

12

y

4

R

L

L L

4

R

L

L

2

3

2

2

2

1

1 2

2

1

2

1

2 2

y



 













 

 















  . (14)

So that:

  2

2

2

x

2

geo

2

Qx L

2 σ

















  . (15)

  2

2

2

y

2

geo

2

Qy L

2 σ

















  .

This is the first part of the Q resolution variance.

Wavelength Spread Contribution to the Q Resolution

The neutron wavelength is assumed to obey a triangular distribution peaked around  and of 

full-width at half maximum . 

Figure 3: Triangular wavelength distribution. 

This is a typical distribution outputted by a velocity selector. For simplicity of notation, the 

same symbol  is used to denote both the wavelength variable  and the average wavelength 

<>. The average over this wavelength distribution can be readily calculated as:





























    

2

2 2

λ

Δλ

6

1

1 . (16)











 




153

Note that if we had assumed a square (also called “box”) wavelength distribution, the factor 

of 1/6 would be replaced by 1/12. 

The wavelength variance is therefore:

 

2

2

2 2

2

2

6

1





















      







. (17)

The wavelength spread contribution to the Q resolution variance is therefore as follows:

 

2

2

wav x

2

Qx λ

Δλ

6

1

Q 











 

(18)

 

2

2

wav y

2

Qy λ

Δλ

6

1

Q 











 

This is the second part of the Q resolution variance. 

Neutron Trajectories

Gravity affects neutron trajectories. Consider neutrons of wavelength  and wavelength 

spread  incident on the source aperture. The initial neutron velocity is v0 with components 

v0y and v0z along the vertical and horizontal directions. 
154

-1000

-800

-600

-400

-200

0

200

0 1 0 2 0 3 0 4 0 

Figure 4: Parabolic neutron trajectory under gravity effect. Neutrons must cross the source 

and sample apertures. This figure is not to scale. 

Under the effect of gravity, neutrons follow the following trajectories:

z v t  0z

(19)

gt v t

2

1

y y 0y

2

 0   .

Here g is the gravity constant (g = 9.81 m/s2) and t is time. Neutrons are assumed to be at the 

horizontal axis origin at time zero. In order to obtain the neutron trajectories equation, the 

time variable is eliminated using the fact that neutrons must cross the source and sample 

apertures; i.e., the condition y = y0 for z = 0 and for z = L1. This gives:

L v t

1  0z 

0z

1

v

L

t 

(20)

gt v t

2

1

y 0y

2

0    

0z

1

0y 0

v

L

g

2

1

v  y  .

The horizontal neutron speed v0z is related to the neutron wavelength  by: 

source

aperture

sample

aperture

L1 L2

2D area 

detector

y

z

v0
155

mλ

h

v0z . (21)

Here also, h is Planck’s constant and m is the neutron mass. At any other position along the 

neutron path (other than z = 0 and z = L1), the parabolic variation followed is:

(z z L ) y B (z z L )

h

mλ

g

2

1

y(z) y 1

2 2

1 0

2

2

0      











  (22)

where:

2

2

2h

gm B  . (23)

The neutron fall trajectory is characterized by a parabolic variation with respect to z and with 

respect to . 

For z = L1+L2, neutrons fall by the distance y(L1+L2) = y0 - B 

2 L2(L1+L2). 

Effect of Gravity on the Q Resolution

Gravity affects the fall of the neutron and therefore the resolution in the y direction. Neutron 

trajectories follow a parabola:

2

y  y0  A

with 

A BL (L L )  2 1  2

and 

2

2

2h

gm B  . (24)

g is the gravitation constant (g = 9.81 m/s2), m is the neutron mass and h is Plank’s constant 

(h/m=3995 Å.m/s). A= 3.073*10-7 L2(L1+L2) given in units of m/Å2 where L1 and L2 are the 

source-to-sample and sample-to-detector distances given in meters. 

The gravity contribution to the Qy variance is given by:

  2

2

grav

2

y

2

grav

2

Qy L

2 [ ]

















 

(25)

  (y y ) (y y ) A ( )

2 2 4 2 2

0

2

0 grav

2

y                .

The two averages over the triangular wavelength distribution are performed as follows:
156





























    

2

2 2

λ

Δλ

6

1

1

(26)





























 











    

2 4

4 4

λ

Δλ

15

1

λ

Δλ 1 .

Therefore:

2

4 2 2 4

λ

Δλ

3

2













         . (27)

So that:

 

2

2 4

grav

2

y

λ

Δλ

3

2

A 











   . (28)

and finally:

 

2

2 2

2

2

grav

2

Qy 3

2

A

L

2

















 













 

  . (29)

This term is added in quadrature with the other two contributions (geometry and wavelength 

spread) to the Q resolution variance Q

2

. 

Summary of the Q Resolution

Putting the geometry contribution, the wavelength spread contribution and the gravity 

contribution together yields:

2

2

x

2

3

2

2

2

1

1 2

2

1

2

1

2

2

2

2

Qx λ

Δλ

6

1

Q

2

x

3

1

4

R

L

L L

4

R

L

L

L

2









































 

 













 

 



































  (30)

2

2

y

2

2 4

2

3

2

2

2

1

1 2

2

1

2

1

2

2

2

2

Qy λ

Δλ

6

1

Q

3

2

A

2

y

3

1

4

R

L

L L

4

R

L

L

L

2















































  









 

 













 

 



































 

2

2

2 1 2

2h

gm A  L (L  L )

where:
157

R1: source aperture radius

R2: sample aperture radius

x3 and y3: sides of the detector cell

L1: source-to-sample distance

L2: sample-to-detector distance

: wavelength spread, FWHM of triangular distribution function

g: gravity constant

m: neutron mass

h: Planck’s constant.

This result was obtained assuming a uniform neutron distribution within the apertures and a 

triangular wavelength distribution. 

4. MINIMUM Q

A figure of merit for SANS instruments is the minimum value of the scattering variable Q 

(also called Qmin) that can be reached for a given configuration. This value is imposed by the 

neutron spot size on the area detector and dictates the size of the beamstop to be used. In 

order to minimize the spot size, one has to minimize the “umbra” and “penumbra” of the 

neutron beam. 

Figure 5: Converging collimation geometry to minimize spot size. This figure is not to scale. 

The penumbra is the maximum spot size to be blocked by the beamstop. 

source 

aperture

sample

aperture

L1 L2

2D area 

detectorumbra

penumbra

y
158

Given the standard SANS geometry, the extent of the penumbra in the horizontal direction is 

given by:

2

x

R

L

L L

R

L

L

X

3

2

1

1 2

1

1

2

min







  . (31)

And the minimum Q in the horizontal direction is therefore Qmin

X = (2)(Xmin/L2). 

In the vertical direction, the effect of gravity plays a role. The upper edge of the penumbra 

moves down by A(

2

because it corresponds to faster neutrons with wavelength -. 

The lower edge of the penumbra drops down by more; i.e., by A(

2

because it 

corresponds to slower neutrons with wavelength + This results in a distorted beam spot 

at the detector. To first order in wavelength spread, one obtains:

















 







 

3 2

2

1

1 2

1

1

2

min 2A

2

y

R

L

L L

R

L

L

Y . (32)

Note that Qmin is determined by the spot size in the vertical direction where the beam is the 

broadest Qmin = Qmin

Y = (2)(Ymin/L2).

































 







 





















3 2

2

1

1 2

1

1

2

2

min 2A

2

y

R

L

L L

R

L

L

L

2

Q . (33)

Figure 6: Neutron spot on the detector. The effect of gravity is to drop both the upper edge 

and the lower edge of the penumbra. The lower edge drops more resulting in distorted isointensity contours. penumbra

y

-A(

2

- A(

2

penumbra with gravity effect

-A

2
159

5. MEASURED SANS RESOLUTION

Specific Instrument Configuration

Consider the following low-Q instrument configuration. 

L1 = 16.14 m

L2 = 13.19 m

R1 = 0.715 cm

R2 = 0.635 cm

x3 = y3 = 0.5 cm

λ

Δλ

= 0.13.

This gives a gravity fall parameter of A = 0.01189 cm/Å2. This configuration does not strictly 

obey the “cone rule” whereby the beam spot umbra at the detector is minimized. 

Assuming a neutron wavelength of  = 6 Å, the variance Q

2

has the following Q 

dependence: 

2.76*10 0.0028Q (Å )

2 7 2 2

Q

     . (34)

The minimum scattering variable is:

-1 Qmin  0.0017Å . (35)

Gravity effects are small for 6 Å neutrons. Neutrons fall by only 0.428 cm. 

The focus here will be on empty beam measurement (i.e., with no sample in the beam). This 

corresponds to the resolution limit of Q = 0. 
160

10-9

10-8

10-7

10-6

10-5

0.0001

0.001 0.01 0.1

Resolution Variance



Q

2

[

Q

2

]

geo

 + [

Q

2

]

grav

[

Q

2

]

wav

Variance of the Q Resolution 

Q

2 (Å

-2

)

Q (Å-1)

Qmin

Figure 7: Variation of Q

2 with Q plotted on a log-log scale. The main contributions 

(geometry, wavelength spread and gravity effect) are added in quadrature. 

Empty Beam Measurements

Empty beam measurements were made using the above instrument configuration and varying 

the neutron wavelength. 

Predicted and measured resolution characteristics are compared in a series of figures. First, 

the position of the beam spot on the detector is plotted for increasing wavelength. 
161

-5

-4

-3

-2

-1

0

1

5 10 15 20

Beam Spot Position

measured y position

calculated y position

measured x position

Beam Spot Position (cm)

Neutron Wavelength (Å)

Figure 8: Variation of the horizontal and vertical neutron beam spot positions with 

wavelength.

Next, the standard deviations x and y of the neutron spot size are plotted with increasing 

neutron wavelength. The measured values were obtained by performing non-linear leastsquares fits to a Gaussian function in the x and in the y directions. Fits were performed on 

cuts through the beam spot center, both horizontally and vertically. Data recorded by two 

adjacent detector cells (normal to the cut) were added in each case in order to improve 

statistics. A scaling factor of 

1.45

= 1.2 was used to scale the measured data. This scaling 

factor gave good agreement between the measured and calculated values for x. The same 

scaling factor was used for y. 

This necessary scaling factor of 1.2 is probably related to the procedure used to obtain 

measured beam spot widths. (1) Slice cuts were performed in the horizontal and vertical 

directions. (2) Gaussian fits were performed on these slices even though the beam profile is 

known to be close to a trapezoidal (not Gaussian) shape. (3) Lastly, the measured beam spots 

were so small (covering only a few detector cells) that Gaussian fits were performed with 

four to eight points only. 
162

0.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

5 10 15 20

Standard Deviation

calculated 

x

calculated 

y

measured 

x

measured 

y Spot Size Standard Deviation (cm)

Neutron Wavelength (Å)

Figure 9: Variation of the measured and calculated neutron beam spot size standard 

deviations x and y with increasing wavelength. 

The minimum spot sizes Xmin and Ymin were obtained experimentally as the values where the 

intensity (of the horizontal or vertical cuts across the beam spot) goes to zero. This method is 

conservative and overestimates the measured values for Xmin. It is not precise, yielding poor 

agreement between measured and calculated values. Our calculated values neglect for 

instance diffuse scattering from the beam defining sample aperture and from the pre-sample 

and post-sample neutron windows. Such scattering tends to broaden the neutron beam. At 

long wavelengths, the gravity effect broadens the neutron spot in the vertical direction with 

the extra difference Ymin-Xmin given by the term 2A

2

(). 
163

1.8

2

2.2

2.4

2.6

2.8

3

3.2

5 10 15 20

Minimum Spot Size

calculated Y

min

calculated X

min

measured Y

min

measured X

min

Y

min and Xmin (cm)

Neutron Wavelength (Å)

Figure 10: Variation of the neutron beam spot sizes in the horizontal and vertical directions 

with increasing wavelength. 

6. DISCUSSION

The choice of a SANS instrument configuration is always a compromise between high 

intensity and good resolution. The instrumental resolution is the main source of data 

smearing. Estimation of the SANS resolution is an integral part of the data reduction process. 

Reduced SANS data include not only the scattering variable Q and the scattered intensity 

I(Q), but also the resolution standard deviation Q. Q is needed to smear models before 

fitting to the data. 

Corrections for smearing due to gravity are never made because they are small and deemed 

to be complex manipulations of the 2D data. The effect of gravity smearing is small except at 

long neutron wavelengths. Fortunately, the wide majority of experiments maximize flux by 

using low wavelengths (5 Å or 6 Å). 
164

REFERENCES

D.F.R. Mildner, J.M. Carpenter, ”Optimization of the Experimental Resolution for SAS”, J. 

Appl. Cryst. 17, 249-256 (1984).

D.F.R. Mildner, B. Hammouda, and S.R. Kline, “A Refractive Focusing Lens System for 

SANS”, J. Appl. Cryst. 38, 979-987 (2005).

QUESTIONS

1. What is the relationship between the standard deviation and the variance of a peaked 

function?

2. What function best describes the wavelength distribution function after the velocity 

selector?

3. What is the shape of the penumbra of the neutron beam spot on the detector?

4. Given a Gaussian function, what is the relationship between its FWHM and its standard 

deviation ?

5. Calculate the following average <

2> over a triangular wavelength distribution. Calculate 

<

2> over a Gaussian wavelength distribution of standard deviation . 

6. What are the various contributions to the SANS instrumental resolution?

7. The gravity effect is worse at what wavelength range?

8. What is the shape of the neutron beam spot on the detector for long wavelengths?

9. Cold neutrons of 20 Å wavelength fall by how much over a distance of 30 m?

10. Name the main “figures of merit” for a SANS instrument. 

11. How would you obtain a lower Qmin?

12. If it takes 4 seconds for a pebble to reach the water level of a well, how deep is the well?

ANSWERS

1. The variance Q

2

is the square of the standard deviation Q. 

2. The wavelength distribution after the velocity selector is best described by a triangular 

function. 

3. The neutron beam spot on the detector has a shape close to trapezoidal. 

4. For a Gaussian distribution, the following relationship holds FWHM = 2

2ln(2)

. In 

order to derive this relation, consider a Gaussian function P(x) = (1/2

2

)

1/2 exp(-x2

/2

2

) 

with standard deviation . Setting P(x) = 1⁄2, two solutions can be found for x = 

 2ln(2)

. 

This yields a band FWHM = 2

2ln(2)

 = 2.355. 

5. The integrations are simple. Only the results are given. 

<

2> = 2































2

λ

Δλ

6

1

1

for triangular distribution of FWHM . 
165

<

2> = 2































2

λ

λ

σ

1

for Gaussian distribution of standard deviation . 

6. The SANS instrumental resolution contains contributions from (1) “geometry” (source, 

sample aperture and detector cell sizes and source, sample and detector inter-distances), (2) 

from “wavelength spread” and (3) from “gravity” effect. Remember that [Q

2

]geo ~ constant, 

[Q

2

]wav ~ Q2()

2

and [Q

2

]grav ~ 

4

()

2

. 

7. The effect of gravity is worse for longer wavelengths. 

8. Neutrons fall more at the bottom of the neutron beam than at the top. For this reason, beam 

spot iso-intensity contour maps are weakly elliptical (weakly oval actually). 

9. Cold neutrons of 20 Å wavelength fall by about 4 cm over a distance of 30 m (see Figure 

8). 

10. Typical figures of merit for SANS instrument include: resolution Q, Qmin, flux-onsample, Q-range (called Q) and background level. 

11. A lower Qmin could be obtained by increasing the sample-to-detector distance. When this 

distance is at its maximum, then one could increase the neutron wavelength. The reason for 

this is that the beam intensity (1) decreases as sample-to-detector distance square but (2) it 

decreases as neutron wavelength to the fourth power. 

12. The pebble falls according to the law of gravity 

y gt / 2

2



where g = 9.81 m/s2is the 

gravity constant and t is time. After a time t = 4 s, the pebble would have fallen 

y 9.81*4 / 2 78.5m

2

  . 
166

Chapter 16 - NEUTRON FOCUSING LENSES

Neutron lenses are used to focus neutron beams. They increase intensity on the sample and 

shrink the neutron spot size on the detector therefore reducing the minimum Q. The effects of 

focusing lenses on SANS resolution are discussed.

1. FOCUSSING LENSES’ BASIC EQUATIONS

The focusing lenses’ basic equations are described here (Mildner et al, 2005; HammoudaMildner, 2007). The focal length for a set of N lenses of radius of curvature R and index of 

refraction n is given by:

2N(1- n)

R

f  . (1)

The index of refraction n is related to the material atomic density , neutron scattering length 

b, and neutron wavelength  as:

2

2

b

n 1 





  . (2)

The focal length f is also related to the source-to-lenses distance L1 and lenses-to-image 

distance L4 as:

1 L4

1

L

1

f

1

  . (3)

Combining the above two equations gives a relationship between the number N of lenses 

used and the neutron wavelength  for an optimized instrument configuration where the 

detector is located at the focal spot. 

1 4

1 4

2

c L L

L L

N

R

b 



 



. (4)
167

Figure 1: Schematic representation of a focusing lens system showing an object (the neutron 

source aperture) and its image (on the detector plane). L1 and L4 are the source-to-sample and 

sample-to-detector distances and f is the focal length. In practice, neutron focusing devices 

comprise many lenses used together. 

For MgF2 lenses, one has:

6 -2 b / 1.632*10 Å

  

so that:

5 2

1 4

1 4

2

6.13*10 Å

L L b

L L

R

Nλ

























. (5)

Consider lenses of radius of curvature R = 2.5 cm and height H = 2.5 cm that are thin at the 

center (1 mm thickness) in order to keep neutron transmission high. Source-to-sample and 

sample-to-image distances corresponding to the following SANS instrument configuration 

(L1 = 16.14 m, L4 = 13.19 m) give a focal length of 

726 cm

L L

L L

f

1 4

1 4



















 . (6)

This gives N

2 = 2111 Å2

. The use of 7 consecutive lenses (N = 7) focuses neutrons of 

wavelength  = 17.36 Å with a focal distance of 726 cm. The use of 30 consecutive lenses 

focuses neutrons of wavelength  = 8.39 Å down to the same focal spot. The use of 14 

consecutive lenses corresponds to a focusing wavelength  = 12.20 Å. 

For MgF2, the index of refraction is:

6 2

n 1 0.816*10 



. 

L1 L4

f

object

image

 lens

system
168

Note that the index of refraction of MgF2 for neutrons is less than unity so that concave 

lenses focus neutrons whereas convex lenses defocus them. This is opposite to basic optics 

for light whereby the index of refraction is greater than unity. 

2. RESOLUTION WITH FOCUSSING LENSES

Consider a neutron beam with a triangular wavelength distribution and a focusing lens 

system optimized for the main wavelength 

0

in that distribution. The main focal length is 

noted f0 and corresponds to object-to-lens and lens-to-image distances of L1 and L2

respectively. Moreover, consider another wavelength  within the same distribution and its 

corresponding focal length f. The object-to-lens and lens-to-image distances are L1 and L4

respectively for this wavelength. 

2

0

0 1 2 2π

ρb

R

2N

L

1

L

1

f

1

     (8)

2

1 4 2π

ρb

R

2N

L

1

L

1

f

1

    

Figure 2: SANS focusing system showing the main image of the neutron source 

corresponding to the main neutron wavelength 

0

and another image corresponding to 

another wavelength . 

In order to calculate the resolution with the lens system, the “geometry” contribution contains 

three terms: one that corresponds to the image of the source aperture; another that 

L1 L2

f0

source

w

detector

 lens

system

L4

f

image 

for 

0

image 

for 


169

corresponds to the sample aperture and a term that corresponds to averaging over a detector 

cell. 

Figure 3: Schematic representation of the three main vertical planes containing the sample 

aperture, the area detector (source aperture image for 

0

) and the source aperture image for 

another wavelength . 

Projection of the “geometry” part of the spatial resolution onto the detector plane in the 

horizontal direction is expressed as 



σx

:

 

2

3

2

2

2

4

4 2

2

4

2

4

2

geo

2

x

2

x

3

1

4

R

L

L L

4

R

L

L











 

 













 

 















  . (9)

Here R4 is the radius of the image of the source aperture for the focal length f at wavelength 

. In order to see how the two scale factors were derived, consider the case R2 = 0 for which 

R3 = 

4

2

L

L

R4, then the case of R4 = 0 for which R3 = 















 

4

4 2

L

L L

R2. x3 is the detector cell 

horizontal size. The image of the source aperture is given by R4 = 

1

4

L

L

R1. 

From the focusing equations, one obtains:

L2

R4

detector

 sample

aperture

L4

f

image 

for 

0

image 

for 



 lens

system

R2

r3

R3

f0
170

































     

2

2 4 0 0

4 2

0 2 4

λ

λ

1

f

1

L L

L - L

L

1

L

1

f

1

f

1

. (10)

Therefore:

 

12

Δx

4

R

λ

λ

1

L

L L

4

R

L

L

2

3

2

2

2

2

0

2

1

1 2

2

1

2

1

2

geo

2

x 

































 













 

 















  . (11)

This is the result valid for any wavelength . Around the focal wavelength 0, the averaging 

over the triangular wavelength distribution yields for the square term 

2

2

0

λ

λ

1

































 the result 

of 

3

2

2

λ

λ











 

. Even though the subscript on 0 is dropped, it should be remembered that these 

results are valid only for the focusing wavelength. 

Finally:

 

12

Δx

4

R

3

2

L

L L

4

R

L

L

2

3

2

2

2 2

1

1 2

2

1

2

1

2

geo

2

y  





























 

 















  . (12)

The spatial resolution in the vertical direction 



σy

involves the same terms as 



σx

along 

with contributions due to the gravity effect. 

Neutrons follow a parabolic trajectory, which at the detector position (for z = L1+L2) is given 

by:

2

y(L1  L2)  y0  A

where 

2

2

2 1 2

2h

gm A  L (L  L ) . (13)

The effect of gravity and wavelength spread contribute terms of the following form to y

2

:

 

2

2 2 2 2 2 2 4

grav

2

Qy λ

λ

3

2

y(z) y(z) A [ ] A 









 

             . (14)

 

2

2

wav

2

Qy λ

Δλ

6

1

Q 











  .

In summary, the Q resolution is then obtained as: 
171      grav

2

wav Qy

2

Qy geo

2

Qy

2

Qy      

(15)

2

2

x

2

3

2

2

2 2

1

1 2

2

1

2

1

2

2

2

2

Qx λ

Δλ

6

1

Q

2

x

3

1

4

R

3

2

L

L L

4

R

L

L

L

2









































 

 





























 

 



































 

2

2

y

2

2 4

2

3

2

2

2 2

1

1 2

2

1

2

1

2

2

2

2

Qy λ

Δλ

6

1

Q

3

2

A

2

y

3

1

4

R

3

2

L

L L

4

R

L

L

L

2















































  









 

 





























 

 



































 

Using focusing lenses modifies the “sample” term only (second term proportional to R2

2

). 

This term becomes much smaller with lenses. When lenses are used, the sample aperture R2

can be made larger without much resolution penalty. 

3. MINIMUM Q WITH FOCUSSING LENSES

The minimum reachable value of Q starts at the edge of the beam spot. The geometry with 

focusing lenses is characterized by an umbra only (with no penumbra). The neutron beam 

spot at the detector is therefore characterized by a box (not a trapezoidal) profile. A simple 

optics argument gives for the edge of the beam umbra in the horizontal and vertical 

directions for each wavelength  the following:

2

x

1 R

L

L L

R

L

L

X ( )

3

2

2

1 0

1 2

1

1

2

min



 



















 













 

  

(16)

















 



 



















 













 

  

3 2

2

2

1 0

1 2

1

1

2

min 2A

2

y

1 R

L

L L

R

L

L

Y ( ) .

The last term in Ymin() is due to gravity effect. Now the minimum achievable spot sizes are 

obtained by considering the part of the spot due to a wavelength spread . 

2

x

1 1 R

L

L L

R

L

L

X ( )

3

2

2

0

0

2

0

0

1

1 2

1

1

2

min









































 



































  

 













 

   . (17)

The magnitude part reduces to:



















































 































 

 2

λ

λ

1

λ

λ Δλ 1

2

0

0

2

0

0

. (18)
172

Now that the wavelength averaging has been performed, the 0 subscript in 0 is dropped for 

simplicity. The horizontal and vertical beam spot sizes are:

2

x

2 R

L

L L

R

L

L

X

3

2

1

1 2

1

1

2

min



 













 

 

(19)

















 



 













 

 

3 2

2

1

1 2

1

1

2

min 2A

2

y

2 R

L

L L

R

L

L

Y .

The corresponding values of the minimum Q are:

2

X min

min L

2 X

Q







(20)

2

Y min

min L

2 Y

Q







4. MEASURED SANS RESOLUTION

Specific Instrument Configuration

Consider the following instrument configuration: 

L1 = 16.14 m

L2 = 13.19 m

R1 = 0.715 cm

R2 = 0.635 cm

x3 = y3 = 0.5 cm

λ

Δλ

= 0.13. (21)

This gives A = 0.01189 cm/ Å2. 

Measurements with Focusing Lenses

Neutron optics measurements were made using a set of 7 consecutive biconcave MgF2 lenses 

(described in a previous chapter) inserted in the beam just before the sample aperture. This 

set corresponds to a focal wavelength 0 around 17.36 Å. 

The measured position of the neutron beam spot on the detector agrees with predictions. 
173

-5

-4

-3

-2

-1

0

1

5 10 15 20

With Neutron Lenses

calculated y position

measured y position

measured x position

Beam Spot Position (cm)

Neutron Wavelength (Å)

Figure 4: Variation of the neutron beam spot positions with wavelength. 

The beam spot resolution has strong (parabolic) wavelength dependence both in the x and in 

the y directions. The minimum resolution in the horizontal direction corresponds to a focal 

wavelength 0. The minimum in the x direction (0 = 17.2 Å) is taken to be the focal 

wavelength for our focusing arrangement since the x direction is independent of gravity 

effects. A procedure of using slice cuts across the beam spot was used to obtain these plotted 

results (including the 

1.45

scaling discussed in a previous chapter). The calculated trends 

agree with the measured ones. 
174

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

5 10 15 20

With Neutron Lenses

calculated 

x

calculated 

y

measured 

x

measured 

y Spot Size Standard Deviation (cm)

Neutron Wavelength (Å)

Figure 5: Variation of the spot size standard deviation with wavelength in the horizontal and 

vertical directions. 

Variation of the minimum spot sizes as a function of increasing wavelength is characterized 

by a minimum around 0 = 17.2 Å. The measured values have been chosen conservatively 

and are found to be overestimates that are higher than the calculated values. 
175

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

5 10 15 20

With Neutron Lenses

calculated Y

min

calculated X

min

measured Y

min

measured X

min

Y

min and Xmin (cm)

Neutron Wavelength (Å)

Figure 6: Variation of the minimum spot sizes with increasing wavelength. 

Discussion

The use of converging lenses has the advantage of allowing the opening up of the sample 

aperture (i.e., increasing R2) without penalty in resolution. This happens because the 

penumbra is minimized when lenses are used. The main effect is increased neutron current 

on sample. 

Refractive lenses are characterized by chromatic aberrations that show up as a dependence of 

both the variance x

2

and Xmin on (). In order to reduce these chromatic aberrations, 

() could be made smaller; which would result in a penalty in neutron current on sample. 

Focusing devices that use reflection (rather then refraction) optics (such as elliptical or 

torroidal mirrors) are not hampered by such chromatic aberrations. 
176

5. LENS TRANSMISSION

The transmission of a set of 7 concave spherical lenses is calculated and compared to 

transmission measurements. Consider a lens of spherical radius R and thickness 2h at the 

center and assume that the beam defining aperture has a radius of B. 

Figure 7: Schematics of the lens geometry. 

The transmission of one focusing lens averaged over the beam aperture is given by:



    





B

0

1 2 t dy2 yexp[ 2 (h R z)]

B

1

T

(22)

Here y is the vertical coordinate, z is the horizontal coordinate obeying 

2 2

z  R  y

and t

is the macroscopic cross section for MgF2. Note that t varies with neutron wavelength as t

= 0.000513  where  is in Å and t in mm-1. This variation was measured using a uniform 

thickness slab of MgF2. 

Performing the simple integration, one obtains:

 

1 2 R B exp2 t R B  1 2 Rexp2 R

2 B

exp[ 2 (h R)]

T t t

2 2 2 2

2 t

t

t

1         



  



(23)

The transmission of a set of 7 focusing lenses is given by T7 = T1

7

. 

The calculated and measured transmissions for the 7-lens system are compared for increasing 

neutron wavelength. 

2h

R B

y

z


177

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

5 10 15 20

7 Lens System

calculated

measured

Transmission

Neutron Wavelength (Å)

Figure 8: Calculated and measured neutron transmissions for a 7-lens system. 

The calculated and measured transmissions agree only partially. 

REFERENCES

D.F.R. Mildner, B. Hammouda, and S.R. Kline, “A Refractive Focusing Lens System for 

SANS”, J. Appl. Cryst. 38, 979-987 (2005).

B. Hammouda and D.F.R. Mildner, “SANS Resolution with Refractive Optics”, J. Appl. 

Cryst. 40, 250-259 (2007).

QUESTIONS

1. What is the main difference between focusing lenses for neutrons and focusing lenses for 

light?

2. Name a typical neutron focusing lens material.

3. When using neutron focusing lenses, what term of the instrumental resolution variance is 

modified? What is the advantage of this? 

4. What are chromatic aberrations? 
178

5. Do reflective optical devices suffer from chromatic aberrations? Name a refractive optics 

focusing device. 

6. Given the transmission T1 of one focusing lens, calculate the transmission T7 of a 7-lens 

system. 

7. Using many lens systems, could one build a neutron microscope?

8. What are the two main figures of merit for making a good refractive material to be used 

for making neutron lenses?

ANSWERS

1. Focusing lenses for neutrons are concave. Focusing lenses for light are convex. This is due 

to the fact that the index of refraction for neutrons is less than one while that for light is 

greater than one for most typical focusing materials. This is due to the fact that the scattering 

length for most materials is positive. Exceptions include hydrogen which has a negative 

scattering length. 

2. MgF2 is a commonly used neutron focusing lens material. 

3. The use of focusing lenses modifies the “sample aperture” term of the resolution variance. 

This term becomes much smaller even for larger source apertures. The advantage is a larger 

neutron current on sample. 

4. Chromatic aberrations correspond to the de-focusing effect for different wavelengths. The 

position of the source aperture image changes with wavelength thereby “blurring” the 

“image”. 

5. There are no chromatic aberrations with refractive optics. Torroidal or elliptical mirrors 

are typical refractive optics focusing devices. 

6. The transmission of a 7-lens system is given by T7 = T1

7 where T1 is the transmission of 

one lens. 

7. If one had lenses after the sample, one could obtain magnification using a neutron beam 

(neutron microscope). Given the low neutron wavelengths  (compared to light) the focal 

length f is very long (

2

f  R Nb

). Chromatic aberrations, the required long flight paths 

and coarse detector resolution give only modest magnification and a fuzzy picture. Note that 

the magnification factor can be worked out to be 

L f

f M

1



where f is the focal length and 

L1 is the object (sample in this case)-to-lenses distance. Note that L1 = f would yield high 

magnification. However, this condition would require that the lenses-to-image distance L4 be 

infinite (recall that 

4 1 L1 1 L 1 f  ). This is not realistic. 

8. The two figures of merit for refractive materials for making neutron lenses are as follows. 

(1) High density  and high coherent scattering length b in order to make the index of 

refraction n as small as possible. Recall that 

2

2

b

1 n 





 

. Making 1-n large (i.e., n small) 

reduces the focal distance f since 

f  R 2N(1 n)

where R is the lens radius and N is the 

number of lenses. (2) One would want to minimize the incoherent and absorption scattering 

cross sections i and a in order to minimize background and maximize lens transmission. 
179

Chapter 17 - GRAVITY CORRECTING PRISMS

Prisms are used to deflect the neutron beam upward thereby correcting for neutron fall due to 

gravity at long wavelengths. Prisms contribution to SANS resolution and Qmin are discussed 

here (Hammouda-Mildner, 2007). 

1. NEUTRON TRAJECTORY

The parabolic neutron trajectory equation in the pre-sample collimation follows.

y(z) B z(L z)

1

2

   0 



z 



L1 (1)

with:

9 1 -2

2

2

3.073*10 cm Å

2h

gm B

 

  . (2)

The z-direction is along the neutron beam and the y axis is in the vertical direction. L1 is the 

source-to-sample distance. The vertical component of the neutron trajectory slope 

y'(z)

is 

therefore:

y'(z) B (L 2z)

1

2

   0 



z 



L1 (3)

1

2

y'(L1)  B L

z = L1.

This neutron trajectory holds between the sample and detector. The addition of a prism 

changes the neutron trajectory by introducing an upward deflection of angle .

Figure 1: Schematics of a prism showing the deflected neutron trajectory in the simple case 

of minimum deviation. 

The slope of the neutron trajectory is changed to 








180

   1  

2

y'(L1) B L

z = L1. (4)

The neutron trajectory is therefore changed between the sample and detector to the following 

form:

y(z) B (z L ) [ B L ](z L )

1 1

2 2

1

2

          L1



z 



L1+L2. (5)

2 1 2 2

2

y(L1  L2)  B L (L  L )  L

z = L1+L2.

The use of a prism with deflection angle  = B

2

(L1+L2) would correct for the gravity effect 

exactly. 

2. THE PRISM DEFLECTION ANGLE

The “prism equation” (case of minimum deflection where the refracted beam is parallel to the 

prism base) relates the deflection angle  the prism angle  and the index of refraction n as:























 



2

α

sin

2

α δ

sin

n . (6)

This is the Snell’s law of refraction (also referred to as the Descartes law). The deflection 

angle is expressed as:























 

   



2

2sin n sin 1

 (7)

The wavelength dependence of the deflection angle enters through the index of refraction.

c 2

2π

ρb

n  1  . (8)

For MgF2 prisms, b/ = 1.632*10-6 Å

-2

so that n = 1-0.816*10-6

2

(where  is the neutron 

wavelength in Å). 

In the small deviation angle approximation, one can expand the prism formula with << to 

obtain:

2 2 C

2

tan

b

( )   









 

















   . (9)
181

This is an easier (approximate) expression to use in order to obtain analytical results. 

3. CONTRIBUTION TO THE Q RESOLUTION

The Q resolution at the detector (where z = L1+L2) involves the spatial variance y

2

. 

 

2

1 2

2

1 2 geo

2

y

2

y      y(L  L )    y(L  L ) 

(10)

2 1 2 2

2

y(L1  L2)  B L (L  L )  ()L . z = L1+L2

With the deviation angle given by  = C

2

, where C depends on the prism material and 

apex angle the following result is obtained. 

  [A L C] [ ]

2 4 2 2

2 geo

2

y

2

y           . (11)

Here the gravity variable A = BL2(L1+L2) has been used.

Assuming a triangular wavelength distribution, the wavelength averages are calculated as 

follows:

2

4 2 2 4

3

2

[ ] 















         . (12)

Therefore:

 

2

2 4

2 geo

2

y

2

y

λ

Δλ

3

2

[A L C] 











     

(13)

This is the variance of the neutron spot spatial resolution at the detector in the vertical 

direction. By analogy, the case without prisms is obtained for C = 0. The familiar “geometry” 

contribution is given in terms of the source aperture radius R1, sample aperture radius R2 and 

detector cell size y3. 

 

12

Δy

4

R

L

L L

4

R

L

L

2

3

2

2

2

1

1 2

2

1

2

1

2

geo

2

y  













 

 















 

(14)

Since most often x3 = y3, [



σx

]geo =[



σy

]geo . The standard deviation of the Q resolution 

Qy is related to the spatial standard deviation y as Qy = 





















L2

2

y. 
182

4. CONTRIBUTION TO QMIN

Qmin has contributions from geometry, gravity effect and the addition of a prism.

 

 

2

( ) ( )

Y Y A L C

2 2

min min geo 2

      

   . (15)

The wavelength term can be expressed (to first order) as:

 

















 

       2

2 2

2

2

( ) ( )

(16)

Therefore:

















  







 

2

2

3

2

1

1 2

1

1

2

min A L C2

2

y

R

L

L L

R

L

L

Y . (17)

Note that the same factor |A- L2C| enters in the resolution variance y

2

and in Ymin. Qymin is 

obtained by multiplying Ymin by the factor (2/L2). 

5. MEASUREMENTS WITH GRAVITY CORRECTING PRISMS

A prism cassette containing a row of five prisms is used for neutron optics measurements. 

Each prism is made out of single-crystal MgF2 and has a base of 3 cm*3 cm and a height of 

0.5 cm. In order to correct fully for the effect of gravity, between one and two prism cassettes 

would have to be used. Here only one cassette is used for the sake of simplicity.

Figure 2: Representation of the prism cassette containing a row of 5 MgF2 prisms. 

3 cm

3 cm

0.5 cm
183

The prism angle is equal to  = 2tan-1(1.5/0.5) = 143.13o. The prism variable is C = 

4.896*10-6 Å

-2

yielding an estimate for the factor L2C = 6.458 *10-3cm.Å-2and for the factor 

|A - L2C| = 0.00543 cm/Å2. 

A set of neutron optics measurements have been performed using the following instrument 

configuration:

L1 = 16.14 m (18)

L2 = 13.19 m

R1 = 0.715 cm

R2 = 0.635 cm

x3 = y3 = 0.5 cm

λ

Δλ

= 0.13.

This gives A = 0.01189 cm/ Å2.

The vertical position of the neutron beam varies with the neutron wavelength l following the 

parabola: 

2

1 2 2 y(L  L )  (A  L C)

z = L1+L2 (19)

When no prisms are used (C = 0), neutrons fall due to gravity. When prisms are used, falling 

neutrons are deflected upward. This is plotted as a function of wavelength and compared to 

the measured values. The value corresponding to  = 6 Å has been subtracted in each case for 

simplicity. 
184

-5

-4

-3

-2

-1

0

1

5 10 15 20

With Prisms

measured y position

measured x position

calculated y position

measured y position without prisms

Beam Spot Position (cm)

Neutron Wavelength (Å)

Figure 3: Variation of the neutron beam spot positions with wavelength. Statistical error bars 

are smaller than the plotting symbols. 

The variance of the neutron beam spot at the detector has also been measured in each case 

and compared to the calculated value. A figure shows the square root of the difference in the 

variances of the beam spot in the orthogonal directions as a function of wavelength. The 

measured values are obtained using the same procedure described in previous chapters 

(taking horizontal and vertical slice cuts across the neutron beam spot). The prediction for the 

case without prisms is also shown. The measured values are seen to be systematically higher 

than the calculated ones. This is believed to be caused principally by neutrons leaking 

between the apex and the base of adjacent prisms. 
185

0

0.1

0.2

0.3

0.4

0.5

5 10 15 20

With Prisms

measured (s

y

2

 - s

x

2

)

1/2

calculated (s

y

2

 - s

x

2

)

1/2

calculated (s

y

2

 - s

x

2

)

1/2 without prisms



y

2 - 

x

2

)1/2

(cm)

Neutron Wavelength (Å)

Figure 4: Variation of the variance of the neutron spot at the detector with wavelength. 

Statistical error bars have been included. Discrepancy between measured and calculated 

values is likely due to neutron leakage between adjacent prisms. 

6. PRISMS TRANSMISSION

Consider a prism system containing a row of prisms of width W and height H and assume a 

beam defining aperture of radius B. 
186

Figure 5: Transmission through one row of prisms. 

The transmission through one row of prisms using a source aperture of diameter 2R2 can be 

calculated as follows. 

2B

dyexp[ 2z]

T

2B

0



t

 (20)

Here 2z is the path across a prism at a height y. For 0



y



H, it is 2z = (H-y)W/H. The 

integration is performed for y covering each prism. When 2B is not a multiple of H, the result 

is:

2BW/ H

[2 3exp( W) exp( 3 W 2BW/ H)]

T

t

t t t



      



(21)

Note that this expression is for the transmission of one row of prisms. When 2B is a multiple 

of H, T is given simply by Tm:

W

[1 exp( W)]

T

t

t

m



 



(22)

This result assumes that the beam defining aperture is rectangular. The total cross section for 

MgF2 is estimated empirically as t (cm-1) = 0.00513  (Å). A transmission measurement of 

the cassette containing two rows of prisms has been made using a sample aperture of 2B = 

1.27 cm and a neutron wavelength of  = 17.2 Å. The measured transmission was found to be 

T = 0.70 while the predicted transmission is T = 0.75. This result is not reliable due to the 

neutron streaming in-between the prisms.

2B

W

H

z

y
187

7. DISCUSSION

Prisms correct for the neutron fall by deflecting the direct neutron beam back up. They also 

correct for the anisotropy of the neutron beam spot on the detector. Gravity deforms this spot 

to an oval shape. Prisms correct this shape back to a circular shape. Analytical expressions 

for the spatial resolution have been presented. Neutron beam optics measurements using a 

prism cassette have been made. Good agreement was found between calculated and 

measured beam spot positions. However, no good agreement was found for the instrumental 

resolution variance due to neutron leakage between adjacent prisms. 

REFERENCE

B. Hammouda and D.F.R. Mildner, “SANS Resolution with Refractive Optics”, J. Appl. 

Cryst. 40, 250-259 (2007).

QUESTIONS

1. What is the advantage of using prisms in neutron optics?

2. Prisms refract incident white light into what?

3. The use of gravity correcting prisms affects what part of the instrumental resolution 

variance?

4. What is the wavelength dependence of the prism deflection angle?

5. What is the prisms minimum deviation approximation?

6. Could a prism system be used for all neutron wavelengths?

ANSWERS

1. Prisms correct for gravity effects. At long wavelengths the effect of neutron fall (due to 

gravity) is to lower the neutron beam spot and deform it into an oval shape. The use of 

prisms kicks the neutron spot back up and corrects it back to a circular shape. 

2. Prisms refract incident white light into the rainbow spectrum. 

3. The use of gravity correcting prisms affects the wavelength spread part of the instrumental 

resolution variance. 

4. The prism deflection angle varies like the square of the wavelength. 

5. The prism’s minimum deviation approximation corresponds to the case where the 

refracted beam (inside the prism) is parallel to the prism’s base. 

6. Since the gravity correction factor (A-L2C) is independent of neutron wavelength , the 

same prism system can be used to correct for gravity at all wavelengths. 
188

Chapter 18 - NEUTRON BEAM CURRENT

1. VARIATION OF DETECTOR COUNT RATE WITH SPOT SIZE

One important figure of merit relevant to the performance of SANS instruments is the 

neutron beam current on sample (i.e., number of neutrons per second reaching the sample 

aperture) as measured by the area detector; this is related to the total detector count rate 

summed up over the beam spot (Hammouda-Mildner, 2007). Since the detector is a “1/v” 

absorber, the beam current is proportional to the count rate/wavelength. 

Using the same notation as before, consider the following SANS instrument configuration. 

L1 = 16.14 m (1)

L2 = 13.19 m

R1 = 0.715 cm

R2 = 0.635 cm

x3 = y3 = 0.5 cm

λ

Δλ

= 0.13.

Total detector count rates are included here for the 3 cases considered previously: empty 

beam configuration, the use of focusing lenses and the use of gravity correcting prisms. A 

figure summarizes the variation of the total detector count rate with the minimum neutron 

spot size (in the vertical direction) Ymin for each case. Note that Ymin (not Xmin) is what 

determines Qmin. The beam current depends on the ratio (R1R2/L1)

2 which was the same in all 

three cases, and also on the neutron source spectrum. Each point corresponds to a different 

neutron wavelength (from 6 Å to 20 Å). No corrections have been made to account for the 

area detector efficiency (“1/v” absorber) or dead time losses. These effects are strongly 

wavelength dependent and are outside of the scope of the present discussion. The main 

observation is that increasing the neutron wavelength increases Ymin (i.e., broadens the 

neutron beam) for regular SANS optics but decreases Ymin when focusing lenses are used. 

Using prisms corrects for the beam broadening with increasing wavelength. 

Using neutron lenses lowers Ymin substantially without too much loss in neutron current 

whereas using prisms is accompanied by neutron losses because of the prisms’ low neutron 

transmission and due to the un-optimized first generation prisms system used here. The 

lowest Ymin obtained with the lenses (at 17.2 Å neutrons) has low detector count rate (1,252 

counts per second). Note that the sample aperture radius R2 can be increased considerably 

without degrading the overall resolution. The use of lenses with a large sample area can 

enable a great increase in count rate on sample, with small penalty on resolution. The prisms 

used here correct only partially for the effect of gravity. If we had used prisms that correct 

“exactly” for the gravity effect, the prisms curve in the figure would have been vertical and 

shown no wavelength dependence. 
189

100

1000

104

105

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

All Cases

lenses

empty beam

prisms

Detector Counts per Second

Y

min

 (cm)

Figure 1: Variation of the total detector counts rate within the neutron spot with the extent of 

the beam size in the vertical direction Ymin. The 3 curves correspond to the 3 cases: (1) empty 

beam, (2) with lenses and (3) with prisms. Each data point corresponds to a different neutron 

wavelength. 

2. VARIATION OF DETECTOR COUNT RATE WITH Qmin

Another slightly modified plot uses QY

min = (2Ymin)/(L2) instead for the horizontal axis. 

Here  is the neutron wavelength, L2 is the sample-to-detector distance, Ymin is the vertical 

size of the neutron spot on the detector. The detector count rate is seen to drop quickly for 

lower Qmin. A log-log plot shows no simple power law behavior between the detector count 

rate and Qmin. Note that this measured detector count rate is not the neutron current since the 

detector absorption cross section is proportional to the neutron wavelength (“1/v” absorber).
190

100

1000

104

105

0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018

All Cases

lenses

empty beam

prisms

Detector Counts per Second

Q

Y

min

 (Å-1)

Figure 2: Variation of the detector count rate with QY

min for the same 3 cases. 

REFERENCE

B. Hammouda and D.F.R. Mildner, “SANS Resolution with Refractive Optics”, J. Appl. 

Cryst. 40, 250-259 (2007).

QUESTIONS

1. What is the neutron flux? What is the neutron beam current?

2. Why does the neutron beam spot on the detector broaden in the vertical direction when the 

wavelength is increased?

3. Why is there a minimum in the variation of the neutron beam current with neutron 

wavelength when a focusing lens system is used?
191

ANSWERS

1. The neutron flux is the number of neutrons per cm2per second. The neutron beam current 

is the number of neutrons per second. The neutron flux is also called the current density. 

2. The neutron beam spot on the detector broadens in the vertical direction when the 

wavelength is increase due to the larger gravity effect. The neutron fall increases with 

wavelength square. 

3. When a specific focusing lens system is used, there is a specific “focusing” wavelength 

whereby the source image occurs exactly at the detector position. This corresponds to the 

smallest focal spot on the detector. 
192

Chapter 19 - THE SMEARING EFFECT

In analyzing SANS data, smearing of the model function used is necessary before performing 

nonlinear least-squares fits. The smearing procedure involves a convolution integral between 

the resolution function and the scattering cross section for the scattering model. 

1. THE RESOLUTION FUNCTION

Consider a 1D Gaussian resolution function (Barker-Pedersen, 1995):



































 2

Qx

2

x

1/ 2

2

Qx

1D x

2σ

Q

exp

2πσ

1

P (Q )

(1)

This distribution is normalized to 1.







dQxP1D(Qx)  1.

In order to show this normalization, make a variable change to 

2 X  Qx

so that 

dX  2Qx dQx

and the normalization integral becomes as follows. 

  





 





































 

0 0

2

Q x

1/ 2

2

Q x

x 1D x x 1D x

2

X

exp

2πσ

1

2 X

1

dQ P (Q ) 2 dQ P (Q ) 2 dX . (2)

The following integral is used:





 

0

a

π

exp[ aX]

X

1

dX

(for a>0) (3)

This verifies that the P1D(Qx) distribution is normalized. The Qy distribution is similar.

Consider a 2D Gaussian resolution function:

P (Q) P (Q )P (Q ) 2D 1D x 1D y

(4)

















 

















 

















 2

Q y

2

y

2

Q x

2

x

2

1

2

Q y

2

1

2

Q x

2σ

Q

2σ

Q

exp

2 σ

1

2 σ

1

.

This distribution is also normalized to 1.
193  



 

0

2D

2π

0

QdQ d P (Q) 1

(5)

In order to show this, make a variable change to 

2 R  Q

and 

dR  2QdQ

so that the 

normalization integral becomes as follows.



































 





















  

















 

















  

 

2

Q

2

Q

2

Q

0 0

2

Q

2D

2π

0

0

(2π) exp exp

σ

R

exp

πσ

1

dR

2

1

QdQ d P (Q)

=1.

(6)

2. THE RESOLUTION CORRECTION

The smeared 1D cross section corresponds to radially averaged SANS data and is given by 

the following integral (using polar coordinates):

 



 

 













 

0

1D

smeared d

d (Q Q')

dQ'P (Q')

d

d (Q)



 



 







































d

d (Q Q')

0

2σ

2 Q'

exp

1/2

2 σ

1

dQ' 2

Q'

2

Q'

. (7)

The smeared 2D cross section integral corresponds to 2D SANS data and is given by the 

following expression:





  

 



 





 d

d (Q Q',Q Q' )

dQ' P (Q' ) dQ' P (Q' )

d

d (Q ,Q )

x x y y

x 1D x y 1D y

x y

(8)

Note that (Qx,Qy) are in Cartesian coordinates. In cases where radial averaging of the data is 

not possible, the 

2

Qx

and 

2

Qy

variances are needed. Note that the variance 

2

Qy

2

Qx

2

Q    

is never used. 
194

Figure 1: Parametrization in the detector plane.

3. ISO-INTENSITY CONTOUR MAPS WITH GRAVITY EFFECT

Gravity effect on the neutron trajectory distorts the iso-intensity contour maps from 

concentric circles to concentric oval shapes. The following parametric equation describes an 

elliptical shape: 

1

(a b)

y

a

x

2

2

2

2





 . (9)

Here a is the minor (horizontal) axis and a+b is the major (vertical) axis of the elliptical 

shape. If we consider different major axes for the top and bottom parts, an oval shape is 

obtained.

b A(2 sign(y) )

2

    . (10)

The top and bottom parts have been represented using the sign function. The x and y 

coordinates can be expressed as:

Q



Q Q'

 



Q'


195

x  r()cos()

(11)

y  r()sin() .

 is the azimuthal angle for binning in the detector plane. Combining these equations, one 

obtains the following parametric equation:

2

2

2

2

2

(a b)

sin

a

cos

1

r ( )









  . (12)

Note that this applies to Q = 0 only. 

4. NUMERICAL APPLICATION

Consider the following realistic case:

L1 = 16.14 m (13)

L2 = 13.19 m

 18 Å

0.13

λ

Δλ



2 A  0.01189 cm/Å

This gives 

min = 15.66 

Å , max = 20.34 Å . 

The following beam spot characteristics are obtained:

ymin  2.916 cm , ymax

 4.919 cm y  3.852 cm ,  y  3.863cm ytop

 ymax  y 1.0667 cm ybot  y  ymin  0.9365cm

Here 

y

is the spot height corresponding to the mean wavelength 



and 

 y 

is the vertical 

location of the beam center. Note that for any practical purpose 

y  y 

and the difference 

ytop  ybot  0.130 cm

is so small that the oval shapes are really elliptical. 
196

The beam standard deviation in the vertical direction is estimated to be 

y 0.409 cm

using 

both the numerical integration over y and the analytical averaging over  (formula given 

above). 

-12

-8

-4

0

4

8

12

-12 -8 -4 0 4 8 12

 Contour Plot 

y pos k=1 a=0.5 cm

y pos k=5

y-pos k=10

y-pos k=15

ypos k=20

y pos

 x pos

Figure 2: Iso-intensity contour map when neutrons are under the influence of gravity; i.e., at 

long wavelength ( = 18 Å) and typical wavelength spread (

Δλ λ 13%

). Contours 

corresponding to a = 0.5 cm and k = 1, 5, 10, 15 and 20 are shown. The x and y axes are in 

channel numbers (each detector channel corresponds to 0.5 cm). 

5. SMEARING FOR HARD SPHERES

Consider idealized scattering from hard spheres and compare it to the smeared case. The 

form factor for a hard sphere of radius R = 50 Å is given by the following function:
1972

2 QR

cos(QR)

(QR)

sin(QR)

QR

3

P(Q) 



























 

















(14)

Consider the following high-Q configuration:

R1 = 2.5 cm (15)

R2 = 0.5 cm

x3 = y3 = 0.5 cm

L1 = 1.5 m

L2 = 1.5 m

 = 6 Å

λ

Δλ

= 15 %.

The direct beam spatial resolution on the detector plane is:

x

2 = 1.83 cm2

(16)

y

2 = 1.83 cm2

.

The variance of the Q resolution is: 

2

x

2 5 Qx  8.94*10  0.0037 Q



(in units of Å-2) (17)

2

y

2 5 Qy  8.94*10  0.0037 Q



(in units of Å-2) .

The wavelength spread contribution dominates for this high-Q configuration. The gravity 

contribution is negligible for the 6 Å wavelength. 

For this high-Q configuration, 

Qmin

X = QminY = 0.028 Å-1

. (18)
198

10-6

10-5

0.0001

0.001

0.01

0.1

1

0 0.05 0.1 0.15 0.2 0.25 0.3

Sphere Form Factor 

unsmeared

smeared

Model Form Factor

Q (Å-1)

Figure 3: Plot of the form factor for a sphere of radius R = 50 Å before and after smearing 

produced by the high-Q configuration. 
199

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022

0 0.05 0.1 0.15 0.2 0.25 0.3



Q

 calculated



Q

Q (Å-1)

Qmin

Figure 4: Variation of the standard deviation of the Q resolution vs Q. 

Consider the following low-Q instrument configuration and spheres of radius R = 500 Å. 

R1 = 2.5 cm (19)

R2 = 0.5 cm

x3 = y3 = 0.5 cm

L1 = 15 m

L2 = 15 m

 = 12 Å

λ

Δλ

= 15 %.

Therefore:

A = 0.0138 cm/Å2(20)

x

2 = 1.83 cm2

y

2 = 1.83 cm2

So that:

2

x

2 7 Qx  2.23*10  0.0037 Q



(in units of Å-2) (21)
200 2

y

2 7 Qx  2.31*10  0.0037 Q



(in units of Å-2). 

The first term is slightly different for Qx and Qy because of the small gravity contribution. 

For this configuration, the geometry part dominates at low-Q, the wavelength-spread part 

contributes at higher Q, and the gravity term is small. 

For this low-Q configuration, 

Qmin

X = 0.0014 Å-1

, (22)

Qmin

Y = 0.0016 Å-1

. 

10-7

10-6

10-5

0.0001

0.001

0.01

0.1

1

0 0.005 0.01 0.015 0.02

smeared_sphere_r=500a

unsmeared

smeared

Model Form Factor

Q (Å-1)

Figure 5: Plot of the form factor for a sphere of radius R = 500 Å before and after smearing 

produced by the low-Q configuration.
201

0.0001

0.001

0.01

0.1

0.001 0.01 0.1 1

high-Q

low-Q

Resolution Standard Deviation

Q

Q (Å-1)

Qmin

Qmin

Figure 6: Plot of the standard deviation of the Q resolution for both the low-Q and the high-Q 

configurations. The values of Qmin are also indicated. 

6. SANS FROM SILICA PARTICLES

SANS data have been taken from a dilute solution of monodisperse silica particles in D2O 

(volume fraction of 0.1 %) and fit to the sphere model. Fit results gave a sphere radius of R = 

563.51 ± 0.45 Å. SANS data were taken using a low-Q instrument configuration. 
202

0.01

0.1

1

1 0

100

1000

0.001 0.01 0.1

Silica Particles in D2O 

SANS Data 

Smeared Model Fit 

Unsmearted Model Fit 

Scattered Intensity (cm-1

)

Q (Å-1)

Figure 7: SANS data from a dilute solution of monodisperse silica particles in D2O along 

with the fit to the sphere model. 

REFERENCE

J.G. Barker, and J.S. Pedersen, “Instrumental Smearing Effects in Radially Symmetric SANS 

by Numerical and Analytical Methods”, J. Appl. Cryst. 28, 105-114 (1995). 

QUESTIONS

1. What are the two ways of accounting for instrumental resolution?

2. Is it OK to perform a 1D smearing convolution integral on 2D SANS data? 

3. What is the effect of instrumental smearing on the radius of gyration obtained from a 

Guinier fit?

4. What are the two ways of correcting for the effect of gravity?
203

ANSWERS

1. Instrumental resolution is included either (1) by smearing of the model used to fit the data 

or (2) by desmearing the data through an iterative process. Method (1) is the most reliable 

and the most used. Method (2) does not work well when sharp peaks appear in the data. 

2. It is OK to perform a 1D smearing convolution integral if the 2D SANS data are 

azimuthally symmetric (scattering is isotropic). 

3. Instrumental resolution tends to broaden peaks. The Guinier region is the tail of a peak at 

Q = 0. Broadening implies a lower slope and therefore a lower radius of gyration. The 

smeared radius of gyration is lower than the real value. 

4. Gravity correction can be made (1) through a software method by defining constant-Q 

elliptical bins or (2) through a hardware method using gravity-correcting prisms. 
204

Part E – SANS DATA CORRECTIONS AND DATA REDUCTION

Chapter 20. SANS Data Corrections

20.1 The Solid Angle Correction

20.2 The Jacobian Correction

20.3 Absolute Intensity

20.4 Wide-Angle Correction

20.5 Estimation of Incoherent Scattering

20.6 Dead Time Correction

References

Questions

Answers

Chapter 21. SANS Data Reduction

21.1 Instrument Configurations

21.2 SANS Data Acquisition

21.3 The SANS Data Correction Steps

21.4 Typical Reduced SANS Data

References

Questions

Answers
205

Chapter 20 - SANS DATA CORRECTIONS

The 2D SANS data undergo a number of corrections during the data reduction process. Some 

of these corrections are described here (Berk-Glinka, 1985; Krueger-Hammouda, 1993). 

1. THE SOLID ANGLE CORRECTION

The scattering geometry is represented by an Ewald sphere in general terms. For SANS, 

scattering angles are small and the detector is planar. This brings about a solid angle 

correction performed early on in the data reduction process. 

The corrected macroscopic scattering cross section is related to the measured one by:

dΩ corrected

dΣ















= 

dΩ measured

dΣ













dΩ

d





. (1)

The ratio of the two (corrected and measured) solid angles is expressed as:

dΩ

d





= 

2

r

A

A'

r'2

= 

A'

A

2

2

r

r'

. (2)

Here A and A’ are the solid angle base areas on the sphere and on the detector plane 

respectively and r and r’ are the magnitudes of the vectors subtending these bases. These 

ratios can be calculated by inspecting the scattering geometry involved. The bottom part of 

the figure (which is a projection onto the vertical scattering plane) shows that 

2

2

r

r'

= cos2() and 

A'

A

= cos(). (3)

Therefore, the solid angle correction factor is:

dΩ

d





= cos3(). (4)

This correction is performed as the first manipulation of the 2D data. 
206

Figure 1: Schematic representation of the scattering geometry. The bottom part represents a 

projection onto the vertical scattering plane. The scales are grossly exaggerated. The 

horizontal scale is of order meters while the vertical scale is of order centimeters.

2. THE JACOBIAN CORRECTION

Some SANS instruments use neutron area detector that use the delay line method. This 

method uses only two detection (anode) wires that wind their way horizontally for y and 

vertically for x. The advantage of this method is that it uses only 4 signals (2 for x and 2 for 

y) from the detector. The time difference between the two x signals determines the position 

sample

position

area

detector

d’ d



r’

r



r’

r

r’

A

A’

A’

A

A’


207

of the detection event. One disadvantage is the low count rate saturation level. 

Figure 2: Schematic representation of a delay line detector showing only 16 wire portions. A 

more realistic detector contains 64 wire for x and 64 wires for y. 

This type of area detector introduces a systematic distortion of the detection event 

coordinates due to the nonlinearity of the process. The coordinate of a detected event x in 

channel number space corresponds to a coordinate x’ in real position space. This nonlinear 

mapping corresponds to the transformation:

x’ = B tan(x/B). (5)

Here B is the nonlinearity detector constant in x. The x-dependence of the scattering cross 

section d(x)/d obeys the following conservation relation:

dx'

d

d (x')

dx

d

d (x)











. (6)

The cross section in real position space is therefore given by:























B

x

cos

d

d (x)

d

d (x') 2

. (7)

Pre-amplifier

y-detection

wire

x-detection

wire
208

Here the Jacobian of the (x,x’) transformation has been obtained from the following 

derivative:















B

x

cos

dx'

dx 2

. (8)

This correction is made on the 2D SANS data. The detector nonlinearity constants B for x 

and C for y are measured using the cadmium mask method. This method consists in using a 

large cadmium mask with equally spaced holes right in front of the detector. The measured 

(distorted) positions and the equally spaced (known) positions are then used to obtain the 

nonlinearity constants. 

In practice, most SANS instruments nowadays use area detectors that are based on the 

coincidence method with grids of wires for x and y positioning. These detector types are not 

hampered by the described distortion and therefore do not require any Jacobian correction. 

3. ABSOLUTE INTENSITY

SANS data are rescaled to form a macroscopic scattering cross section (units of cm-1). This 

rescaling involves a measurement from the scattering sample Is(Q) and a measurement from 

the empty beam transmission I0() which is the incident neutron beam current. Here  is the 

neutron wavelength. The measured SANS scattered intensity can be expressed as:

Is(Q) = I0() T() d 

dΩ

dΣ(Q)

 (9)

I0() is given by:

I0() =  A2  t. (10)

Here:

 is the neutron flux on sample,

A2 is the illuminated sample area,

T() is the sample transmission,

d is the sample thickness,

d(Q)/d is the macroscopic scattering cross section,

 is the solid angle that subtends a detector cell,

() is the detector efficiency for the neutron wavelength used

t is the effective counting time normalized to a fixed number of monitor counts 

(108cps). 

Is(Q) is the number of neutrons detected in a unit detector cell in time t. The scattering cross 

section can be measured as the ratio:
209

dΩ

dΣ(Q)

= 

I (λ)T()d

I (Q)

0

s

. (11)

I0() is measured as an empty beam transmission run and Is(Q) is the usual SANS scattering 

run. This absolute intensity rescaling is performed on the 2D SANS data. 

4. WIDE-ANGLE CORRECTION

Note that when the small-angle approximation is not valid, there are angular corrections to 

the detector efficiency (,) and to the sample transmission T(,) (here  is the scattering 

angle). Correction to the detector efficiency is taken care of by measuring the sensitivity 

correction sample (either water or plexiglass) placed in the same stringent angular condition 

as the sample. 

For finite scattering angles, the transmission angular correction involves the following 

integral:





















 





 

cos( )

(d z)

exp

d

d (Q)

dz exp( z) t

d

0

t

(12)



















   

















 







cos( )

z

dz exp( z)exp

cos( )

d

exp

d

d (Q)

t

d

0

t t

.

Part of the angular correction to the transmission is contained in the second term and is due 

to the longer neutron path inside the sample at large scattering angles. The remaining 

correction is taken into account by the integral. The following result is obtained:





















 





 

cos( )

(d z)

exp

d

d (Q)

dz exp( z) t

d

0

t

( d)a

(1 T )

Td

d

d (Q)

t

a









 . (13)

Here 

T exp( d)  t

is the regular sample transmission given in terms of the macroscopic 

(total) scattering cross section t and 





















 1

cos( )

1

a

is the angular correction factor. This 

angular correction is performed at the stage of radially averaging the SANS data. 
210

Figure 3: Geometry of the angular dependence of the sample transmission when the 

scattering angle is not small. 

5. ESTIMATION OF INCOHERENT SCATTERING 

Incoherent scattering is a flat (i.e., Q-independent background) mostly due to hydrogen 

scattering. When only single scattering events are taken into account, the isotropic incoherent 

scattering cross section can be estimated as:

dΩ

dΣi

= 







I (λ)T()d 4

I i

0

i

. (14)

When multiple scattering events are included, this cross section becomes:

dΩ

dΣi

= 

  











  n 1

n

i

0

i

d

n!

1

4 d

1

I (λ)T( )d

I

. (15)

This n-scattering events summation can be resumed to yield:

dΩ

dΣi

= 

exp( d)

1 exp( d)

4 d

1

i

i



 



. (16)

If incoherent scattering dominates (

t  i

), the transmission is approximated as 

T exp( d) exp( d)  t  i

and the cross section is simplified to be (Shibayama et al, 2009):

T

1 T

4 d

1

dΩ

dΣi 



 . (17)

z

dz

d


211

This simple relation is approximate since it neglects inelasticity corrections (among others). 

Another more practical way of estimating incoherent scattering is by preparing and 

measuring an incoherent scattering sample for each measured sample. The incoherent 

scattering sample contains the same number density of hydrogen atoms as the measured 

sample. It could consist of a mixture of H2O and D2O that would reproduce the level of flat 

(mostly incoherent) SANS scattering from aqueous solutions. It could also consist of pure 

hydrogenated polymers that would reproduce the incoherent level in mixtures of deuterated 

and non-deuterated polymer blends. Subtraction of the scattering from the incoherent 

scattering sample is best performed on the 2D data just in case there is anisotropy in the data. 

Actually, some prefer to consider the incoherent scattering run as their “empty” run; it 

replaces the empty cell run. Another alternative is to estimate the incoherent scattering level 

as a Q-independent constant in the nonlinear least squares fitting methods.

6. DEAD TIME CORRECTION

Processing of the detector signal takes a finite amount of time. The neutron area detector time 

response constant  was measured and found to be between 3 and 5 sec. The “measured” 

count rate NM and the “true” count rate NT are related by the following relation:

NT = 

1 N τ

N

M

M



. (18)

This assume a non-paralyzable detector response; i.e., that the detector keeps on counting 

even when it is processing signals. 

This dead time detector correction could be performed at any stage of the data reduction 

process (Brulet et al, 2007). It depends solely on the total detector count rate NM. 

REFERENCES

N. Berk and C. Glinka, “Data Reduction and Imaging System Manual”, NIST Center for 

Neutron Research (1985).

S. Krueger and B. Hammouda “SANS Data Acquisition Manual” and “NIST Center for 

Neutron Research Data Reduction Manual”, NIST Center for Neutron Research (1993, 

2000). 

A. Brulet, D. Lairez, A. Lapp, and J.P. Cotton, “Improvement of Data Treatment in SANS”, 

J. Appl. Cryst. 40, 165-177 (2007).

M. Shibayama, T. Matsunaga, and M. Nagao, “Evaluation of Incoherent Scattering Intensity 

by Transmission and Sample Thickness”, J. Appl. Cryst. 42, 621-628 (2009)
212

QUESTIONS

1. What is the origin of the solid angle correction?

2. What is the absolute scattering cross section?

3. Why is the scaling to an absolute cross section necessary?

4. What is the difference between a delay line detector and a coincidence detector?

5. Why is it important to perform the various corrections on the 2D data (before radial 

averaging)?

6. Write down an estimate of the macroscopic incoherent scattering cross section. 

ANSWERS

1. The solid angle correction is due to the fact that scattering occurs on the Ewald sphere 

whereas SANS area detectors are flat. 

2. The absolute cross section is the macroscopic scattering cross section d(Q)/d (units of 

cm-1). It is related to the microscopic scattering cross section d(Q)/d (units of barns) by 

the number density of the scattering objects (N/V). 

3. When the SANS data are reduced to an absolute cross section, the number density of the 

scattering objects can be obtained. For example, if the scattering objects are aggregates, one 

could obtain the number of molecules per aggregate. 

4. A delay line detector uses one wire for x and one wire for y detection. These wires wind 

their way to form a grid.

5. It is important to perform the various corrections on the 2D data (before radial averaging) 

so as to preserve any scattering anisotropy in the data. 

6. The macroscopic incoherent scattering cross section can be estimated to be 

T

1 T

4 d

1

dΩ

dΣi 





where T is the sample transmission and d is its thickness. This was 

published by Shibayama. 
213

Chapter 21 - SANS DATA REDUCTION

The SANS data reduction procedures used at the NIST Center for neutron research are 

described here after a brief summary of the data acquisition process (Krueger-Hammouda, 

1993; Kline, 2006). 

1. INSTRUMENT CONFIGURATIONS

Every SANS experiment begins with the choice of one (or more) instrument configurations 

to be measured. Decisions are made based on the characteristic features of the investigated 

system (characteristic sizes and scattering level). A program is used to “simulate” 

configurations by choosing a neutron wavelength and wavelength spread, source and sample 

aperture sizes, source-to-sample and sample-to-detector distances. The Q range, neutron 

beam current and beamstop size are obtained for each configuration. 

The following high-Q configuration was obtained from the SASCALC program for the NG3 

SANS instrument.

Input:

Neutron Wavelength  = 6.0 Å

Wavelength spread:  = 0.15 (FWHM)

Detector Offset: 25 cm

Source aperture diameter: D1 = 5.0 cm

Sample Aperture diameter: D2 = 1.27 cm

Source aperture to sample aperture distance: L1 = 537 cm

Sample-to-detector distance: L2 = 133 cm

Output:

Total Qmin = 0.0179 A-1

Total Qmax = 0.4742 A-1, Horizontal Qmax = 0.4211 A-1, Vertical Qmax = 0.2467 A-1

Beam diameter: 2.39 cm, Beamstop diameter: 2.54 cm, Umbra/Penumbra: 0.414

Attenuator transmission: 0.0003 (Attenuator number 8 for transmission runs.)

Neutron beam current at the sample: 7.5543*105counts/sec

2. SANS DATA ACQUISITION

The data acquisition software package consists of a set of menus and tables within a 

graphical user interface. The main menu contains “single run” and “multiple runs” modes as 

well as a “manual operations” mode. This last mode contains all aspects of hardware control. 

Typically two to three configurations are chosen and saved as templates at the beginning of 

every new set of measurements (i.e., for every new user group). A set of SANS 

measurements includes scattering runs and transmission runs from every sample as well as 
214

from the empty cell. Transmission runs are performed by attenuating the beam and moving 

the beamstop out. Note that the transmission measurements depend on neutron wavelength 

but are independent of collimation parameters. When the same wavelength is chosen for the 

low-Q and the high-Q configurations, only one transmission run is needed. A scattering run 

is taken for the blocked beam background measurement. The neutron beam is blocked using 

a neutron absorber (lithiated compounds are preferred because they do not produce gammas). 

Every run is taken for a fixed amount of time. The total measurement time is divided into a 

number of prefactors. This allows the recording of the beam monitor and total detector 

counts for each prefactor. A change of the total detector counts points to changes occurring in 

the sample. A change in the beam monitor counts points to changes in the neutron source. A 

transmission run is also taken from the empty beam for every instrument configuration. This 

run measures the neutron beam current and is used to scale the scattering intensity to an 

absolute cross section (unit of cm-1). This run replaces the use of a secondary standard 

sample. Scattering from a flat hydrogen-containing scatterer (such as water or plexiglass) is 

taken on a regular basis by the instrument staff and made available to users to correct for 

slight difference in detector cell efficiency. The data acquisition sequence is programmed 

into a series of runs that are performed automatically. Stepping among the samples as well as 

varying sample environment conditions (such as temperature) is also programmed. Real time 

imaging of the data is helpful in monitoring the data acquisition process. 

The SANS data files are in the binary format. The data header (256 bytes) contains all 

instrumental parameters along with sample information (label, thickness, etc). The main data 

part consists of 16,384 values (= 14-bit) comprising data from 128*128 detector cells. 

Historically, binary SANS data have been compressed to keep file sizes small. 

3. THE SANS DATA CORRECTION STEPS

Calculation of the various sample (and empty cell) transmissions is performed. This is 

performed through linking of the various transmission runs and the empty beam transmission 

run. The calculated transmissions are then used in the data reduction protocol. 

SANS raw data files are loaded into “work files” and normalized to a fixed (= 108) number of 

monitor counts. The empty cell and blocked beam runs are subtracted and rescaled following 

a specific recipe:

   

c

c b

s c

s c b

s

T

I (Q) I

T

I (Q) I

I (Q)













. (1)

Is+c(Q) is the scattering run from the sample + cell, Ic(Q) is the scattering from the empty cell, 

Ib is the scattering with beam blocked, Ts+c is the transmission for the sample inside the cell 

and Tc is the empty cell transmission. Note that Ts+c = Ts.Tc. 

Using the neutron beam counts (empty beam transmission) I0(), the scattering intensity is 

scaled to an absolute cross section (units of cm-1) as:
215

dΩ

dΣ(Q)

= 

I (λ)T()d

I (Q)

0

s

. (2)

T, d and  are the sample transmission and thickness and the solid angle subtending one 

detector cell. SANS data in peripheral detector cells and those close to the beamstop are 

masked out in order to keep only the reliable data. Then the 2D corrected and scaled data are 

radially averaged to produce 1D data. Circular binning is the norm for isotropic scattering. 

Sector or rectangular averaging is used for scattering with anisotropic features. 

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.08 -0.04 0.00 0.04 0.08

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

Figure 1: Typical SANS data image. Radial averaging consists in forming circular bins in 

which data are summed up. 
216 120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.08 -0.04 0.00 0.04

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

2.5

2.0

1.5

1.0

0.5

Figure 2: Typical SANS data from an anisotropic scatterer (collagen from kangaroo tail 

tendon). Sector averaging is performed on this asymmetric data. 

The radially averaged data format contains the following columns:

Q, d(Q)/d, I, <Q>, Q, BS (3)

I is the statistical uncertainty on the scattering intensity, <Q> is the average Q over the 

neutron beam spot, Q is the standard deviation of the Q resolution function, and BS is the 

beam shadowing factor which is equal to zero for cells under the beamstop and to one for 

cells far from it. 

The last step in the data reduction process consists in merging data taken from the two (or 

three) instrument configurations together into one combined data file. 

4. TYPICAL REDUCED SANS DATA

Typical SANS data taken from 4 % poly(ethylene oxide) of molecular weight Mw = 42,900 

g/mol in deuterated ethanol are shown. First, data taken using a high-Q configuration data are 

shown. 
217

0.01

0.1

1

10

100

1000

104

0.001 0.01 0.1 1

4% PEO/d-ethanol, Mw = 42,900 g/mole, T = 25 0C

high-Q

Scattered Intensity (cm-1

)

Q (Å-1)

Figure 1: Reduced SANS data taken with a high-Q configuration. 

Then data taken using both a low-Q and a high-Q configuration are plotted together. 
218

0.01

0.1

1

10

100

1000

104

0.001 0.01 0.1 1

4% PEO/d-ethanol, Mw = 42,900 g/mole, T = 25 0C

high-Q

low-Q

Scattered Intensity (cm-1

)

Q (Å-1)

Figure 2: Reduced SANS data taken with a low-Q and the high-Q configurations spliced 

together. 

This figure shows scattering from a large lamellar structure (at low-Q) and local crystalline 

ordering (at high-Q). 

REFERENCES

S. Krueger and B. Hammouda “SANS Data Acquisition Manual” and “SANS Data 

Reduction Manual”, NIST Center for Neutron Research (1993, 2000). 

S. Kline, “Reduction and analysis of SANS and USANS data using IGOR Pro”, J. Appl. 

Cryst. 39, 895-900 (2006). 

QUESTIONS

1. What are the required runs for a complete set of SANS measurements?

2. What are the main steps in the SANS data reduction process?

3. Is there any information about the instrumental resolution in the averaged 1D data file? 

4. What type of radial averaging is required for anisotropic scattering? 

5. What produces the blocked beam background?
219

ANSWERS

1. In order to obtain a complete set of SANS measurements the required runs are: 

transmission and scattering from the sample, transmission and scattering from the empty cell, 

scattering run from the blocked beam, transmission run from the empty beam and the 

detector sensitivity run from plexiglass. 

2. The data reduction process involves the following steps: calculating the various 

transmissions, subtracting the empty cell and blocked beam, rescaling the 2D data to an 

absolute cross section, masking the unwanted detector cells, and radially averaging to obtain 

1D data. Merging (“sorting”) of data from multiple configurations is also performed. 

3. The averaged 1D data file contains the standard deviation of the Q resolution function Q

in the 5th data column. 

4. Anisotropic scattering yields asymmetric (oriented) 2D data. It requires either sector or 

rectangular averaging. 

5. The blocked beam background is produced by electronics noise on the detector electronics 

and stray neutrons outside of the neutron beam collimation. 
220

Part F – SIMPLE SANS DATA INTERPRETATION

Chapter 22. Standard Plots

22.1 The Guinier Plot

22.2 The Guinier Plot for Elongated Objects

22.3 The Porod Law

22.4 The Porod Plot

22.5 The Zimm Plot

22.6 The Kratky Plot

References

Questions

Answers

Chapter 23. Empirical Models

23.1 The Correlation Length Model

23.2 The Broad Peak Model

23.3 The Teubner-Strey Model

23.4 The Debye-Bueche Model

23.5 The Guinier-Porod Model

References

Questions

Answers

Chapter 24. Representative SANS Data 

24.1 Increasing Fluctuations

24.2 Ordered Structures

24.3 Concentrated Systems

24.4 Dilute Solution of Monodisperse Particles

References

Questions

Answers

Chapter 25. SANS Data from Oriented Samples

25.1 Oriented Fiber

25.2 Single Nanocrystal

25.3 Multilayer Vesicles

25.4 Oriented Membrane

25.5 Magnetic Scattering

References

Questions

Answers
221

Chapter 22 - STANDARD PLOTS

The first tool used to understand SANS data consists of a set of standard plots that yield 

results right after data reduction. These are linear plots of functions of the scattered intensity 

I(Q) plotted against functions of the scattering variable Q. Note that the absolute intensity 

I(Q) is a short hand notation for the macroscopic scattering cross section d(Q)/d. 

1. THE GUINIER PLOT

The Guinier plot involves plotting 

LnIQ

vs Q

2

(Ln refers to natural logarithm) in order to 

obtain the slope 

R 3

2

g

(Rg is the radius of gyration of the scattering objects). The expansion 

is as follows: 

















 

3

Q R

I(Q) I exp

2

g

2

0

(1)

    

3

Q R

Ln I Q Ln I

2

g

2

 0  .

The radius of gyration represents the effective size of the scattering "particle" whether it is a 

polymer chain, part of a protein, a micelle, or a domain in a multiphase system. The 

usefulness of this plot stems from the fact that the obtained particle “size” Rgis independent 

of the absolute intensity I0and of any model. Instrumental smearing as well as polydispersity 

and multiple scattering appear to decrease the effective Rg. Inter-particle effects also 

contribute to Rg except at the infinite dilution limit (case of an isolated particle). 

Consider the Guinier plot for a solution of Pluronic P85 in D2O. Pluronics are triblock 

copolymers of poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide), i.e., PEOPPO-PEO. At low temperature, both PEO and PPO dissolve in water so that SANS observes 

isolated polymer chains. This is the case for 20 oC. The radius of gyration obtained from the 

Guinier plot gives an estimate of polymer chain dimension. A Guinier plot is shown for 10 % 

(g/g) P85 in D2O measured at 20 oC. 
222

0

0.2

0.4

0.6

0.8

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016

Pluronics/d-water

Data

Ln(I) 

Q

2

 (Å-2)

Rg = 34 Å

Figure 1: Guinier plot for SANS data taken from 10 % (g/g) P85 Pluronic in D2O at 20 oC. 

The slope of the Guiner plot is Rg

2

/3. 

Note that at higher temperatures, PPO does not dissolve in water so that P85 forms micelles 

with PPO forming the core and PEO forming an outside shell. An inter-particle peak forms 

and the Guinier plot can no longer be used. Other methods used to analyze such SANS data 

will be described later. 

Another example of a Guinier plot is for SANS data from a solution of PAMAM dendrimers 

formed of seven generations and dissolved in D2O. The dendrimer fraction (g/g) is varied in 

the dilute solution range. No acid or salt has been added. The apparent radius of gyration is 

seen to decrease with dendrimer fraction. 
223

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 1 2 3 4 5

Dendrimer/d-water 

3.81 %, Rg = 25.6 Å 

1.82 %, Rg = 30.0 Å 

0.88 %, Rg = 31.6 Å 

0.36 %, Rg = 33.3 Å 

Ln(I)

Q

2

 *1000 (Å-2)

Figure 2: Guinier plot for SANS data taken from seventh-generation PAMAM dendrimers in 

D2O. The dendrimer fraction is varied. 

The range of a Guinier plot corresponds to 

QRg  3

. This is obtained when the probed 

range (2/Q) is larger than the particle size. 

Guinier region

Guinier region
224

Figure 3: Scattering particles are smaller than the probed range in the Guinier region shown 

for isolated particles and for single polymer coils. 

2. THE GUINIER PLOT FOR ELONGATED OBJECTS

The Guinier plot is modified when the scattering objects are elongated (Glatter-Kratky, 

1982). For instance, for a cylinder of length L and radius R, the low-Q Guinier 

approximation remains:

















 

3

Q R

I(Q) I(0) exp

2

g

2

where 

2

R

12

L

R

2 2

2

g

  . (2)

The low-Q Guinier plot is still 

LnIQ

vs Q

2

. The intermediate-Q Guinier approximation is 

different:

















 

2

Q R

exp

Q

I(0)

I(Q)

2

g

2

where 

2

R

R

2

2

g

 . (3)

The intermediate-Q Guinier plot becomes 

LnQIQ

vs Q

2

. A figure shows the form factor 

for a cylinder of length L = 345 Å (

R L 12 R 2 100 Å)

2 2

g2   

and radius R = 14 Å 

(

R R 2 10 Å)

2

g1   . 
225

0.0001

0.001

0.01

0.1

1

0.001 0.01 0.1 1

Cylinder with Rg2 = 100 Å and Rg1 = 10 Å

Form Factor 

Scattering Variable Q (Å -1)

1/Q1

1/Q4

Q2 Q1

 

1/Q0 Low-Q 

Guinier 

Region

Intermediate-Q 

Guinier Region

Porod 

Region

Figure 4: Form factor for a cylinder showing the low-Q Guinier region, the intermediate-Q 

Guinier region and the high-Q Porod region. 

Similarly for a lamella (flat object) of thickness T, the intermediate-Q Guinier approximation 

becomes:

















 

1

Q R

exp

Q

I(0)

I(Q)

2

g

2

2

where 

12

T

R

2

2

g

 . (4)

The intermediate-Q Guinier plot becomes 

LnQ IQ

2

vs Q

2

.

3. THE POROD LAW

Consider the case of an infinitely dilute solution of spheres of radius R and smooth surfaces. 

The scattering intensity is given by:

V F (QR)

V

N

I(Q)

2 2

P

2













 . (5)
226

The standard characteristic parameters have been defined as: (N/V) is the spheres number 

density, 

2

is the contrast factor, VP is the sphere volume and F(QR) is the single-sphere 

form factor amplitude given as follows:

















  

QR)

cos(QR)

(QR)

sin(QR)

QR

3

QR

3j (QR)

F(QR)

2

1

. (6)

Note that the single-sphere form factor P(QR) = F2(QR) is also defined as:

P(r)

Qr

sin(Qr)

dr4 r

V

1

P(QR) d r exp[ iQ.r] P(r)

2

P 0

3

     

  



. (7)

Here the pair correlation function 

P(r)



has been defined. The pair correlation function 

P(r)



is the probability of finding a scatterer at a vector distance 

r



inside the sphere knowing that 

there is another scatterer at the origin. P(r) is the equivalent 1D probability distribution 

defined radially. Consider a sphere of radius R and a scatterer located at a radial distance r’ 

from the sphere origin. Draw another sphere of radius r. P(r) represents the relative fraction 

of area of the second sphere located inside the large sphere integrated over all possible 

locations. Defining that relative fraction as p(r,r’), the following two cases can be considered: 

p(r,r’) = 1 R-r > r’ (8)











  

 

rr'

R r r'

4

1

2

1

p(r,r')

2 2 2

R-r 



r’.

The radial pair correlation function for a sphere is therefore (Stein et al, 1963):

3

R

0

2

R

0

2

R

r

16

1

R

r

4

3

1

dr'4 r'

dr'4 r' p(r,r')

P(r)

A

A













 











 

 

 

 . (9)

r

r’

R
227

Figure 4: Representation of the geometry used to calculate the radial pair correlation function 

for a sphere. 

The pair correlation function P(r) is the 3D Fourier transform of the single particle scattering 

factor P(Q). The 1D sine Fourier transform of P(Q) is rP(r). 

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

 Sphere 

Pr

r



Pr

rPr  Pair Correlation Function 

r/R

Figure 5: Plot of the pair correlation function P(r), of r2P(r) and of rP(r). 

Using this form, P(QR) can be expressed as follows:





























 











   

 3

0

2

P R

r

16

1

R

r

4

3

1

Qr

sin(Qr)

dr4 r

V

1

P(QR) . (10)

Note that this is the well known form factor for a sphere P(QR) = [3j1(QR)/QR]2introduced 

earlier. The interest here is in the high-Q expansion. The highest order in this expansion is 

obtained by integrating by parts three times:

4

P

P

2 4 3

P

P

4

P Q

1

V

S

2R

3

2Q R

3

V R

S

Q

2 '(0)

V

4

P(QR) ~ 















 

 

 . (11)

(SP/VP) is the surface to volume ratio. This is the so-called Porod law. 
228

The scattering intensity can simply be expressed as I(Q) = A/Q4 + B where B is the constant 

(mostly incoherent) scattering background. 

4. THE POROD PLOT

The Porod region corresponds to a probed range smaller than the scattering objects so that 

the scattering radiation is probing the local structure. The Porod plot Log(I) vs Log(Q) (Log 

is base-10 logarithm) yields information about the so-called "fractal dimension" of the 

scattering objects. At high-Q, one can approximate:

B

Q

A

I(Q)

n

  or 

LogIQB  LogA nLogQ. (12)

A Porod slope n = 1 is obtained for scattering from rigid rods; a slope n = 4 represents a 

smooth surface for the scattering particle; whereas a slope n between 3 and 4 characterizes 

rough interfaces of fractal dimension D with n = 6-D. This is called a surface fractal. 

Moreover, in the case of polymer coils, the Porod slope n is related to the excluded volume 

parameter  as its inverse n = 1/. A slope n = 2 is a signature of Gaussian chains in a dilute 

environment, a slope n = 5/3 is for fully swollen coils and a slope n = 3 is for collapsed 

polymer coils. A slope between 2 and 3 is for “mass fractals” such as branched systems 

(gels) or networks.

An example of a Porod plot is shown for SANS data from a 4 % (g/g) solution of salmon 

DNA in d-ethyelene glycol) at a temperature of 50 oC. At this temperature, the helical 

structure has melted into coil conformation.1 M NaCl salt has been added in order to screen 

charge interactions. The slope of the Porod plot of n = 1.76 is close to the value n = 5/3 = 

1.667 which is a signature for fully swollen coils. 
229

-2.3

-2.2

-2.1

-2

-1.9

-1.8

-1.7

-1.6

-0.75 -0.7 -0.65 -0.6 -0.55 -0.5 -0.45 -0.4 -0.35

DNA in d-ethylene glycol 

Data 

Log[I(Q) - B]

Log(Q)

slope = -1.76 

Figure 6: Porod plot for SANS data taken from 4 % (g/g) DNA coils in d-ethylene glycol at 

50 oC (above the helix-to-coil transition temperature). 0.1 M NaCl was added to screen 

charge interactions. 
230

Figure 7: Assortment of Porod law behaviors for different shape objects. 

5. THE ZIMM PLOT

Another well known plot is the Zimm plot (1/I vs Q2) which found wide use in light 

scattering from dilute polymer solutions where extrapolation to zero Q and zero 

concentration yields the molecular weight, the radius of gyration and the second virial 

coefficient. The Zimm plot is also useful in polymer blends (in the single-phase region) 

where the slope is proportional to the correlation length, which is proportional to the FloryHuggins interaction parameter (incompressible RPA model) to be described later. 

Assume a Lorentzian form for the Q-dependence of the scattering intensity: 

2 2

0

1 Q

I

I(Q)

 

 . (13)

1D object 2D object 3D object

Q-5/3 Q-2 Q-3

MASS FRACTALS SURFACE FRACTALS

I(Q) ~ Q-1 Q-2 Q-4

Q-3 Q-4

Porod region
231

Here  is the correlation length. A plot of 1/I(Q) vs Q

2

yields 1/I0as intercept and 

2

/I0as 

slope. The correlation length is obtained as  = (slope/intercept)1/2. In the low-Q region, one 

can also expand:

















  



 ...

3

Q R

I 1

3

Q R

1

I

I(Q)

2

g

2

2 0

g

2

0

. (14)

Therefore yielding 

  Rg/ 3

for low-Q. The Zimm plot applies, however, beyond the lowQ region. In the high-Q region where Q2



2

<1, one can approximate:

0

2 2

I

Q

I(Q)

1 

 . (15)

In this region, the single polymer chain form factor behaves as 2/Q2R

g

2

(high-Q expansion of 

the Debye function) so that 

  Rg/ 2

is identified for high-Q. In the case of polymer 

solutions with excluded volume interactions, the high-Q expansion is, instead:

 





1/

g

0

QR

2

I(Q) I . (16)

Here  is the excluded volume exponent ( = 3/5 for fully swollen chains,  = 1/2 for theta 

chains and  = 1/3 for collapsed chains). 

Low-Q departure from linear behavior of the Zimm plot is a signature of non-homogeneity in 

the sample or of chain-branching. A negative value of the intercept I0 (obtained through 

extrapolation) is a sign of phase separation. 

An example of a Zimm plot is shown for SANS data taken from a blend mixture of 

poly(ethyl butylene) and deuterated poly(methyl butylene); i.e., hPEB/dPMB. The molecular 

weights for hPEB/dPMB are Mw = 40,100 g/mol and 88,400 g/mol respectively. The volume 

fraction of the represented sample corresponds to 57 % hPEB. This blend mixture was 

measured at a temperature of 10 oC. The Zimm plot is linear pointing to Gaussian chains. The 

slope yields an apparent radius of gyration which depends on the polymer/polymer 

interaction parameter. These issues will be described in detail when the Random Phase 

Approximation (RPA) model is introduced. 
232

0

0.05

0.1

0.15

0.2

0 0.0005 0.001 0.0015 0.002 0.0025

hPEB/dPMB 

Data 

1/I(Q) 

Q

2

 (Å-2)

intercept gives I(Q=0)

slope gives apparent 

 radius of gyration

Figure 8: Zimm plot for a polymer blend mixture of hPEB and dPMB with Mw = 40,100 

g/mol, and 88,400 g/mol respectively. The hPEB fraction is 57 % (g/g) and the measurement 

temperature is 10 oC (single-phase region). 

A more detailed Zimm plot is for SANS data from a polymer blend mixture of deuterated 

polystyrene and poly(vinyl methyl ether); i.e., dPS/PVME (Briber et al, 1994). Four dilute 

dPS volume fractions were measured at a temperature of 140 oC. The dPS/PVME blend 

system is characterized by a Lower Critical Spinodal temperature (LCST) and 140 oC 

corresponds to the single-phase region. Extrapolation to zero volume fraction yields a slope 

and intercept which give the degree of polymerization for polystyrene and the radius of 

gyration respectively.
233

1

1.5

2

2.5

3

3.5

4

0 0.2 0.4 0.6 0.8 1

 dPS/PVME 

0 % dPS 

1 % dPS 

1.8 % dPS 

3.8 % dPS 

5.4 % dPS 

dPS/I(Q)*10

3

Q

2

 *103 (Å-2)

Figure 9: Zimm plot for a deuterated polystyrene/polyvinylmethylether blend (Mw= 

1.88*105g/mol and 3.98*105g/mol respectively) mixture for four dilute polystyrene volume 

fractions of dPS = 1 %, 1.8 %, 3.8 % and 5.4 % at a temperature of 140 oC. 

6. THE KRATKY PLOT

Kratky plots emphasize deviation from the high-Q behavior of the scattering intensity I(Q). 

For polymer chains, the Kratky plot (Q2I(Q) vs Q) emphasizes the Gaussian chain nature or 

departure from it. Since the form factor for Gaussian chains varies like I(Q) ~ 1/Q2at high-Q, 

this plot tends to a horizontal asymptote. Inter-chain contributions affect only the constant 

multiplying this term and not the 1/Q2scaling behavior. Deviation from a horizontal 

asymptotic behavior indicates a non-Gaussian characteristic for the scattering chains. 

For instance, for rigid rods this plot would go to a linearly increasing asymptote Q2I = A + 

BQ because the form factor for a rod varies like I(Q) ~ 1/Q at high Q and one has to use a 

more suitable Kratky plot for a rod (QI vs Q) in order to recover the horizontal asymptote. In 

order to illustrate this in simple terms, three functions that die out differently at high Q are 

considered. These three cases are (1) for rigid rods where I(x) = I0/(1+x), (2) for Gaussian 

chains where I(x) = I0/(1+x2), and (3) for branched systems (or mass fractals) where I(x) = 
234

I0/(1+x3). Here x is the dimensionless variable x = Q where  is a characteristic length 

(radius of gyration or correlation length). These functions reproduce the proper low x and 

high x limits. 

0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2 2.5 3 3.5 4

 Kratky Plot 

I(x)=I

0

/(1+x) 

I(x)=I

0

/(1+x2) 

I(x)=I

0

/(1+x3) 

x

2*I(x)/I

0

x 

Figure 10: Symbolic representation of the Kratky plot for the three cases of a rigid rod, a 

Gaussian chain and a mass fractal. 

Gaussian chains tend to the Kratky plot limit of 1. Stiff chains (for example rigid rods) 

increase linearly at high x and branched systems (mass fractals) reach a maximum then 

decrease as 1/x at high x. 

An example of a Kratky plot is shown for SANS data taken from an isotopic blend mixture 

of deuterated polystyrene with non-deuterated polystyrene, i.e., dPS/hPS with Mw = 174,000 

g/mol and 195,000 g/mol respectively at 50 % fraction (g/g) and measured at ambient 

temperature. This plot represents the Gaussian nature of polymer chains in isotopic blends 

and tends to the asymptote of 1 at high Q. 
235

0.2

0.4

0.6

0.8

1

0 0.01 0.02 0.03 0.04 0.05

dPS/hPS 

Data 

 Q

2*I(Q)/I

0

Q (Å-1)

Figure 11: Kratky plot for an isotopic blend mixture of dPS and hPS with Mw = 174,000 

g/mol and 195,000 g/mol, 50 % fraction (g/g) measured at ambient temperature. The line is a 

smoothing fit as a guide to the eye. 

Another Kratky plot is shown for a seventh generation PAMAM dendrimer in D2O. SANS 

data were taken from a series of dilute solutions and extrapolated to the infinite dilution limit 

(Hammouda, 1992). Measurements were taken at ambient temperature. This plot represents 

the branched character of this scattering system. It has not been rescaled at high Q. 
236

0

0.0002

0.0004

0.0006

0.0008

0.001

0 0.02 0.04 0.06 0.08 0.1

Dendrimer/d-water 

Data

Q

2*I(Q)

Q (Å-1)

Figure 12: Kratky plot for seventh generation PAMAM dendrimer solution in D2O 

extrapolated to the infinite dilution limit (zero concentration). The Katky plot reaches a 

maximum then tends to a constant level at high Q. 

The manner in which the asymptote of a Kratky plot is reached yields information about 

chain branching. For instance, in a plot of Q2I vs 1/Q2(Q2I = A + B/Q2) the intercept B is 

related to the crosslink density in branched gels and networks (Benoit et al, 1993).

REFERENCES

O. Glatter, and O. Kratky, “Small-Angle X-Ray Scattering”, Academic Press (1982). 

R.S. Stein, P.R. Wilson and S.N. Stidham, “Scattering of Light by Heterogeneous Spheres”, 

J. Applied Physics 34, 46-50 (1963)

B. Hammouda, "Structure Factor for Starburst Dendrimers”, J. Polym. Sci., Polym. Phys. 

Ed., 30, 1387-1390 (1992)

R.M. Briber, B.J. Bauer and B. Hammouda, “SANS from Polymer Blends in the Dilute 

Concentration Limit“, J. Chem. Phys. 101, 2592-2599 (1994)
237

H. Benoit, J.F. Joanny, G. Hadziioannou and B. Hammouda, Macromolecules 26, 5790 

(1993)

QUESTIONS

1. What is a Guinier Plot? What can be obtained from it? What can be obtained from the 

intercept? 

2. Do scattering inhomogeneities have to be spherical for a radius of gyration to be defined 

and measured through a Guinier plot?

3. What information could be obtained using a Porod plot for smooth interfaces?

4. How does polydispersity and instrumental smearing affect the Guinier plot and the Porod 

plot? 

5. Consider the pair correlation function for a sphere of radius RA, given by: 

3

R A

r

16

1

R

r

4

3

p(r) 1

















 











  . Explain the limit p(r=2RA) = 0.

6. A Zimm plot is linear for what scattering objects?

7. What information can be obtained from a Kratky plot?

ANSWERS

1. A Guinier plot is a plot of Ln(I) vs Q

2

. The radius of gyration (Rg) can be obtained from 

the slope of a Guinier plot (slope = Rg

2

/3). The intercept of a Guinier plot is I(0) which can 

yield the aggregation number which is the number of basic scattering units per scattering 

“particle”. A scattering unit could be a monomer and a scattering particle could be a polymer. 

2. The Guinier plot Ln(I) vs Q

2 measures a radius of gyration from any shape objects. These 

do not have to be globular. 

3. The Porod plot Log(I) vs Log(Q) for scattering objects with smooth interfaces yields an 

exponent from the slope and a surface-to-volume ratio from the intercept. 

4. Polydispersity and instrumental smearing yield broader forward scattering peaks and 

therefore a lower radius of gyration from the Guinier plot. These, however, do not affect the 

Porod exponent which remains unchanged. 

5. Consider a scatterer inside a sphere of radius RA and draw another sphere of radius r. 

Choosing the first scatterer on the surface of the sphere and choosing a second sphere of 

radius r = RA covers the maximum correlation range of 2RA. Beyond that range, scatterers are 

not correlated. 

6. A Zimm plot 1/I vs Q

2

is linear for Gaussian polymer coils. 

7. A Kratky plot Log(Q2I) vs Q saturates to a constant level at high-Q for flexible polymer 

coils but increase linearly for rigid rods. The break between the constant and the linear 

behaviors yields an estimate of the so-called persistence length which is a measure of chain 

stiffness. 
238

Chapter 23 - EMPIRICAL MODELS

Standard plots give the first order interpretation of SANS data. Precise models give a more 

detailed approach at obtaining results. Precise molecular models are however not always 

available or too complex to use. An intermediate approach consists in using empirical models 

that reproduce the main trends observed in the SANS data. Some of these models are 

described here.

1. THE CORRELATION LENGTH MODEL

Oftentimes when the scattering intensity I(Q) is a decreasing function with Q, it is modeled 

using the following functional form:

 

B

1 Q

C

I(Q)

m



 

 . (1)

Here C and B are (Q-independent) constants obtained for I(Q0) = C + B and I(Q



) = 

B,  is a correlation length and m is a Porod exponent. Note that when m = 2, this functional 

form becomes the familiar Lorentzian function. The Fourier transform of a Lorentzian 

function corresponds to correlations dying out as exp(-r/)/r. The correlation length  is large 

for systems that are highly correlated like polymers and gels. For example,  is equal to the 

entanglement distance for a semi-dilute polymer solution and it is equal to the end-to-end 

distance for very dilute polymers. Note that the low-Q limit of this empirical form reproduces 

the Guinier law only when m = 2. 

A figure shows SANS data from 4 % (g/g) solution of poly(ethylene oxide) or PEO for short 

of Mw = 41,500 g/mol in D2O at a temperature of 20 oC (Hammouda et al, 2004). Fit to the 

correlation length model gave the following parameters: C = 0.52 cm-1,  = 17.47 Å, m = 

1.93 and B = 0.069 cm-1. The fit is good except for the very low-Q points where statistics are 

poor. The correlation length  gives a good estimate of the average entanglement length for 

this semi-dilute polymer solution. The Porod exponent m points to a “mass fractal” for 

dissolved polymer chains close to the theta condition. The fractal exponent for chains in a 

good solvent is m = 5/3 and that for chains in theta condition is m = 2. The theta condition is 

defined when the monomer-solvent, monomer-monomer and solvent-solvent molecular 

interactions are comparable. 
239

0.1

1

0.01 0.1

4 % Poly(ethylene oxide)/d-water 

SANS Data

Model Fit

Scattering Intensity (cm-1

)

Scattering Variable Q (Å-1)

Figure 1: SANS data from 4 % solution of PEO (Mw = 41,500 g/mol) in D2O at 20 oC 

temperature and fit to the correlation length model. 

2. THE MODIFIED CORRELATION LENGTH MODEL

In some cases where there is a low-Q power law behavior, the correlation length model is 

model to incorporate a “stretching” feature in the low-Q mass fractal structure.

 

B

1 Q

C

Q

1

I(Q)

s m



 



(2)

Here also  is a characteristic length and m is a high-Q Porod exponent. Parameter s is a 

“stretching” factor (s = 0 for isotropic structure and s = 1 for fully stretched mass fractal 

(think network) structure. Note that when m = 2 and s = 0, this functional form becomes the 

Lorentzien and  is the correlation length (1/e-folding length in r-space). 

SANS data and fit to the correlation length model are included for a POSS (generation 3) 

dendrimer in d-THF solution at 100 mg/ml mass fraction and ambient temperature (25 oC). 

Fit results yielded the following parameters s = 0.68 and  = 14.51 (Yuan et al, 2016). 
240

0.1

1

10

100

1000

0.001 0.01 0.1 1

poss_6-1

SANS Data Model Fit

Scattering Intensity (cm-1) 

Scattering Variable Q (Å-1)

Figure 2: SANS data and fit to the modified correlation length model for a POSS dendrimer 

(POSS65-PEG392) in d-THF at 100 mg/ml mass fraction and ambient temperature (25 oC). 

3. THE BROAD PEAK MODEL

Many SANS spectra are characterized by a broad peak even though they are from amorphous 

soft materials. The d-spacing corresponding to the broad peak is a characteristic distance 

between the scattering inhomogeneities (such as in lamellar, cylindrical, or spherical 

morphologies or for bicontinuous structures). The following simple functional form 

reproduces the broad peak feature:

 

B

1 | Q Q |

C

I(Q)

m

0



  

 . (3)

Here the peak position is related to the d-spacing as Q0 = 2/d0. Soft systems that show a 

SANS peak include copolymers, polyelectrolytes, multiphase systems, layered structures, etc. 

A figure shows SANS data from 4 % poly(lysine) polyelectrolyte solution in D2O at 25 oC 

temperature. Poly(lysine) is a poly(amino acid). Fit to the broad peak model gave the 

following parameters: C = 0.075 cm-1,  = 13.10 Å, Q0 = 0.099 Å-1, m = 1.05 and B = 0.064 

cm-1. Here again, the fit is good except for the low-Q points where statistics are poor. The dspacing is d0 = 2/Q0 = 63.47 Å. This is an average inter-distance between charged 

polyelectrolyte domains. 
241

0.06

0.08

0.1

0.12

0.14

0.01 0.1

4 % Poly(lysine)/d-water 

SANS Data

Model Fit

Scattering Intensity (cm-1

)

Scattering Variable Q (Å-1)

Figure 3: SANS data from 4 % poly(lysine) poly(amino acid) solution in D2O at 25 oC 

temperature and fit to the broad peak model. 

4. THE TEUBNER-STREY MODEL

The Teubner-Strey model (Teubner-Strey, 1987) was originally introduced to represent the 

structure of micellar systems. These are characterized by a peak representing inter-micellar 

interactions. This model assumes a pair correlation function of the form:











 























 

d

2 r

sin rexp

2 r

d

(r) . (4)

Here  is a correlation length (length beyond which correlations die out) and d is a d-spacing 

(characteristic of a domain size or periodicity). Recall that the coherent macroscopic 

scattering cross section is given by:

      



 

0

2 2

P I

2

(r)

Qr

sin(Qr)

V P(Q)S (Q) drr 4

d

d (Q)

(5)
242

 is the sample volume fraction, VP is the scattering “particle” volume, 

2

is the contrast 

factor, P(Q) is the form factor and SI(Q) is the structure factor. Performing this integration 

yields:

2 2 4 4

2

2

2

2

3

2

2 Q Q

d

2

2

d

2

1

8

d

d (Q)

 

















  









 

  



























 





 





. (6)

The functional form for the scattering intensity can therefore be presented in the form:

B

Q Q

c

c

c

a

8

B

d

d (Q)

I(Q)

2 4

2

1

2

2

2















 





















  





 . (7)

B is a Q-independent incoherent scattering background. The various parameters a2, c1 and c2

are defined as:

4

2

2

2

2

d

2

1

c

a





























 





4

2

2

2

2

1

2

d

2

2

c

c



















  









 

 

 . (8)

These are considered as fitting parameters. The correlation length  and the d-spacing d can 

be expressed as:

2

1

2

2

c

c

4

1

c

a

2

1

1



  (9)

2

1

2

2

c

c

4

1

c

a

2

1

2

d





 . 

A factor 

a 1 2 2

f  c 4a c

is defined to represent the amphiphile “strength” which dictates 

the microstructure. For example, the ordered lamellar phase corresponds to fa = -1 while the 

disordered phase corresponds to fa = 1. 
243

Consider SANS data from 10 % P85 Pluronic (triblock copolymer of PEO-PPO-PEO) 

measured in D2O at 60 oC (temperature for which the micelles are well formed). Fits of the 

SANS data to the Teubner-Strey model yields the following fitting results. 

a2 = 0.038 (10)

c1 = -51.23

c2 = 24,929

B = 0.118. 

Note that for the functional form to produce a peak, parameter c1 has to be negative. These 

parameters give the following value for the two characteristic lengths:

 = 96 Å (11)

d = 186.6 Å.

Adding a constant incoherent level B, the scattering intensity (coherent and incoherent 

contributions) are 

I(Q)  d(Q) d  B.

0.1

1

10

100

0.01 0.1

10 % P85 Pluronic/d-Water, T = 60 oC 

SANS Data

Model Fit

Scattering Intensity (cm-1

)

Scattering Variable Q (Å-1)

Figure 4: SANS data from 10 % P85 Pluronic in D2O at 60 oC plotted along with the fit to the 

Teubner-Strey model. 
244

-0.2

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350

The Teubner-Strey Model 

Model 

(r)

r

Figure 5: The pair correlation function (r) for the Teubner-Strey model. 

Note that a peaked behavior in I(Q) results in a pair correlation function (r) going negative 

then positive. This is referred to as the “correlation hole” effect. This happens in block 

copolymers, in polyelectrolytes and in concentrated systems. 

The Teubner-Strey model applies to concentrated solutions of particles (spheres, cylinders, 

etc) and to the bicontinuous structure. It does not do well for lamellar systems and for other 

highly ordered morphologies (for example, ordered diblock copolymers). It misses the higher 

order oscillations completely. Moreover, the Teubner-Strey model was developed for 

water/oil/surfactant ternary mixtures in the micelle-formation region. Using it for 

polymer/copolymer mixtures requires some adjustments. 

5. THE DEBYE-BUECHE MODEL

The Debye-Bueche model is used to describe scattering from phase-separated (two-phase) 

systems. Here also correlations are characterized by an e-folding length . The pair 

correlation function is give by (Debye-Bueche, 1949):



















  

r

(r) exp . (12)
245

The scattering cross section is obtained by taking the Fourier transform to obtain:

   

2

2

1 Q

C

d

d (Q)

 







. (13)

The prefactor can be expressed in terms of the volume fraction  and contrast factor 

2

as:

2 3 C  8  . (14)

The Debye-Bueche model is obtained as a special case of the Teubner-Strey model for very 

large d-spacing (d>>). 

6. THE GUINIER-POROD MODEL

An empirical Guinier-Porod model is useful for analyzing SANS data (Hammouda, 2010). 

The scattering intensity is given by the two contributions:

1

2

g

2

 for Q Q

3

Q R

I(Q) Gexp 

















 (15)

m

Q Q1 Bfor 

Q

D

I(Q)    .

Note that the incoherent scattering has been added as a constant (Q-independent) term. 

Imposing that the values of the Guinier and Porod terms and their slopes (derivatives) be 

continuous at a value Q1 yields the following relationships:

2

3m

R

1

Q

g

1 

(16)

m

g

2

m

m

1

2

g

2

1

R

1

2

3m

2

m

Q Gexp

3

Q R

D Gexp 























 

















 .

The Guinier form is used for 

Q  Q1

and the Porod form is used for 

Q  Q1

. Note that the 

value of Q1 does not have to be set; it is calculated internally using Eq. 2. This model is 

general and should apply in the entire range of Porod parameters. It is completely empirical.
246

0.1

1

10

0.01 0.1

 0.5 % P85/d-water, T = 50 oC 

SANS Data

Model Fit

Scattering Intensity (cm-1

)

Scattering Variable Q (Å-1)

Q1 

Figure 6: SANS data and fit to the Guinier-Porod model for 0.5% P85/d-water at 50 oC 

where spherical micelles are formed. Note that the high-Q background has been excluded 

from the fit. The fitting region is delimited by arrows. 

This model is generalized to account for asymmetric scattering objects (such as rods or 

lamellae) as follows: 

1

2

g

2

s

 for Q Q

3 s

Q R

exp

Q

G

P(Q) 





















 (17)

m

Q Q1 B for 

Q

D

P(Q)    .

This is based on the generalized Guinier law for such elongated objects (Glatter-Kratky, 

1982). The same scaling factor G has been kept even though it has different units. For 3D 

globular objects (such as spheres), s = 0 and one recovers the previous results. For 2D 

symmetry (such as for rods) s = 1 and for 1D symmetry (such as for lamellae or platelets) s = 

2. The dimensionality parameter corresponds to 3-s. 

Applying the same continuity of the Guinier and Porod functions and their derivatives yields 

to the generalized Guinier-Porod model yields:
247

2

(m s)(3 s)

R

1

Q

g

1

 



(18)

2

(m s)

(m s)

g

(m s)

1

2

g

2

1

2

(m s)(3 s)

2

(m s)

exp

R

G

Q

3 s

Q R

D Gexp

















  











 

 























This empirical model is used to analyze SANS data from a Pluronic P85 which consists of 

the following block sequence EO26PO40EO26 where EO and PO represent ethylene oxide and 

propylene oxide monomers respectively. A 0.5 % P85/d-water is known to form micelles 

upon heating. It forms spherical micelles at 50 oC, cylindrical micelles at 70 oC and lamellar 

micelles at 90 oC. 

0.1

1

10

100

0.01 0.1

 0.5 % P85/d-water, T = 70 oC 

SANS Data

Model Fit

Scattering Intensity (cm-1

)

Scattering Variable Q (Å-1)

Figure 7: Fit of the generalized Guinier-Porod model to the SANS data from 0.5% P85/dwater at 70 oC where cylindrical micelles are formed. 

Nonlinear least squares fit yields the following parameters for the scale factor G, 

dimensionality parameter 3-s, the radius of gyration Rg and the Porod exponent m. 

G = 0.32 (19)
248

3-s = 2.06

Rg = 37.1 Å

m = 4.82. 

Note that this single model can fit SANS data from spherical micelles (s = 0), cylindrical 

micelles (s = 1) and lamellar micelles (s =2) as well as intermediate structures. 

REFERENCES

B. Hammouda, D. Ho and S. Kline, “Insight into Clustering in Poly(ethylene oxide) 

Solutions”, Macromolecules 37, 6932-6937 (2004)

M. Teubner and R. Strey, “Origin of the Scattering Peak in Microemulsions”, J. Chem. Phys. 

87, 3195-3200 (1987)

O. Glatter, and O. Kratky, “Small-Angle X-Ray Scattering”, Academic Press. Check out 

Chapter 4 on Data Treatment, pages 155-156 (1982). 

P. Debye, A.M. Bueche, "Scattering by an Inhomogeneous Solid", J. Appl. Phys. 20, 518-

525 (1949).

B. Hammouda, “A New Guinier-Porod Model”, J. Appl. Cryst. 43, 716-719 (2010).

G Yuan, X. Wang, D. Wu and B. Hammouda, “Structural Analysis of Dendrimers Based on 

Polyhedral Oligomeric Silsesquioxane and their Self-Assembly in Solution by Small-Angle 

Neutron Scattering: Fits to a Modified Two Correlation Lengths Model”, Polymer (to be 

published, 2016). 

 

QUESTIONS

1. What are the three main methods used to analyze SANS data?

2. What is referred to as the Ornstein Zernike functional form? What parameter can be 

obtained from a fit to that form?

3. What is the meaning of a peak in SANS data (at Q0 for example)?

4. What type of scattering does the Teubner-Strey model apply to?

5. What are the main parameters for the Guinier-Porod model for elongated scattering 

objects? 

ANSWERS

1. The three main ways used to analyze SANS data are: (1) standard plots (linear plots of 

functions of I(Q) vs functions of Q), (2) non-linear least squares fits to reasonable models 
249

including empirical models and (3) molecularly realistic complex methods for particle shape 

reconstruction and molecular simulation. 

2. The Ornstein-Zernike functional form is a Lorentzian. A correlation length can be 

obtained.

3. A peak in SANS data (at Q0) means that there is a structure with a characteristic repeat 

distance d = 2/Q0. 

4. The Teubner-Strey model applies to scattering data with a peak and that decay as 1/Q4at 

high Q. These are concentrated systems with inter-particle separation distance comparable to 

particle size or to bicontinuous structures. 

5. Data fitting to the Guinier-Porod model yields an intercept G, a radius of gyration Rg, a 

Porod exponent m and a dimensionality factor for elongated scattering objects 3-s. s=0 for 

spherical symmetry, s=1 for rods and s=2 for lamellae. 
250

Chapter 24 - REPRESENTATIVE SANS DATA

SANS data come in many trends and shapes. Most data sets show a forward scattering peak. 

Some show a peaked behavior at finite Q. Only a few show a multitude of peaks. The SANS 

technique is not abundant in peaks like other characterization methods. Representative SANS 

data are presented here. 

1. INCREASING FLUCTUATIONS

Increase in SANS intensity is a signature of an increase in density or composition 

fluctuations. This is observed for systems undergoing phase transition, aggregation or 

crystallization. SANS is a good monitor of phase separation.

A figure summarizes SANS data taken from a phase separating polymer solution as 

temperature is increased (Hammouda et al, 2002). The polymer solution is made of 4 % 

poly(ethylene oxide) of Mw = 41,800 g/mol in d-water. This high-Q signal represents 

solvent-polymer interactions (though hydrogen bonding in this case). As temperature is 

increased, hydrogen bonding breaks leading to the onset of a lower critical solution 

temperature. The upturn at low Q is due to a clustering effect characterizing most watersoluble systems. 

0.1

1

10

0.01 0.1

 Poly(ethylene oxide)/d-water 

85 oC 

65 oC 

50 oC 

37.5 oC 

25 oC 

10 oC 

 Scattering Intensity (cm-1) 

Scattering Variable Q (Å-1)
251

Figure 1: SANS data taken from a 4 % poly(ethylene oxide) solution in d-water. The polymer 

molecular weight is Mw = 41,800 g/mol. 

2. ORDERED STRUCTURES

A class of SANS spectra is characterized by a sharp peak. The peak is either due to a welldefined repeat distance (in lamellar systems for example) or due to the correlation hole 

effect. 

Figure 2: Representation of the two cases that can give a SANS peak. 

SANS data from a polystyrene-polyisoprene diblock copolymer are included. The SANS 

peak is due to the correlation hole. At low-temperatures, the morphology formed is highly 

ordered (lamellar). 

Scattering 

Factor

Periodic

Density

Q r

Scattering 

Factor

Pair 

Correlation

Q r
252

0

5

10

15

20

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Polystyrene-Polyisoprene Diblock Copolymer 

20 oC 

30 oC 

40 oC 

50 oC 

60 oC 

70 oC 

 Scattering Intensity (cm-1) 

Scattering Variable Q (Å-1)

Figure 3: SANS data from a polystyrene-polyisoprene diblock copolymer solution in DOP 

solvent. 

3. CONCENTRATED SYSTEMS

Another class of SANS spectra is when an inter-particle peak is formed. This is the case 

where the inter-particle spacing is comparable to the particle size and is characteristic of 

“concentrated” systems. The case of a 25 % Pluronic P85 (PEO-PPO-PEO triblock 

copolymer) micelles in d-water is included. Micelles form above ambient temperature due to 

the hydrophobic nature of PPO.
253

0.1

1

10

100

0.01 0.1

P85/d-water 

60 oC

50 oC

40 oC

30 oC

Scattered Intensity (cm-1

)

Scattering Variable Q (Å-1)

Figure 4: SANS data from P85 Pluronic micelles. The inter-particle interaction peak is 

clearly observed. 

4. DILUTE SOLUTION OF MONODISPERSE PARTICLES

The case of dilute solutions is characterized by scattering from “isolated” particles. A 0.1 % 

solution of silica particles in d-water is included here. The higher order peaks are a signature 

of monodispersity and are limited by instrumental resolution. The fitted sphere radius is R = 

563 Å. 
254

0.01

0.1

1

10

100

1000

0.001 0.01 0.1

Silica Particles/d-water

Data 

Model Fit

Scattered Intensity (cm-1

)

 Scattering Variable Q (Å-1) 

Figure 5: SANS data from a dilute solution of silica particles (0.1 % mass fraction) in dwater and fit to sphere model. The fitted sphere radius is R = 563 Å. 

REFERENCES

B. Hammouda, D. Ho and S. Kline, “Neutron Scattering from Associating PEO Systems”, 

Macromolecules 35, 8578-8585 (2002)

QUESTIONS

1. Why does scattering increase when a phase transition line is approached?

2. What is the “correlation hole” effect in block copolymers?

3. Why does the block copolymer peak broaden when the mixed-phase region is entered?

4. Why are SANS data from concentrated systems characterized by a peak?

5. What is the origin of the higher order peaks observed in SANS data from monodisperse 

dilute solution of particles. 
255

ANSWERS

1. The approach to phase transition conditions is accompanied by composition fluctuations. 

Likewise molecules attract each other whereas unlike molecules repel each other. 

2. Copolymers are formed of blocks that are covalently bonded. Consider, say, a diblock AB. Around each A block, there is a region where another A block cannot reside because of 

crowding from B blocks. This region empty of A blocks is referred to as a “correlation hole” 

effect. 

3. The mixed phase (also called disordered phase) region is obtained by dissolving the 

macrodomain morphology formed in the ordered phase. The SANS peak in the ordered phase 

is sharp due to the characteristic (lamellar, cylindrical or spherical) morphology. The SANS 

peak in the disordered phase is due to the correlation hole effect. 

4. SANS data from concentrated systems are characterized by a peak because the interparticle d-spacing becomes comparable to the size of the particles. The SANS peak position 

characterizes the nearest neighbor inter-distance. It is due to the inter-particle structure factor 

SI(Q). 

5. The single-particle scattering factor for a single spherical particle is given by the spherical 

Bessel function P(Q) = [3j1(QR)/QR]2 where R is the sphere radius and j1(X) = sin(X)/X2-

cos(X)/X. This is an oscillatory function with many higher order peaks. 
256

Chapter 25 - SANS FROM ORIENTED SAMPLES

Scattering from oriented samples is rich in peaks and anisotropic features as observed on the 

2D detector. A series of these “interesting” spectra is included here in order to sample the 

wide variety of possibilities. 

1. ORIENTED FIBERS

Collagen from a kangaroo tail tendon is characterized by a fiber-like structure along the 

oriented fibers with repeat spacing of 667 Å. 

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.08 -0.04 0.00 0.04

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08 3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Figure 1: Scattering pattern from highly ordered biopolymer (collagen from a kangaroo tail 

tendon) showing the strong first and third Bragg peaks as well as weak higher order peaks; 

the second peak is not allowed. The ordered structure is along the fibers and has a d-spacing 

of 667 Å.
257

2. SINGLE NANOCRYSTAL

Anisotropic SANS data from P85 Pluronic micelles sheared in a Couette shear cell are 

included. The characteristic hexagonal peak pattern (six fold symmetry) points to a cubic 

structure formed by the spherical micelles for 25 % mass fraction P85 in D2O solutions. P85 

is a triblock copolymer of poly(propylene) which is hydrophobic in the middle of the 

molecule and poly(ethylene oxide) which is hydrophilic on the outside of the molecule 

(PEO-PPO-PEO). P85 micelles are well formed at ambient temperature. Shearing helps the 

packing of the spherical micelles into a face centered cubic structure (Slawecki et al, 1998). 

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.05 0.00 0.05

-0.05

0.00

0.05

4.0

3.5

3.0

2.5

2.0

1.5

Figure 2: SANS data from 25 % P85 Pluronic (PEO-PPO-PEO triblock copolymer) in D2O 

under Couette shear (5 Hz frequency) at 40 oC. The micelles form a cubic “single crystal” 

structure. 

3. MULTILAYER VESICLES

Multilayer vesicles are formed by mixing AOT surfactant with a brine/D2O solution. A brine 

solution contains more than 100 g/l sodium chloride salt. Shearing the multilayer vesicles in 

an in-situ Couette shear cell shows orientation along the shearing direction (i.e., 

horizontally). The sheared structure resembles a horizontally elongated “onion skin” 
258

structure. Two weaker spots along the equatorial axis show weak orientation of the vesicle 

layers parallel to the shearing cell walls as well. This SANS image was obtained with 

oscillatory (i.e., reciprocating) shear and with the neutron beam incident tangentially to the 

shear cell through a vertical beam defining slit (Bergenholtz-Wagner, 1996). 

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.10 -0.05 0.00 0.05 0.10

-0.10

-0.05

0.00

0.05

0.10

3.0

2.5

2.0

1.5

1.0

Figure 3: SANS data from multilayer vesicles formed of AOT in brine/d-water solution 

under oscillatory Couette shear. This view corresponds to a configuration where the neutron 

beam is incident tangentially to the shear cell. 

4. ORIENTED MEMBRANE

Highly oriented samples containing supramolecular peptide assemblies imbedded in 

multilayer membranes have been investigated. Samples were prepared in the form of 

orientated multilayers that were held between several parallel quartz plates. Deuterated water 

fills the inter-layer space for enhanced neutron contrast. Peptides form inter-layer “pores” 

that can be clearly observed. Temperature and relative humidity were controlled during 

SANS data acquisition. Fully hydrated samples show no inter-layer correlation. Dehydrated 

samples show strong such correlation that shows up as rich anisotropy in the SANS pattern 

characteristic of “single crystal” structure. The shown data set was taken from magainin 

peptide in dimyristoyl phosphatidylcholine (DMPC) bilayers at the peptide-to-lipid ratio of 1 
259

to 30. In order to sample both the in-plane and the out-of-plane structures, the oriented 

membrane was tilted with respect to the neutron beam (Yang et al, 1998). 

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.2 -0.1 0.0 0.1

-0.2

-0.1

0.0

0.1

4

3

2

1

0

Figure 4: SANS data from oriented DMPC/DMPG membranes containing magainin peptides 

in DMPC bilayers and D2O. The sample was oriented at 60 oto the neutron beam direction in

order to observe structures both parallel and perpendicular to the membrane surface. 

5. MAGNETIC MATERIAL

SANS from a single-crystal of NdBa2Cu3O7 (high Tc superconductor) at 100 K is shown. 

This sample is twinned (i.e., is formed of two orthogonally orientated crystals) and the 

nuclear scattering overwhelms the magnetic scattering. Crystal boundaries occur when two 

crystals intergrow with a highly symmetrical interface, often with one crystal being the 

mirror image of the other; atoms are shared by the two crystals at regular intervals. Detwinning of the crystal is necessary in order to reduce the nuclear scattering thereby 

enhancing the magnetic scattering component. A similar system YBa2Cu3O7 is also 

superconductor (Keimer et al, 1993). 
260 120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.02 -0.01 0.00 0.01 0.02

0.02

0.01

0.00

-0.01

-0.02

5.5

5.0

4.5

4.0

3.5

3.0

2.5

Figure 5: SANS data from the high Tc superconducting NdBa2Cu3O7 cuprate at 100 K. Most 

of the scattering is due to the nuclear structure. The oriented structures characterizing the two 

crystals forming the twinned crystal are orthogonal (with orthorhombic symmetry) yielding 

the cross-like SANS patterns. 

REFERENCES

T. Slawecki, C. Glinka and B. Hammouda, “Shear-Induced Micellar Crystal Structures in an 

Aqueous Triblock Copolymer Solution”, Phys Rev E 58, R4084-4086 (1998)

J Bergenholtz and N.J Wagner, “Formation of AOT/Brine Multilamellar Vesicles”, 

Langmuir 12, 3122-3126 (1996)

L. Yang, T.A. Harroun, W.T. Heller, T.M. Weiss and H.W. Huang, “Neutron Off-Plane 

Scattering of Aligned Membranes, Method of Measurement”, Biophysical Journal 75, 641-

645 (1998). 
261

B. Keimer, F. Dogan, I.A. Aksay, R.W. Erwin, J.W. Lynn, and S. Sarikaya, “ Inclined-Field 

Structure, Morphology, and Pinning of the Vortex Lattice in Microtwinned YBa2Cu3O7”, 

Science 262, 83-86 (1993).

QUESTIONS

1. If a sample is oriented along the vertical direction, is the most likely axis of symmetry of 

the iso-intensity contour maps in the vertical or horizontal direction? Is this always the case? 

What is the exception? 

2. When scattering contains bright peaks, why are peaks at high Q broader than peaks at low 

Q?

3. Thinks whether you could learn more about a sample of your research interests by aligning 

it (either though shear, rubbery stretch, or by applying a magnetic field).

ANSWERS

1. If a sample is oriented along the vertical direction, the most likely axis of symmetry of the 

iso-intensity contour maps is in the horizontal direction. Direct space and reciprocal space 

form a conjugate pair and are therefore characterized by orthogonal asymmetry. This applies 

in most cases. The known exception is the case of the “butterfly” pattern whereby the 

asymmetry in the scattering plane and the direct space plane are along the same direction. 

2. Bright spots in the 2D SANS image are usually broader at high Q due to the increase of the 

instrumental resolution with Q. Recall that the variance of the resolution function has two 

contributions: (1) one due to geometry which is independent of Q and (2) one that varies like 

the square of the relative wavelength spread ()

2

.

3. SANS data from aligned samples always contains more information than from randomly 

oriented samples. If the SANS image contains spots, the d-spacing in the two orthogonal 

directions could be different pointing to anisotropic structures. Bright spots turn into 

scattering rings when the sample orientation is random. 
262

Part G – SANS DATA MODELING

Chapter 26. Radius of Gyration Calculations

26.1 Simple Shapes

26.2 Circular Rod and Rectangular Beam

26.3 Comments

26.4 Twisted Ribbon

26.5 Gaussian Polymer Coil

26.6 The Excluded Volume Parameter Approach

References

Questions

Answers

Chapter 27. Single-Particle Form Factors

27.1 Definition of Scattering Factors

27.2 Form Factor for a Uniform Sphere

27.3 Spherical Core-Shell

27.4 Form Factors for Other Spheroid Shapes

27.5 Form Factors for Cylindrical Shapes

27.6 Form Factor for a Parallelepiped

27.7 Twisted Ribbon Form Factor

27.8 Pair Correlation Functions

References

Questions

Answers

Chapter 28. Form Factors for Polymer Systems

28.1 The Debye Function for Gaussian Chains

28.2 Single-Chain Form Factor for Gaussian Chains

28.3 Other Polymer Chain Architectures

28.4 Star Polymer Architecture

28.5 Polymer Rings

28.6 More Complex Ring-Containing Architectures

References

Questions

Answers

Chapter 29. Effect of Polydispersity

29.1 Scattering from a Monodisperse System

29.2 Effect of Polydispersity

29.3 The Gaussian Polydispersity Distribution

29.4 The Log-Normal Polydispersity Distribution

29.5 The Schulz Polydispersity Distribution

References

Questions

Answers
263

Chapter 30. Scattering from Dilute Polydisperse Systems

30.1 Infinitely Dilute Solution of Monodisperse Spheres

30.2 Infinitely Dilute Solution of Polydisperse Spheres

30.3 Dilute Solution of Non-Interacting Monodisperse Polymer Coils

30.4 Solution of Non-Interacting Polydisperse Coils

References

Questions

Answers

Chapter 31. Structure Factors for Polymer Systems

31.1 Scattering from Incompressible Systems

31.2 Inter-Particle Interactions

31.3 The Pair Correlation Function

31.4 Polymer Solutions

31.5 The Zero Contrast Method

31.6 The Random Phase Approximation

31.7 The Isothermal Compressibility Factor

References

Questions

Answers

Chapter 32: Structure Factors for Particulate Systems

32.1 The Ornstein-Zernike Equation

32.2 The Percus-Yevick Approximation

32.3 The Mean Spherical Approximation

32.4 The Random Phase Approximation

References

Questions

Answers

Chapter 33. Scattering from Fractal Systems

33.1 Mass Fractal

33.2 Surface Fractal

33.3 Fractal Porod Exponents

References

Questions

Answers

Chapter 34. The Multi-Component Random Phase Approximation

34.1 Compressible Polymer Mixture

34.2 Incompressible Polymer Mixture

34.3 The Single-Chain Form Factors

34.4 Binary Homopolymer Blend Mixture

34.5 Ternary Homopolymer Blend Mixture

34.6 Blend Mixture of a Copolymer and a Homopolymer
264

34.7 The Diblock Copolymer Case

34.8 The Triblock Copolymer Case

34.9 Mixture of Polyelectrolytes

34.10 Discussion

References

Questions

Answers
265

Chapter 26 - RADIUS OF GYRATION CALCULATIONS

The radius of gyration is a measure of the size of an object of arbitrary shape. It can be 

obtained directly from the Guinier plot [ln(I(Q)] vs Q

2

] for SANS data. The radius of 

gyration squared Rg

2

is the second moment in 3D. 

1. SIMPLE SHAPES

First consider some simple shape objects. 

Figure 1: Representation of the polar coordinate system for a disk.

For an infinitely thin disk of radius R, Rg

2

is given by the following integral using polar 

coordinates. 

Rgx

2 = 

 

 



 

2π

0

R1

0

2π

0

2

R1

0

2

rdrd

r cos ( )rdrd

= 

 

 



 

1

1

R

0

2π

0

R

0

2π

0

3 2

rdr d

r dr cos ( )d

= 

4

R

2

. (1)

Similarly for Rgy

2 = 

4

R

2

. For an infinitely thin disk Rg

2 = Rgx2+ Rgy2=

2

R

2

.

x

y

r cos()

R r


266

Figure 2: Representation of the spherical coordinate system for a sphere.

In the case of a full sphere, the integration is performed with spherical coordinates. 

Rg

2 = 

 

 





 

 

0

R

0

2

0

R

0

2 2

sin( )d r dr

sin( )d r dr r

= 

5

3R2

. (2)

The radius of gyration (squared) for the spherical shell of radii R1 and R2 is given by:

 

 



2

1

R

R

4

3

1

3

2

2

g 4 r dr

4 (R R )

3

R

(3)

 

(R R )

(R R )

5

3

3

1

3

2

5

1

5

2





 .

x

y

z

R

r




267

Figure 3: Representation of the Cartesian coordinate system for a rectangular plate.

For an infinitely thin rectangular object of sides W and H, the integration is performed in 

Cartesian coordinates. 

Rgx

2 = 









W/2

W/2

W/2

W/2

2

dx

dx x

= 

2

2

W

3

1













. (4)

Similarly for Rgy

2 =

2

2

H

3

1













. The sum gives Rg

2 =





























 











2 2

2

H

2

W

3

1

. 

Note that the moment of inertia I for a plate of width W, height H and mass M is also given 

by the second moment. 

I = Ixx + Iyy = 





























 











2 2

2

H

2

W

3

M . (5)

x

y

W

H
268

2. CIRCULAR ROD AND RECTANGULAR BEAM

Figure 4: Representation of the cylindrical rod and rectangular beam. 

The radius of gyration for a cylindrical rod of length L and radius R is given by:

2

2 2 2

2

g

12

L

2

R

2

L

3

1

2

R

R   











  . (6)

The radius of gyration for a rectangular beam of width W, height H and length L is given by:





























 











 













2 2 2

2

g

2

L

2

H

2

W

3

1

R . (7)

This formula holds for a straight “ribbon” where W<H<<L. 

The value of Rg

2

for a cylindrical rod with radius R = 10 (diameter D = 20) and length L = 10 

is Rg

2 = 58.3. This value is to be compared with the case of a rectangular beam with sides W 

= L = 20 and length L = 10 for which Rg

2 = 75. 

3. COMMENTS

The radius of gyration squared can be calculated for other more complicated shapes as the 

second moment for each of the symmetry direction. 

Note that Rgx

2

for a horizontal strip is the same as that for the whole square plate Rgx

2 = 

2

2

W

3

1













. Rgx

2

is independent of the height of the object. Of course Rgy

2

depends of the height 

but not on the width. 

Circular Rod

Rectangular Beam
269

Figure 5: Case of a horizontal flat strip. 

Figure 6: Case of a circular ring. 

The radius of gyration for an infinitely thin circular ring of radius R is Rgz

2 = R2

. This is 

obtained by spinning the ring in the horizontal plane (around the z-axis). Note that it is the 

same value for an infinitely thin spherical shell of radius R. 

4. TWISTED RIBBON

The radius of gyration for rigid twisted shape objects are worked out here. Consider the 

simple case of a rigid helical wire, then the case of a rigid twisted ribbon with finite size 

thickness. 

Helical Wire

Consider a very thin helically twisted wire aligned along the vertical z axis. Choose the 

origin of the Cartesian coordinate system at the center-of mass of the twisted wire. The helix 

has a radius R and a height L so that -L/2 



z 



L/2. The parametric equation of the helix is:

X = R cos() (8)

x

y

z

R

x

y

W
270

Y = R sin()

Z = p/2

Here p is the helix pitch and  is the azimuthal angle in the horizontal plane. The wire 

position along the helix is represented by the vector 

r()



. Note that by definition of the 

center-of-mass, the average of this vector is null, <

r()



> = 0. 

Figure 7: Schematic representation of the twisted wire. 

The radius of gyration (squared) Rg

2

is defined as follows:

Rg

2 = <r2

()> = 







 

d

d r ( )

2

. (9)

Here r2() = X2+Y2+Z2 = R2 + (p/2)

2

. The azimuthal angle  varies in the range: 

-L/p 







L/p. 

The  integration is readily performed to give:

Rg

2 = R2 + 

2

2

L

3

1













. (10)

Note that this is the same result as for a cylindrical shell of radius R and height L. This is not 

surprising since a cylinder could be built by a number of twisted wires stacked vertically. 

Twisted Wire

x

y

z

r

L 

R
271

Thin Twisted Ribbon

The case of a thin twisted helical ribbon of width W can be worked out similarly using a twovariable parametric notation r2

(,z) where  is the azimuthal angle and z is the vertical 

ribbon width with –W/2 



z 



W/2. 

Figure 8: Schematic representation of the thin twisted ribbon.

Here, the variable Z is replaced by Z+z. The radius of gyration (squared) is therefore given 

by:

Rg

2 = <r2

(,z) = 

 

 



 

d dz

d dz r ( , z)

2

(11)

r

2

(,z) is now given by 

2

2 2

z

2

p

r ( , z) R 

















  

. The integrations can here also be readily 

performed to give:

Rg

2 = R2 + 

2

2

L

3

1













+ 

2

2

W

3

1













. (12)

Twisted Ribbon

x

y

z

r

L 

R

W
272

These involve contributions from <Z2> and <z2>. The cross term gives no contribution 

because it involves the null average <z> = 0. 

Thick Twisted Ribbon

For the case of a twisted ribbon of horizontal thickness T, the variable R is replaced by R+

where –T/2 







T. 

Figure 9: Top view of a thick twisted ribbon. 

The calculation of the second moment proceeds as before:

X =  cos() (13)

Y =  sin()

Z = p/2.

Here  is the polar coordinate variable in the horizontal plane with limits: R-T/2 







R+T/2. In this case r2(z,) = 

2 + (Z+z)2 where z is the same parameter as before. Rg2 = 

< 

2> + <(Z+z)2> involves two averages. The first average is:

<

2> = 

 

 













  

   

R T / 2

R T / 2

2

0

2

R T / 2

R T / 2

2

0

d d

d d

= 





























  









































  













2 2

4 4

2

T

R

2

T

R

2

1

2

T

R

2

T

R

4

1

= 

2

2

2

T

R 











 .

(14)

The final result involving both (horizontal and vertical) averages is:

Thick Twisted Ribbon

Top View

R

T

y

x
273

Rg

2 = R2 + 

2

2

T













+ 

2

2

L

3

1













+ 

2

2

W

3

1













. (15)

Note that all terms add up in quadrature since all cross terms (first moments) average to zero. 

5. GAUSSIAN POLYMER COIL

The radius of gyration (squared) for a polymer coil is defined as:

  

n

i

2

i

2

g S

n

1

R . (16)

Si refers to the position of monomer i with respect to the center-of-mass of the polymer coil 

and n is the total number of monomers per coil. The inter-distance vector between two 

monomers within the same macromolecule is defined as 

Sij Sj Si

  

  . Consider the following 

relation:

      

n

i,j

i j

n

j

2

j

n

i

2

i

n

i,j

2

Sij n S n S 2 S .S

    

. (17)

The last summation is null 

S .S S . S 0

n

j

j

n

i

i

n

i,j

 i j    

   

since by definition of the center-ofmass 

S 0

n

i

 i 



. The radius of gyration (squared) is therefore simplified as:

     

n

i,j

2

2 ij

n

i,j

2

2 ij

2

g

r

2n

1

S

2n

1

R . (18)

The vectorial notation has been dropped for simplicity. 
274

Figure 10: Schematic representation of a Gaussian coil showing monomers i and j and their 

inter-distance rij. Note that 

ij ij S r







in the notation used. 

S a | i j|

2 2

 ij   . (19)

Here a is the statistical segment length, and <...> is an average over monomers. The 

following formulae for the summation of arithmetic progressions are used:

2

n(n 1)

k

n

k 1



 



(20)

6

n(n 1)(2n 1)

k

n

k 1

2  

 



.

The radius of gyration squared becomes:

  

n

i,j

2

2

2

g

| i j|

2n

a

R

= 

)k

n

k

(1

n

a

n

k

2

  (21)

 = 

6

a n

n

(n 1)

6

a

2 2 2





for n >> 1.

Note that taking the n >> 1 limit early on allows us to replace the summations by 

integrations. Using the variable x = k/n, one obtains:

6

a n

3

1

2

1

R a n dx(1 x)x a n

2

2

1

0

2 2

g  











    

. (22)

Similarly, the end-to-end distance squared R1n

2

for a Gaussian polymer coil is given by:

Si



ij ij S r







center 

of mass

i

j

i

r



j

r



Sj


275

R a n

2 2

1n 

for n >> 1. (23)

These results are for Gaussian coils that follow random walk statistics (Flory, 1969). 

6. THE EXCLUDED VOLUME PARAMETER APPROACH

The Flory mean field theory of polymer solutions describes chain statistics as a random walk 

process along chain segments. For Gaussian chain statistics, the monomer-monomer interdistance is proportional to the number of steps:



  

2 2 2

ij S a | i j|  

Here a is the statistical segment length,  is the excluded volume parameter, Sij represents an 

inter-segment distance and <...> is an average over monomers. The radius of gyration 

squared for Gaussian chains is expressed as:

 



    

n

i,j

2

2

n 2

i,j

2

2 ij

2

g

| i j|

2n

a

S

2n

1

R

(25)

 

 

   

2

2

2

n

k

2

n

(2ν 1)(2ν 2)

a

) k

n

k

(1

n

a

.

i and j are a pair of monomers and n is the number of chain segments per chain. Three cases 

are relevant:

(1) Self-avoiding walk corresponds to swollen chains with  = 3/5, for which 

2 2 6 5

g

a n

176

25 R  . 

(2) Pure random walk corresponds to chains in theta conditions (where solvent-solvent, 

monomer-monomer and solvent-monomer interactions are equivalent) with  = 1⁄2, for which 

a n

2

1

R

2 2

g

 . 

(3) Self attracting walk corresponds to collapsed chains with  = 1/3, for which 

2 2 2 3

g

a n

40

9

R  .

Note that the renormalization group estimate of the excluded volume parameter for the fully 

swollen chain is  = 0.588 (instead of the 0.6 mean field value). 

Note also that the radius of gyration for a thin rigid rod can be recovered from this excluded 

volume approach by setting  = 1 and defining the rod length as L = na. 
276

12

L

12

a n

n

(2ν 1)(2ν 2)

a

R

2 2 2

2

2

2

g

 

 





. (26)

This is the same result derived earlier for a thin rod. 

REFERENCES

http://en.wikipedia.org/wiki/List_of_moments_of_inertia

P.J. Flory, “Statistical Mechanics of Chain Molecules”, Interscience Publishers (1969)

QUESTIONS

1. How is the radius of gyration measured by SANS?

2. How is the center-of-mass of an object defined?

3. Why is the radius of gyration squared for an object related to the moment of inertia for that 

object?

4. Calculate Rg

2

for a full sphere of radius R. Calculate Rg

2

for a thin spherical shell of radius 

R. 

5. What is the value of Rg

2

for a Gaussian coil of segment length a and degree of 

polymerization n? How about the end-to-end distance?

6. What is the radius of gyration squared for a rod of length L and radius R?

ANSWERS

1. The radius of gyration is measured by performing a Guinier plot on SANS data. The slope 

of the linear variation of ln[I(Q)] vs Q

2

is Rg

2

/3. 

2. The center-of-mass of an object is defined as the spot where the first moment is zero. 

3. The radius of gyration squared and the moment of inertia for that object are both expressed 

in terms of the second moment. 

4. Rg

2

for a full sphere of radius R is given by: 

Rg

2 = 

















 

















     

 

0

R

0

2

0

R

0

2 2

sin( )d r drr / sin( )d r dr

= 

5

3R2

. Rg

2

for a thin spherical shell is simply 

given by: Rg

2 = R2

. 

5. For a Gaussian coil of segment length a and degree of polymerization n, one can calculate 

the radius of gyration squared as Rg

2 = 

a n / 6

2

and the end-to-end distance squared as R1n

2 = 

a n

2

. 

6. The radius of gyration squared for a rod of length L and radius R is given by: 

2 2

2

g

2

L

3

1

2

R

R 











  . 
277

Chapter 27 – SINGLE PARTICLE FORM FACTORS

1. DEFINITION OF SCATTERING FACTORS

Consider a scattering object consisting of n scatterers occupying a volume VP. The scattering 

density and its Fourier transform are defined as: 



  

n

i 1

i n(r) (r r )

 

(1)

  



n

i 1

i n(Q) exp[ iQ.r ]





.

Note that these quantities vary randomly with position 

r



and scattering vector 

Q



. The 

average density being constant (<n(r)> = 

n

= n/Vp), a fluctuating density and its Fourier 

transform are defined as:

    

n

i

n(r) (r r

i

) n

 

(2)

n(Q) exp[ iQ.r ] (2 ) n (Q)

3

n

i

i

 

       . 

Here 

(Q)





is the Dirac Delta function which does not contribute except at 

Q 0

 



(along the 

very forward scattering direction) which is experimentally irrelevant. The static form factor 

for the scattering “particles” is defined as the density-density correlation function summed up 

(or integrated) over the particle volume:

   

  



n

i,j

2 2 i j

exp[iQ.(r r )]

n

1

n

n( Q)n(Q)

P(Q)

 



(3)

 exp[iQ.(r' r)]

n

n(r)n(r')

dr dr'

2

 

  

  

 

 .

The form factor of various shape objects are worked out next (Guinier-Fournet, 1955; 

Glatter-Kratky, 1982; Higgins-Benoit, 1994; Hammouda, 1995; Roe, 2000).

2. FORM FACTOR FOR A UNIFORM SPHERE

Consider a sphere of radius R and uniform density (this could be a spherical domain in a 

microphase separated block copolymer or a latex particle in a colloidal suspension). The 

single particle form factor P(Q) involves integrations over the volume VP of the sphere (in 

spherical coordinates):
278

exp[iQ.(r' r)]

n

n(r)n(r')

dr dr'

n

n( Q)n(Q)

P(Q)

2 2

 

  



 

  

  

 B. (4)

Since the scattering elements are not correlated, the average of the product <n(r)n(r')> is 

equal to the product of the averages <n(r)><n(r')> and therefore:

2

P(Q) | F(Q) | . (5)

Here the amplitude of the form factor F(Q) has been defined as:

exp[ iQ.r]

n

n(r)

F(Q) dr



 

 

 

 . (6)

For uniform density, the average over configurations <n(r)> becomes trivial:

n

V

n

n(r)

P

  

if rR (7)

 n(r)  0 . if r>R

Therefore:

       







1 

1

2

0

R

0

2

3

r dr d exp[ iQr ] d

4 R

3

F(Q)

(8)

QR

3j (QR)

Qr

sin(Qr)

r dr

R

3 1

R

0

2

3

  

. (9)

Here the spherical Bessel function j1(x) has been defined as: 

J (x)

x 2x

cos(x)

x

sin(x)

j (x) 1 2 3 2



   . (10)

The spherical Bessel function jB1B(x) is related to the cylindrical Bessel function JB3/2B(x) as 

shown. It is also related to jB0B(x) as follows:

j (x)

dx

d

j (x)

1   0

and 

x

sin(x)

j (x)

0  . (11)

The form factor for the sphere is therefore:

2

2

2

1

QR

cos(QR)

(QR)

sin(QR)

QR

3

QR

3j (QR)

P(Q) 



























  











 . (12)
279

Note the following normalization 

P(Q  0) 1

and recall the calculation of the radius of 

gyration squared for a uniform density sphere of radius R as

R 3R 5

2 2

g

 .

The low-Q Guinier expansion follows: 

    4

2 2

2

2 4

(QR)

175

3

5

Q R

1

280

QR

10

QR P(Q) 1    











  

(13)

21

Q R

3

Q R

1

4

g

2 4

g

2

  

for QP

2

PRBgPB

2

P < 1

-8

-6

-4

-2

0

0 2 4 6 8 10 12

Form Factor for a Sphere 

calculated 

Log[P(Q)]

QR

QR = 4.48

QR = 7.66 

QR =10.91 

Figure 1: Plot of Log[P(Q)] vs QR for a uniform sphere showing many order oscillations.

3. SPHERICAL CORE-SHELL

Consider a sphere with an inner core and an outer shell. Three regions can be defined 

corresponding to the inner core, the outer shell and the solvent. Three cases are considered 

where (1) the shell is visible (with matched core and solvent scattering length densities), (2) 
280

the core and shell scattering length densities are matched and (3) the core is visible (with 

matched shell and solvent scattering length densities). 

Note that the “correlation hole” peak is enhanced in case 1 for which the shell is visible 

whereas the core is not (i.e., it is matched to the solvent). Polydispersity (/R = 0.3) has been 

included in order to damp higher order oscillations. This level of polydispersity was enough 

to damp oscillations for case 3 but not enough for case 2.

0.001

0.01

0.1

1

10

100

1000

0.001 0.01 0.1 1

polydisperse core-shell sphere 

case 1

case 2

case 3

 Scattering Factors

Scattering Variable Q (Å-1) 

Figure 2: Scattering factors for a core-shell sphere of inner radius R = 20 Å and radial shell 

thickness T = 20 Å. Case 1 corresponds to the core scattering length density matched to the 

solvent. Case 2 corresponds to matched scattering length densities for the core and shell. 

Case 3 corresponds to the shell scattering length density matched to the solvent. The vertical 

scale is arbitrary and a constant background value of 0.001 has been added. 

4. FORM FACTORS FOR OTHER SPHEROID SHAPES

Following the same procedure, the form factor for a spherical shell between radii R1 and R2

(and hollow for r<R1) can be calculated as follows:

R T

case 2

case 1

case 3
281

      

 







1 

1

2

0

R

R

2

3

1

3

2

r dr d exp[ iQr ] d

4 (R R )

3

F(Q)

2

1

 (14)





























 



















3

1

1

3 1 1

2

2

1 2

3

1

3

2

R

QR

3j (QR R

QR

3j (QR

(R R )

1

.

For an ellipsoid of half axes a, b, c oriented so that its axes make angles , ,  with the 

Q



direction, an effective radius Reis defined as: 

R a cos ( ) b cos ( ) c cos ( )

2 2 2 2 2 2 2

e

      . (15)

The form factor amplitude is the same as the one for a sphere of radius Re: 

e

1 e

QR

3j (QR )

F(Q,)  . (16)

The form factor (for a randomly oriented sample) is an average over all possible orientations 

of the ellipsoid:







  

1

1

2

d | F(Q, ) |

2

1

P(Q) . (17)

 = cos() and  is the angle between the major axis of the ellipsoid and the 

Q



direction. It is 

straightforward to extend these results to an ellipsoidal shell.

5. FORM FACTORS FOR CYLINDRICAL SHAPES

The form factor amplitude F(Q) for a uniform cylinder (rod) of radius R and length L 

oriented at an angle  from the 

Q



direction is the product of a longitudinal (z along the rod) 

and a transverse (



perpendicular to the rod) contributions in cylindrical coordinates:

F(Q, ) F (Q, )F (Q, )  

z   

(18)

Q L/ 2

sin(Q L/ 2)

dz exp[ iQ z]

L

1

F (Q, )

L/ 2

L/ 2

z





     



        



 





R

0

2

0

2

2

d d exp[ iQ 1 cos( ) ]

R

1

F (Q, ) .
282

Here  = cos() and  is the inclination angle. The following definition of the cylindrical 

Bessel functions are used (Abromowitz-Stegun, 1972):





 





0

0 d exp[iz cos( )] 1

J (z) . (19)

d exp[iz cos( )]cos( )

i

1

J (z)

0

1    







One obtains:

       

R

0

2

2 0 d J (Q 1 )

R

2

F (Q, ) . (20)

An integration variable change to t = /R is made and the following integral is used:

J (a)

a

1

tdtJ (at)

0 1

1

0

 

. (21)

The following result is obtained:

Q 1 R

2J (Q 1 R)

F (Q, )

2

2

1

 

 

   . (22)

The final result for the form factor amplitude for an oriented rod is:

















 

 

















 

Q 1 R

2J (Q 1 R)

Q L/ 2

sin(Q L/ 2)

F(Q, )

2

2

1

. (23)

The form factor for a randomly oriented rod is therefore given by the following orientation 

average:

2

1

1

d | F(Q, ) |

2

1

P(Q)





   . (24)

In order to model the scattering from very dilute solutions of rods, the last integral (over ) is 

performed numerically. 
283

Figure 3: Geometry of the uniform rod.

-7

-6

-5

-4

-3

-2

-1

0

1

0 1 2 3 4 5 6 7 8

 Form Factors for a Rod 

 [2J

1

(X)/X]2

[sin(X)/X]2 Log Scale 

X 

X = 3.78

X = 7.02 

Figure 4: Plots of the two functions [2J1(X)/X]2and [sin(X)/X]2that give the variations of the 

form factor perpendicular and parallel to the rod axis respectively. 

x

y

z





Q
284

10-5

0.0001

0.001

0.01

0.1

1

0.001 0.01 0.1

Cylinder with R = 20 Å, L = 400 Å 

cylinder 

Form Factor P(Q)

Scattering Variable Q (Å-1)

1/Q

1/Q4

Figure 5: Form factor P(Q) for a cylinder with radius R = 20 Å and length L = 400 Å. 

Note that the result for a rod of length L applies also to a disk of thickness L. 

For a disk of radius R and negligible thickness, the 

L  0

limit in the general result is taken 

so that:

        



 





R

0

2

0

2

2

d d exp[ iQ 1 cos( ) ]

R

1

F (Q, )

(25)

 

Q 1 R

2J (Q 1 R)

2

2

1

 

 

 .

Averaging over orientations is performed as follows:



 















 

 

 

1

1

2

2

2

1

Q 1 R

2J (Q 1 R)

d

2

1

P(Q)

(26)
285













 

QR

J (2QR)

1

(QR)

2 1

2

.

To obtain the form factor for an infinitely thin rod of length L, we take the 

R  0

limit 

instead, and obtain:

2 1

1 Q L/ 2

sin(Q L/ 2)

d

2

1

P(Q)





















  . (27)

Integrate by part once to obtain:



















 

  















  









1

1

1

1

2

2

(QL / 2)

2sin(QL / 2) cos(QL / 2)

d

(QL/ 2)

sin (QL / 2)

2

1

P(Q)

(28)

2

2

(QL/ 2)

sin (QL/ 2)

Si(QL)

QL

2

 















 .

Si(x) is the sine integral function defined as:





x

0

u

sin(u)

Si(x) du . (29)

6. FORM FACTOR FOR A PARALLELEPIPED

Consider a uniform density rectangular parallelepiped of sides a, b, c. In Cartesian 

coordinates, the form factor amplitude can be split into the product of three parts that depend 

on the three coordinates respectively:













  











  











  

  

c / 2

c / 2

z

b / 2

b / 2

y

a / 2

a / 2

x dx exp[ iQ x) dyexp[ iQ y) dz exp[ iQ z)

abc

1

F(Q)

(30)











































(Q c / 2)

sin(Q c / 2)

(Q b / 2)

sin(Q b / 2)

(Q a / 2)

sin(Q a / 2)

z

z

y

y

x

x

.

The form factor is, here also, an average over orientations:

2

1

1

d | F(Q, ) |

2

1

P(Q)





   . (31)
286

 = cos() and  is the orientation angle between 

Q



and one of the symmetry axes of the 

parallelepiped.

7. TWISTED RIBBON FORM FACTOR

The parametrization of the twisted ribbon was described in an earlier section when 

calculating the radius of gyration. Consider a helically twisted ribbon aligned along the 

vertical z axis with a helical radius R, height L, width W and ribbon thickness T. Define the 

helix pitch p and the azimuthal angle  in the horizontal plane. Define also the polar

coordinate variable in the horizontal plane  and the vertical variable z. The parametric 

position along the ribbon is given by:

2

2

z

2

p

r( , z, ) 

















     . (32)

The single twisted ribbon form factor amplitude is given by: 

  

  

 













 

 

  

 

d dz dρ

Qr( , z,ρ)

sin[Qr( , z,ρ) ]

d dz d

F(Q, ) . (33)

All three integrations can be performed numerically using the following limits: -L/p 







L/p, –W/2 



z 



W/2 and R–T/2 







R+T/2. 

Here also, the form factor is given by an average over orientations:

2

1

1

d | F(Q, ) |

2

1

P(Q)





   . (34)

 = cos() and  is the orientation angle between 

Q



and the vertical axis of the ribbon. 

8. PAIR CORRELATION FUNCTIONS

The form factor P(Q) is the Fourier transform of the probability distribution function 

P(r)



:

P(Q)  d rexp(iQ.r)P(r)

3  



. (35)

Given an infinitesimal scattering volume chosen randomly inside the considered "particle",

P(r)



represents the probability of finding another scatterer within the particle a distance 

r



away. 
287

Figure 7: Density-density correlation function

Usually, a one-dimensional probability distribution p(r) (also referred to as "distance 

distribution function") is defined instead:

p(r)

Qr

sin(Qr)

dr

R

1

P(Q)

R

0



 . (36)

p(r) is available for some of the common shape objects. For a sphere of radius R:













 











 













2R

r

2

2R

r

1

2R

r

p(r) 12

2 2

(37)





























  













2 3

R

r

16

1

R

r

4

3

1

R

r

3 .

Note the other definition 

p(r) 3r R (r)

2

 

where (r) is the radial pair correlation function 

given by 





























   

3

R

r

16

1

R

r

4

3

(r) 1 . 
288

Figure 8: Radial pair correlation function for uniform sphere

For a disk of radius R, the distance distribution function is given by:





























  















2

2R

r

1

2R

r

2R

r

arccos

R

8 r

p(r) . (38)

For an infinitely thin rod of length L, the integration is performed from 0 to L and the 

normalization constant is 1/L so that:













 

L

r

p(r) 2 1 . (39)

Note that the probability distribution function P(

r



) is better known when defined for the 

"inter-particle" structure factor SI(Q) and is often referred to as pair correlation function 

g(

r



)=VP(

r



) (where V is the sample volume):

  dr exp iQ.rg(r) 1

V

1

S (Q)

I

 

 

. (40)

Here the following constant term:

 dr exp iQ.r  2 Q

3









(41)
289

has been subtracted from 

g(r)



. This term has no contribution except in the (experimentally 

irrelevant) forward scattering direction (for which 

Q 0

 



). 

REFERENCES

A. Guinier, and G. Fournet, "Small-Angle Scattering of X-Rays", John Wiley and Sons, New 

York, (1955).

M. Abromowitz and I.A. Stegun, “Handbook of Mathematical Functions”, Dover 

Publications (New York, 1972)

O Glatter and O Kratky, “Small Angle X-Ray Scattering”, Academic Press (1982).

J.S. Higgins and H. Benoit, "Polymers and Neutron Scattering", Oxford (1994). 

B. Hammouda, “SANS from Polymers Tutorial”, NIST Center for Neutron Research Report 

(1995)

R.J. Roe, "Methods of X-Ray and Neutron Scattering in Polymer Science", Oxford 

University Press, New York, (2000).

QUESTIONS

1. What is the relationship between the form factor P(Q) and its amplitude F(Q) for the case 

of a uniform sphere? How about the case of a Gaussian polymer coil? 

2. What is the form factor for a uniform sphere of radius R?

3. What is the form factor for a disk of radius R with its axis of rotation oriented parallel to 

the 

Q



direction?

4. What is the form factor for a disklike lamella of thickness L with its normal axis oriented 

parallel to the 

Q



direction?

5. What is the form factor for a cylinder of radius R and length L oriented perpendicular to 

the 

Q



direction?

6. How is the averaging over random orientations performed for the calculation of the form 

factor?

7. Write down the radial pair correlation function (r) for a uniform sphere of radius R. (r) is 

defined through the following 1D Fourier transform: 

(r)

Qr

sin(Qr)

dr4 r

V

1

P(Q)

R

0

2

P

    . 

8. What are the various parts that are used to calculate the SANS macroscopic scattering 

cross section for a solution of compact scatterers? 

9. What is the Porod exponent for an infinitely thin rod of length L?
290

10. Define the spherical Bessel function of first order j1(x). What is J1(x)?

ANSWERS

1. For a uniform sphere P(Q) = |F(Q)|2. For a Gaussian coil, there is no uniform density and 

the form factor amplitude cannot be defined. 

2. The form factor for a uniform sphere is given as 

 

2

P(Q)  3j

1

(QR) QR

where j1(QR) is 

the spherical Bessel function. 

3. The form factor for a disk of radius R with its axis of rotation oriented parallel to the 

Q



direction is given by 

 

2

P(Q)  sin(QR) QR . 

4. The form factor for a disklike lamella of thickness L with its normal axis oriented parallel 

to the 

Q



direction is given by 

 

2

P(Q)  2J1(QL QL

where J1(QL) is the cylindrical Bessel 

function. 

5. The form factor for a cylinder of radius R and length L oriented perpendicular to the 

Q



direction is given by 

 

2

P(Q)  2J1(QR) QR . 

6. The form factor for a randomly oriented object with its symmetry axis along the zdirection is calculated as 

 





  

1

1

P(Q) 1 2 d P(Q, )

where P(Q,) is the form factor for the 

object oriented at an angle  from the 

Q



direction ( = cos(). 

7. The radial pair correlation function for a uniform sphere of radius R is given as 





























   

3

R

r

16

1

R

r

4

3

(r) 1

. Note that (r=2R) = 0. 

8. The SANS macroscopic scattering cross section for a solution of compact scatterers is the 

product of (1) the contrast factor, (2) the number density of scatterers, (3) the scatterer’s 

volume squared, (4) the form factor and (4) the structure factor. 

9. Since P(Q) for an infinitely thin rod of length L is given by 



 















 

 

 

1

1

2

2

2

1

Q 1 R

2J (Q 1 R)

d

2

1

P(Q)

, one would think that the Porod law gives 

2

P(Q) 1 Q . 

However after orientational averaging, one obtains the following 

2

2

(QL/ 2)

sin (QL/ 2)

Si(QL)

QL

2

P(Q)  

















, so that 

P(Q) 1 Q

. The Porod exponent for an 

infinitely thin rod is 1. 

10. The spherical Bessel function of first order is given by 

x

cos(x)

x

sin(x)

j (x)

1 2

  . J1(x) is 

the cylindrical Bessel function. 
291

Chapter 28 - FORM FACTORS FOR POLYMER SYSTEMS

1. THE DEBYE FUNCTION FOR GAUSSIAN CHAINS

Polymer coils in theta solvents or in the melt state follow Gaussian chain statistics whereby 

the inter-monomer distance rij is given by the following Gaussian distribution function:

















 



















 



2

ij

2

ij

3/2

2

ij

ij 2 r

3r

exp

2π r

3

P(r ) . (1)

Here <rij

2> is the variance given in terms of the statistical segment length a as:

r a | i j|

2 2

 ij   . (2)

The single-chain form factor is given by:

   

n

i, j

2 ij exp[ iQ.r ]

n

1

P(Q)





(3)

  ij ij

n

i,j

2 ij dr P(r )exp iQ.r

n

1 

 

   

 













  

 

n

i,j

2

ij

2

2

6

Q r

exp

n

1

 









 

 

n

i,j

2 2

2

6

Q a | i j|

exp

n

1

.

The following property of the Gaussian distribution has been used:















  

 















  

   

6

Q r

exp

2

Q x

exp[ iQ x ] exp

2

ij

2

x

2

ij

2

x

x ij

(4)















  

   

6

Q r

exp[ iQ.r ] exp

2

ij

2

ij





.

The following general identity is used:
292  

   

n

i,j

n

k 1

F(| i j|) n 2 (n k)F(k). (5)

Assuming that the number of chain segments n is large (n>>1), one obtains the Debye 

function (Debye, 1947):

























    



n

k 1

2 2

2

6

Q a k

n 2 (n k)exp

n

1

P(Q)

(6)

 

2

g

2

2

g

2

4

g

4

exp( Q R ) 1 Q R

Q R

2

P(Q)     . 

The radius of gyration is given by 

R a n / 6

2

g

 .

Small-Q and high-Q expansions of the Debye function are:

3

Q R

P(QR 1) 1

2

g

2

g    (7)

2

g

2

g

Q R

2

P(QR 1)  . 

Two approximations are included here for the Debye function:





















3

Q R

1

1

P(Q)

2

g

2

(8)





















2

Q R

1

1

P(Q)

2

g

2

.

The first form agrees better at low-Q and the second form agrees better at high Q. 
293

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7

Debye Function

1/(1+Q2Rg

2

/3)

1/(1+Q2Rg

2

/2)

 Debye Function P(Q)

Q

2Rg2

Figure 5: Variation of the Debye function P(Q) along with two approximations that bracket 

its variation. The form 

1 1 Q R / 3

2

g

2



is a good approximation at low-Q and the form 

1 1 Q R / 2

2

g

2



is a good approximation at high-Q. 

Polymer chains are not characterized by uniform density. The form factor (Debye function) is 

not a square and cannot therefore be expressed as a square of the amplitudes. 

2. SINGLE-CHAIN FORM FACTOR FOR GAUSSIAN CHAINS

Consider a flexible polymer coil where each monomer pair located a distance 

ij r



apart obeys 

the Gaussian distribution:

















 



















 



2

ij

2

ij

3/2

2

ij

ij 2 r

3r

exp

2π r

3

P(r )



. (9)

The average of the segment inter-distances squared is kept in the general form:
294 

  

2 2 2

ij r a | i j|  (10)

 is the excluded volume parameter. Note that 

ij ij S r







in the notation used where 

ij r



is in the 

laboratory reference frame and 

Sij



is in the center-of-mass reference frame. Within this 

approach, the single-chain form factor is expressed as: 

   

n

i, j

2 ij exp[ iQ.r ]

n

1

P(Q)





(11)

  ij ij

n

i,j

2 ij dr P(r )exp iQ.r

n

1 

 

   

 











  

n

i,j

2ν

2 2

2

| i j|

6

Q a

exp

n

1

.

Note that the monomer pair is always correlated through chain connectivity so that the 

simplifying approximation

2

P(Q) | F(Q) |

(which is made for uniform density objects) is not 

valid for polymers. The typical manipulations (as in the case of the Debye function described 

previously) are performed. 

Assuming that the number of chain segments n is large (n >> 1), one obtains:













  



2ν

n 2 2

k 1

2

k

6

Q a

2 (n - k)exp

n

1

P(Q) . (12)

Going to the continuous limit:













  





2 2ν

1 2 2

0

n x

6

Q a

P(Q) 2 dx (1 x) exp . (13)

This integral is “almost” analytical and can be expressed in terms of the incomplete gamma 

function:

d 1

U

0

(d,U) dt exp( t)t



   

. (14)

The result is:

,U)

ν

1

γ(

νU

1

,U)- 

2ν

1

γ(

νU

1

P(Q)

1/2ν 1 /ν

 . (15)
295

The modified variable is 

6

Q a n

U

2 2 2ν

 . 

The high-Q limit of this form is given by:

)

ν

1

(

νU

1

) - 

2ν

1

(

νU

1

P(Q )

1/2ν 1 /ν

     . (16)

Here (x) = (x,) is the gamma function. The asymptotic limit is dominated by the first 

term 

)

2ν

1

(

νU

1

1/2ν



which varies like 

1 2 U ~ 

1  Q . 

Polymer chains follow Gaussian statistics in polymer solutions: they are swollen in good 

solvents, are thermally relaxed in “theta” solvents and partially precipitate in poor solvents. 

The familiar Debye function is recovered when  = 1⁄2. 

3. OTHER POLYMER CHAIN ARCHITECTURES

Many polymer chain architectures exist: "stars" consist of many equal size branches 

connected to a central core, "combs" consist of side branches grafted onto a main chain, 

"rings" consist of looped chains, "gels" consist of highly branched structures that are grown 

outwardly (dendrimers are the most regular gels), "networks" consist of crosslinked systems 

that contain a large number of inter-connected structures, etc. These various polymer systems 

are made in the homopolymer form (all monomers are chemically identical) or copolymer 

form (each chain portion consists of blocks of monomers that are chemically different). 

Single-chain form factors for such architectures have been worked out and are summarized 

elsewhere (Burchard, 1983; Hammouda, 1993; Higgins-Benoit, 1994). Basic elements are 

included here.

In the same spirit used to derive the form factor for an isolated polymer chain (Debye 

function):

 

2

g

2

2

g

2

4

g

4

exp( Q R ) 1 Q R

Q R

2

P(Q)     , (17)

one can also derive the form factor amplitude for a polymer chain anchored at one end. In 

this case:

 









 

 

n

i

2 2

6

Q a | i 1|

exp

n

1

F(Q)

(18)
296  

2

g

2

2

g

2

Q R

1 exp  Q R

 .

Similarly, a propagation factor can be defined (involving no summation):

E(Q) exp[ Q R ]

2

g

2   . (19)

Figure 6: Schematic representation of the summation variables for the various scattering 

factors for Gaussian polymer chains. 

The three scattering factors E(Q), F(Q), and P(Q) can be used to work out the form factors 

for many polymer architectures. 

Consider the simple case of a diblock copolymer A-B consisting of two blocks with NA-NB

segments. The various partial form factors follow:

P(Q) n P (Q) n P (Q) 2n n P (Q) BB A B AB

2

AA B

2

 A  

(20)

P (Q) F (Q)F (Q) AB  A B

.

Consider now an A-B-C triblock copolymer with nA-nB-nC segments. The form factor 

involves many terms:

P(Q) n P (Q) n P (Q) n P (Q) CC

2

BB C

2

AA B

2

 A  

(21)

2n n P (Q) 2n n P (Q) 2n n P (Q)  A B AB  B C BC  A C AC P (Q) F (Q)F (Q) AB  A B

P (Q) F (Q)F (Q) BC 

B C

P (Q) F (Q)E (Q)F (Q) AC  A B C

.

P(Q) F(Q) E(Q)

i

j

1

j

1

N
297

The scattering lengths have been omitted for convenience. They have all been assumed to be 

equal. In order to calculate the SANS scattering cross section, one would have to include the 

contrast factors, the segment volumes, the polymer macromolecules number densities, and 

the inter-polymer structure factors. 

Other more complex architectures can be handled this way. 

Figure 7: Various possible polymer architectures exist. 

4. STAR POLYMER ARCHITECTURE

The simplest case of polymer chain branching is the star polymer which is considered here. 

Figure 8: Representation of a star polymer with 5 branches. 

The form factor for a star polymer containing nb branches and n statistical segments per 

branch is given by:

Star Polymer Comb Polymer Dendrimer
298 n n P(n) n (n 1)n F (n)

n n

1

P(Q)

2 2

b b

2

b 2 2

b

   . (22)

P(n) is the form factor for a chain with n segments (Debye function) and F(n) is the form 

factor amplitude. Consider the following relationship (identity):

(2n) P(2n) 2n P(n) 2n F (n)

2 2 2 2

  . (23)

Therefore:

2P(2n) (n 2)F (n)

n

1

P(Q)

2

b

b

   . (24)

This is the result for the form factor for a Gaussian polymer star. More complicated 

architectures (comb, dendrimers, arborescent structures, etc) can be handled this way. 

5. POLYMER RINGS

The form factor for a polymer ring can be calculated using a multivariate Gaussian 

distribution approach. For a Gaussian polymer ring, P(Q) can be calculated as follows:

 













  

 

n

i,j

2

ij

2

2

6

Q r

exp

n

1

P(Q) . (25)

In order to evaluate <rij

2>, construct the ring from a linear chain which is then closed.

Figure 9: A polymer ring can be constructed by closing a linear chain. 

A bivariate Gaussian distribution is defined as:



































 

 

   

2

,

2

3

1 2 2

r .D .r

2a

3

exp

1

2 a

3

P(r ,r )

   

. (26)

2

r



1 ij r r

 



i

j
299

Here 

1 ij r r

 

 ,  is the determinant of the correlation matrix C , D

is the inverse (

1 D C



) 

and the 4 elements of 

C

are given by: 

2 C  r .r  / a    

 

with {=1,2}. The ring closing 

step is formed by setting 

r

2  0



. This leaves a univariate Gaussian distribution:





dr P(r ,0)

P(r ,0)

P(r )

1 1

1

1  





(27)













 















2

2 1 1 1

2

3

1 1

2

3

2 D r

2a

3

D exp

2 a

3 

.

The average mean square distance between 2 monomers i and j that belong to the blocks of 

length n is therefore given by:

11

2

2

ij

D

1

a

r



 

. (28)

More specifically, in this case:

(j i)

a

r

C

2

2

1

11  

 



(29)

n

a

r

C

2

2

2

22 

 

 .

So that:

| i j| n | i j|

n

D11

  



(30)











 

   

n

| i j|

r a | i j| 1

2 2

ij

.

The form factor for the polymer ring is therefore:

 





















 





 

n

i,j

2 2

2

n

| i j|

1

6

Q a | i j|

exp

n

1

P(Q)

(31)





































  











   



n

k 1

2 2

2

n

k

1

6

Q a k

exp

n

k

n 2n 1

n

1

.
300

The first term is dropped for n >>1. In order to simplify this equation, we take the continuous 

chain limit (whereby Q2a

2

/6<<1 and n >>1 but keeping Q2a

2

n/6 finite) and change the 

summations into integrations: 

   











    s 1 s

6

Q a n

P(Q) 2 ds 1 s exp

1 2 2

0

. (32)

We notice the following identity:

   











   











    s 1 s

6

Q a n

s 1 s ds exp

6

Q a n

2 ds(1 s)exp

1 2 2

0

1 2 2

0

. (33)

Therefore:

 











   s 1 s

6

Q a n

P(Q) ds exp

1 2 2

0

. (34)

After integration variable changes and a few manipulations, one obtains the final result:

U

D(U)

P(Q)  . (35)

Here D(U) is Dawson’s integral:

    

 

U

0

2 2 D(U) exp U dt exp t . (36)

The variable U is given by

U Q a n 6 2 QRg 2

2 2

  . 

The method described here for a single ring can be generalized to calculate more complex 

structures containing looping features. 

6. MORE COMPLEX RING-CONTAINING ARCHITECTURES

Another case involving correlations between 2 blocks (n monomers each) separated by 3 

linear chain portions (n1, n2and n3 monomers respectively) that are joined at the extremities 

of the 2 blocks is considered here. This structure can be constructed using a long linear chain 

(with 2n+n1+n2+n3 monomers) that includes 2 crosslinks (corresponding to 

r

2  0



and 

r

3  0



). All segment lengths are assumed to be equal to a for simplicity. 
301

Figure 10: Correlations between two (outer) blocks for a particular polymer chain 

architecture.

A trivariate Gaussian distribution describing this structure is given by:



































 

 

   

3

,

2

2

3

2

9

1 2 3 2

r .D .r

2a

3

exp

1

2 a

3

P(r ,r ,r )

    

. (37)

Here also 

1 ij r r

 



and

C

has 9 elements. The two crosslinks are formed by setting 

r

2  r3  0

 

leading to 

P(r )

1



. 

In this case:

C11 = (n-i+j+n1+n2+n3) (38)

C12 = C21 = (n2+n3)

C13 = C31 = (n1+n2)

C22 = (n2+n3)

C23 = C32 = n2

C33 = (n1+n2).

Therefore:

  













 

     

 

1 2 1 3 2 3

2 2 1 2 1 3 2 3 1 2 3

ij

n n n n n n

i j n n n n n n n n n n

r a . (39)

The partial form factor describing correlations between the two outer blocks is given by:

3

r



2

r



1 ij r r

 



i

j

1

1 n

2 n 3 n

n

n
302

 













  

 

n

i,j

2

ij

2

2

6

Q r

exp

n

1

P(Q)

(40)

which can be written simply as:

  

 

2 2 2

2 2 2

1 2 3

2 2

Q a n / 6

1 exp Q a n / 6

n

1

n

1

n

1

6

Q a

P(Q) exp

 





























    . (41)

In summary, this method consists in forming the correlation diagram using one single chain 

and choosing judiciously the location of crosslinks. All elements of the correlation matrix 

C

need to be calculated so that the first element (recall that 

1 ij r r

 



) of its inverse, D11=11/

(where 11 is the cofactor of element C11 and  is the determinant of 

C

) is obtained therefore 

yielding <rij

2

>/a2=/11. This procedure is useful for the calculation of correlations needed in 

the modeling of more complicated architectures ("olympic rings", regular networks, etc).

7. FORM FACTORS FOR BRANCHED POLYMERS WITH EXCLUDED VOLUME

An assortment of form factors branched polymers that incorporate excluded volume 

following the Flory approach have been compiled (Hammouda, 2016). These include the 

form factors for star-branched polymers with linear branches or with looping branches, the 

form factor for ring polymers and for dendrimers. The effect of chain swelling excluded 

volume is incorporated through an excluded volume parameter. Analytical results are 

possible in the Gaussian chain case, but the calculations are left in a form involving 

summations over monomers in the general case; these summations are performed 

numerically. 

REFERENCES

P. Debye, J. Phys. Colloid. Chem. 51, 18 (1947)

W. Burchard, "Static and Dynamic Light Scattering from Branched Polymers and 

Biopolymers", Advances in Polymer Science 48, 1 (1983)

B. Hammouda, "SANS from Homogeneous Polymer Mixtures: A Unified Overview", 

Advances in Polymer Science U106U, 87 (1993).

B. Hammouda, “Form Factors for Branched Polymers with Excluded Volume”, NIST 

Journal of Research NIST, 121, 139-165 (2016). 

J.S. Higgins and H. Benoit, "Polymers and Neutron Scattering", Oxford (1994).
303

QUESTIONS

1. What is the form factor for a Gaussian polymer coil of radius of gyration Rg? 

2. Calculate the form factor PAC(Q) between the two outer blocks for a triblock copolymer AB-C. 

3. What is the form factor for a Gaussian polymer ring?

4. Calculate the radius of gyration for a Gaussian ring polymer.

ANSWERS

1. The form factor for a Gaussian polymer coil is given by the Debye function 

   

4

g

2 4

g

2 2

g

2

P(Q)  2 exp Q R 1 Q R /Q R

where Rg is the radius of gyration. 

2. The form factor PAC(Q) between the two outer blocks for a triblock copolymer A-B-C is 

given by 

     

2

g C

2

2

g C

2

2

g B

2

2

g B

2

2

g A

2

2

g A

2

AC Q R

1 exp Q R

Q R

exp Q R

Q R

1 exp Q R

P (Q)

    



where Rg’s are the 

radii of gyration of the blocks. 

3. The form factor for a Gaussian polymer ring is given by 

      

 

U

0

2 2

P(Q) exp U U dt exp t

where U = Rg/2. 

4. The radius of gyration squared for a Gaussian ring polymer is given by: 

6

na

2

1

12

na

4

s

3

s

2

2

s

R na dss(1 s) na ds(s 2s s ) na

1 2 2

0

1

0

2 3 4

2 2 3 2

1

0

2 2 2

g   











         

. 

Recall that for a linear polymer 

6

na R

2

2

g



. Here n is the degree of polymerization and a is 

the statistical segment length. 
304

Chapter 29 - EFFECT OF POLYDISPERSITY

Most scattering systems are characterized by a size distribution regardless of their shapes and 

sizes. Polydispersity is discussed briefly here by introducing the main polydispersity size 

distributions used in the literature and presenting averages over these distributions for a few 

quantities relevant to SANS measurements. 

1. SCATTERING FROM A MONODISPERSE SYSTEM 

Consider the following scattering cross section for an infinitely dilute solution of 

monodisperse particles (think spheres) containing N particles of radius R.

R P(QR)

3

4

V

N

V P(QR)

V

N

d

d (Q) 6

2

2 2

P

2











 













 

















(1a)

R P(QR)

3

4

V P(QR)

V

2 2 3

P

2

P











 

  



 . (1b)

 is the particles’ volume fraction (

  NVP V

), 

2

is the contrast factor, V is the sample 

volume, VP is the particle volume and 

2

P(Q) | F(Q) |

is the form factor for the scattering 

particles.

The forward scattering cross section is obtained for Q = 0; i.e., for P(QR) = 1. The radius of 

gyration (also called Guinier radius) is given by the low-Q expansion:

5

Q R

1

3

Q R

P(QR) 1

2 2

2

g

2

    . (2)

2. EFFECT OF POLYDISPERSITY

Consider polydisperse size particles with distribution f(R) which is normalized as 

dRf(R) 1

0

 



. The polydispersity averaged cross section can have one of two forms: 

dRf(R)R P(QR)

3

4

V

N

d

d (Q) 6

0

2

2















 































(3a)

dRf(R)R P(QR)

3

4

d

d (Q) 3

0

2















 

 

















. (3b)
305

These two forms apply depending on whether the number density (N/V) or the volume 

fraction  is independent of polydispersity. 

The first formula is used in the literature (Kotlarchyk-Chen, 1983; Walter et al, 1985). 

The low-Q Guinier expansion for 

P(QR) 1 Q R 5

2 2

  yields either one of the two 

following cases:

    











 









 













 















 























5

Q R

dRf(R)R 1

3

4

V

N

1 Q R 3

d

d (Q 0)

d

d (Q)

2 2

6

0

2

2 2

g

2

(4a)

 

6

2

2 R

3

4

V

N

d

d (Q 0)











 





























 

 

 

 

6

8

2

g

R

R

5

3

R 

    











 









 

  















 























5

Q R

dRf(R)R 1

3

4

1 Q R 3

d

d (Q 0)

d

d (Q)

2 2

3

0

2 2

g

2

(4b)

 

2 3 R

3

4

d

d (Q 0)











 

 















 

 

 

 

3

5

2

g

R

R

5

3

R  .

Note that these results do not apply only to spherical particles but to other arbitrary shapes. 

3. THE GAUSSIAN POLYDISPERSITY DISTRIBUTION

The Gaussian polydispersity distribution is given by:

f(R) = 

σ 2π

1

exp











 

 2

2

av

2σ

(R R )

. (5)

 is the standard deviation and Rav is the average radius (Rav=<R>). Their ratio is defined as 

p = /Rav. 

The Nth moment is given by:
306 R  R (1 A p 3A p 3*5A p )

6

6

4

4

2

2

N

av

N

    

(6)

The following coefficients have been used:

A2 = 

2

N(N 1)

(7)

A4 = 

4!

N(N 1)(N - 2)(N - 3)

= 

(N - 4)!4!

N!

A6 = 

(N - 6)!4!

N!

.

N! represents “N factorial” given by N! = N(N-1)(N-2)...3.2.1. The A2k are the even 

coefficients of the Nth-nomial expansion:

(1+R)N = 1 + A1R + A2R

2 +... ANRN

(8)

Ak = 

k!

N(N 1)(N - 2)...(N - k 1)

= 

(N - k)!k!

N!

.

The average scattering particle volume is given by:

     

3 2

av

3

P R 1 3p

3

4

R

3

4

V 







 . (9)

The low Q (Guinier) expansion involves the following averages:

   

3 2

av

3 R  R 1 3p

(10)

   

5 2 4

av

5 R  R 110p 15p

   

6 2 4 6

av

6 R  R 115p  45p 15p

   

8 2 4 6 8

av

8 R  R 1 28p  210p  420p 105p .

4. THE LOG-NORMAL POLYDISPERSITY DISTRIBUTION

The log-normal polydispersity distribution is identical to the Gaussian distribution but with 

the variable ln(R) instead or R. It is sometime used to describe bicontinuous (non-particulate) 

structures and is given by:
307

f(R) = 

σR 2π

1

exp











 

 2

2

2σ

(ln(R) μ)

. (11)

 = ln(Rmed) and Rmed is the median radius. The various relevant averages are as follows:

  













 

  

2

N

R exp N

2 2

N

(12)

  











   

2

2

1

R exp     











  









3 2

2

9

exp 3

3

4

R

3

4

V

 

2

peak R  exp    .

The low Q (Guinier) expansion involves the following averages:

  











   

3 2

2

9

R exp 3

(13)

  











   

5 2

2

25 R exp 5

  











   

6 2

2

36 R exp 6

  











   

8 2

2

64 R exp 8 .

5. THE SCHULZ POLYDISPERSITY DISTRIBUTION

The Schulz polydispersity distribution (Schulz, 1939) was introduced to describe the

molecular weight distribution of synthetic polymers. It is given by:

 

R Γ(z 1)

exp[ (z 1)x] f(R) z 1 x

av

z 1 z



 

 



for z > -1 (14)

Rav is the average radius and x is the scaled variable 

x  R Rav

, p is given by the ratio of the 

standard deviation  to the average radius as 

p   Rav

and 

z 1 p 1

2

  is the width 
308

parameter. (z+1) is the Gamma function. Note that (z+1) = z(z) = z! (factorial z). The 

Schulz distribution tends to the Gaussian distribution at large z (i.e., for the highly 

monodisperse case). 

The various relevant averages are calculated through the Nth moment:

  





0

N N R dRR f(R)

(15)

 

R Γ(z 1)

exp[ (z 1)R R ]

R

R

dRR z 1

av

av

z

av

z 1

0

N



 

















 







 

R Γ(z 1)

exp[ (z 1)R R ]

R

R

dR z 1

av

av

z

av

z N

z 1

0



 

 









.

Integrate by parts once:

 

 

 

 

R Γ(z 1)

exp[ (z 1)R R ]

R

R

R dR z 1

z 1

z N

R

av

av

z

av

z N 1

z 1

0

av

N



 









 







. (16)

It is noted that 

R.exp (z 1)R Rav 0 0   



. Integrate by parts N times to obtain the final 

result:

 

 

 

R Γ(z 1)

exp[ (z 1)R R ]

R

R

R dR z 1

z 1

(z N)! R

av

av

z

av

z N 1

z 1

0

N av

N



 









 







(17)

 

z!

(z N)!

(z 1)

R

R N

N

N av 



 .

This can be used to obtain:

    2

3

av

3

(z 1)

(z 3)(z 2) R

3

4

R

3

4

V



  





 .

The low Q (Guinier) expansion involves the following averages:

  2

3

av

3

(z 1)

(z 3)(z 2) R R



 



(18)
309  

(z 1) z!

(z 5)! R R 5

5

av

5







 

(z 1) z!

(z 6)! R R 6

6

av

6







 

(z 1) z!

(z 8)! R R 8

8

av

8





 .

These averages are used to calculate 















 

d

d (Q 0)

and 

 

2

Rg

. 

A figure compares the Gaussian and the Schulz polydispersity distributions. 

-0.01

0

0.01

0.02

0.03

0.04

0.05

-20 0 20 40 60 80 100 120

polydispersity distributions 

Gaussian

Schulz

Polydispersity Distributions

R

Figure 1: Comparison of the Gaussian and Schulz polydispersity distributions for Rav = 50 Å 

and  = 10 Å. The Schulz distribution is skewed toward larger sizes (it crosses over the 

Gaussian distribution for R around 70 Å). 
310

REFERENCES

G.V. Schulz, Z. Phys. Chem. 43, 25-46 (1939).

M. Kotlarchyk and S.H. Chen, J. Chem. Phys. 79, 2461-2459 (1983).

G. Walter, R. Kranold, T. Gerber, J. Baldrian and M. Steinhart, “Particle-Size Distribution 

from Small-Angle X-Ray-Scattering Data”, J. of Appl. Cryst. 18, 205-213 (1985)

QUESTIONS

1. What is the effect of polydispersity on SANS data?

2. Which of the polydispersity distributions is used for polymers?

3. Calculate <R> using any distribution. 

4. Calculate <R3> using the Schulz distribution.

5. How to calculate the polydispersity averaged scattering cross section for a dilute system?

ANSWERS

1. Polydispersity has the same effect as instrumental smearing. It tends to broaden peaks and 

fill in valleys. 

2. The Schulz distribution is the best choice for describing polydispersity effects in polymers. 

It yields compact analytical results for characteristic chain properties (radius of gyration and 

form factor). 

3. <R> = Rav by definition regardless of the distribution used. 

4. For the Schulz distribution 

2

3

av

3

(z 1)

(z 3)(z 2) R R



 

 

. In order to demonstrate this, 

integrate by parts three times. 

5. The polydispersity averaged scattering cross section for a dilute system is calculated 

through the following integration: 

dRf(R)R P(QR)

3

4

V

N

d

d (Q) 6

0

2

2















 































.
311

Chapter 30 - SCATTERING FROM DILUTE

POLYDISPERSE SYSTEMS

As the next step towards calculating the scattering cross section for realistic systems, we 

introduce the case of non-interacting systems. A good example for this could be a noninteracting (think infinitely dilute) solution of spheres or a non-interacting solution of 

polymer coils. Consider the simple monodisperse cases first then introduce the effect of 

polydispersity. 

1. INIFINITELY DILUTE SOLUTION OF MONODISPERSE SPHERES

The macroscopic scattering cross section for a solution of non-interacting monodisperse 

spheres (of radius R) contains the spheres number density 

N V

, the contract factor 

2

, the 

particle volume squared 

2

3 R

3

4











 

and the single-particle form factor P(QR) given by: 

P(QR) = 

2

1

(QR)

3j (QR)













= 

2

3

(QR)

3[sin(QR) (QR) cos(QR)











 

. (1)

Putting these terms together, one obtains:





d

d (Q)

= 

2

V

N













2

3 R

3

4











 

2

1

(QR)

3j (QR)













. (2)

Note that this cross section has units of cm-1and that the single-particle form factor is 

normalized to unity, i.e., P(QR=0) = 1. This result is for non-interacting spheres only (in the 

so-called infinite dilute limit). The effect of polydispersity is included next. 

2. INFINITELY DILUTE SOLUTION OF POLYDISPERSE SPHERES

Consider a solution of non-interacting polydisperse spheres (or radius R) and include 

polydispersity to the sphere size by choosing the Schulz distribution for the sphere radius 

with Rav as the mean radius and  as the standard deviation. Defining the polydispersity 

parameter z = 

1

Rav 



, the macroscopic cross section becomes:





d

d (Q)

= 

A 2

V

N













2

0

1

2

3

QR

3j (QR)

R

3

4π

dRf(R)





























(3)
312

Where f(R) is taken to be the Schulz distribution:

f(R) = (z+1)z+1

z

Rav

R





























 

av  Rav

R

exp (z 1)

R Γ(z 1)

1

for z > -1 (4)

Therefore:





d

d (Q)

= 

A 2

V

N













2

3

4











 

3

2

*

















6 Q

1

 







 











 

 



















0

2

av av

z

av

z 1

sin(QR) (QR)cos(QR)

R

R

exp (z 1)

R (z 1)

1

R

R

dR(z 1) .

(5)

This long formula could not fit in one line. The integral in the second line involves Laplace 

transforms of trigonometric functions multiplied by powers of the variable R. The integration 

steps are too tedious to report here. The final result is (Bartlett-Ottewill, 1992):





d

d (Q)

=

2

V

N













2

3

4











 

3

2

*

G(2QR) cos[(z 1)F(2QR)]

2

1

(QR)

(z 1)

(z 2)

2

1

2

1

2

(z 1)

2

 















(QR)G(2QR) 2 sin[(z 2)F(2QR)]

(z 2)

 



  















G(2QR) cos[(z 3)F(2QR)]

(z 1)

(z 2)

(QR)

2

1

2

(z 3)

2

. (6)

The following functions have been defined:

G(2QR) = 

2 2

2

(z 1) (2QR)

(z 1)

 



(7)

F(2QR) = 





















(z 1)

2QR

tan 1

Tan-1is the inverse trigonometric function sometime referred to as Arctan. This result is for a 

solution of non-interacting polydisperse spheres. 
313

3. DILUTE SOLUTION OF NON-INTERACTING MONODISPERSE POLYMER 

COILS

The macroscopic scattering cross section for a solution of non-interacting polymer coils (of 

radius of gyration Rg) contains the coils number density 

N V

, the contract factor 

2

, the 

coils volume squared 

 

2

nv

where n is the degree of polymerization (number of monomers 

per coil) and the single-particle form factor 

P(QRg) = 















   

2 2

g

2

2

g

2 2

g

2

(Q R )

exp[ Q R ] 1 Q R

2 . (8)

P(QRg) is also referred to as the Debye function. Note that in this case, the single-coil form 

factor cannot be written as a square like in the spheres case. Putting all terms together gives:















   



















2 2

g

2

2

g

2 2

g

2

2 2

(Q R )

exp[ Q R ] 1 Q R

(nv) 2

V

N

d

d (Q)

. (9)

Defining the polymer volume fraction as  = Nnv/V, this result can also be written as:

n vP(Q)

d

d (Q) 2

  





. (10)

This non-interacting coils’ result applies to very dilute solutions only. Concentration effects 

are not included here at all. Instead, polydispersity effects are discussed next. 

4. DILUTE SOLUTION OF NON-INTERTACTING POLYDISPERSE COILS

Consider polydisperse coils that follow the Schulz distribution f(n) with average number of 

monomers per coil (degree of polymerization) n = {n} and standard deviation  = 

   

2 2

n  n

. The polydispersity variable z is related to  and to the degree of polydispersity 

 

 

2

2

n

n

as:

 

 

2

2 2

n

n

1

z 1 n

z 2

  









 







. (11)

The scattering cross section for a solution of non-interacting coils is given by:
314





d

d (Q)

= 

2

V

N













v

2

2

0

dnf(n)n 

















   

2

( n)

exp[ n] 1 n

2 . (12)

Another scattering variable is introduced as  = Q2a

2

/6 (a being the statistical segment 

length). The polydispersity integral can be performed to obtain the cross section for 

polydisperse coils (in the infinite dilution limit):





d

d (Q)

= 

2

V

N













v

2

.

















   

















 



 

1 n

(z 1)

n

1

2

(z 1)

2

(13)

n is related to the number average molecular weight as:

Mn = m{n} = mn (14)

m is the molar mass of one monomer. The weight average molecular weight is defined as:

Mw = m

 

n

n

2

= mn

z 1

z 2





. (15)

The polydispersity index is defined as the ratio:

n

w

M

M = 

 

 

2

2

n

n

=

z 1

z 2





. (16)

The results derived in this section apply to polydisperse non-interacting coils (AragonPecora, 1976; Higgins-Benoit, 1996).

REFERENCES

S.R. Aragon and R. Pecora, “Theory of Dynamic Light Scattering from Polydisperse

Systems”, J. Chem. Phys. 64, 2395-2404 (1976). 

P. Bartlett and R.H. Ottewill, “A Neutron Scattering Study of the Structure of a Bimodal 

Colloidal Crystal”, J. Chem. Phys. 96, 3306-3318 (1992).

J.S. Higgins, and H.C. Benoit, “Polymers and Neutron Scattering”, Oxford Science 

Publications (1996). 
315

QUESTIONS

1. Write down the scattering cross section for a dilute polydisperse solution of spheres. There 

is no need performing the integral. 

2. Write down the scattering cross section for a dilute polydisperse solution of polymer coils 

using the Schulz distribution. 

3. Write down the degree of polydispersity for polymers using the Schulz distribution. 

ANSWERS

1. The scattering cross section for a dilute polydisperse solution of spheres is given by the 

following integral 





d

d (Q)

= 

A 2

V

N













2

0

1

2

3

QR

3j (QR)

R

3

4π

dRf(R)





























. 

2. The scattering cross section for a dilute polydisperse solution of polymer coils is given by 

the following expression 





d

d (Q)

= 

2

V

N













v

2

.

















   

















 



 

1 n

(z 1)

n

1

2

(z 1)

2

where  = 

Q

2

a

2

/6 and standard notation has been used. 

3. Using the Schulz distribution, the degree of polydispersity is given by 

n

w

M

M = 

 

 

2

2

n

n

. 
316

Chapter 31 – STRUCTURE FACTORS FOR POLYMER SYSTEMS

Up to now, this effort has focused on infinitely dilute systems only. Such systems are noninteracting and require solely the calculation of the form factor P(Q) for isolated particles. 

More concentrated (or interacting) systems require the calculation of the structure factor 

S(Q). Structure factors for fully interacting polymer systems are considered here. These 

apply to semi-dilute and concentrated polymer solutions and polymer blend mixtures in the 

homogeneous phase. 

1. SCATTERING FROM INCOMPRESSIBLE SYSTEMS

Consider a system consisting of N “particles” of scattering length bP occupying the sample 

volume V. The following would still hold if the word “polymers” were substituted for the 

word “particles”. The scattering cross section is proportional to the density-density 

correlation function as follows:

       





n ( Q)n (Q)

V

1

exp iQ.r b

V

1

b

d

d (Q)

P P

2

P

N

i,j

ij

2

P





. (1)

Here nP(Q) is the fluctuating particle density in Fourier space. The cross section for particles 

in solution is given by:

       





n ( Q)n (Q)

V

1

n ( Q)n (Q) b

V

1

b

d

d (Q)

S S

2

P P S

2

P

  n (Q)n (Q) 

V

1

2bPbS P S

. (2)

The subscripts P and S stand for particle and solvent respectively. For the sake of 

convenience, the following scattering factors are defined: 

  n (Q)n (Q) 

V

v

S (Q) P P

2

P

PP

(3)

  n (Q)n (Q) 

V

v

S (Q) S S

2

S

SS

  n (Q)n (Q) 

V

v v

S (Q)

P S

P S

PS

.
317

The volumes vP and vS and scattering length densities 

P bP vP

and 

S  bS vS

are 

defined for the polymer and the solvent respectively. To clarify, vP is the monomer volume 

and vS is the volume of the solvent molecule. The scattering cross section becomes:

S (Q) S (Q) 2 S (Q)

d

d (Q)

SS P S PS

2

PP S

2

 P     





. (4)

Most scattering systems are incompressible. It is often convenient to make the following 

incompressibility assumption:

vPn P(Q)  vSnS(Q)  0. (5)

This introduces the following simplification:

v  n (Q)n (Q)   v  nS(Q)nS(Q) 

2

P P S

2

P

(6)

 vP vS  n P(Q)nS(Q) .

In other words:

S (Q) S (Q) S (Q) S (Q)

PP

 SS   PS   SP

(7)

This simplifies the cross section to the following form:

( ) S (Q) S (Q)

d

d (Q)

PP

2

PP

2

 P  S  





. (8)

This is reasonable since the contrast factor 

2

is always calculated relative to a 

“background” scattering length density value. Here, the solvent’s scattering length density is 

taken to be that reference value. 

2. INTER-PARTICLE INTERACTIONS

Consider a system consisting of N polymers of contrast factor 

2

occupying volume V. 

Each polymer comprises n monomers of volume v each so that the polymer volume is vP = 

nv. Let us separate out the intra-polymer and the inter-polymer terms in the scattering cross 

section as follows:

    











          





 

   

N n

i,j

N n

i,j

i j i j

2

2

exp iQ.r exp iQ.r

V

v

d

d (Q) 

 

. (9)
318

The indices  and  run over the polymer chains and the indices i and j run over the 

monomers in a specific polymer chain. Consider a pair of polymer coils (called 1 and 2) and 

sum over all pairs.

    











         



 n

i,j

1i2 j

n

i,j

1i1j

2

2 N exp iQ.r N(N 1) exp iQ.r

V

v

d

d (Q) 

 

. (10)

Note that this formalism holds if the word “particles” were to be substituted for the word 

“polymers” assuming (of course) that the particles have internal structure (think monomers). 

Figure 1: Schematic representation of the coordinate system showing a pair of scatterers that 

belong to two different polymer coils.

The inter-distance between the scattering pair 

1i2 j

r



can be expressed as 

1i2 j S1i S2 j R12 r



  

   

and the inter-particle average can be split into the following parts: 

 exp iQ.r

1i2j    expiQ.S1i  exp iQ.S2j  exp iQ.R12  





     

. (11)

The first two averages are within single particles and the third average is across particles. The 

summations become:

              

n

j

2 j

n

i

12 1i

n

i,j

exp iQ.S1i2 jexp iQ.R exp iQ.S exp iQ.S

       

.(12)

The form factor amplitude is defined as:

2 j

r



R1



1i2 j

r



coil 

coil 

S1i



S2 j



1i

r



R2



R12


319

         

n

j

2 j

n

i

1i

exp iQ.S

n

1

exp iQ.S

n

1

F(Q)

   

. (13)

The single-particle form factor itself is defined as:

    

n

i,j

2

S1i1jexp iQ.

n

1

P(Q)

 

. (14)

For uniform density particles, the following relation holds 

2

P(Q) | F(Q) |

. This is not true, 

however, for non-uniform density objects such as polymer coils. 

An inter-particle structure factor is defined as:

    

 



N

,

I

exp iQ.R

N

1

S (Q)

 

. (15)

The cross section can therefore be written as follows:

P(Q) | F(Q) | S (Q) 1

V

v n N

d

d (Q)

I

2

2 2

2

   





. (16)

Note that the statistical average 

 expiQ.r

1i2 j 





involves integration over the following 

probability distribution 

P(r ,r ,R )

1i 2 j 12

 



which can be split to show a conditional probability 

P(r ,r ,R ) P(r ,r | R )P(R )

1i 2 j 12 1i 2 j 12 12

 



 

 



. For compact scatterers which do not interfere with 

each other’s rotation 

P(r ,r | R )

1i 2 j 12

 



is independent of 

R12



. P(R )

12



is the probability of 

finding the centers of mass of polymer coils 1 and 2 a distance 

R12



apart. 

    I 12 12 12 R12 dR P(R )exp iQ.

V

N

S (Q) N exp iQ.R

     

      . (17)

The cross section for systems in this case is given by:

 











   





S (Q) 1

P(Q)

| F(Q) |

P(Q) 1

V

v n N

d

d (Q)

I

2 2 2

2

. (18)

This result applies to systems with non-spherical symmetry and non-uniform density such as 

polymers. Polymer are, however, so highly entangled that an inter-chain structure factor 

SI(Q) is meaningless except for dilute solutions whereby polymer coils do not overlap. Interchain interactions for polymer systems are better handled using other methods described 

below. 
320

Uniform density scatterers (such as particles) are characterized by 

2

P(Q) | F(Q) |

, so that:

P(Q)S (Q)

V

v n N

d

d (Q)

I

2 2

2

 





. (19)

Defining a particles’ volume fraction as  = Nnv/V, the following result is obtained:

S(Q)

d

d (Q) 2

 





(20)

S(Q) n vP(Q)S (Q)   I

.

This is a well-known result. It is included here even-though it does not apply to polymer 

systems so that the derivation does not have to be repeated when covering scattering from 

particulate systems later. Note that the scattering factor S(Q) and the inter-particle structure 

factor SI(Q) should not be confused; S(Q) has the dimension of a volume whereas SI(Q) is 

dimensionless. 

3. THE PAIR CORRELATION FUNCTION

Recall the definition for the inter-particle structure factor for a pair of particles (named 1 and 

2):

        

 



N

,

I 12 exp iQ.R

N

1

S (Q) N exp iQ.R

   

(21)

   

 

  

N

,

3

d R exp iQ.R P(R )

NV

1   

.

P(R ) 



is the probability of finding particle  in volume d3R a distance 

R



away given 

that particle  at the origin. When the self term ( = ) is omitted, this result becomes:

   



        

N

3

I d R exp iQ.R P(R )

NV

1

S (Q) 1

   

(22)

  d R exp iQ.R P(R )

V

N

12 12 12

3

   

.

The probability 

P(R )

12



is referred to as the pair correlation function and is often called 

g(R )

12



. Removing the forward scattering term yields the following well known result:

d R exp iQ.R g(R ) 1 (2 ) (Q)

V

N

S (Q) 1

3

12 12 12

3

I        

  

. (23)
321

The last term (containing the Dirac Delta function) is irrelevant and can be neglected. This 

last equation shows that 

SI(Q) 1

and 

g(R12 ) 1



are a Fourier transform pair. Note that 

g(R )

12



peaks at the first nearest-neighbor shell and goes asymptotically to unity at large 

distances. The total correlation function is introduced as 

h(R12 )  g(R12 ) 1

 

. 

4. POLYMER SOLUTIONS

In the case of polymer solutions, the Zimm single-contact approximation (Zimm, 1946; 

Zimm, 1948) is a simple way of expressing the inter-polymer structure factor. Within that 

approximation, the first order term in a “concentration” expansion is as follows:

  n P(Q) ...

V

v

V

v

exp iQ.r

2

2

2

e x

n

i,j

1i2 j 

















    





(24)

vex is a dimensionless factor representing interactions. The cross section becomes an 

expansion:

 













   





S (Q) ...

V

v

S (Q)

d

d (Q) 2

0

ex

0

2

. (25)

This expansion can be resumed as follows 

1 x x ... 1 1 x

2

   

to yield:

S (Q)

V

v

1

S (Q)

d

d (Q)

0

ex

2 0



 





. (26)

The bare structure factor for non-interacting polymers has been defined as:

P(Q) n vP(Q)

V

Nn v

S (Q)

2 2

0    . (27)

Resuming the series extends the single-contact approximation’s applicability range to a wide 

concentration regime. The single-contact approximation applies best to semi-dilute solutions. 
322

Figure 2: Typical interactions that are included and those that are excluded within the singlecontact approximation. 

5. THE ZERO CONTRAST METHOD

The zero contrast (or scattering length density match) method also called the high 

concentration method for polymer systems consists of using a mixture of deuterated and nondeuterated polymers and deuterated and non-deuterated solvents in order to isolate the singlechain form factor; i.e., in order to cancel out the inter-chain interaction terms. The scattering 

cross section for a polymer solution containing both deuterated and non-deuterated polymers 

is given by:

S (Q) S (Q) 2 S (Q)

d

d (Q)

HH D H HD

2

DD H

2

 D     





. (28)

The scattering length density differences between the deuterated (or hydrogenated) polymer 

and the solvent are:

Excluded Interactions 

Included Interactions
323

 

















      

S

S

D

D

D D S

v

b

v

b

(29)

 

















      

S

S

H

H

H H S

v

b

v

b

.

The various partial scattering factors are split into single-chain parts and inter-chain parts as 

follows:

S (Q) n v P (Q) P DD (Q)

I

D D

S

DD



DD D  

(30)

S (Q) n v P (Q) P HH (Q)

I

H H

S

HH



HH H  

S (Q) n v n v P HD (Q)

I

HD



HH H DD D HD

.

Note that the inter-chain structure factors could be negative depending on the volume 

fraction. Assume that deuterated and hydrogenated polymers have the same degree of 

polymerization (

n D n H n P

), and the same specific volume (

vD vH vP

), and define 

the polymer volume fraction as 

P D  H

. The contrast match method consists in varying 

the relative deuterated to hydrogenated volume fraction but keeping the total polymer volume 

fraction constant. 

Define the following “average of the square” and “square of the average” polymer contrast 

factors:

  















 





  

P

2 H

H

P

2 D

D

2

BP

(31)

 

2

P

H

H

P

D

D

2

BP 















 





   .

The scattering cross section becomes:

 B n v P (Q)  B  n v P (Q)

d

d (Q)

P P P I

2

P P P S P

2

  P    





 B   B  n v P (Q)  B  n v P (Q)

P P P T

2

P P P S P

2

P

2

  P       . (32)

The following definition has been used:

P (Q) P (Q) P (Q) T S  P I

. (33)

Note the following simplifications:
324      

2

P

2 D H

D H

2

P

2

BP B



 

       . (34)

  S

P

H

H

P

D

D

P

H

H

P

D BP D  





 





 





 





   .

The final result follows:

  n v P (Q) n v P (Q)

d

d (Q)

P P P T

2

S

P

H

H

P

D

2 P P P S D

P

2 D H

D H 

















 





 





  



 

   





. (35)

Setting the second contrast factor (between the polymer and the solvent) to zero cancels out 

the second term containing PT(Q) leaving only the first term containing the single-chain form 

factor PS(Q). This zero contrast condition is therefore:

S

P

H

H

P

D

D

 





 





 . (36)

Note that in general in order to achieve this condition, the solvent must also consist of 

mixtures of deuterated and non-deuterated solvents. Defining the following four indices DP, 

HP, DS, and HS for deuterated polymer, non-deuterated (hydrogenated) polymer, deuterated 

solvent and non-deuterated solvent, the contrast match condition becomes in the general 

case:

S

HS

HS

S

DS

DS

P

HP

HP

P

DP

DP 



 





 





 





 . (37)

Note that DP + HP = P , DS + HS = P and P + S = 1. 

6. THE RANDOM PHASE APPROXIMATION

The random phase approximation (de Gennes, 1979, Akcasu-Tombakoglu, 1990; 

Hammouda, 1993; Higgins-Benoit, 1994) is a simple mean-field approach used to calculate 

the linear response of a homogeneous polymer mixture following a thermodynamic 

fluctuation. Consider a binary mixture consisting of a mixture of polymers 1 and 2 with 

fluctuating densities n1(Q) and n2(Q). The interaction potentials between monomers 1 and 2 

are W11, W12, W21 and W22. Assume an external perturbation represented by potentials U1

and U2 and a constraint u that helps apply the incompressibility assumption. The parameter u 

can be thought of as a Lagrange multiplier in an optimization problem with constraints. The 

constraint here is the incompressibility condition. The linear response equations follow:
325











   

 

k T

U u W v n (Q) W v n (Q)

v n (Q) S (Q)

B

0 1 1 1 1 1 1 2 2 2

1 1 1 1

(38a)











   

 

k T

U u W v n (Q) W v n (Q)

v n (Q) S (Q)

B

0 2 2 1 1 1 2 2 2 2

2 2 2 2

(38b)

v1n1(Q)  v2n2(Q)  0 . (38c)

The last equation represents the incompressibility constraint. The non-interacting or “bare” 

structure factors 

S (Q)

0

11

and 

S (Q)

0

22

have been defined. These equations have assumed that 

no copolymers are present in the homogeneous mixture; i.e., that 

S (Q) S (Q) 0

0

21

0

12   . 

In order to solve the set of linear equations, we extract the perturbing potential u from the 

second equation and substitute it into the first equation to obtain:

















  v (Q)n (Q)

k T

U U

v n (Q) S (Q)

1 1 1

B

0 1 2

1 1 1 1

. (39)

This applies along with the following equation representing the response of the fully 

interacting system:











 

 

k T

U U

v n (Q) S (Q)

B

1 2

1 1 11

. (40)

The factor v11(Q) and the Flory-Huggins interaction parameter 12 are defined as:

0

12

0

22

11

v

2

S (Q)

1

v (Q)



  (41)















 

  

2k T

W W

k T

W

B

11 22

B

12

12

.

Here v0 is a reference volume (often taken to be 

v0  v1v2

). 

The RPA result for a homogeneous binary blend mixture follows:

0

1 2

0

2 2

0

1 1 1 1 v

2

S (Q) S (Q) S (Q)

1 











(42)

S (Q) n v P (Q)

1 1 1 1

0

11   .
326  

2

g1

2

2

g1

2

4

g1

4

1

exp( Q R ) 1 Q R

Q R

2

P (Q)    

Rg1 is the radius of gyration. The incompressibility assumption yields the simplifying 

relations:

S (Q) S (Q) S (Q) S(Q)

11  22   12  . (43)

The scattering cross section is given by:

( ) S(Q)

d

d (Q) 2

 1  2 



(44)

 

0

1 2

0

2 2

0

1 1

2

1 2

v

2

S (Q) S (Q)

d

d (Q)

















  

.

This is the so-called de Gennes formula representing the scattering cross section for polymer 

blends in the single-phase (mixed phase) region. This is based on the Random Phase 

Approximation that applies for long degree of polymerizations (n1>>1 and n2>>2) and far 

from the phase boundary condition. This approach does not apply inside the demixed phase 

region. 

This formalism also applies to polymer solutions by replacing one of the polymers (say 

component 2) by solvent; i.e., by setting n2 = 1 and P2(Q) = 1. In the case of polymer 

solutions, the excluded volume effect is included in the polymer form factor P1(Q). Note that 

the second virial coefficient can be defined for polymer solutions as 

2 1 1 2 2 2 1 2 v0 A  v (Q  0) 2  (1 2n  v )   . 

The phase separation condition is achieved when the scattering intensity “blows up”; i.e., in 

the limit 

S11(Q  0)  

. This is achieved for

S (0)S (0) 0

v

2

S (0) S (0)

0

2 2

0

1 1

0

0 1 2

2 2

0

1 1 



  . (45)

This is the so-called spinodal condition. Note that with the simplifying assumptions that n1 = 

n2 = n, v1 = v2 = v0 and 1 = 2 = 0.5, the spinodal condition for polymer blends simplifies to 

12n  2 . 
327

7. THE ISOTHERMAL COMPRESSIBILITY FACTOR

Most mixed polymer systems have finite compressibility. The scattering cross section 

consists of a Q-dependent coherent scattering term which is a good monitor of the structure, a 

Q-independent incoherent scattering term (mostly from hydrogen scattering), and another Qindependent “isothermal compressibility” term expressed as:

B T

2

iso comp

k T

d

d

   

















. (46)

Here 

2

is the contrast factor, kBT is the temperature in energy units and T is the isothermal 

compressibility which is defined as:

T

T

P

V

V

1

















   . (47)

The isothermal compressibility term is usually small compared to the other terms. For 

example, T = 4.57*10-4cm3/J for pure water at 25 oC and atmospheric pressure (Weast, 

1984). T is set equal to zero altogether for incompressible mixtures. 

REFERENCES

B. Zimm, “Application of the Methods of Molecular Distribution to Solutions of Large 

Molecules”, J. Chem. Phys. 14, 164 (1946); and B. Zimm, “The Scattering of Light and the 

Radial Distribution Function of High Polymer Solutions”, J. Chem. Phys. 16, 1093 (1948). 

P.G. de Gennes, "Scaling Concepts in Polymer Physics", Cornell University Press, New 

York (1979). 

A.Z. Akcasu and M. Tombakoglu, “Dynamics of Copolymer and Homopolymer Mixtures in 

Bulk and in Solution via the Random Phase Approximation”, Macromolecules 23, 607-612 

(1990)

B. Hammouda, "SANS from Homogeneous polymer Mixtures: A Unified Overview", 

Advances in Polymer Science 106, 87 (1993). 

J.S. Higgins and H. Benoit, "Polymers and Neutron Scattering", Oxford (1994). 

R.C. Weast, Editor-in-Chief, “CRC Handbook of Chemistry and Physics”, CRC Press (1984)
328

QUESTIONS 

1. What is the primary effect of the incompressibility assumption on the scattering cross 

section?

2. If an incompressible polymer solution is characterized by one (independent) structure 

factor, how many structure factors describe the equivalent compressible solution?

3. What is the Zimm single-contact approximation?

4. Does the inter-chain structure factor (with excluded volume) for dilute polymer solutions 

tend to increase or decrease the scattering intensity at low-Q?

5. What is the use of the zero contrast condition in concentrated polymer systems? What is 

the procedure to follow?

6. The Random Phase Approximation applies in what conditions?

7. What is the origin of monomer/monomer interactions in polymer mixtures?

8. Are polymer chains in mixed polymer blends characterized by excluded volume; i.e., are 

they swollen?

9. What is the pair correlation function g(r)?

10. Estimate kBTT (T is the isothermal compressibility) for pure water for 25 oC and 1 

atmosphere pressure. 

ANSWERS

1. The primary effect of the incompressibility assumption is to simplify the scattering cross 

section from its full form 

S (Q) S (Q) 2 S (Q)

d

d (Q)

SS P S PS

2

PP S

2

 P     





(where P and S 

represent the polymer and the solvent respectively) to its simplified form 

( ) S (Q) S (Q)

d

d (Q)

PP

2

PP

2

 P  S  





. This is due to the incompressibility condition 

relating the various partial structure factors 

S (Q) S (Q) S (Q) S (Q)

PP

 SS   PS   SP

. 

2. An incompressible polymer solution is characterized by one structure factor SPP(Q). The 

equivalent compressible polymer solution is described by three structure factors: SPP(Q), 

SSS(Q) and SPS(Q). 

3. The Zimm single-contact approximation assumes that inter-chain interactions occur only 

through single contacts or chains of single contacts. Double contacts within the same chain or 

between two different chains or higher order contacts are not included. 

4. The inter-chain structure factor (with excluded volume) for dilute solutions decreases the 

scattering intensity at low-Q. Recall the negative sign in Zimm’s single-contact 

approximation formula: 

 













   





S (Q) ...

V

v

S (Q)

d

d (Q) 2

0

ex

0

2

. 

5. The contrast match method is a way to extract single-chain properties (such as the radius 

of gyration) from concentrated polymer systems. This method consists in using a mixture of 

deuterated and non-deuterated polymers and deuterated and non-deuterated solvents in the 

zero average contrast condition. This involves varying the deuterated to non-deuterated 

polymer fraction but keeping the total polymer fraction constant. 
329

6. The Random Phase Approximation applies for high molecular weight polymers in the 

single-phase (mixed phase) region. It does not apply in the demixed phase region. 

7. Monomers interact with each other and with organic solvent molecules due to Van der 

Waals interactions mostly. Hydrogen bonding dominates in water-soluble polymers.

8. Polymer coils follow random walk statistics in mixed polymer blends. They are not 

swollen like in polymer solutions. Their form factor is the well-known Debye function. 

9. The pair correlation function g(r) is the probability of finding a scatterer at a radial 

distance r from another scatterer at the origin. 

10. kBT = 1.38*10-23 [J.K-1]*295 [K] = 4.112*10-21 [J] and T = 4.57*10-4cm3/J so that 

kBT T = 1.879*10-24 cm3. 
330

Chapter 32 - STRUCTURE FACTORS FOR PARTICULATE SYSTEMS

Scattering factors from nanostructures consisting primarily of particles (think spheres) in a 

background medium (think solvent) are described here in the case of non-dilute systems. The 

Ornstein-Zernike approach is a suitable way to describe inter-particle contributions. 

1. THE ORNSTEIN-ZERNIKE EQUATION

The radial distribution function for a pair of scattering particles with no internal structure 

separated by a distance r is called g(r). It is the probability of finding a scatterer at radial 

distance r provided that there is a scatterer at the origin. g(r) is related to the inter-particle 

interaction potential U(r) as follows:

g(r) = exp[-U(r)/kBT] . (1)

Since the potential of mean-force U(r) contains contributions from many-body interactions, it 

is expanded in terms of binary (wij), ternary (wijk), and higher order interactions:

U(r) = 

  

i,j,k

ijk

i,j

ij w (r) w (r) ... (2)

Note that g(r) is zero for very short distances since two particles cannot occupy the same 

space and is equal to one for large distances since at far enough distance, a particle can be 

located for sure. 

Direct interactions between the pair of interacting particles are represented by the direct 

correlation function c(r) whereas interactions through other particles are represented by the 

total correlation function h(r) = g(r) -1. 

Figure 1: Direct and indirect inter-particle interactions.

The Ornstein-Zernike integral equation (Ornstein-Zernike, 1918; Hansen-McDonald, 1986) 

is a relation between the direct correlation function c(r) and the total correlation function h(r). 

c(r) h(r)
331 

h(r)  c(r)  N dr'c(r  r')h(r')

     

. (3)

N  N V

is the particle number density. 

In the Fourier variable space, this equation reads:

H(Q)  C(Q)  N C(Q) H(Q) . (4)

The inter-particle structure factor is defined as:

1- NC(Q)

1

S (Q) 1 N H(Q) I   . (5)

Note that the Ornstein-Zernike equation contains two unknowns (h(r) and c(r)). It can be 

solved only if another (so called "closure") relation is added. Many of these closure relations 

have been introduced (hypernetted chains, Born-Green, Percus-Yevick, Mean Spherical 

Approximation, etc). Using one such closure relation, numerical solutions of the OrnsteinZernike equation yield realistic inter-particle structure factors. The last two closure relations 

(Percus-Yevick and Mean Spherical Approximation) are discussed here because they permit 

simple analytical solutions to the integral equation. 

Three inter-particle interaction potentials can be considered: hard sphere, screened Coulomb 

and square well. The hard sphere potential is used with the Percus-Yevick closure relation 

and the screened Coulomb potential is used with the Mean Spherical Approximation. 

Figure 2: Representation of the various inter-particle interaction potentials. 

U(r)

r

Hard Sphere

Screened Coulomb

Square Well
332

2. THE PERCUS-YEVICK APPROXIMATION

The Percus-Yevick approximation (Percus-Yevick, 1958) uses the following closure relation 

in order to solve the Ornstein-Zernike integral equation:





























  

k T

w(r)

c(r) g(r) 1 exp

B

. (6)

Assume the following hard sphere interaction potential between particles:

w(r)  0

for 

r  D

(7)

w(r)  

for 

r  D .

Here D is the sphere diameter. Solution to the Ornstein-Zernike equation is analytical:

c(r)  0

for 

r  D

(8)

3

3

1 2 1 D

r

D 2

r

c(r) 6 



     for 

r  D .

The following parameters have been defined:

 

 

4

2

1

1

1 2

 

 

 

(9)

 

 

4

2

2

1

1 2

 

  

  .

 is the particle volume fraction (

ND 6

3

  

), 

N

is the density of scattering particles and 

D is the "effective" particle diameter. 

The Fourier transform of the direct correlation function can be calculated as:

     

 

         

   

















   

  









 

    4

2

1 3 2

QD

QD cos QD 2 QD sin QD 2cos QD 2

6

QD

sin QD QD cos QD NC(Q) 24                  

  





















      

 

6

4 3 2

1

QD

QD cos QD 4 QD sin QD 12 QD cos QD 24 QD sin QD 24cos QD 24

2

(10)

The structure factor for a liquid of structureless particles is given by:
333

1 NC(Q)

1

S (Q)

I





(11)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 5 10 15

Percus Yevick Model

Calculated 

S

I(Q)

QR

Figure 3: Inter-particle structure factor SI(QR) vs QR prediction from the Percus-Yevick 

model (with hard sphere potential) for  = 0.30. Note that the sphere radius is R = D/2.

The scattering cross section involves the product of the form factor and the structure factor:

(Q)

I

V P(Q)S

2

d

d (Q)

   P 



. (12)

Here 

2

is the contrast factor,  is the volume fraction and VP is the particle volume 

(

V 4 R 3

3

P

 

). Note that in this simple “hard sphere” interaction potential model, the sphere 

diameter that enters in the form factor is taken to be the same as the hard sphere radius used 

in the structure factor. 
334

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

Percus Yevick Model 

P(Q)

P(Q)*S

I

(Q) 

Form Factor and Structure Factor 

QR

Figure 4: Form factor P(Q) for isolated spheres (infinite dilution limit), and product 

P(Q)SI(Q) for a solution of spheres with a volume fraction of  = 0.30. The Percus-Yevick 

model (hard sphere potential) has been used to model the inter-particle structure factor SI(Q). 

3. THE MEAN SPHERICAL APPROXIMATION

When Coulomb interactions are present, another closure relation to the OZ equation is 

applied; the Mean Spherical Approximation (Hayter-Penfold, 1981). Consider a scattering 

system consisting of macroions (charged positive), counter ions (charged negative) and 

solvent. The Coulomb interaction potential is defined as:

 

r

exp (r D)

U(r) D

2

0

2

0

  

   

for

r  D . (13)

The macroion surface interaction potential is given by:

2 D

z

0

m

0

   

  . (14)

The following parameters have been defined:
335

0: Permittivity of free vacuum

: Dielectric constant

D: Macroion diameter



: Debye-Huckel inverse screening length

zm: Macroion electric charge (number of electrons).

The Debye-Huckel screening parameter (inverse length) squared is expressed as follows:

z N

k T

e

m

B

2

2

 

(15)

Here e is the electron charge, zme is the macroion charge, 

N

is the macroion number density 

(number per unit volume) and kBT is the sample temperature in absolute units.

Dimensionless parameters are defined:

D

r

x  , k  D, K  QD

and 

k T

1

B

  . (16)

Along with the following contact potential (for r = 2D) a`s:

 

2

k 0 0 exp     . (17)

The Mean Spherical Approximation (MSA) closure relation to the Ornstein-Zernike equation 

is given by:

c(r)  U(r)

for r > D (18)

h(r)  1

for r < D.

Note that the limiting case for which 

  0

or 

k  

yields the Percus-Yevick result. 

The MSA closure is used to solve for c(r):

 

x

F cosh(kx) 1

x

Csinh(kx)

Ax

2

1

c(r) A Bx 3      

for x<1

x

exp( kx)

c(r)



 

for x>1. (19)

The structure factor is obtained as:

1 NC(Q)

1

S (Q)

I





(20)
336

With:

 

3

2

3 K

K

2

1 Kcos(K) 2sin(K)

K

2

B

K

A sin(K) Kcos(K)

24

NC(Q)













  





















3

3 2 2 4

2K

Kcos(K)

K

24

K

12 sin(K) 1

K

6

4 1

K

24 A 























   











  



 

 

2 2 K K k

C k cosh(k)sin(K) Ksinh(k)cos(K)







  

 

2 2 K K k

F k sinh(k)sin(K) K cosh(k)cos(K) 1



 



    

 

2 2 2 K K k

exp k k sin(K) Kcos(K)

K

F cos(K) 1



  





 . (21)

The macroion volume fraction 

ND 6

3

  

has been expressed in terms of the macroion 

number density 

N

. The forward scattering limit is given by SI(0) = -1/A. 

Note that expressions for the constants A, B, C, and F are too lengthy to reproduce here. 

They can be found in the original publication (Hayter-Penfold, 1981). F is the solution of a 

4

th power polynomial equation. 
337

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.05 0.1 0.15 0.2

 D = 40 Å, zm = 20,  = 0.01, T = 25 oC 

Calculated 

Structure Factor S

I(Q)

Scattering Variable Q (Å-1)

Figure 5: Variation of the structure factor SI(Q) obtained from the MSA for a spherical 

macroion diameter of D = 40 Å, macroion charge of zm = 20 electrons, a volume fraction of 

= 0.01 and at T = 25 oC. The dielectric constant  = 78 is for D2O at 25 oC. 

4. THE RANDOM PHASE APPROXIMATION

Consider now particles with internal structure or polymers made out of spherical monomeric 

units. Note that spheres are assumed to fill the particles or replace the monomers in 

polymers. The Random Phase Approximation (RPA) provides another closure relation used 

to solve the OZ equation. The RPA assumes that 

c(r)  w(r) kBT

. Note that within the 

RPA, different notation is used for interaction potentials. By convention, these are called w(r) 

for polymers and U(r) for particulate systems. 

The intra-particle contributions are included in the Ornstein-Zernike equation as follows (in 

Fourier space):

H(Q) S (Q)C(Q)S (Q) S (Q)C(Q)NH(Q)  0 0  0

. (22)

Along with the RPA closure relation:
338

k T

W(Q)

NC(Q)

B

  . (23)

This closure is reminiscent of the MSA closure relation for r > . Note the following 

relations for particles with internal structure (or polymers with spherical monomers):

1-S (Q)NC(Q)

1

S (Q)

0

I



(24)

1-S (Q)NC(Q)

S (Q) S(Q) S (Q)S (Q)

0

0

 0 I

It follows that:

k T

W(Q)

S (Q) S (Q)

B

1

0

1

 

 

. (25)

This is the Random Phase Approximation result obtained for compressible polymer mixtures. 

The scattering cross section is given by:

S(Q)

2

d

d (Q)

 





. (26)

Note that this approach can be extended to the multi-component case by changing the various 

structure factors to matrices. 

The scattering factor for polymer mixtures S(Q) and the structure factor for particulate 

systems SI(Q) are related by the relationship:

S(Q) V P (Q)S (Q)  1 1 1 I

. (27)

Here 1, V1 and P1(Q) are the volume fraction, polymer volume and form factor for polymer 

component 1. Recall that within the incompressible RPA, the structure factor SI(Q) is given 

by: 

S (Q)S (Q)

v

S (Q) S (Q) 2

S (Q)

S (Q)

0

2 2

0

1 1

0

0 1 2

2 2

0

1 1

0

2 2

I



 

 . (28)

The scattering factor S(Q) is therefore proportional to 12 as it should.

Note also that the mean field approximation does not model the local interactions properly 

for inter-particle inter-distances smaller than particle sizes since packing effects on 
339

thermodynamics and phase separation are neglected. For this reason, the g(r) obtained from 

such a mean field approach does not show realistic oscillations for the neighboring 

coordination shells. The appeal of this approach, however, is that it gives simple analytical 

results.

REFERENCES

J.P. Hansen and I.R. McDonald, "Theory of Simple Liquids" Academic Press, London 

(1986)

L.S. Ornstein and F. Zernike, Phys. Z. 19, 134 (1918)

J.K. Percus, G.J. Yevick, “Analysis of Classical Statistical Mechanics by Means of 

Collective Coordinates”, Phys. Rev. 110 1 (1958). 

J.H. Hayter and J. Penfold, “An Analytic Structure Factor for Macroion Solutions”, 

Molecular Physics 42, 109-118 (1981)

QUESTIONS

1. Does a numerical solution to the Ornstein-Zernike integral equation (with a realistic 

closure relation) describe local packing adequately? How about a mean field analytical 

solution (using the mean spherical approximation)?

2. Can the scattering cross section for a concentrated solution of particles (colloidal 

suspension for example) be described as the product of a single-particle and an inter-particle 

structure factors?

3. Name the closure relation that yields an analytical solution to the OZ integral equation in 

the case of hard sphere interaction potential. 

4. What approach gives the most realistic solution to the Ornstein-Zernike equation?

5. What is the Mean Spherical Approximation (so called MSA)? What systems are well 

described by the MSA?

6. Are the Random Phase Approximation (used to describe polymer systems) and the 

Ornstein-Zernike equation (used to describe particulate systems) related at all? 

ANSWERS

1. A numerical solution to the Ornstein-Zernike integral equation along with a realistic 

closure relation describes local packing well. A mean field analytical solution is too 

simplistic and yields correct overall trends but incorrect local packing information. 

2. The scattering cross section for a concentrated solution of particles can be described as the 

product of a single-particle and an inter-particle structure factors provided that the particles 

are not elongated (i.e., are isotropic). 
340

3. The Percus-Yevick closure relation yields a simple analytical solution to the OZ integral 

equation for the hard sphere interaction potential. 

4. A numerical solution to the Ornstein-Zernike equation along with one of the closure 

relations gives more realistic results than highly approximated analytical solutions. 

5. The Mean Spherical Approximation (MSA) is a closure relation used to solve the 

Ornstein-Zernike equation. Charged systems are well described by the MSA since Coulomb 

interactions are included. The MSA yields analytical (albeit lengthy) results. 

6. The Random Phase Approximation (used to describe polymer systems) is a mean-field 

closure relation to the Ornstein-Zernike equation. The RPA closure is a simplified form of 

the MSA closure. 
341

Chapter 33 - SCATTERING FROM FRACTAL SYSTEMS

Consider a system of interacting particles in a medium. The particles could have fractal 

(rough) surfaces or they could form a mass fractal structure through clustering. In general 

terms, the scattering cross section is given by:

V P(Q)S (Q)

d

d (Q)

P I

2

 





. (1)

  N  N V   VP

is the particle number density, VP is the particle volume,  is the particle 

volume fraction, P(Q) is the form factor, 

2

is the contrast factor and SI(Q) is the structure 

factor. The two types of fractal behavior (mass fractal and surface fractal) have been 

investigated (Bale-Schmidt, 1984; Teixeira, 1988) and will be discussed in turn. 

1. MASS FRACTAL

A mass fractal is a structure containing branching and crosslinking to form a 3D network.

Figure 1: Schematic representation of a mass fractal structure containing branching points

and crosslinks. This structure is made out of monomeric units or small particles that are 

clustered. 

The inter-particle structure factor is given by:

      



0

2

I Qr

sin(Qr)

S (Q) 1 4 N dr r g(r) 1 . (2)



r0
342

Here g(r) is the pair correlation function. It is the probability of finding another scatterer at 

position 

r



given that there is a scatterer at the origin. Defining a mass fractal dimension Dm, 

g(r) can be modeled as follows:

 























 

 r

r exp

4 r

D

N g(r) 1

D 3

D

0

m m

m

. (3)

This comes from a particle number density that varies like a mass fractal:

Dm

0

r

0

2

r

r

N(r) N dr g(r)4 r

















    . (4)

The parameter  is a characteristic size for the mass fractal and r0 is the radius of the 

individual particles making up the fractal object. Performing the Fourier transform, one 

obtains:

    

 

    

 

 

 





sin D 1 tan Q

1 1 Q

D (D 1)

Qr

1

S (Q) 1

1

D 1 2 m

2 2

m m

D

0

I

m m

. (5)

Note that tan-1(z) is also called arctan(z). The small-Q limit is obtained using standard 

expansions:

...

3

z

tan (z 0) z

3

1   

(6)

...

3!

z

sin(z 0) z

3

  

















 













      

m m 2 2

D

0

I m Q

6

D (D 1)

1

r

S (Q 0) 1 (D 1)

m

.

This gives an estimate of the radius of gyration for a mass fractal as:

2

D (D 1)

R

2

2 m m

g

 

 . (7)

The high-Q limit is obtained using the following expansion which yields the asymptotic Qdependence:

...

z

1

2

tan (z )

1





  



(8)
343



























 

  









 

      

Q

(D 1)

2

cos (D 1)

2

D (D 1) sin (D 1)

Q r

1

S (Q ) 1

m

D m m m m

0

I Dm m

.

for 

1

0

1 Q r 

  

. This is a modified 

Q

Dm 1/

behavior. The mass fractal dimension Dm varies 

between 2 and 3 and is equivalent to the Porod exponent. Note that when Dm = 2, 

0

2

cos (Dm 1) 









 

 so that the asymptotic behavior varies with a Porod exponent Dm. When 

Dm = 3, 

0

2

sin (Dm 1) 









 

 instead and the Porod exponent is Dm+1. 

1 0-8

1 0-7

1 0-6

1 0-5

0.0001

0.001

0.01

0.1

1

0.001 0.01 0.1 1

Mass Fractal Model, Rg = 100 Å, Dm = 3 

r

0

 = 10 Å, without P(Q)

r

0

 = 10 Å, with P(Q)

r

0

 = 20 Å, with P(Q)

I(Q)/I(0)

Q

Figure 2: Normalized scattering intensity for the mass fractal model with and without the 

form factor P(Q) and with Rg = 100 Å and Dm = 3. 

Note that the form factor P(Q) for the individual particles that make up the mass fractal was 

modeled here by spheres or radius r0 with smooth surface. The case of particles with a fractal 

(i.e., rough) surface is considered next. 

2. SURFACE FRACTAL

Consider a particle with fractal (rough) surface of fractal dimension Ds between 3 and 4. 
344

Figure 3: Schematic representation of a surface fractal structure of intermediate roughness. 

The Porod law can be generalized to fractal surfaces through the following scaling for the 

surface:

Ds

0

P

r

r

S (r) ~ 















. (9)

The form factor for the particle with fractal surface becomes at high-Q:

 

6 Ds

s

s

P

P

P

Q

1

2

D 1

(5 D )sin

V

S

V P(Q )

 









 

       . (10)

Note that this result yields zero for Ds = 3. In the case of the mass fractal model, a similar 

inconsistency was avoided by going to a higher term in the high-Q expansion. 

A Porod plot (Log[I(Q)] vs Q) yields a slope of -6+Ds. A surface fractal dimension Ds = 2 

corresponds to a smooth surface which, for high-Q, gives:

4

P

P

3

0 Q

1

V

S

2r

3

P(Q  )  . (11)

SP and VP are the particle surface and volume. This is the well known Porod law for smooth 

surfaces. 

3. FRACTAL POROD EXPONENTS
345

Some figures summarize the various fractal Porod law exponents for mass fractal systems 

such as polymer chains and networks and for fractal surfaces. 

Figure 4: Assortment of fractal Porod exponents. 

Q-5/3 Q-2 Q-3

MASS FRACTALS SURFACE FRACTALS

Q-3 Q-4

rough 

surface

smooth

surface

clustered 

network

Gaussian 

chains

swollen 

chains
346
347

Figure 5: Assortment of more Porod exponents

REFERENCES

H.D. Bale and P.W. Schmidt, “Small-Angle X-Ray-Scattering Investigation of 

Submicroscopic Porosity with Fractal Properties”, Phys. Rev. Lett. 53, 596-599 (1984)

J. Teixeira, “Small-Angle Scattering by Fractal Systems”, J. Appl. Cryst. 21, 781-785 (1988)

QUESTIONS

1. What is the Porod exponent for scattering from a fully swollen polymer coil?

2. What is the Porod exponent for scattering from a very rough surface? How about from a 

smooth surface?

3. What is the range of mass fractal Porod exponents for scattering from a clustered network?

ANSWERS

1. The Porod exponent for scattering from a fully swollen polymer coil is 5/3. 

2. The Porod exponent for scattering from a very rough surface is 3. For a smooth surface, 

the Porod exponent is 4. 
348

3. Scattering from a clustered network has a range of mass fractal Porod exponents between 2 

and 3. 
349

Chapter 34 –THE MULTI-COMPONENT

RANDOM PHASE APPROXIMATION

1. COMPRESSIBLE POLYMER MIXTURE

Consider a homogeneous mixture consisting of m polymer components. Components are 

homopolymers, blocks in copolymers or solvents. Within the Random Phase Approximation 

formalism (De Gennes, 1979; Benmouna et al, 1987; Akcasu-Tombakoglu, 1990) for 

compressible mixtures, the scattering cross section for this system is expressed in general 

terms as follows:

.S(Q).

d

d (Q) T

  





. (1)

Here 

S(Q)

is an m*m matrix and 



is a column vector containing the m scattering length 

densities. 

T



is the “transpose” row vector. The fully interacting scattering factors 

S(Q)

are 

expressed in terms of the non-interacting (so called “bare”) scattering factors 

S (Q) 0

and 

inter-monomer interaction potentials 

W

as follows:

k T

W

S (Q) S (Q)

B

1

0

1

 

 

. (2)

This equation can easily be derived using the linear response approach without the 

incompressibility constraint. The incompressibility condition simplifies things as described 

next. 

2. INCOMPRESSIBLE POLYMER MIXTURE

Now consider one of the m polymer components as the “background” component. This can 

be a homopolymer or a solvent. It cannot be a block that belongs to a copolymer. Imposing 

the incompressibility condition eliminates the background component. This leaves (m-1) 

explicit components. Even though this formalism is general, it is described here explicitly for 

four components (m = 4). Component 4 is referred to as the “background” component. The 

Random Phase Approximation formalism for incompressible mixtures yields the following 

macroscopic scattering cross section:

.S(Q).

d

d (Q) T

  





(3)

S (Q) S (Q) S (Q) 2 S (Q) 2 S (Q) 2 S (Q)

3 3 1 2 1 2 1 3 1 3 2 3 2 3

2

2 2 3

2

1 1 2

2

 1             
350

In this general RPA formalism (Akcasu-Tombakoglu, 1990; Hammouda, 1993) 

S(Q)

is an 

(m-1)*(m-1) matrix and 



is an (m-1) column vector for the scattering length density 

differences (the scattering length density of the background component is subtracted for each 

component). The new general relation follows:

S (Q) S (Q) v(Q)

1

0

1

 

 

(4)

S(Q) 1 S (Q).v(Q) .S (Q)

0

1

0



  .

The “bare” scattering factor 

S (Q) 0

is a diagonal matrix for homopolymer blends and 

homopolymer solutions. Mixtures containing copolymers contain off-diagonal elements. 























S (Q) S (Q) S (Q)

S (Q) S (Q) S (Q)

S (Q) S (Q) S (Q)

S (Q)

0

3 3

0

3 2

0

3 1

0

2 3

0

2 2

0

2 1

0

1 3

0

1 2

0

1 1

0

. (5)

The interaction matrix is expressed in terms of the bare scattering factor for the background 

component and the various Flory-Huggins interaction parameters.

0

i4

0

44

ii

v

2

S (Q)

1

v (Q)



  for i = 1,2,3 (6)

0

j4

0

i4

0

ij

0

4 4

ij S (Q) v v v

1

v (Q)











 

for i,j = 1,2,3. 

The various scattering length densities for the various components are given by:

i  i  4

for i = 1,2,3. (7)

The incompressibility assumption in this case becomes:

v1n1(Q) + v2n2(Q) + v3n3(Q) + v4n4(Q) = 0. (8)

S (Q) v n ( Q).v n (Q) S (Q) S (Q) S (Q)

1 1  1 1  1 1   1 2  1 3  1 4

.

Note that the spinodal condition is obtained for

Det1 S0(0).v(0) 0 . (9)

Here 

1

is the identity matrix and 

Det...

denotes the determinant of a matrix. 
351

3. THE SINGLE-CHAIN FORM FACTORS

The various single-chain form factors for homopolymers and block copolymers are expressed 

as follows:

S (Q) n v P (Q)

i i i i

0

ii  

(10)

S (Q) n v n v P (Q) i i i j j j ij

0

ij   

 

2

g i

2

2

g i

2

4

g i

4

i

exp( Q R ) 1 Q R

Q R

2

P (Q)    

 

 

 















  

















  

 

2

g j

2

2

g j

2

2

g k

2

2

g i

2

2

g i

2

ij i k j

Q R

1 exp Q R

exp( Q R )

Q R

1 exp Q R

P (Q) F (Q)E (Q)F (Q) .

For Gaussian chains, the radii of gyration are given in terms of the degree of polymerization 

ni, and statistical segment lengths ai as follows:

6

n a

R

2

2 i i

gi  . (11)

The last expression is better explained through examples. This is done next. Consider 

examples of block copolymer sequences.

Figure 1: Examples of block copolymer sequences. 

F(Q) is used for the two blocks under consideration and E(Q) is used for the blocks inbetween. Some inter-block form factors are given here:

P (Q) F (Q)F (Q)

12  1 2

(12)

P (Q) F (Q)E (Q)F (Q)

35  3 4 5

P (Q) F (Q)E (Q)E (Q)F (Q)

69



6 7 8 9

.

1

2

3

4 5

6 7

8

10 9

0

11

0

12

0
352

These results are valid for Gaussian chains following a random walk (theta condition). For 

fully swollen chains, the excluded volume parameter approach could be used. 

4. BINARY HOMOPOLYMER BLEND MIXTURE

The simplest case to consider is that of a binary blend mixture of two homopolymers 

(components 1 and 2). In this case, component 2 is taken to be the background component. 

The results are: 

S (Q) n v P (Q)

1 1 1 1

0

11  

(13)

0

12

0

22

11

v

2

S (Q)

1

v (Q)



 

v (Q)

S (Q)

1

S (Q)

1

0 11

11 11

 

1 v (Q)S (Q)

S (Q)

S (Q)

0

11 11

0

11

11



 . 

This is the so-called de Gennes formula (De Gennes, 1979) used to describe binary polymer 

blends in the mixed-phase region. 

5. TERNARY HOMOPOLYMER BLEND MIXTURE

The case of a ternary homopolymer mixture is worked out similarly. Component 3 is taken to 

be the background component. The results are:

S (Q) n v P (Q)

1 1 1 1

0

11  

(14)

S (Q) n v P (Q)

2 2 2 2

0

22  

S (Q) n v P (Q) 3 3 3 3

0

33  

0

13

0

33

11

v

2

S (Q)

1

v (Q)



  (15)

0

23

0

33

22

v

2

S (Q)

1

v (Q)



 

0

2 3

0

1 3

0

1 2

0

3 3

1 2 S (Q) v v v

1

v (Q)











 

The partial scattering factors for the fully interacting mixture are as follow:
353  

1 v S (Q)1 v S (Q) v S (Q)S (Q)

S (Q) 1 v S (Q)

S (Q)

0

2 2

0

1 1

2

1 2

0

2 2 2 2

0

1 1 1 1

0

2 2 2 2

0

1 1

1 1

  





(16)

 

1 v S (Q)1 v S (Q) v S (Q)S (Q)

S (Q) 1 v S (Q)

S (Q)

0

2 2

0

1 1

2

1 2

0

2 2 2 2

0

1 1 1 1

0

1 1 1 1

0

2 2

2 2

  





1 v S (Q)1 v S (Q) v S (Q)S (Q)

S (Q)v S (Q)

S (Q)

0

2 2

0

1 1

2

1 2

0

2 2 2 2

0

1 1 1 1

0

1 2 2 2

0

1 1

1 2

  





Recall that the cross section for an incompressible ternary blend mixture is given by 

(Benmouna et al, 1987; Akcasu-Tombakoglu, 1990):

S (Q) S (Q) 2 S (Q)

d

d (Q)

2 2 1 2 1 2

2

1 1 2

2

 1     





. (17)

This case applies to a ternary polymer mixture in the homogeneous phase region.

6. BLEND MIXTURE OF A COPOLYMER AND A HOMOPOLYMER

The case of a homopolymer and a copolymer mixture is also readily obtained from the RPA 

formalism. In this case, the background component is taken to be the homopolymer 

(component 3). The diblock is formed of components 1-2. The results follow. 

   

     

0

2 2 2 1

0

2 1 1 1

0

1 2 2 2

0

1 1 1 2

0

2 2 2 2

0

2 1 1 2

0

1 2 2 1

0

1 1 1 1

0

2 2 2 1

0

2 1 1 1

0

1 2

0

2 2 2 2

0

2 1 1 2

0

1 1

1 1 1 v S v S 1 v S v S v S v S v S v S

S 1 v S v S S v S v S

S (Q)

      

   



(18)

   

     

0

2 2 2 1

0

2 1 1 1

0

1 2 2 2

0

1 1 1 2

0

2 2 2 2

0

2 1 1 2

0

1 2 2 1

0

1 1 1 1

0

1 1 1 2

0

1 2 2 2

0

2 1

0

1 1 1 1

0

1 2 2 1

0

2 2

2 2 1 v S v S 1 v S v S v S v S v S v S

S 1 v S v S S v S v S

S (Q)

      

   



   

     

0

2 2 2 1

0

2 1 1 1

0

1 2 2 2

0

1 1 1 2

0

2 2 2 2

0

2 1 1 2

0

1 2 2 1

0

1 1 1 1

0

1 2 2 1

0

1 1 1 1

0

1 2

0

1 2 2 2

0

1 1 1 2

0

1 1

1 2 1 v S v S 1 v S v S v S v S v S v S

S v S v S S 1 v S v S

S (Q)

      

    



The (Q) dependence has been dropped to lighten the notation. 

7. THE DIBLOCK COPOLYMER CASE

The RPA result for polymer mixtures containing only copolymers (no homopolymers or 

solvent) is more complex and will not be included here. It has, however, been worked out 
354

explicitly using a matrix notation and assuming one of the blocks as the background 

component. The result for the simple case of a diblock copolymer is included here. 

  S (Q)S (Q) S (Q)S (Q)

v

2

S (Q) S (Q) 2S (Q)

S (Q)S (Q) S (Q)S (Q)

S (Q)

0

2 1

0

1 2

0

2 2

0

1 1

0

0 1 2

1 2

0

2 2

0

1 1

0

2 1

0

1 2

0

2 2

0

1 1

1 1





  



 . (19)

This is the so-called Leibler formula (Leibler, 1980). This formula can be derived using the 

linear response approach. Note that the binary blend case is recovered by setting the bare 

scattering factor cross term 

S (Q) 0

0

12  . 

8. THE TRIBLOCK COPOLYMER CASE

Consider a triblock copolymer in solution. In our terminology, this is a four “component” 

case with the triblock as components 1-2-3 and the solvent as component 4. Block 1 is 

connected to block 2 which is connected to block 3. Block 1 is connected to block 3 only 

through Block 2 (not directly). Consider component 4 as the “background” component and 

apply the multi-component RPA formula:























S (Q) S (Q) S (Q)

S (Q) S (Q) S (Q)

S (Q) S (Q) S (Q)

S (Q)

0

3 3

0

3 2

0

3 1

0

2 3

0

2 2

0

2 1

0

1 3

0

1 2

0

1 1

0

. (20)

In order to work out the various elements Sij(Q), 3*3 matrix inversion and matrix 

multiplication are needed. The result is too lengthy to reproduce here. The derivation is, 

however, straightforward (Akcasu et al, 1993). 

9. MIXTURE OF POLYELECTROLYTES 

Consider a binary mixture containing a charged polymer (polyelectrolyte). The RPA 

formalism can be adapted to include charge effects. The scattering equations for a binary 

mixture (where component 1 is a polyelectrolyte) are summarized here (Benmouna-Vilgis, 

1991). 

v (Q)

~

S (Q)

1

S (Q)

1

0 11

11 11

 

(21)

v (Q)S (Q)

~ 1

S (Q)

S (Q)

0

11 11

0

11

11





v (Q) v (Q) v (Q)

~

11  11  DH
355

0

12

0

22

11

v

2

S (Q)

1

v (Q)



 

2 2

B

2

1

DH Q

4 l

v

1

v (Q)

 





)

v v

4 l (f

salt

salt

1

1

B

2 





   .

vDH(Q) is the extra “Debye-Huckel” term that accounts for Coulomb interactions, lB is the 

Bjerrum length given by 

k T

e

l

B

2

B





, f is the electron charge fraction per monomer and salt

is the salt volume fraction. v11 is the interaction factor (v1

2

v11 is the so-called “excluded 

volume”). As described previously for incompressible neutral polymer mixtures, the 

scattering cross section for polyelectrolyte mixtures is given by:

S (Q)

d

d (Q)

11

2

 1 



(22)

S (Q) n v P (Q)

1 1 1 1

0

11  

S (Q) n v P (Q)

2 2 2 2

0

22  

 

2

g1

2 2

g1

2

4

g1

4

1

exp( Q R ) 1 Q R

Q R

2

P (Q)     .

n1, v1, 1, n2, v2, 2, are the degree of polymerization, the specific volume and the volume 

fraction for components 1 and 2 respectively. P1(Q) and P2(Q) are the familiar Debye 

functions for Gaussian coils. This describes the general case of a polymer blend. If the binary 

mixture is a polyelectrolyte solution instead, then n2 = 1, P2(Q) = 1 and v11 becomes 

independent of Q. 

Consider the following parameters for a polyelectrolyte solution:

f = 0.5 (23)

n1 = 1000

1 = 0.04

v1 = 100 Å3

salt = either 0.01 (small salt addition) or 0.1 (large salt addition)

vsalt = 100 Å3

lB = 10 Å 

v11 = 0.05 Å-3
356

Rg1 = 100 Å.

The scattering factor S11(Q) is plotted for two salt conditions. The polyelectrolyte peak is 

observed when small amount of salt is added. When lots of salt is added, the “interaction” 

peak disappears due to the screening of Coulomb interactions. 

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1

Polyelectrolyte Solution



salt

= 0.01



salt

 = 0.1

Scattering Factor S11(Q)

Scattering Variable Q (Å -1)

Figure 2: Variation of the scattering factor S11(Q) for two salt conditions. 

The polyelectrolyte interaction peak position obtained corresponds to the maximum in the 

S11(Q) function. The Debye function is approximated by a simple form and the notation is 

modified for convenience.

2

Q R

1

1

P (Q)

2

g1

2

1



 (24)

1 [v G(Q)]F(Q)

F(Q)

1 [v v (Q)]S (Q)

S (Q)

S (Q)

1 1

0

1 1 DH 1 1

0

1 1

1 1

 



 



 

 

2

1 1

1 1

1 1

1 1

1 [v G(Q)]F(Q)

F(Q) v F'(Q) G'(Q)F(Q) G(Q)F'(Q)

1 [v G(Q)]F(Q)

F'(Q)

Q

S (Q)

 

 



 







.
357

The peak position is obtained for the condition:

0

Q

S (Q)

Q Qmax

11

 

















. (25)

This implies: 

F'(Q ) F (Q )G'(Q ) max max

2

max



(26)

2

1 g1

2 2 1 1 B

max

v R

n 8 l

Q

 

   .

The so-called Lifshitz line corresponds to the condition for which the polyelectrolyte 

interaction peak disappears (i.e., Qmax = 0). 

The spinodal temperature corresponds to the “blowing up” of the scattered intensity; i.e., 

when the denominator becomes equal to zero.

1 [v v (Q 0)]S (Q 0) 0

0

 11  DH 11   . (27)

The interaction factor v11 is plotted as function of salt volume fraction salt. v11 is related to 

the Flory-Huggins interaction parameter which depends (inversely) on temperature. 
358

-0.5

-0.4

-0.3

-0.2

-0.1

0

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.02 0.04 0.06 0.08 0.1

Polyelectrolyte Solution 

v

11

Qmax

v11 for Spinodal Line

Qmax (Å-1

)



salt

Lifshitz Line mixed phase region 

phase separated region

spinodal line

Figure 3: Variation of the interaction factor v11 and the of polyelectrolyte peak position Qmax

with increasing salt volume fraction salt. The spinodal line and the Lifshitz line are included. 

The polyelectrolyte peak feature is due to the modified monomer-monomer interactions that 

become characterized by two sizes: the actual monomer size and the screened Coulomb 

interaction distance (represented by 

1



). This produces a “correlation hole” effect. The 

screened Debye-Huckel interaction potential varies like 

V (r) ~ exp( r)/r DH 

. Charged 

interactions tend to stabilize the phase diagram and favor mixing. Adding salt tends to favor 

demixing. The interaction factor v11 is related to the Flory-Huggins interaction parameter 12

which is inversely proportional to temperature. The RPA approach outlined here can apply to 

more complex polymer mixtures containing polyelectrolytes and neutral polymers. 

10. DISCUSSION

The RPA approach described here can handle more complex polymer mixtures containing 

complex architectures and blockiness (Hammouda, 1993). It amounts to inverting and 

multiplying larger matrices. The effect of chain stiffness has also been included in limited 

cases. Mixtures of flexible and stiff polymer chains are characterized by the familiar spinodal 

condition as well as by the isotropic-to-nematic phase transition (Hammouda, 1993). 
359

It should be emphasized, however, that this mean-field approach applies strictly in the 

mixed-phase region (not too close to the phase boundary line). Non-mean field corrections 

have been worked out. These are, however, outside the scope of this effort. 

REFERENCES

P.G. de Gennes, "Scaling Concepts in Polymer Physics", Cornell University Press, New 

York (1979). 

L. Leibler, “Theory of Microphase Separation in Block Copolymers”, Macromolecules 13, 

1602-1617 (1980).

M. Benmouna, H. Benoit, R. Borsali, M. Duval, “Theory of Dynamic Scattering from 

Copolymer Solutions using the Random Phase Approximation”, Macromolecules 20, 2620-

2624 (1987).

A.Z. Akcasu and M. Tombakoglu, “Dynamics of Copolymer and Homopolymer Mixtures in 

Bulk and in Solution via the Random Phase Approximation”, Macromolecules 23, 607-612 

(1990).

A.Z. Akcasu, R. Klein and B. Hammouda, “Dynamics of Multicomponent Polymer Mixtures 

via the Random Phase Approximation Including Hydrodynamic Onteractions”, 

Macromolecules 26, 4136-4143 (1993). 

B. Hammouda, "SANS from Homogeneous polymer Mixtures: A Unified Overview", 

Advances in Polymer Science 106, 87 (1993). B. Hammouda, "Scattering from Mixtures of 

Flexible and Stiff Polymers", J. Chem. Phys. 98, 3439-3444 (1993).

M. Benmouna and T. Vilgis, “Scattering from Multicomponent Polymer Mixtures: Weakly 

Charged Polymers”, Macromolecules 24, 3866-3872 (1991). 

QUESTIONS

1. Write down the generalized Random Phase Approximation formula for multi-component 

incompressible polymer mixtures. Does it apply to pure copolymers?

2. Write down the scattering cross section for a multi-component polymer mixture of 

arbitrary composition in matrix form. 

3. What is the size of the matrix to be inverted for an incompressible mixture with four 

polymer components?

4. Write down the so-called de Gennes formula for scattering from a binary polymer blend 

mixture. Define the various terms. 

5. Write down the S11(Q) interacting scattering factor for a ternary polymer blend mixture. 

6. Write down the so-called Leibler formula for scattering from a diblock copolymer. 
360

7. Stiff (rodlike) polymers are characterized by orientational phase transitions beside the 

spinodal and binodal lines. Name the two best known phase transitions.

8. What is the extra interaction term which is added to account for screened charge-charge 

interactions and thereby extend the multicomponent RPA approach to included 

polyelectrolytes? 

ANSWERS

1. The Random Phase Approximation general formula for multi-component incompressible 

polymer mixtures expresses the fully interacting scattering factor 

S(Q)

in terms of the “bare” 

(non-interacting) scattering factor 

S (Q)

0

and the various interaction factors 

v(Q)

as 

S (Q) S (Q) v(Q)

1

0

1

 

 

. This does not apply to pure copolymers since a “background” 

component (either a homopolymer or a solvent) is required. 

2. The scattering cross section for a multi-component polymer mixture of arbitrary 

composition is expressed as .S(Q).

d

d (Q) T

  



  

where 

.



is a column vector containing all 

of the scattering length densities and 

S(Q)

is a matrix containing all of the scattering factors. 

3. A homogeneous polymer mixture with four components generates a 3*3 RPA matrix to be 

inverted. The fourth component is taken to be the background component. 

4. The de Gennes formula for scattering from a binary polymer blend mixture is expressed as 

0

1 2

2 2

0

1 1

0

1 1 v

2

S (Q)

1

S (Q)

1

S (Q)

1 

   . Here 

S 11(Q)

0

and 

S 22 (Q)

0

are the bare scattering 

factors, 

12

is the Flory-Huggins interaction parameter and 

0 v

is a reference volume usually 

expressed as 

v0  v1v2

where v1 and v2 are the specific monomer volumes. 

5. The scattering factor for a ternary polymer blend mixture is given by 

 

1 v S (Q)1 v S (Q) v S (Q)S (Q)

S (Q) 1 v S (Q)

S (Q)

2 2

0

1 1

2 0

1 2 2 2

0

2 2 1 1

0

1 1

2 2

0

1 1 2 2

0

1 1

  





. The various factors are 

defined in the text. 

6. The Leibler formula is expressed as 











 

  W(Q)

v

2

S (Q) W(Q) S(Q)

0

1 2

1 1

where 

W(Q) S (Q)S (Q) S (Q)S 2 1(Q)

0

1 2

0

2 2

0

1 1

0

  and S(Q) S (Q) S (Q) 2S 12 (Q)

0

22

0

11

0

   . 

7. Stiff polymers are characterized by the spinodal and binodal temperatures as well as 

orientational transitions leading from the isotropic to the nematic or smectic phases. 

8. The extra interaction term added to extend the multicomponent RPA approach to include 

polyelectrolytes is the Debye-Huckel factor 

2 2

B

2

1

DH Q

4 l

v

1

v (Q)

 





where v1 is the 

monomer volume, 

1



is the screening length, and lB is the Bjerrum length given by 

k T

e

l

B

2

B



 . 
361
362

Part H – SANS FROM POLYMERS

Chapter 35. Introduction to Polymers

35.1 What Are Polymers?

35.2 SANS from Polymers

35.3 Polymer Chain Conformations

35.4 Thermodynamics of Polymer Miscibility

35.5 Crystalline Polymers

References

Questions

Answers

Chapter 36. Polymer Contrast Factors

36.1 Coherent Scattering Lengths for a Few Monomers 

36.2 Coherent Scattering Lengths for a Few Solvents

36.3 A Few Neutron Contrast Factors for Polymer Mixtures

References

Questions

Answers

Chapter 37. SANS from Polymer Solutions

37.1 Polymer Solutions Basics

37.2 Case of a Simple Polymer Solution

37.3 Fit to a Simple Model

37.4 The Correlation length

37.5 The Spinodal Temperature

37.6 The Excluded Volume Parameter 

37.7 Branch of the Phase Diagram

37.8 Polymer Solution Thermodynamics

37.9 the Zero Average Contrast Method

References

Questions

Answers

Chapter 38. SANS from Polymer Blends

38.1 The Flory-Huggins Model

38.2 Binodal and Spinodal Lines

38.3 The Random Phase Approximation

38.4 Polyolefin Blends

38.5 A Polyolefin Blend

38.6 Gibbs Free Energy Density and Phase Diagram

References

Questions

Answers
363

Chapter 39. SANS from Block Copolymers

39.1 Polystyrene-Polyisoprene Copolymer in DOP Solution

39.2 SANS Data

39.3 The RPA Model for a Diblock Copolymer Solution

39.4 SANS Data Analysis

39.5 The Ordered Phase

References

Questions

Answers

Chapter 40. SANS from Ternary Polymer Blends

40.1 SANS from A Ternary Polymer Blend

40.2 The Ternary RPA Model

40.3 Fitting of the SANS Data

40.4 Spinodal Temperature

40.5 Discussion

References

Questions

Answers

Chapter 41: SANS from Polymers Literature Review

41.1. Branched Polymers

41.2 Copolymers

41.3 Crystalline Polymers

41.4 Ionomers

41.5 Polyelectrolytes

41.6 Polymer Blends

41.7 Polymer Clustering 

41.8 Polymer Nanocomposites

41.9 Polymer Gels

41.10 Polymer Micelles

41.11 Polymer Networks

41.12 Polymer Solutions

41.13 Polymers under Pressure

41.14 Polymers under Shear

41.15 Thin Polymer Films

References
364

Chapter 35 - INTRODUCTION TO POLYMERS

Polymer research makes up a good fraction of beamtime use on SANS instruments. Using 

the partial deuteration method, SANS has been useful for investigations of chain 

conformations and miscibility in polymer solutions and polymer blend mixtures as well as in 

other polymeric systems. Many aspects of polymer research have benefited from the SANS 

technique. 

1. WHAT ARE POLYMERS?

Polymers are synthetic macromolecules that make up a great deal of what we use in our daily 

lives. They include low-cost disposable conveniences (like milk containers or soda bottles) to 

high-tech structural materials (like hip joint replacement or computer CDs). Polymers are 

split into “bunching” categories like solutions or blends (alloyed polymers), thermoplastics 

(that can deform) or thermosets (that are hard to deform), with linear or branched 

architectures, that are amorphous or crystalline, etc. (Bandrup-Immergut 1975; Kawakatsu, 

2004). Liquid crystal polymers contain stiff mesogen groups that increase material 

toughness. Their high degree of alignment makes them useful in optical devices. Polyolefins 

(polymers containing C=C double bonds) are at the heart of petroleum chemistry. Much 

SANS research has been performed on polyolefins. SANS from polymers research is broad 

and deep. Only the simplest aspects of this research are covered here. 

Polymer research has bloomed over the past fifty years (Flory, 1969; de Gennes, 1979; 

Higgins-Benoit, 1994). The development of light scattering in the 1960s and of neutron 

scattering in the 1970s along with advances in polymer synthesis and computational power 

have greatly benefited polymer research. Impact of the SANS technique has been substantial 

in many areas of polymer research. 

2. SANS FROM POLYMERS

Polymer research has benefited greatly from the SANS technique which matured in the 

1970s and 1980s. SANS was first developed to the scale of a user program at the ILL 

(Grenoble, France) then spread to most neutron scattering facilities. Polymer research 

accounted for the largest share of SANS beamtime. The advent of judicious sample 

environments brought about renewed interest. These include temperature and pressure 

control, the application of in-situ shear, etc. The development of scattering theory for 

polymer systems such as the Random Phase Approximation helped promote growth in the 

use of the technique. Most neutron scattering facilities maintain SANS instruments that are 

overbooked. SANS from polymers research has developed from cutting edge research for 

hardcore users into a routine characterization method for laboratories that have access to the 

technique. For example, the Exxon Mobil company has maintained constant use of the SANS 

technique and its constant funding at the NCNR for over twenty years. Moreover, the 

National Science Foundation has copiously funded the SANS program at many US facilities. 
365

3. POLYMER CHAIN CONFORMATIONS

The partial deuteration method helps observe the conformation of polymer chains in the 

environment of a solvent or of other polymers. This is similar to the staining method in 

microscopy. An apparent radius of gyration is often estimated from SANS data in the Guinier 

region. The Porod region yields chain conformation details such as the degree of chain 

swelling or solvent quality in polymer solutions. This region also shows the onset of chain 

stiffness whereby the polymer chain persistence length can be measured. 

The SANS technique has permitted measurements of the radius of gyration of polymer 

chains in various polymer systems whether in solution or in blends. The contrast match 

method has helped the separation of single-chain properties even in concentrated mixtures. It 

was found for example, that polymer chains follow random walk statistics in “theta” 

solvents, in concentrated solutions as well as in polymer melts. 

4. THERMODYNAMICS OF POLYMER MISCIBILITY

SANS intensity increases close to phase separation lines due to enhanced composition 

fluctuations. This makes SANS an effective tool for the investigation of miscibility 

thermodynamics. Polymeric systems phase separate either through heating and are 

characterized by a lower critical solution (or spinodal) temperature (LCST) or through 

cooling and are characterized by an upper critical solution temperature (UCST). Some 

polymer solutions or polymer blend mixtures are known to phase separate through both 

heating and cooling and are characterized by a miscibility gap. Others phase separate only 

within a specific temperature region and are characterized by a closed loop immiscibility 

island. Polymer solutions in organic solvents tend to be characterized by UCSTs whereas 

water soluble polymers tend to be characterized by LCSTs. The four major types of phase 

diagrams are summarized in a figure. Combination of these basic types is also possible; for 

example UCST at low temperature and closed loop immiscibility at high temperature. 
366

Figure 1: The four main types of phase separation diagrams for polymer blends showing the 

variation of the phase separation temperature with polymer composition. Upper left: LCST, 

upper right: UCST, lower left: miscibility gap, lower right: closed loop immiscibility island. 

The SANS technique has permitted the estimation of Flory-Huggins interaction parameters 

and the mapping out of miscibility phase diagrams. The advent of in-situ pressure and 

temperature control has brought about a better understanding of polymer thermodynamics. 

5. CRYSTALLINE POLYMERS

The SANS technique probes density fluctuations (just like SAXS using x-rays) as well as 

composition fluctuations. Crystalline polymers are characterized by a strong low-Q signal 

below the crystallization temperature. This is due to the density fluctuations component. The 

use of partially deuterated chains allows the monitoring of chain conformation inside 

crystallites. Early findings have found, for instance, that polymer chains follow Gaussian 

chain statistics in the crystalline lamellae as well as in the melt state. It was also found that 

melt crystallization is characterized by “random re-entry” of the polymer chains to form 

mixed phase

phase

separated

composition

T

e

m

p

LCST

phase

separated

mixed phase

composition

T

e

m

p

UCST

phase

separated

phase

separated

mixed phase

composition

T

e

m

p

MISCIBILITY GAP

phase

separated

mixed phase

composition

T

e

m

p

IMMISCIBILITY ISLAND

mixed phase
367

lamellae. Solution crystallized polymers, however, are characterized by an “adjacent reentry” scheme. 

Due to the various folding and structural defects of crystalline polymers, it is not possible to 

grow perfect single crystals. 

REFERENCES

P.J. Flory, “Statistical Mechanics of Chain Molecules”, Wiley & Sons (1969)

J. Bandrup and E.H. Immergut, Editors, “Polymer Handbook”, John Wiley & Sons (1975). 

P.G. de Gennes, "Scaling Concepts in Polymer Physics", Cornell University Press, New 

York (1979). 

J.S. Higgins and H. Benoit, "Polymers and Neutron Scattering", Oxford (1994).

T. Kawakatsu, “Statistical Physics of Polymers: an Introduction”, Springer (2004)

QUESTIONS

1. When was the “golden age” for polymers research?

2. What are thermoplastics? How about thermosets?

3. Name some contributions of the SANS technique to polymer research. 

4. Why is the SANS technique a good probe for thermodynamics investigations? How is this 

manifested? 

ANSWERS

1. Polymers research was very strong for the past 50 years. Lots of progress was made in the 

1960s and 1970s. 

2. Thermoplastics can recover their original shapes when they are deformed. Thermosets are 

highly cross linked. It is hard to deform them. 

3. The SANS technique has had broad impact on polymer research. A few examples follow: 

single-chain conformations in polymeric materials, phase separation thermodynamics, chain 

properties in crystalline polymers, clustering in water-soluble polymer solutions, etc. 

4. The SANS technique is a good thermodynamics probe since it can monitor density and 

composition fluctuation. The intensity increases close to phase transition boundaries.
368

Chapter 36 - POLYMER CONTRAST FACTORS

The SANS scattering intensity is proportional to the neutron scattering contrast factor. The 

contrast factor for a polymer in solution or a polymer blend mixture is defined as (bM/vM –

bS/vS)

2

. Here bM and vP are the neutron scattering length and specific volume for the 

monomer that forms the polymer. bS and vS are for the solvent molecule or for another 

monomer in the polymer mixture.

1. COHERENT SCATTERING LENGTHS FOR A FEW MONOMERS

The following table summarizes scattering lengths for a few monomers (Bandrup-Immergut, 

1975). These have been calculated using tabulated values for the scattering lengths of the 

various elements and their relative amounts. 

Table 1: Coherent scattering lengths for a few synthetic monomers in fm (1 fermi=10-13 cm). 

Polymer Name Formula Hydrogenated

(fm)

Deuterated

(fm)

Polystyrene [CH2-CH(C6H5)] 23.30 106.62

Polymethylmethacrylate [CH2-C(CH3)(CO2CH3)] 14.95 98.27

Polymethylacrylate [CH2-CH(CO2CH3)] 15.78 78.27

Polyvinylchloride [CH2CH(Cl)] 13.78 45.03

Polyethylene [CH2-CH2] -1.66 40.0

Polycarbonate [C6H4-C(CH3)

2C6H4

-O-CO2] 71.50 217.30

Polyvinylmethylether [CH2OH(OCH3)] 3.32 65.81

Polytetrahydrofuran [C4OH6] 9.97 72.46

Poly  chlorostyrene [CH2

-CH(C6H4Cl)] 38.74 111.64

Polyurethane

(Ethylcarbonate)

[NH-CO2-CH2-CH2] 22.23 74.31

Polyethylene oxide [CH2CH2O] 4.139 45.78
369

Figure 1: Chemical formulas for a few monomers. 

2. COHERENT SCATTERING LENGTHS FOR A FEW SOLVENTS

The following table summarizes scattering lengths for a few commonly used solvents.

Table 2: Coherent scattering lengths for a few solvents.

Solvent Name Formula Hydrogenated

(fm)

Deuterated

(fm)

Toluene C6H5CH316.64 99.96

Benzene C6H617.47 79.96

Cyclohexane C6H12 -4.97 120.01

Acetone CH3-COCH33.32 65.821

Chloroform CHCl331.60 42.05

Methylene Chloride CH2Cl222.57 43.40

Carbon Disulfide CS212.26 ------

Tetrahydrofurane C4OH82.47 85.81

Tri-m-Tolylphosphate CH3-C6H2P343.26 95.53

Trimethylbenzene C6H3(CH3)

3

14.98 139.96

Water H2O -1.675 19.145

Polystyrene

Polyethylene oxide Polyethylene 1,4 Polyisoprene

Polyvinyl methyl ether 

Poly-n-butyl 

methacrylate

CH2 CH

H

CH2 CH2 CH2 CH2 O CH2 C CH CH2

CH3

Polyacrylic acid 

CH2 CH

C

O OH

CH2 CH

O CH3

CH2 CH

CH2

C

O CH2

CH2

CH2

CH3
370

Figure 2: Chemical formulas for a few solvent molecules. 

Table 3: Coherent scattering length densities for a few deuterated solvents.

Solvent Name Formula

Molar Mass

(g/mol)

Density

(g/cm3)

Scattering Length 

Density

(Å

-2

)

d-Ethylene Glycol C2O2D6 66 1.113 4.47*10-6

d-Dioxane1-4 C4D8O2 96 1.129 6.46*10-6

d-Toluene C7D8 100 0.943 5.66*10-6

d-Benzene C6D6 84 0.950 5.43*10-6

d-Ethanol C2D6O 52 0.888 6.07*10-6

d-Methanol CD4O 36 0.888 5.8*10-6

d-Water D2O 20 1.107 6.37*10-6

d-Chloroform CDCl3 120.35 1.50 6.16*10-6

d-DMF NC3D7O 80 1.03 6.33*10-6

d-Xylene C8D10 116 0.953 5.92*10-6

3. A FEW NEUTRON CONTRAST FACTORS FOR POLYMER MIXTURES

Consider a two-component polymer system (say component 1 homogeneously mixed with 

component 2). The neutron contrast is defined as the square of the difference between two 

scattering length densities (b1/v1- b2/v2)

2

where b1and b2are the scattering lengths for 

monomers 1 and 2 and v1and v2are the monomer molar volumes for the two components. 

Component 2 could represent a solvent for polymer solutions. A few contrast factors have 

been calculated for the following polymer mixtures.

Table 4: Poly(ethylene oxide)/Deuterated Water (hPEO/D2O) Solution.

Substance Notation Formula Scattering Length Specific Volume

Toluene Benzene

Methanol

Chloroform Water

CH3

CH3 OH

Ethylene glycol

CH3 CH2 OH

Ethanol

HO CH2 CH2 OH

O

H H

CH

Cl

Cl Cl
371

(cm) (cm3/mol)

Poly(ethylene oxide) hPEO C2H4O 4.139*10-13 38.94

Deuterated water d-water D2O 19.14*10-13 18

Contrast Factor: 

-3 4

av

2

D O

D O

hPEO

hPEO N 5.498*10 mol/cm

v

b

v

b

2

2





















Navis Avogadro's number

Table 5: Deuterated Poly(ethylene oxide)/Water (dPEO/H2O) Solution.

Substance Notation Formula Scattering Length

(cm)

Specific Volume

(cm3/mol)

Deuterated 

Poly(ethylene oxide)

dPEO C2D4O 45.78*10-13 38.94

Water h-water H2O -1.67*10-13 18

Contrast Factor: 

-3 4

av

2

H O

H O

dPEO

dPEO N 9.657 *10 mol/cm

v

b

v

b

2

2





















Table 6: Deuterated Polystyrene/Polyvinylmethyether (dPS/PVME) Blend.

Substance Notation Formula Scattering Length

(cm)

Specific Volume

(cm3/mol)

Deuterated 

Polystyrene

dPS C8D8 1.06*10-11 100

Poly(vinyl methyl 

ether)

PVME C3H6O 3.30*10-13 55.4

Contrast Factor: 

-3 4

av

2

PVME

PVME

dPS

dPS N 6.07 *10 mol/cm

v

b

v

b

 

















Table 7: Deuterated Polystyrene/Hydrogenated Polystyrene (dPS/hPS) Blend.

Substance Notation Formula Scattering Length

(cm)

Specific Volume

(cm3/mol)

Deuterated dPS C8D8 1.06*10-11 100
372

Polystyrene

Polystyrene hPS C8H8 0.23*10-11 100

Contrast Factor: 

-3 4

av

2

hPS

hPS

dPS

dPS N 4.18*10 mol/cm

v

b

v

b

 

















Table 8: Deuterated Polystyrene/Polybutylmethacrylate (dPS/PBMA) Blend.

Substance Notation Formula Scattering Length

(cm)

Specific Volume

(cm3/mol)

Deuterated 

Polystyrene

dPS C8D8 1.06*10-11 100

Poly(butyl 

methacrylate)

hPBMA C8H14O2 1.24*10-12 133

Contrast Factor: 

-3 4

av

2

hPBMA

hPBMA

dPS

dPS N 5.61*10 mol/cm

v

b

v

b

 

















Table 9: Polystyrene/Polyisoprene (PS/PI) Blend.

Substance Notation Formula Scattering Length

(cm)

Specific Volume

(cm3/mol)

Deuterated 

Polystyrene

dPS C8D8 1.06*10-11 100

Polystyrene hPS C8H14 0.23*10-11 100

Poly(isoprene) hPBMA C5H8 0.33*10-12 76

Contrast Factor: 

-4 4

av

2

hPI

hPI

hPS

hPS N 2.09*10 mol/cm

v

b

v

b

 

















Contrast Factor: 

-3 4

av

2

hPI

hPI

dPS

dPS N 6.20*10 mol/cm

v

b

v

b

 

















Table 10: Deuterated Polystyrene/Dioctylphthalate (dPS/DOP) Solution.

Substance Notation Formula Scattering Length Specific Volume
373

(cm) (cm3/mol)

Deuterated 

Polystyrene

dPS C8D8 1.06*10-11 100

Dioctylphthalate DOP C24H38O4 4.07*10-12 390

Contrast Factor: 

-3 4

av

2

DOP

DOP

dPS

dPS N 5.48*10 mol/cm

v

b

v

b

 

















REFERENCES

http://www.ncnr.nist.gov/resources/sldcalc.html contains a scattering length density 

calculator. 

J. Bandrup and E.H. Immergut, Editors, “Polymer Handbook”, John Wiley & Sons (1975). 

QUESTIONS

1. Consider a polymer dissolved in a specific solvent. What would happen to the contrast 

factor if one were to “switch” the scattering length densities for the polymer and the solvent?

2. Which one of the two following dilute solution samples would you rather prepare for a 

SANS experiment: a deuterated polymer in hydrogenated solvent or a hydrogenated polymer 

in deuterated solvent?

3. Scattering from a non-deuterated polymer mixture is dominated by what type of 

fluctuations?

ANSWERS

1. If one were to “switch” the scattering length densities for the polymer and the solvent, the 

contrast factor would remain the same. This is the so-called Babinet principle. 

2. The best sample to prepare for a SANS experiment from a dilute solution is a 

hydrogenated polymer in deuterated solvent. This minimizes the amount of hydrogen (and 

therefore incoherent background) in the sample. This enhances the signal (coherent 

scattering) to noise (incoherent scattering). 

3. Scattering from a non-deuterated polymer mixture is dominated by density fluctuations. 

The same would be observed using x-ray scattering. 
374

Chapter 37 - SANS FROM POLYMER SOLUTIONS

Solubility is a determining factor in the synthesis, mixing ability and end-use of polymers. A 

general model for describing solubility (Flory, 1953) is discussed here with an emphasis on 

what information is obtained from SANS measurements from polymer solutions. SANS data 

from specific polymer solutions are discussed in some detail. 

1. POLYMER SOLUTIONS BASICS

Most non-polar polymers dissolve in organic solvents and some polar polymers dissolve in 

water. Concentration ranges vary from dilute to semi-dilute to concentrated solutions. The 

borderline between the dilute and the semi-dilute regimes is referred to as the overlap 

concentration (c*) which is estimated as 

 

3

w 4π 3 Rg

c* ~ M /

(Mw is the molecular weight 

and Rg is the radius of gyration). The polymer concentration c is related to the volume 

fraction  through the density d as c = d. 

2. CASE OF A SIMPLE POLYMER SOLUTION

As an example of a polymer solution, the case of poly(ethylene oxide) PEO in water is 

discussed (Hammouda et al, 2002; Hammouda-Ho, 2007). The PEO monomeric unit -

(CH2CH2O)- is the simplest one for a water-soluble polymer . When dissolved in water, PEO 

is characterized by hydrophilic interactions (hydrogen bonding of water molecules to the 

oxygen atoms on the polymer) and hydrophobic interactions (the CH2CH2 groups repel 

water). PEO dissolves in water for a wide range of temperatures and concentrations. Its 

homologues, PMO (-CH2O-) and PPO (-CH(CH3)CH2O-) do not dissolve in water an 

ambient temperature. This may be due to the fact that the oxygen-oxygen inter-distance on 

the PEO chain matches the oxygen-oxygen inter-distance in the structure of pure water. 

A typical SANS spectrum from a 4 % PEO/d-water (weight average and number average 

molecular weights of Mw = 100,000 and Mn = 96,000 g/mol respectively) is plotted for the T 

= 10o C temperature. This sample is located well in the semi-dilute region (c* is estimated to 

be 0.0275 g/cm3 which corresponds to 2.4 % volume fraction). The low-Q feature 

characterizes large size clusters (of no interest to us here) and the high-Q feature 

characterizes the polymer chains. 
375

0.1

1

0.01 0.1

4% PEO/d-water, Mw=100,000 g/mole, T=10oC

SANS Data

Scattering Intensity (cm-1

)

Scattering Variable Q (Å-1)

Figure 1: SANS data for a PEO/d-water sample over a wide Q range showing a low-Q 

feature and a high-Q feature. Only the tail of the low-Q feature is observed. Focus here is on 

the high-Q feature. 

3. FIT TO A SIMPLE MODEL

In order to characterize our results, the following empirical functional form is fitted to the 

data:

 

B

1 Q

C

Q

A

I(Q)

n m



 

  . (1)

The first term describes Porod scattering from clusters and the second term describes 

scattering from polymer chains. This second term characterizes the polymer/solvent 

interactions and therefore the thermodynamics and is of interest here. The two multiplicative 

factors A and C, the incoherent background B and the two exponents n and m are used as 

fitting parameters. The final parameter  is a correlation length for the entangled polymer 

chains. It gives an estimate of the entanglement length (average distance between 

entanglements). Non-linear least squares fits to the empirical functional form yield  = 20 Å, 
376

and m = 1.9 for the 4 % PEO/d-water sample. This empirical model should be used with 

caution since it does not reproduce the Guinier limit properly (except for m = 2). 

4. THE CORRELATION LENGTH

The correlation length  decreases with increasing polymer volume fraction because the 

entanglement length increases.  goes from close to 80 Å at low polymer volume fraction to 

under 10 Å at high volume fraction. At low polymer volume fraction and at high-Q, the 

chains radius of gyration is given by Rg = 2 = 113 Å and the end-to-end chain distance is 

R1n = 

6

Rg = 277 Å. 

20

40

60

80

100

0.02 0.04 0.06 0.08 0.1 0.12

PEO/D2O, Mw = 100,000 g/mole

20 oC

Correlation Length (Å)

PEO Weight Fraction

Y = M0*XM 1

M0 1.5291

M1 -0.74994

R 0.99863

Figure 2: Variation of the correlation length with polymer volume fraction. 

5. THE SPINODAL TEMPERATURE

The correlation length  and the coefficient C increase with increasing temperature T due to 

increased composition fluctuations when approaching phase separation. The PEO/d-water 

system is characterized by a lower critical solution temperature (LCST), i.e., it phase 

separates upon heating. The spinodal (phase separation) temperature Ts is obtained when C 
377

diverges; it can be accurately estimated from the intercept of a C-1vs T

-1

plot of data taken at 

various temperatures. In this case of 1 % PEO/d-water, one finds Ts = 127 oC. 

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.0024 0.0026 0.0028 0.003 0.0032 0.0034 0.0036

PEO/D2O, Mw = 100,000 g/mole

1 %

1/C 

Inverse Temperature (K-1)

T

s

 = 127 oC

Figure 3: Variation of the high-Q inverse intensity C-1 with inverse temperature T-1. The 

intercept represents the spinodal (phase separation) boundary line. 

6. THE EXCLUDED VOLUME PARAMETER

Our fitting results (high-Q Porod exponent) for the 1 % PEO/d-water solution show that 

chains are mostly swollen at low temperatures (m = 1.85 which corresponds to an excluded 

volume parameter around  = 1/m = 0.54) and change to theta conditions at high 

temperatures (m = 2.02 which corresponds to an excluded volume parameter around  = 0.5) 

as the spinodal temperature is approached. 
378

1.8

1.85

1.9

1.95

2

2.05

0 20 40 60 80 100

PEO/D2O, Mw = 100,000 g/mole

1 %

Porod Exponent m 

Temperature (oC)

Figure 4: Variation of the Porod exponent with temperature. Polymer chains change from a 

swollen state to a “theta” condition as the spinodal temperature is approached. 

7. BRANCH OF THE PHASE DIAGRAM

The spinodal temperature Ts was obtained from the various PEO volume fraction samples 

that were measured. A branch of the LCST phase diagram was obtained. What is interesting 

is that the phase boundary line Ts is estimated through extrapolation (i.e., before reaching it). 

For some of our samples, Ts happens to be above the boiling temperature of water (and 

therefore unreachable except when measurements are made inside a pressure cell). The 

SANS technique is a good monitor of phase separation because it is sensitive to composition 

fluctuations which get enhanced close to phase boundary lines. 
379

50

100

150

200

250

0 5 10 15 20 25

PEO/D2O, Mw = 100,000 g/mole

Data and Interpolation

Spinodal Temperature (

oC)

PEO Volume Fraction (%)

Two-phase region

One-phase region

Figure 5: Limited branch of the phase diagram for the PEO/d-water polymer solution system. 

Phase separation is obtained upon heating (LCST behavior). 

8. POLYMER SOLUTION THERMODYNAMICS

Polymer solutions can phase separate upon heating (LCST behavior) or upon cooling (UCST 

behavior). Polymers that dissolve in organic solvents tend to be characterized by a UCST 

whereas water-soluble polymers tend to follow LCST thermodynamics. The Flory-Huggins 

approach is a mean-field theoretical model that predicts phase separation behavior. This 

model will be discussed later for polymer blends. 

9. THE ZERO AVERAGE CONTRAST METHOD

The zero average contrast method (also called “high concentration” method) uses variation of 

the fraction of deuterated polymer and deuterated solvent but keeping the total polymer 

concentration (or volume fraction) constant to measure the single-chain form factor even at 

high concentrations because the interchain contribution cancels out. 

A series of PEO/water solutions were prepared whereby the total polymer fraction was kept 

constant (volume fraction of 4 %) but the relative amount of dPEO/hPEO was varied. In 
380

order to isolate the single-chain contribution, we used mixtures of D2O and H2O solvent 

molecules that match the average polymer scattering-length density in each case. For such 

hPEO/dPEO/H2O/D2O mixtures, the scattering intensity is given by:

 B   B  n v P (Q)  B  n v P (Q)

d

d (Q)

P P P T

2

P P P S P

2

P

2

  P      





(2)

  















 





  

P

2 H

H

P

2 D

D

2

BP

 

2

P

H

H

P

D

D

2

BP 















 





  

 

















      

S

S

D

D

D D S

v

b

v

b

 

















      

S

S

H

H

H H S

v

b

v

b

.

Here, bH and bD are the scattering lengths for the hPEO and dPEO monomers, vH and vD are 

the corresponding specific volumes, and H and D are the corresponding polymer volume 

fractions (and similarly for the solvent scattering length density bS/vS). In order to arrive at 

this formula, it was assumed that the protonated and deuterated polymer degrees of 

polymerization and specific volumes are matched. The degree of polymerization used here, 

nP, represents the value for the two mixed polymer species (nH = nD = nP). The total polymer 

volume fraction P (P = H+D) and polymer specific volume vP (vP = vH = vD) have also 

been defined. The single-chain form factor PS(Q) and the total-chain structure factor 

(including intra-chain and inter-chain contributions) PT(Q) have also been defined. The 

average contrast match condition zeroes the second term in the cross section equation leaving 

only the first term proportional to PS(Q).

This formula assumes that deuteration does not affect chain structure or interactions. This is 

obviously an assumption for our hydrogen-bonded system. 
381

Figure 6: The series of PEO/water solutions for the average contrast match series. Mixtures 

of deuterated and non-deuterated polymers (dPEO and hPEO) and solvents (D2O and H2O) 

are used. 

Specific values for the defined parameters for our system are as follows. 

nhPEO = 2273, ndPEO = 2125, (3)

bhPEO = 4.14*10-13 cm, bdPEO = 45.78*10-13 cm, 

bH2O = -1.67*10-13 cm, bD2O = 19.14*10-13 cm, 

vhPEO = vdPEO = 38.94 cm3/mol, 

vH2O = vD2O = 18 cm3/mol. 

The four possible contrast factors corresponding to the 4 corners in the figure are as follows: 

3 4

av

2

H2O

H2O

dPEO

dPEO N 9.66*10 mol / cm

V

b

V

b 

 















 (4)

3 4

av

2

D2O

D2O

hPEO

hPEO N 5.50*10 mol / cm

V

b

V

b 

 

















4 4

av

2

H2O

H2O

hPEO

hPEO N 2.38*10 mol / cm

V

b

V

b 

 

















5 4

av

2

D2O

D2O

dPEO

dPEO N 7.53*10 mol / cm

V

b

V

b 

 















 .

4 % dPEO/H2O

4 % hPEO/D2O 4 % hPEO/H2O

4 % dPEO/D2O

average contrast

match line

63.5 % D2O /

36.5 % H2O

17.2 % D2O /

82.8 % H2O

50 % dPEO /

50 % hPEO
382

Here, we have multiplied by Avogadro’s number (Nav = 6.022*1023 molecules/mol) for 

convenience. The strongest neutron contrasts correspond to the two mixtures: dPEO/H2O and 

hPEO/D2O. Contrasts corresponding to the other two mixtures are much lower. The higher 

incoherent background is found in the samples with the most hydrogen (i.e., with nondeuterated solvent). 

Data from a specific 4 % PEO/water mixture with 50 % dPEO/50 % hPEO and 63.5 % D2O / 

36.5 % H2O are shown in a figure. This mixture is represented by a circle on the average 

contrast match line in the same figure. This sample is characterized by the single-chain 

scattering feature only. The low-Q feature representing clustering has mostly disappeared. 

This method is useful for isolating single-chain properties in semi-dilute (and even 

concentrated) polymer solutions. 

1

10

0.01 0.1

SANS Data

Scattering Intensity (cm-1

)

Scattering Variable Q (Å-1)

4% PEO/water, T=10oC

50 % dPEO/50 % hPEO

63.5 % D2O/36.5 % H2O

Figure 7: SANS data from a PEO/water sample on the average contrast match line. Mixtures 

of dPEO/hPEO and D2O/H2O are used to cancel out scattering from the clusters leaving 

scattering from single polymer chains only. Mw = 100,000 g/mol for both the dPEO 

(deuterated) and hPEO (non-deuterated) polymers. 

Nonlinear least-squares fit of this data to the Gaussian chain model with excluded volume 

(described before) gave a segment length of aPEO = 6.7 Å and an excluded volume parameter 

of  = 0.51. Based on these numbers, the radius of gyration can be estimated as 
383

137 Å

(2 1)(2 2)

a n

R

0.5 1

PEO PEO

g



   



. The PEO chain degree of polymerization nPEO = 2200 has 

been used. 

Fit to the simple “empirical model” form (described before) gave a correlation length  = 

92.8 Å and a Porod exponent m = 2.06. The radius of gyration can be estimated here also as 

Rg = 

2

 = 131 Å. This shows acceptable agreement. 

REFERENCES

P. Flory, “Principles of Polymer Chemistry”, Cornell University Press, Ithaca, 1953.

B. Hammouda, D. Ho, and S. Kline, “SANS from Poly(ethylene Oxide)/Water Systems”, 

Macromolecules, 35, 8578-8585 (2002).

B. Hammouda and D. Ho, “Insight into Chain Dimensions in PEO/Water Solutions”, J. 

Polym. Sci., Polym. Phys. Ed. 45, 2196-2200 (2007).

QUESTIONS

1. What is the high-Q expansion of the Debye function (form factor for Gaussian coil)?

2. What standard plot is used to obtain the radius of gyration, the correlation length, the 

persistence length?

3. What is the meaning of the correlation length?

4. What does it mean to refer to the PEO/water solution as characterized by an LCST phase 

diagram?

5. A Porod exponent of 5/3 is an indication of what type of polymer chains?

6. The high-Q SANS data is characteristic of what type of interactions in polymer solutions?

ANSWERS

1. The high-Q expansion of the Debye function is: 

Lim P(Q ) 2/(Q R )

2

g

2    . 

2. The Guinier plot is used to obtain the radius of gyration, the Zimm plot is used to obtain 

the correlation length, and the Kratky-Porod plot is used to obtain the persistence length. 

3. The correlation length is the average distance between entanglement points. 

4. The PEO/water solution is characterized by a Lower Critical Solution Temperature phase 

diagram means that phase separation occurs upon heating. 

5. A Porod exponent of 5/3 is an indication of fully swollen polymer chains. 

6. The high-Q SANS data is characteristic of solvent/polymer interactions (the so-called 

solvation shell) and therefore of the thermodynamics of mixing. 
384

Chapter 38 - SANS FROM POLYMER BLENDS

Polymer blends are mixtures of homopolymers for which mixing happens at the molecular 

level. Two polymers mix only when they are compatible with each other (i.e., when their 

chemical units are similar) and when they correspond to conditions in the mixed region of the

phase diagram. The process of polymer blends demixing (phase separation) is discussed. 

1. THE FLORY-HUGGINS MODEL

Polymer thermodynamics are described by the Gibbs free energy density which consists of 

two main contributions: an entropic part (combinatorial mixing of the monomers) and an 

enthalpic part (interactions between monomers). The Flory-Huggins equation for polymer 

blends gives a simple expression for the Gibbs free energy density:

1 2

0

2

2 2

2

1

1 1

1

B v

(T)

ln( )

n v

ln( )

k T n v

G

 



 



 





 

. (1)

n1, 1 and v1 are the degree of polymerization (i.e., number of monomers per chain), volume 

fraction and specific volume for monomer 1 and n2, 2 and v2 are the equivalent quantities for 

monomer 2. 12 is the usual Flory-Huggins interaction parameter and v0 is a “reference” 

volume. Note that 1 + 2 = 1. 

The SANS technique is sensitive to composition fluctuations and is therefore a good monitor 

of phase separation thermodynamics. The scattering intensity is related to the second 

derivative of the Gibbs free energy density with respect to composition. The scattering 

function in the forward direction (Q = 0) is given by:

1 1 1 2 2 2 0

2

1

2

B

1

v

(T)

2

n v n v

G

k T

S (Q 0)

  

















 

  . (2)

Phase separation thermodynamics are buried in the temperature-dependent Flory-Huggins 

parameter 12. 
385

Figure 1: Schematic representation of a polymer blend mixture. 

2. BINODAL AND SPINODAL LINES

The phase separation thermodynamics of polymer mixtures is characterized by two lines: the 

binodal and the spinodal lines. The binodal line corresponds to generalized phase separation 

(at large length scales) and the spinodal line corresponds to the onset of density fluctuations 

(at local length scales) leading the phase separation. The region between the binodal and 

spinodal lines is referred to as the nucleation-and-growth region. 

The chemical potential 1 is the first derivative of the Gibbs free energy density G with 

respect to the volume fraction 1:

B B 1

1 G

k T

1

k T 







(3)

1

0

1

2 2 0

1

1 1 2 2

1

B 1 1

1

v

(T)

(1 )

v

(T)

n v

1

log(1 )

n v n v

1

log( )

k T n v





  



   



  





    

The binodal phase separation line is obtained when the two minima of G() have the same 

tangent slope line. 

The osmotic pressure 1 corresponds to the second derivative of G with respect to 1:

1 1 1 2 1 2 0

2

1

2

B B

1

v

(T)

2

n v n (1 )v

G

k T

1

k T

  

 



















. (4)

Polymers Blend
386

This expression also represents the inverse of the structure factor in the forward direction S

-

1

(Q=0). The spinodal phase transition line is obtained when the condition 1=0 (or S(Q=0) –> 

) is met, i.e., it corresponds to the inflection points on the G(1) curve. 

3. THE RANDOM PHASE APPROXIMATION

The Random Phase Approximation (RPA) model is a mean field theory that describes 

polymer mixtures in the homogeneous phase region. It is a generalization of the FloryHiggins equation to finite momentum transfer Q.

1 1 1 1 2 1 2 2 0

1

v

(T)

2

n v P (Q) n (1 )v P (Q)

S (Q)

   

 











(5)

P1(Q) is the so-called Debye function for Gaussian random polymer coils. 

 

2

g1

2

2

g1

2

4

g1

4

1

exp( Q R ) 1 Q R

Q R

2

P (Q)     . (6)

Rg1 is the radius of gyration for component 1. The radius of gyration can be expressed in 

terms of the statistical segment length a1 (think monomer size) as Rg1

2 = na12

/6. Precisely, the 

statistical segment length is the length for which the polymer chain is represented by a 

random walk. The Debye function is used because chains in compatible polymer blends do 

not swell or collapse. 

The SANS scattering cross section is given in terms of an absolute macroscopic cross section 

(units of cm-1) as:

S(Q)

v

b

v

b

d

d (Q)

2

2

2

1

1

















 





(7)

where b1/v1 is the scattering length density for monomer 1 and the term (b1/v1-b2/v2)

2

is the 

contrast factor. This scattering cross section is also referred to as the scattering intensity I(Q).

4. POLYOLEFIN BLENDS

Polyolefins are a wide class of polymeric materials that find uses in many aspects of our 

daily lives. These are polymers containing C=C double bonds either in an aromatic ring or in 

the aliphatic chain. These double bonds can be saturated (i.e., broken) through the 

hydrogenation process. Gentle interactions among polyolefin blend mixtures make them 

ideal “mean field” systems to investigate phase separation thermodynamics. The mixed onephase region is easily accessible for a wide range of temperatures and compositions. 
387

Polyolefin blends are characterized by an Upper Critical Spinodal Temperature (UCST); i.e., 

they phase separate upon cooling. Phase separation occurs either by spinodal decomposition 

(sinusoidal composition fluctuations) or by nucleation and growth (phase coarsening around 

nucleating centers). The SANS method is used here to map out the spinodal temperature for a 

polyolefin blend mixture. 

5. A POLYOLEFIN BLEND

The polyolefin blend system considered here is polyethylbutylene mixed with deuterated 

polymethylbutylene (hPEB/dPMB) (Lefebvre et al, 2000). The PMB polymer is deuterated 

for enhanced neutron contrast. The chemical formula for hPEB is [-CH2-CH(C2H5)-CH2-

CH2]- and for dPMB is [-CHD-CD(CH3)-CD2-CHD]-. An hPEB/dPMB polymer blend with 

molecular weights of Mw = 40,100 g/mol and 88,400 g/mol and composition of hPEB = 0.57 

and dPMB = 0.43 was measured by SANS. The stoichiometry of this sample is C6H12 for 

hPEB and close to C5H4.5D5.5 for dPMB as measured by NMR.

The scattering intensity at various temperatures is shown below. Scattering increases for 

decreasing temperature, due to the approach to the phase transition (spinodal) line. 

0

20

40

60

80

100

120

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

hPEB/dPMB, Mw=40,100 g/mole/88,400 g/mole

10 oC

20 oC

30 oC

40 oC

50 oC Scattering Intensity (cm-1

)

Scattering Variable Q (Å-1)
388

Figure 2: SANS scattered intensity for the hPEB/dPMB sample at various temperatures. The 

intensity increases as the temperature is lowered (UCST system)

The Zimm plot (I-1(Q) vs Q2) yields an extrapolated value of the forward scattering I(Q = 0). 

0

0.05

0.1

0.15

0.2

0 0.0005 0.001 0.0015 0.002 0.0025

hPEB/dPMB, Mw=40,100 g/mole/88,400 g/mole

Data at 10 oC 1/d(Q)/d

Q

2

 (Å-2)

intercept gives d(Q=0)/d

 slope gives 

radius of gyration

Figure 3: The Zimm plot yields I(Q=0) for the intercept. 

A plot of I-1(Q=0) vs T-1(T is in K) yields the spinodal temperature from the intercept; i.e., 

when I(Q=0) diverges or I-1(Q=0) is equal to zero. An estimated spinodal temperature of 220 

K = -53 oC is obtained for this sample. 
389

0.005

0.006

0.007

0.008

0.009

0.01

0.003 0.0031 0.0032 0.0033 0.0034 0.0035 0.0036

hPEB/dPMB, Mw=40,100 g/mole/88,400 g/mole

1/I(0) 

1/I(0) (cm)

Inverse Temperature (K-1)

T

s

 = 220 K

Figure 4: Variation of the inverse intensity (extrapolated to Q = 0) with the inverse 

temperature (in degree K). The spinodal temperature is obtained for 1/I(0)=0. 

The reduced SANS data for every temperature were fit to the RPA functional form for the 

binary blend mixture with the following input parameters: 

Degrees of polymerization: n1 = 525 n2 = 1105

Volume fractions: 1 = 0.57 2 = 0.43

Specific volumes: v1 = 97.5 cm3/mol v2 = 81.9 cm3/mol

Scattering lengths: b1 = -4.99*10-13 cm b2 = 53.1*10-13 cm.

The specific volumes were obtained as the ratios of the monomer molecular weights to their 

densities. Note that the scattering length for hPEB is negative whereas that for dPMB is 

positive.

Fit results are:

Segment lengths: a1 = 8 Å a2 = 8 Å

Interaction parameter:

5

0

1.84*10

T

0.0106

v

(T)   

 



(T in deg K).
390

The approach used here is approximate. The specific volumes were assumed to be 

independent of temperature. The purpose of the argument being made here is to show how 

the Flory-Huggins theory could be used to obtain a phase diagram. Note that in this 

approach, the “reference” volume v0 needs to be specified only if 12(T) needs to be known. 

In this case, v0 is approximated as 

v0  v1v2

.

1.4 10-5

1.5 10-5

1.6 10-5

1.7 10-5

1.8 10-5

1.9 10-5

0.003 0.0031 0.0032 0.0033 0.0034 0.0035 0.0036

hPEB/dPMB, Mw=40,100 g/mole/88,400 g/mole

Data and Fit

hPEB/bPMB/v

0

Inverse Temperature (K-1)

Figure 5: Variation of the Flory-Huggins 

0 v

(T) 

parameter with inverse temperature. 

6. GIBBS FREE ENERGY DENSITY AND PHASE DIAGRAM

Based on the Flory-Huggins parameter obtained from one hPEB/dPMB sample, the Gibbs 

free energy density and the phase diagram can be constructed. This assumes that the FloryHuggins parameter does not depend on polymer blend composition. This is a fairly good 

assumption for polyolefin blends. 

Based on the figure showing the Gibbs free energy density, the 220 K (= -53 oC) temperature 

corresponding to the spinodal temperature of this sample is also the critical temperature of 

the phase diagram. The 270 K temperature is well in the one-phase region and the 200 K 

temperature is well into the spinodal region. The spinodal points are located at the two 
391

inflection points (where the second derivation of the free energy is zero). The two binodal 

points correspond to two-points on the curve that share the same tangent as shown in the 

figure. 

Recall that the spinodal condition is given by:

0

v

(T)

2

n v n v

G

k T 1 1 1 2 2 2 0

2

1

2

B





















   

. (8)

The binodal condition is obtained for:

1

B

1

A

A B

A B G( ) G( ) G( ) G( )



 





 



  

  

. (9)

At fixed T, the two binodal volume fractions are called A and B. 

-6 10-6

-5 10-6

-4 10-6

-3 10-6

-2 10-6

-1 10-6

0

1 10-6

2 10-6

0 0.2 0.4 0.6 0.8 1

hPEB/dPMB, Mw=40,100 g/mole/88,400 g/mole

270 K

220 K

200 K

Gibbs Free Energy Density G(

1)/k

B

T

Volume Fraction 

1

spinodal points

binodal points

critical point
392

Figure 6: Variation of the Gibbs free energy density with volume fraction 1 = hPEB for 

hPEO/dPMB blends. The measured sample with 0.57/0.43 volume fraction is at the critical 

point. 

Once the Gibbs free energy density is known, the demixing phase diagram can be mapped 

out. 

-50

0

50

100

150

200

250

300

0 0.2 0.4 0.6 0.8 1

hPEB/dPMB, Mw=40,100 g/mole/88,400 g/mole

binodal line 

spinodal line 

Temperature (K)

Volume Fraction 

1

 two-phase spinodal region

 one-phase region

spinodal

 line

binodal

 line

nucleation 

and growth 

region

Fig

ure 7: Variation of the spinodal and binodal temperatures with volume fraction 1 = hPEB for 

hPEO/dPMB blends. The spinodal region is underneath the binodal line. Vertical lines 

continue from the previous figure. 

REFERENCES

A.A. Lefebvre, J.H. Lee, N.P. Balsara, B. Hammouda, “Fluctuations in Highly Metastable 

Polymer Blends”, J. Polym. Sci., Polym. Phys. Ed. 38, 1926-1930 (2000).
393

QUESTIONS

1. What scattering method detects the spinodal temperature? How about the binodal 

temperature? 

2. What equation relates scattering methods and polymer mixture thermodynamics?

3. What relation defines the spinodal condition?

4. Based on the Flory-Huggins theory of phase separation, can you guess what would be the 

main difference between phase diagrams for a polymer blend and a polymer solution? 

5. Polymer phase separation occurs either through spinodal decomposition or nucleation and 

growth? Define these two regions of the phase diagram. 

6. Does the RPA apply when the sample is in the two-phase region?

7. What is the morphology characteristic of a fully phase separated polymer blend?

ANSWERS

1. The spinodal temperature corresponds to phase separation through local scale fluctuations. 

It is best detected by the SANS method. The binodal temperature corresponds to phase 

separation at a large length scale. It is best detected by light scattering. The binodal 

temperature can also be detected by SANS. 

2. Scattering and polymer mixture thermodynamics are related by the following equation: 

1 k T G  S (Q 0)

2 1

1

2

B    



. Here G is Gibbs free energy density, 1 is the volume 

fraction of component 1 and S(Q=0) is the scattering factor for Q=0 (i.e., in the 

thermodynamics limit). 

3. The spinodal condition is defined by the inflection point of G(1) vs 1 where G is the 

Gibbs free energy density. This condition is expressed as 

G 0

2

1

2

   .

4. The thermodynamics of phase separation of a polymer solution can be described by the 

Flory Huggins theory with n2 = 1 (assuming that component 2 is the solvent). This tends to 

skew the phase diagram for polymer solutions towards the dilute limit. The critical point for 

polymer solutions takes place at low polymer volume fractions. 

5. The spinodal region is underneath the spinodal line in the phase diagram (temperature T vs

volume fraction 1). The nucleation and growth region is between the binodal and spinodal 

lines. 

6. The RPA breaks down when the sample is in the two-phase region.

7. A fully phase separated polymer blend is characterized by a droplet morphology (droplets 

of one component in the medium of the other). A fully phase separated sample is also “hazy” 

since the droplets’ size becomes comparable to the wavelength of visible light (fraction of a 

micron). 
394

Chapter 39 - SANS FROM BLOCK COPOLYMERS

Scattering from block copolymer systems is characterized by a scattering peak representing 

the average inter-distance between domains in the ordered phase region. The scattering peak 

persists in the disordered phase region where it represents the “correlation hole” effect 

(Leibler, 1980). Scattering from a diblock copolymer solution is discussed here. 

1. POLYSTYRENE-POLYISOPRENE COPOLYMER IN DOP SOLUTION

The polystyrene-polyisoprene (hPS-hPI) diblock copolymer considered here has typical 

molecular weights (Mw=11,000 for hPS and Mw=17,000 g/mol for hPI). Since the order-todisorder phase transition temperature (ODT) for this copolymer is high, it was dissolved in 

dioctyl phthalate (DOP) at a copolymer volume fraction of 0.65. This brought the ODT down 

to an easily reachable value. DOP is a non-volatile good solvent for both hPS and hPI.

Even though none of the blocks is deuterated, the hPS-hPI copolymer is characterized by a 

reasonable “natural” neutron contrast in DOP. 

Table 1: Summary of the scattering lengths and specific volumes. 

Notation Formula Scattering Length

(cm)

Density

(g/cm3)

Specific Volume

(cm3/mol)

Polystyrene hPS C8H8 0.23*10-11 1.04 100

Polyisoprene hPI C5H8 0.33*10-12 0.91 75

Dioctylphthalate DOP C24H38O4 4.07*10-12 0.98 397

The three contrast factors are given by:

N 2.08*10 mol / cm .

v

b

v

b 4 4

av

2

hPI

hPI

hPS

hPS 

 















 (1)

N 9.75*10 mol / cm .

v

b

v

b 5 4

av

2

DOP

DOP

hPS

hPS 

 

















N 2.05*10 mol / cm .

v

b

v

b 5 4

av

2

DOP

DOP

hPI

hPI 

 

















The contrast factors were multiplied by Avogadro’s number Nav in order to keep the order of 

magnitude reasonable. Some of these contrast factors are used in the data analysis process. 
395

2. SANS DATA

A SANS data set was acquired using one instrument configuration for which the prominent 

scattering peak occurs in the middle of the scattering window. Sample temperature was 

varied between 20 oC and 70 oC at 10 oC intervals. 

0

5

10

15

20

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

 hPS-hPI/DOP 

20 oC 

30 oC 

40 oC 

50 oC 

60 oC 

70 oC 

 Scattering Intensity (cm-1) 

Scattering Variable Q (Å-1)

Figure 1: SANS data from the hPS-hPI/DOP diblock copolymer solution at various sample 

temperatures. 

This order-to-disorder transition temperature (ODT) is the equivalent of the spinodal 

temperature for polymer blends displaying a UCST behavior. When temperature is raised, the 

copolymer ordered phase melts into a disordered phase. 

An empirical model characterized by a peaked behavior is used first to fit the SANS data:

 

B

1 | Q Q |

C

I(Q)

m

0



  

 . (2)

The fitting parameter C is a measure of composition fluctuations. C is high in the ordered 

phase and decreases in the disordered phase. A simple way of estimating the ODT is through 

a plot of 1/C vs 1/T (T is the absolute temperature). Extrapolation to the condition where the 
396

scattering peak intensity C blows up yields an ODT of 29 oC. The hPS-hPI/DOP is ordered 

below that temperature. The measured sample phase separates into a lamellar morphology. 

0

0.02

0.04

0.06

0.08

0.1

0.0028 0.0029 0.003 0.0031 0.0032 0.0033

 hPS-hPI/DOP 

T

ODT

=29 oC 

1/C

1/T (K)

Y = M0 + M1*X

M0 0.71318

M1 -215.18

R 0.99054

Figure 2: Determination of the order-to-disorder temperature. 

3. THE RPA MODEL FOR A DIBLOCK COPOLYMER SOLUTION

Consider the DOP solvent (component 3) as the background component in the Random 

Phase Approximation (RPA) model (Akcasu-Tombakoglu, 1990; Hammouda, 1993). The 

incompressible RPA model applies to the disordered phase only. The relevant equations in 

matrix form are reproduced here as a review. 

S (Q) S (Q) v(Q)

1

0

1

 

 

(2)

S(Q) 1 S (Q).v(Q) .S (Q)

0

1

0



 
397















S (Q) S (Q)

S (Q) S (Q)

S (Q)

0

22

0

21

0

12

0

11

0

S (Q) n v P (Q)

i i i i

0

ii  

S (Q) n v n v P (Q) i i i j j j ij

0

ij   

 

2

g i

2

2

g i

2

4

g i

4

i

exp( Q R ) 1 Q R

Q R

2

P (Q)    

   















  















  

 

2

g j

2

2

g j

2

2

g i

2

2

g i

2

ij i j

Q R

1 exp Q R

Q R

1 exp Q R

P (Q) F (Q)F (Q)

6

n a

R

2

2 i i

gi 

0

13

0

33

11

v

2

S (Q)

1

v (Q)



 

0

23

0

33

22

v

2

S (Q)

1

v (Q)



 

0

2 3

0

1 3

0

1 2

0

3 3

1 2 S (Q) v v v

1

v (Q)











 

   

     

0

2 2 2 1

0

2 1 1 1

0

1 2 2 2

0

1 1 1 2

0

2 2 2 2

0

2 1 1 2

0

1 2 2 1

0

1 1 1 1

0

2 2 2 1

0

2 1 1 1

0

1 2

0

2 2 2 2

0

2 1 1 2

0

1 1

1 1 1 v S v S 1 v S v S v S v S v S v S

S 1 v S v S S v S v S

S (Q)

      

   



   

     

0

2 2 2 1

0

2 1 1 1

0

1 2 2 2

0

1 1 1 2

0

2 2 2 2

0

2 1 1 2

0

1 2 2 1

0

1 1 1 1

0

1 1 1 2

0

1 2 2 2

0

2 1

0

1 1 1 1

0

1 2 2 1

0

2 2

2 2 1 v S v S 1 v S v S v S v S v S v S

S 1 v S v S S v S v S

S (Q)

      

   



   

     

0

2 2 2 1

0

2 1 1 1

0

1 2 2 2

0

1 1 1 2

0

2 2 2 2

0

2 1 1 2

0

1 2 2 1

0

1 1 1 1

0

1 2 2 1

0

1 1 1 1

0

1 2

0

1 2 2 2

0

1 1 1 2

0

1 1

1 2 1 v S v S 1 v S v S v S v S v S v S

S v S v S S 1 v S v S

S (Q)

      

    



S (Q) S (Q) 2 S (Q)

d

d (Q)

2 2 1 2 1 2

2

1 1 2

2

 1     





.

Note that a Q-independent constant has been added to the scattering intensity to account for 

the incoherent scattering background. These equations are used to fit the SANS data in the 

disordered phase. 
398

4. SANS DATA ANALYSIS

Analysis of the SANS data was performed using the RPA model and the following input 

parameters:

Component 1: hPS, nhPS=106, hPS=0.23, vhPS=100 cm3/mol (3)

Component 2: hPI, nhPI=250, hPI=0.42, vhPI=75 cm3/mol

Component 3: DOP, nDOP=1, DOP=0.35, vDOP=397 cm3/mol.

The fitting parameters obtained for the 70 oC temperature are included here:

ahPS = 8.45 Å (4)

ahPI = 6.44 Å

aDOP = 29.15 Å

3

0

hPS/ DOP 0.0104 mol/cm

v





3

0

hPI/ DOP 0.040 mol/cm

v





3

0

hPS/ hPI 0.013mol/cm

v





B = 0.828 cm-1.

These parameters are reasonable estimates given the fact that so many parameters were 

floated at the same time. More precise parameters could be obtained if pure hPS/DOP, 

hPI/DOP and/or hPS/hPI samples were measured first. The 70 oC temperature is above the 

ODT (of 29 oC) and well within the single-phase region whereby the copolymer 

microstructure has melted. The observed SANS peak is due to the correlation hole effect. 
399

0

2

4

6

8

10

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

 hPS-hPI/DOP at 70 oC 

RPA Model Fit 

SANS Data 

 Scattering Intensity (cm-1) 

Scattering Variable Q (Å-1)

Figure 3: Superposition of the RPA model fit and the SANS data for the 70 oC temperature. 

5. THE ORDERED PHASE

The ordered phase is characterized by three main morphologies: spherical, cylindrical or 

lamellar domains. Other morphologies such as the gyroid and double-diamond phases are 

also known. The RPA model can predict the ODT line but does not apply in the ordered 

phase region. It yields predictions from the disordered phase region. Other more 

sophisticated models such as the self-consistent field theory can predict the various phases in 

the ordered phase region. The critical condition for a symmetric diblock copolymer is given 

by n = 10.5. Here  is the Flory-Huggins interaction parameter and n is the degree of 

polymerization. Recall that this condition is n = 2 for symmetric polymer blends. Above 

that condition, phase separation occurs. Both the spinodal phase separation in polymer blends 

and the order-to-disorder transition (ODT) in copolymers are second order phase transitions 

since they involve the second derivative of the Gibbs free energy density 

k T/S(0) G 0

2

1

2

B

   

where 1 is the order parameter (polymer composition). The ODT 

is also referred to as a “weak” first order phase transition. 
400

Figure 4: Representation of the disordered phase and the ordered phases.

Disordered Block Copolymer

Ordered Block Copolymer
401

Figure 5: Schematic representation of the copolymer phase boundaries for the main 

morphologies. The copolymer composition is the volume fraction of one of the blocks. 

The position of higher order peaks (relative to the first order peak) varies for the various 

copolymer morphologies. The first order peaks’ sequence occurs as follows:

-- 1, 2, 3, 5for the spherical morphology (cubic lattice).

-- 1, 3, 4, 7for the cylindrical morphology (hexagonal close-packed lattice).

-- 1, 2, 3, 4for the lamellar morphology.

Due to the broad wavelength spread characterizing SANS (and therefore of the standard 

deviation of the Q resolution), higher order peaks are smeared and hard to resolve. 

REFERENCES

L. Leibler, “Theory of Microphase Separation in Block Copolymers”, Macromolecules 13, 

1602-1617 (1980).

A.Z. Akcasu and M. Tombakoglu, “Dynamics of Copolymer and Homopolymer Mixtures in 

Bulk and in Solution via the Random Phase Approximation”, Macromolecules 23, 607-612 

(1990).

B. Hammouda, "SANS from Homogeneous polymer Mixtures: A Unified Overview", 

Advances in Polymer Science 106, 87 (1993). 

Disordered 

(1-phase)

ordered 

(2-phase)

lamellae

cylinders

spheres

Copolymer Composition

Copolymers

cylinders

spheres

lamellae
402

QUESTIONS

1. The SANS data from copolymers are characterized by a peak in what phase (ordered or 

disordered or both)? Why?

2. What is the correlation hole effect?

3. What are the main morphologies formed in the ordered phase region?

4. What is the most likely morphology formed by a symmetric diblock copolymer? Why?

5. What is the critical condition (for phase separation) for n for symmetric diblock 

copolymers. Here  is the Flory-Huggins interaction parameter and n is the degree of 

polymerization. What is that condition for symmetric polymer blends?

6. What is the order of the spinodal and order-to-disorder phase transitions?

ANSWERS

1. The SANS data from copolymers are characterized by a peak in both the ordered and the 

disordered phases. The peak in the ordered phase is due to the periodic microphase separated 

morphology whereas the peak in the disordered phase region is due to the correlation hole 

effect. 

2. The correlation hole effect occurs when the nearest neighbors are excluded from the region 

right next to a particle. In copolymers, the second block excludes regions containing the first 

block. 

3. The main morphologies formed in the ordered phase region are spherical, cylindrical and 

lamellar. 

4. The most likely morphology formed by a symmetric diblock copolymer is lamellar 

because there is 50 % of each block. 

5. The critical condition (for phase separation) for symmetric diblock copolymers is n = 

10.5. That condition is n = 2 for symmetric polymer blends.

6. The spinodal transition for polymer blends and the order-to-disorder phase transition for 

copolymers are both second order phase transitions. The ODT is also referred to as a “weak” 

first order transition. 
403

Chapter 40 - SANS FROM TERNARY POLYMER BLENDS

A ternary polymer blend is a mixture of three homopolymers. The thermodynamics of 

mixing and phase separation can be investigated for ternary blends using the SANS 

technique and the Random Phase Approximation. 

1. SANS FROM A TERNARY POLYMER BLEND

Consider a ternary polymer blend consisting of deuterated polystyrene (dPS), non-deuterated 

polystyrene (hPS) and poly(vinyl methyl ether) (PVME) (Hammouda et al, 1992). The binary 

blend dPS/PVME is characterized by a Lower Critical Spinodal Temperature (LCST) 

whereas the binary blend dPS/hPS is characterized by a weak Upper Critical Spinodal 

Temperature (UCST). Here, the ternary blend was investigated with the goal of testing the 

RPA model for multi-component mixtures and investigating the phase separation of the 

binary hPS/PVME blend. This blend does not contain any deuteration and could not be 

investigated by SANS directly. 

A series of three samples was prepared. The characterization of the polymers used and of the 

volume fractions of the three samples are summarized in two tables. 

Table 1: Characterization of the polymers used.

Mw

(g/mol)

Mw/Mn Degree of 

Polymerization

Specific Volume (cm3/mol)

dPS 1.95*1051.02 1741 100

hPS 1.90*1051.04 1827 100

PVME 1.59*1051.30 2741 55.4

Table 2: Volume fractions for the three samples. 

Sample Number dPS PVME hPS

1 0.484 0.516 0

2 0.360 0.511 0.129

3 0.238 0.506 0.256

SANS data were taken from the three samples for a range of temperatures above the 

polystyrene glass transition temperature (i.e., above 100 oC). SANS data from sample 3 for a 

range of temperatures in the homogeneous phase region are show in a figure. The intensity is 

seen to increase when temperature is increased. This is a sign of LCST behavior. 
404

0

50

100

150

200

250

300

350

0 0.01 0.02 0.03 0.04 0.05 0.06

Ternary Blend 

160 oC 

155 oC 

150 oC 

145 oC 

135 oC 

Scattering Intensity (cm-1

)

Scattering Variable Q (Å-1)

Figure 1: SANS data taken from sample 3 for a range of temperatures. 

Variation of the forward scattering intensity with increasing temperature shows an interesting 

variation for the 3 samples. 
405

0

50

100

150

200

250

300

350

320 340 360 380 400 420 440

Ternary Blends 

sample 1 

sample 2 

sample 3 

Forward Scattering Intensity (cm-1

)

Temperature (K)

Figure 2: Variation of the forward scattering intensity 

d(Q  0) d

with increasing 

temperature for the three ternary blends. The lowest value of Q = 0.0085 Å-1is used to 

represent the forward scattering value. 

Sample 3 shows a decrease in intensity at first then an increase. This is due to the interplay 

between the LCST behavior for dPS/PVME and hPS/PVME and the UCST behavior for the 

dPS/hPS. 

2. THE TERNARY RPA MODEL

The ternary RPA equations are summarized here. Consider a ternary blend consisting of 

polymer 1, polymer 2 and polymer 3. The degrees of polymerization are noted n1, n2 and n3, 

the volume fractions are 1, 2 and 3, the specific monomer volumes are v1, v2 and v3, the 

radii of gyration are Rg1, Rg2 and Rg3, the form factors are P1(Q), P2(Q) and P3(Q) and the 

Flory-Huggins interaction parameters are 12, 23 and 13. 

The RPA equations for an incompressible ternary blend in the homogeneous phase are as 

follows. The non-interacting structure factors are given by:

S (Q) n v P (Q)

1 1 1 1

0

11  

(1)
406

S (Q) n v P (Q)

2 2 2 2

0

22  

S (Q) n v P (Q) 3 3 3 3

0

33   .

The form factors are given by the Debye function:

 

2

g1

2

2

g1

2

4

g1

4

1

exp( Q R ) 1 Q R

Q R

2

P (Q)    

(2)

 

2

g2

2

2

g2

2

4

g2

4

2

exp( Q R ) 1 Q R

Q R

2

P (Q)    

 

2

g3

2

2

g3

2

4

g3

4

3

exp( Q R ) 1 Q R

Q R

2

P (Q)     .

The excluded volume factors are:

0

13

0

33

11

v

2

S (Q)

1

v (Q)



  (3)

0

23

0

33

22

v

2

S (Q)

1

v (Q)



 

0

2 3

0

1 3

0

1 2

0

3 3

1 2 S (Q) v v v

1

v (Q)











  .

The partial structure factors for the fully interacting mixture are given by:

 

1 v S (Q)1 v S (Q) v S (Q)S (Q)

S (Q) 1 v S (Q)

S (Q)

0

2 2

0

1 1

2

1 2

0

2 2 2 2

0

1 1 1 1

0

2 2 2 2

0

1 1

1 1

  





(4)

 

1 v S (Q)1 v S (Q) v S (Q)S (Q)

S (Q) 1 v S (Q)

S (Q)

0

2 2

0

1 1

2

1 2

0

2 2 2 2

0

1 1 1 1

0

1 1 1 1

0

2 2

2 2

  





1 v S (Q)1 v S (Q) v S (Q)S (Q)

S (Q)v S (Q)

S (Q)

0

2 2

0

1 1

2

1 2

0

2 2 2 2

0

1 1 1 1

0

1 2 2 2

0

1 1

1 2

  



 .

The scattering cross section is given by:

S (Q) S (Q) 2 S (Q)

d

d (Q)

2 2 1 2 1 2

2

1 1 2

2

 1     





(5)

The relevant contrast factors are:
407

 

2

3

3

1

2 1

1 3

2

1

v

b

v

b

















       (6)

 

2

3

3

2

2 2

2 3

2

2

v

b

v

b

















       .

Note that the scattering cross section for incompressible blends is independent of the 

 

2

1  2

contrast factor. 

3. FITTING OF THE SANS DATA

The SANS data were fitted to the RPA model for ternary blends with the various known 

parameters (ni, i, vi, i

2

) with index i representing either dPS, hPS or PVME. The radii of 

gyration RgdPS, RghPS and RgPVME and the Flory-Huggins interaction parameters 

dPS/ PVME v0

, 

hPS/ PVME v0

and 

dPS/ hPS v0

were allowed to vary during the nonlinear least-squares fits. 

The various contrast factors are:

4 4

av

2

PVME

PVME

hPS

hPS N 1.79*10 mol / cm

v

b

v

b 

 















 (7)

3 4

av

2

PVME

PVME

dPS

dPS N 6.07 *10 mol / cm

v

b

v

b 

 

















3 4

av

2

hPS

hPS

dPS

dPS N 4.16*10 mol / cm

v

b

v

b 

 















 . 

Only two of these contrast factors are needed. Note that deuteration enhances the contrast 

factor and that Avogadro’s number Nav has been introduced to keep from handling very small 

numbers. 

Results of the fits are included here. The binary dPS/PVME blend (sample 1) gave a reliable 

estimate of the Flory-Huggins interaction parameter:

T

0.416 9.73*10

v

4

0 dPS/ PVME

  













  

. (8)

This interaction parameter decreases with inverse temperature (1/T) which is a signature of 

the LCST behavior. 

Previous measurements from a dPS/hPS binary blend and these measurements from ternary 

blends yielded:
408

T

0.0020 2.9*10

v

6

0 dPS/ hPS

   













  

. (9)

This interaction parameter for an isotopic blend is orders of magnitude smaller than typical 

ones for other blends. It increases with (1/T) which means that the dPS/hPS is characterized 

by a UCST behavior. 

The third interaction parameter obtained from fits of the data taken from samples 2 and 3 are:

T

0.422 10.3*10

v

4

0 hPS/ PVME

  













  

from sample 2 (10)

T

0.436 10.6*10

v

4

0 hPS/ PVME

  













  

from sample 3.

These values agree fairly well. The units of (/v0) are in cm3/mol. 

-30

-25

-20

-15

-10

-5

0

5

2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1

Ternary Blends 

dPS/PVME from sample 1 

hPS/PVME from sample 3 

/v

0

*10

5 mol/cm

3

103/T (K-1) 

Figure 3: Variation of the Flory-Huggins interaction parameters for dPS/PVME obtained 

from sample 1 and for hPS/PVME obtained from sample 3. 
409

The interaction parameters for the dPS/PVME and the hPS/PVME are close but not identical. 

The difference is due to the isotopic effect. 

4. SPINODAL TEMPERATURE

The spinodal temperature is reached when the denominator of the structure factors in the 

thermodynamics limit (i.e., for Q = 0) goes to zero. 

1 v S (0)1 v S (0) v S (0)S (0) 0

0

2 2

0

1 1

2

1 2

0

2 2 2 2

0

 1 1 1 1    . (11)

This gives the following estimates for the spinodal temperatures:

Ts = 433 K = 160 oC for Sample 1

Ts = 421 K = 148 oC for Sample 2

Ts = 416 K = 143 oC for Sample 3.

Note that this method is based on estimating the spinodal temperature from the one-phase 

(i.e., mixed phase) region. 

5. DISCUSSION

Using ternary blend mixtures allowed measurement of the Flory-Huggins interaction 

parameter for a pair of non-deuterated polymers (hPS/PVME). This binary blend mixture 

could not have been measured directly by SANS due to the very small contrast factor and to 

the overwhelming incoherent scattering background. The method presented here is a reliable 

method to measure the small isotopic effect on the polymer-polymer interaction parameter. 

The phase diagram for a ternary blend mixture can be represented by a triangle with the three 

components at the corners and temperature as the vertical axis (out of the page where the 

triangle is drawn). 

When supplemented with the RPA model, the SANS technique is useful for the investigation 

of the demixing (i.e., spinodal) phase transition. This method has been used to investigate 

many homogeneous polymer mixtures. 

REFERENCES

B. Hammouda, R. Briber and B. Bauer, “SANS from dPS/PVME/hPS Ternary Polymer 

Blends”, Polymer 33, 1785-1787 (1992). 
410

QUESTIONS

1. How does the scattering intensity vary for increasing temperature for a Lower Critical 

Spinodal Temperature (LCST) polymer blend mixture? How about for an Upper CST?

2. In the Random Phase Approximation (RPA) model, what is the size of the matrices for an 

incompressible ternary polymer blend?

3. Does the RPA model apply in the two-phase (i.e., phase separated) region?

4. Using the RPA model, could one work out the phase diagram for a ternary blend mixture?

ANSWERS

1. The scattering intensity increases for increasing temperature for an LCST polymer blend. 

The scattering intensity decreases for increasing temperature for a UCST blend. The 

scattering intensity always increases when approaching the spinodal temperature due to the 

buildup of composition fluctuations. 

2. A ternary polymer blend contains three components. The incompressibility assumption 

allows the elimination of one component (referred to as the “background” component). 

Therefore, the size of the RPA matrices is two. 

3. The RPA model does not apply in the two-phase region. It applies only in the 

homogeneous (i.e., mixed phase) region and breaks down at the spinodal line. 

4. The RPA model could be used to map out the phase separation lines (spinodal 

temperatures) for a ternary blend mixture. The phase diagram is triangular with one 

component at each corner and temperature as the third dimension. 
411

Chapter 41 – OTHER SANS FROM POLYMERS PROJECTS

A few more SANS from polymers projects in which this author was involved are described 

here. 

1. CO-SOLVENCY OF PNIPAM IN SOLVENT MIXTURES

Poly(N-isopropylacrylamide) polymer also referred to as PNIPAM is known for cononsolvency when dissolved in water/alcohol mixtures. Co-nonsolvency means that the 

polymer demixes in solvent mixtures while it mixes well in the individual solvents. SANS 

was used to map out the mixing phase diagram for PNIPAM in d-water/d-ethanol mixtures. 

Note that the PNIPAM/d-water solution obeys an LCST phase behavior (phase separates 

upon heating) while the PNIPAM/d-ethanol solution obeys the UCST behavior (phase 

separates upon cooling). Most water-soluble polymers follow the LCST behavior whereby 

water is squeezed out of the hydration region leading to phase separation at high temperature. 

In order to map out the phase diagram for the 4 % PNIPAM in d-water/d-ethanol mixtures 

[1], the scattering intensity in the thermodynamic limit (i.e., for Q = 0) is needed. A simple 

empirical model is used to fit SANS data for samples with d-water fractions.

B

1 (Q )

C

d

d (Q)

m



 







. (1)

Parameters include the solvation intensity C, the correlation length  and the Porod exponent 

m. Variation of C with inverse temperature separates out the LCST and UCST regions of the 

phase diagram. The UCST behavior is characterized by a negative slope while LCST is 

characterized by a positive slope. 
412

-0.5

0

0.5

1

1.5

2

2.5

3

0.0029 0.003 0.0031 0.0032 0.0033 0.0034 0.0035 0.0036 0.0037

0 % d-water

10 % d-water

20 % d-water

30 % d-water

40 % d-water

80 % d-water

90 % d-water

100 % d-water

Inverse Solvation Intensity 1/C

Inverse Temperature (K-1) 

UCST

LCST

Figure 1: Variation of the inverse solvation intensity 1/C with inverse absolute temperature 

for 4 % PNIPAM in various d-ethanol/d-water solvent mixtures. 

Extrapolation of the 1/C vs 1/T linear behavior to 

1 C  

(i.e., intensity blows up!) yields 

estimates of the spinodal temperature. Plotting the spinodal temperature with varying dwater/d-ethanol fraction delimits the two-phase and one-phase regions. This yields the 

PNIPAM co-nonsolvency phase diagram. 

This phase diagram is compared to the co-solvency of poly(ethylene oxide) in d-water/dethanol solutions. Most polymers obey the co-solvency rule; they dissolve better in solvent 

mixtures than in the individual solvents. PNIPAM is a notable exception. 
413

Figure 2: Schematics of the co-solvency and co-nonsolvency phase diagrams for polymers in 

solvent mixtures. 

2. SINGLE-CHAIN CONFORMATION OF PNIPAM IN D-WATER

In order to investigate the PNIPAM polymer single-chain conformations, the so-called zero

average contrast method is applied here. This method consists in measuring a mixture of 

deuterated and non-deuterated (hydrogenated) polymers in a mixture of deuterated and 

hydrogenated water where the average solvent scattering length density matches the average 

scattering length density of the polymers. This matching condition corresponds to the zero 

average contrast.

  n v P (Q)   n v P (Q)

d

d (Q)

P P P T

2

2 P P P S P S

P

2 d P h P

d P h P      



 

   





.

(2)

The scattering length density for the deuterated/hydrogenated polymer mixture is 

P dP dP P  hP hP P

while that for the deuterated/hydrogenated solvent mixture is 

S  dS dS S  hS hS S

. Setting the second contrast factor (between the polymer and 

the solvent) to zero cancels out the second term containing the interchain contributions PT(Q) 

leaving only the first term containing the single-chain form factor PS(Q). Using a series of 

deuterated/hydrogenated solvent mixtures, one reaches a minimum in the cross section at the 

zero average contrast condition. That minimum corresponds to 

P S

. In our case, 

dPNIPAM/hPNIPAM isotopic polymer mixtures are dissolved in d-water/h-water solvent 

mixtures.
414

A series of 12 samples containing 50 % dPNIPAM/50 % hPNIPAM fractions with varying dwater/h-water fractions were prepared [2]. These correspond to 100 % d-water (and 0 % hwater) for sample 1, 10 % d-water for sample 2, 20% d-water for sample 3, etc, till 0 % dwater (and 100 % h-water) for sample 12. An extra sample with 67.5 % d-water (sample 5) 

was also prepared. One of the samples in this series will correspond to the zero average 

contrast condition. 

Figure 3: List of samples prepared. Samples 13 and 14 contain d-water as solvent while 

samples 16 and 15 contain h-water as solvent. 

Plotting the square root of the intensity at low-Q vs the d/water/h-water fraction, one sees 

that it follows a linear behavior as it should (since the contrast factor involves a square). In 

order to obtain an uninterrupted line, the points on the left side were changed from positive to

negative. 

1 9

13

14 15

hPNIPAM 16

dPNIPAM

d-water h-water

2 3 4 5 6 7 8 10 11 12
415

-2

-1

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2

25 oC 

15 oC 

5 oC 

(Relative Scale Factor)1/2

 d-Water Fraction 

Figure 4: Graphical method used to determine the zero average contrast point. The y-axis 

corresponds to the square root of the relative scale factor (relative to sample 8), i.e., equal to 

(Scale  sample8) . 

The following figure compares results for the zero average contrast sample (8) to the 

maximum contrast sample (13). One can clearly see that the intensity for sample 8 does not 

vary much with temperature since only the single-chain contribution has been left. 
416

0.1

1

10

100

1000

104

105

0.01 0.1

4 % PNIPAM in d-water 

sample 13, 35 oC

sample 13, 30 oC

sample 13, 25 oC

sample 13, 20 oC

sample 13, 15 oC

sample 13, 10 oC

sample 13, 5 oC

sample 8, 35 oC

sample 8, 30 oC

sample 8, 25 oC

sample 8, 20 oC

sample 8, 15 oC

sample 8, 10 oC

sample 8, 5 oC Scattering Intensity (cm-1) 

Scattering Variable Q (Å-1) 

Figure 5: SANS data for sample 13 and for sample 8 at the six measured temperatures. SANS 

intensity is enhanced when temperature is increased for sample 13 (separated solid lines) 

while it remains practically constant for sample 8 (overlapping dashed lines) which 

corresponds to the zero average contrast condition

Using the Gaussian chain model with excluded volume, the radius of gyration is extracted for 

both sample 8 and sample 13. One can clearly see that the single-chain radius of gyration 

decreases slightly while the apparent radius of gyration increases a lot due to phase 

separation (chi) effects. The slight decrease of the real (single-chain) radius of gyration is 

due to the onset of chain collapse as temperature is increased. 
417

0

50

100

150

200

250

300

350

0 5 10 15 20 25 30 35

sample 13 

sample 8 

 Radius of Gyration (Å) 

Sample Temperature (oC) 

Figure 6: Variation of the apparent radius of gyration for sample 13 and of the single-chain 

radius of gyration for sample 8. 

3. CONFORMATION OF BOTTLEBRUSH POLYMERS IN SOLUTION

Bottlebrush polymers are highly branched macromolecules containing a linear backbone 

upon which a large number of side chains are grafted. 

Figure 7: Schematic representation of the bottlebrush polymers. 

A series of polystyrene bottlebrush polymers with varying backbone or side-chain lengths 

were prepared in dilute d-toluene solutions, measured by SANS (among other techniques) 

and analyzed [3]. 
418

Scattering Variable Q (A-1) 

Scattering Variable Q (A-1) 
419

Figure 8: SANS data for bottlebrush polymers with similar side-chain molecular weights 

(top) and with long backbones and varying side-chains (bottom). 

Fits to the Guinier-Porod model yielded dimension parameters between 0 (spherically 

symmetric particles) and 1 (elongated particles) with varying side-chain length. Cross 

sectional radii of gyration were found to increase with increasing side-chain length as 

expected. Fits to the flexible cylinder model yielded cylinder radius and lengths that were 

also found to properly increase with increasing backbone and side-chain lengths. A transition 

from compact spherical structures to elongated cylindrical structures was observed with 

increasing backbone length.

4. THE CORE-CHAIN MODEL

A model describing polymer chains that are grafted onto uniform density particles is 

described here [4-5]. This could apply to Pluronic copolymers whereby the hydrophobic 

central block forms the micellar core while the outside hydrophilic block is dissolved in the 

surrounding shell. The uniform core is denoted by c and the grafted polymer is denoted by p. 

The single-particle cross section for this uniform core-grafted polymer chain model 

dS(Q) d

has the following contributions: core-core (Term1), polymer-polymer within the 

same chain (Term2), core-polymer chain cross product (Term3) and polymer- polymer 

correlations across two different grafted chains (Term4). 
420

Figure 9: Schematic representation of the uniform core-grafted polymer chain showing the 

various types of correlations. 

Term1 Term2 Term3 Term4

d

d (Q)

S

   





(3)

Using the familiar notation introduced for the form factor, the form factor amplitude and the 

propagator for both the uniform core part and the grafted polymer part, one obtains:

Term1 V P (QR)

c

2

c

2

 c

Term2 V N P (QR )

p p g

2

p

2

 p

Term3 2 V F (QR)E (QR) V N F (QR )  c c c c p p p p g

Term4 N (N 1)V F (QR )E (QR)

2

g c

2

p

2

p p p

2

 p  . (4)

uniform 

core

grafted 

polymer

Term1

Term2

Term3

Term4
421

Figure 10: Breaking down the contributions to the four correlation terms.

The following parameters have been defined. 

Vc

and 

Vp

are the central core volume and the 

grafted polymer chain volume respectively. 

c

and 

p

are the excess scattering length 

densities (with respect to the solvent) for the core and the polymer respectively. 

Np

is the 

number of grafted polymer chains per particle. Standard notation has been used for the form 

factors 

P (QR)

c

and 

P (QR ) p g

, for the form factor amplitudes 

F (QR)

c

and 

F (QR ) p g

, and 

for the propagators 

E (QR)

c

and 

E (QR ) p g

. Indices c and p refer to the core and the polymer 

respectively. 

Term1=Pc(QR)=Fc

2

(QR)

Term3= Fc(QR) Ec(QR)Fp(QRg)

Term2= Pp(QRg)

Term4= Fp(QRg) Ec

2

(QR)Fp(QRg)
422

 

2

2 1

c c QR

3j (QR)

P (QR) F (QR)

















 

 

2

g

2 2

g

2

2

g

p g

exp( Q R ) 1 Q R

(QR )

2

P (QR )    



















QR

3j (QR)

F (QR)

1

c















  



2

g

2

2

g

2

p

Q R

1 exp( Q R )

F (QR)



















QR

sin[QR]

E (QR) c

. (5)

The case of grafted polymer with Gaussian chain statistics is used. However, the effect of 

excluded volume could be incorporated using the Flory model. 

At finite particle concentration, the single-particle result is multiplied by the particles’ 

number density 

(N V)

and the inter-particle structure factor. 

S (Q)

d

d (Q)

V

N

d

d (Q)

I

S























. (6)

This is assuming that the particles are isotropic and uniform, i.e., that the overall form factor 

could be written as the square of a form factor amplitude. We know that this is true for the 

uniform density core but not true for the grafted polymer. This approximation is used for the 

sake of simplification. One of the analytical structure factors could be used such as for hard 

sphere interactions, for Coulomb interactions, etc. 

The core radius can be assumed to follow a distribution around R with a standard deviation 

R

. Also, in order to avoid interpenetration of the grafted chains onto the central core, the 

grafting shell is moved outwards by 

g dR

with an adjusting parameter 

0  d 1

so that the 

core radius becomes 

(R dR )  g

. The core propagator would therefore be modified as 

E (QR) sin[Q(R dR)] Q(R dR) c  

The uniform distribution inside the core could be 

relaxed by using a peaked distribution. 

This core-chain model will be used in the next section to analyze data from copolymer 

micelles. 
423

5. DISSOCIATION OF BLOCK COPOLYMER MICELLES

Polystyrene-polyisoprene (PD-PI) diblock copolymers form micelles when dissolved in ddecane [6] which is a selective solvent (decane dissolves PI but not PS). The molecular 

weight for the PS-PI diblock was 10 k-17.8 k. Many copolymer fractions (between 0.5 % and 

11 %) in d-decane solutions were measured by SANS at temperatures that varied between 25 

oC and 95 oC. Micelles form above a critical micelle fraction. Increasing the copolymer 

fraction introduces a structure factor peak due to the inter-micelles interactions; this peak 

shifts to higher Q values when the volume fraction increases. 

The core-chain model is used to analyze SANS data. Hard sphere interactions (PercusYevick closure) are assumed for the structure factor. Many fitting parameters were obtained 

such the core radius, the grafted chains radius of gyration. These as well as the aggregation 

number (number of copolymers per micelle) were found to decrease with temperature. As 

temperature is increased, micelles dissociate. 

Figure 11: SANS data for a few copolymer fractions at 25 oC. The core-chain model is seen 

to fit the data well. 

6. ASSOCIATION OF THERMOSENSITIVE COMBLIKE COPOLYMERS IN 

SOLUTION

Thermosensitive comblike polymers composed of a polynorbornene backbone and oligo 

ethylene glycol side chains were investigated in d-water solutions. The side chains dissolve 

in water while the backbone is hydrophobic. The solubility of this polymer could be tuned by 
424

varying the length of the side chains. A 2 % solution was investigated by SANS at different 

temperatures [7]. An association into micelles with a spherical core and a corona formed of

the side chains was observed. The grafted side chains were found to appear a rigid rods. 

Intermicelle interactions are weak as evidenced by the lack of a structure factor peak. The 

aggregation number increases and the overall size of the micelles decreases with increasing 

temperature due to contraction of the side chains since the onset of phase separation expels 

solvent from the corona. 

Above the phase separation (disorder-to-order) transition at 60 oC, a Bragg peak develops 

indicating spatial order inside the micelles. 
425

Figure 12: SANS data from 2 % block comblike copolymer in d-water with varying 

temperature. At the low temperatures, micelles with compact cores and dissolved side chains 

in the corona are observed. The high temperature is above a phase transition whereby water 

is expelled from the corona leaving a spatially ordered phase. 

REFERENCES

1. M. Hore, B. Hammouda, Y. Li and H. Cheng, “Co-Nonsolvency of Poly(nisopropylacrymide) in Deuterated Water/Ethanol Mixtures”, Macromolecules 46, 7894-7901 

(2013).

2. B. Hammouda, D. Jia and H. Cheng, “Single-Chain Conformation for Interacting 

PNIPAM in Aqueous Solution”, Open Access Journal of Science and Technology, 3, 1-8 

(2015). 

3. S. Pesek, X. Li, B. Hammouda, and R. Verduzco “Synthesis and Conformational Analysis 

of Bottlebrush Polymers by SANS”, Macromolecules 46, 6998-7005 (2013).
426

4. J.S. Pedersen, M.C. Gerstenberg, “Scattering Form Factor of Block Copolymer Micelles”, 

Macromolecules 29, 1963-1965 (1996).

5. J.S. Pedersen, M.C. Gerstenberg, “The Structure of P85 Pluronic Copolymer Micelles 

Determined by SANS”, Colloids and Surfaces A 213, 175-187 (2003).

6. G. Cheng, B. Hammouda and D. Perahia “Effects of Intermicellar Interactions on the 

Dissociation of Block Copolymer Micelles: SANS and NMR Studies”, Macromolecular 

Chemistry and Physics 215, 342-350 (2014).

7. G. Cheng, F. Hua, Y.B. Melnichenko, K. Hong, J.W. Mays, B. Hammouda, and G.D. 

Wignall, “Association and Structure of Thermosensitive Comblike Block Copolymers in 

Aqueous Solutions”, Macromolecules 41, 4824-4827 (2008).
427

Part I – SANS FROM COMPLEX FLUIDS

Chapter 42. Phase Diagrams for Micellar Systems

42.1 Basics of Micellar Systems

42.2 Phase Diagram for the Ternary Mixture

42.3 The Critical Points Line

42.4 The Three-Phase Region

42.5 The Fish Phase Diagram

42.6 The Micelle-Formation Phase

42.7 The P85/d-Water Phase Diagram

42.7 Typical Surfactants

42.8 Self-Assembling Systems

References

Questions

Answers

Chapter 43. SANS from Crystalline Lamellae

43.1 A Crystalline Polymer Solution

43.2 Clues about the PEO/d-Ethanol System

43.3 Form Factor for a Single Lamella

43.4 Inter-Lamellae Structure Factor

43.5 The Scattering Factor

43.6 A Crystalline Polymer Solution 

43.5 The Stacked Lamellae Model

43.7 Model Fitting

43.8 The Material Balance Equations

43.9 Numerical Application

References

Questions

Answers

Chapter 44. SANS from a Pluronic

44.1 What Are Pluronics?

44.2 SANS from P85 Pluronic

44.3 The Core-Shell Particle Model

44.4 Concentration Effects

44.5 Fit Results

44.6 Material Balance Equations

44.7 Results for the 10 % PEO in D2O at 40 oC

44.8 Polydispersity Effects

44.9 Temperature Effects

44.10 Discussion

References

Questions

Answers
428

Chapter 45. SANS from Ionic Micelles

45.1 An Ionic Rodlike Micelles System

45.2 Scattering Model

45.3 Fits Of the SANS Data

45.4 Discussion

References

Questions

Answers

Chapter 46. SANS from Complex Fluids Literature Review

46.1 1. Templating and Aggregation

46.2 Asphaltene

46.3 Clay Dispersions

46.4 Colloidal Suspensions

46.5 Gels

46.6 Mesophases

46.7 Micelles

46.8 Microemulsions

46.9 Nanostructures

46.10 Pluronics

46.11 Pressure Effects

46.12 Shear Effects

46.13 Wormlike Micelles

References
429

Chapter 42 – PHASE DIAGRAMS FOR MICELLAR SYSTEMS

A substantial fraction of SANS research is performed on “complex fluids”. These include 

micelles, colloids, gels, networks, etc, and could be referred to as “micellar systems”. Phase 

diagrams for micellar systems are discussed in this chapter (Kahlweit-Strey, 1985). This 

topic is treated at a descriptive level. 

1. BASICS OF MICELLAR SYSTEMS

Micellar systems consist of mixtures of water/oil and an amphiphile surfactant that increases 

their miscibility. The amphiphile surfactant is either not charged (i.e., is nonionic) or consists 

of an ionic (charged) headgroup. The focus here will be on nonionic micellar systems, their 

mixing ability and their demixing phase diagram. 

The phase diagram of the ternary micellar system is represented by a triangle (A: water, B: 

oil and C: surfactant). The three binary systems A-B, A-C and B-C are represented by their 

characteristic mixed-phase (1-phase) and phase separated (2-phase) regions. The oilsurfactant (B-C) binary mixture is characterized by an upper critical solution temperature 

(UCST); i.e., it phase separates upon cooling. The water-oil (A-B) binary mixture is also 

characterized by a UCST behavior but is mostly phase separated (water and oil do not mix). 

The phase diagram for the water-surfactant (A-C) binary solution is more complex and is 

characterized by a UCST behavior at low temperatures and a closed loop immiscibility island 

at high temperatures; i.e., it phase separates both upon cooling and upon heating. The closed 

loop is due to the breaking of hydrogen bonds upon heating and shows up for strong 

amphiphile surfactants. 

water A oil B

surfactant

C

Temp

Temp

Temp

0 oC

T

T

1-phase 1-phase

2-phase

2-phase

2-phase
430

Figure 1: Phase diagram for the three binary mixtures (B-C, A-B, and A-C). The phase 

separation lines and temperatures are shown. The 0 oC water-freezing line has also been 

marked. 

Phase separation occurs upon jumping from the mixed phase (1-phase) region to the phase 

separated (2-phase) region. Phase separation proceeds along tie lines and produces a phase 

rich in the A component (left side) and a phase rich in the C component (right side). The 

points at which the phase separation lines have a horizontal slope are the critical points. The 

A-C binary phase diagram shows three critical points. Note that the lower UCST may lie 

below the freezing point of the mixture. 

Figure 2: The water/surfactant (A-C) binary phase diagram. 

Given the three generic binary mixtures phase diagrams, the phase diagram for the ternary AB-C mixture is discussed next. 

2. PHASE DIAGRAM FOR THE TERNARY MIXTURE

The ABC triangle is obtained by combining the three binary mixtures phase diagrams. It 

contains a micelle-formation phase (m-phase also referred to as “1-phase”) region and a 2-

phase region. The m-phase region contains oil-in-water and water-in-oil micelles. The 2-

phase region contains a phase containing water and surfactant and another phase containing 

mostly oil and happens at low temperature. Note that micelles can form in the 2-phase region 

since binary mixtures (water/surfactant or oil/surfactant) can form micelles. The m-phase 

region is inhomogeneous (contains micelles) in the nanometer size scale but homogeneous in 

Temp

water A surfactant C

T

0

oC

tie line

LCST

UCST

UCST

1-phase

2-phase

2-phase
431

the micrometer (optical range) size scale. The m-phase region is “clear” for light but 

“cloudy” for neutrons. 

Now consider the temperature axis (vertical variable out of the page). A jump from the mphase region to the 2-phase region prompts phase separation that proceeds along tie lines 

(that are no longer horizontal). The critical condition for the ternary mixture is referred to as 

the “plait” point (equivalent to ‘critical” point for binary mixtures). 

Figure 3: Schematic representation of a generic ternary mixture phase diagram. The m-phase 

is the micelle-formation phase.

This simple representation of the ternary phase diagram is valid for low temperatures; i.e., 

before reaching the closed loop region. When the closed loop region is reached and with 

strong amphiphilic surfactants, there is an interplay of interactions which adds two 2-phase 

regions (one water-rich and one oil-rich) and two plait points. The slopes of the tie lines are 

different. This type of phase diagram is characterized by a 3-phase region inside the triangle 

that cuts through both 2-phase regions. 

surfactant C

water A oil B

plait 

point

m-phase

2-phase
432

Figure 4: Ternary phase diagram that shows the 3-phase region. 

There are three types of “Windsor” phase diagrams: type I corresponding to a 2-phase region 

where the surfactant is dissolved mainly in the water phase, type II corresponding to a 2-

phase region where the surfactant is dissolved mainly in the oil phase and type III 

corresponding to a 3-phase region where the surfactant forms a phase of its own between the 

(bottom) water phase and (top) oil phase. 

In order to appreciate the cause of the formation of the 3-phase, the temperature variable is 

added to the phase diagram and explored in more detail. 

3. THE CRITICAL POINTS LINE

Consider first the simple case without a closed loop for the water-surfactant binary mixture 

phase diagram. The 3D phase diagram (referred to as the “prism”) is shown here. The plait 

points form a line that ascends from the oil-rich region. 

water A oil B

surfactant C

m-phase

2-phase

3-phase

plait 

point plait 

point

2-phase
433

Figure 5: The prism phase diagram for a simple ternary mixture. 

When the surfactant becomes more amphiphilic, a closed loop appears. This modifies the 

prism phase diagram to show plait point lines both on the oil-rich (right) side and on the 

water-rich (left) side. Up to now, there are still the m-phase and the 2-phase regions only; the 

3-phase region has not shown up yet. Note that the slopes of the tie lines change with 

temperature for this case.

The water/oil phase volume (relative amount) varies with increasing temperature. At low 

temperature, the 2-phase consists of a water-surfactant phase (called a) and an oil phase 

(called b). As temperature is increased, surfactant is distributed equally between a water 

phase (a) and an oil phase (b). This temperature corresponds to the inflection point in the 

variation of the plait points’ line (also called critical points line). As temperature is increased 

further, the 2-phase mixture becomes an oil-surfactant phase (b) and a water phase (a). 

water A oil B

oil B

surfactant C

plait points line

Temp

T

water A
434

Figure 6: The prism phase diagram when a closed loop is present. The composition of the 2-

phase mixtures is represented for increasing temperature. 

4. THE THREE-PHASE REGION

Depending on the chemical nature of the amphiphile surfactant and oil used, one could obtain 

a 3-phase region. In order to understand its origin, consider the variation of the critical points 

line when the surfactant becomes more amphiphilic or when the oil becomes more 

hydrophobic. The slope at the inflection point becomes steeper until it becomes horizontal. 

This corresponds to a so-called “tricritical” condition. Beyond that, the critical points line 

breaks which leads to the formation of a 3-phase region. 

Figure 7: Variation of the critical points line with increasing amphiphilic character of the 

surfactant and/or increasing the hydrophobic nature of the oil. 

The critical points line changes drastically when a 3-phase region is present. That line 

“breaks” and a “sigmoid” (i.e., s-shaped) behavior appears. The 2-phase regions at low 

temperature and high temperature are almost unchanged but a 3-phase region appears at 

intermediate temperatures. 

water oil

tricritical

point

water oil water oil water oil

Temp 3-phase

water oil

surfactant

water oil

Temp

critical

points

line Temp

a

b

b

a

a

b

b

a

water/oil

phase volume

0 % 100 %

critical

points

line
435

Figure 8: Representation of the prism phase diagram when a 3-phase region is present. 

The surfactant is transferred continuously from the water-rich phase to the oil-rich phase as 

temperature is increased. Consider a triangle phase diagram at a temperature for which a 3-

phase region exists. The 1-phase region is close to the surfactant corner. The 2-phase regions 

are on each of the three sides of the 3-phase region. The 3-phase region itself is a triangle 

within the triangle. The 3-phase triangle size varies depending on the ternary system used 

and on the temperature considered. 

Temp

a

b

b

a

a

b

b

a

c

water/oil

phase volume

0 % 100 %

c

3-phase

2-phase

water oil

Temp

surfactant

water oil
436

Figure 9: The triangle delimiting the 3-phase region. 

Consider a vertical cut MC (referred to as the “isopleth” line) through the ABC triangle. 

Representing the phase diagram along the MC cut gives the “fish” phase diagram. 

5. THE FISH PHASE DIAGRAM

The “fish” phase diagram is obtained when an MC cut is taken across the ABC triangle phase 

diagram (i.e., along the isopleth line). This cut corresponds to increasing the surfactant 

concentration but keeping the amount of water and oil constant. Representation of the 

temperature/surfactant concentration phase diagram comprises the m-phase region at high 

surfactant concentration, two 2-phase regions (at low and high temperatures) and a 3-phase 

region at intermediate temperatures. The name “fish” stems from the shape of the phase 

diagram with the 3-phase as the fish head. 

Temp

b

a

Temp b

a

water/oil

phase volume

0 % 100 %

water/oil

phase volume

0 % 100 %

c

c

water oil

surfactant

3-phase

m-phase

2-phase

surfactant

water oil

A B

C

M
437

Figure 10: The fish phase diagram. 

The m-phase region is the focus of most SANS investigations since it is the region of micelle 

formation and micelles are of nanometer size. The m-phase region is rich in mesophases 

(with various morphologies). It contains spherical, cylindrical (also called wormlike) and 

lamellar micelles depending on the temperature range. Structures for these mesophases 

correspond to cubic (spherical micelles), hexagonal (cylindrical micelles) and lamellar 

symmetry respectively. Note also that the “microemulsion” is also called bicontinuous phase. 

Moreover, oil-in-water micelles are obtained at low temperature and “reverse” (water-in-oil) 

micelles are obtained at high temperatures. 

Water-in-oil and oil-in-water micelles form in the m-phase region. Micelles can also form in 

the 2-phase region as well. These micelles are different from those found in the m-phase 

region and would be formed of surfactant/water (or surfactant/oil) only. Micelles form above 

a critical micelle temperature (CMT) and/or critical micelle concentration (CMC). 

6. THE MICELLE-FORMATION PHASE

SANS data for a nonionic surfactant (Pluronic P85) mixed with d-water are described here. 

This copolymer solution is equivalent to a surfactant/water mixture. Strictly speaking, this is 

not a ternary mixture; its phase diagram is different from the surfactant/water/oil mixture but 

shows some of the same microphases. The Pluronic molecule is a triblock copolymer PEOPPO-PEO which forms micelles above a critical temperature and concentration. PPO is 

hydrophobic and PEO is hydrophilic. P85 is different from a ternary water/oil/surfactant 

system, but the overall trends remain the same. Unimers (dissolved macromolecules) are 

2-phase

3-phase

2-phase

spherical

lamellar

cylindrical

oil in water

water in oil

a

b

c

b

a

b

a

M C m-phase

microemulsion

Surfactant Fraction 
438

obtained at low temperature and micelles form at high temperature. The critical micelle 

temperature and concentration are referred to as the CMT and CMC. At fixed surfactant 

concentration, micelles form as shown by an increase in the low-Q SANS intensity.

0.1

1

10

0 10 20 30 40 50 60 70

1 % P85 in d-water

Intensity at Q = 0.02 A-1

Temperature (oC)

CMT 

Figure 12: Low-Q SANS data from 1% P85 Pluronic in d-water. The sigmoid shape function 

characterizes the critical micelle temperature (CMT). 

Varying temperature for a fixed P85 mass fraction shows a transition from a spherical to a 

cylindrical morphology in the micelle-formation region. 
439

0.1

1

10

100

0.01 0.1

1 % P85 in d-water 

10 oC 

20 oC 

30 oC 

40 oC 

50 oC 

70 oC 

Scattering Intensity (cm-1

)

Scattering Variable Q (Å-1)

spherical micelles

cylindrical micelles

unimers 

Figure 13: Transition from the unimers to the spherical micelles phase to the cylindrical 

micelles phase regions as temperature is increased for 1 % P85 in d-water. 

The transitions from the unimers region to the spherical micelles region, then to the 

cylindrical micelles region are clearly seen in the next figure as well. The heating/cooling 

cycle shows no hysteresis. 
440

0

5

10

15

20

25

0 20 40 60 80

1% P85 in d-water 

Heating 

Cooling 

Intensity at Q = 0.02 Å-1

 Temperature (o C)

spherical micelles

unimers

cylindrical micelles

Figure 14: The unimers-to-spherical micelles and then to cylindrical micelles transitions are 

clearly seen.

Note that the empirical Guinier-Porod model (Hammouda, 2010) could have been used to 

obtain a more precise delimitation of the various phase boundaries. 

7. THE P85/D-WATER PHASE DIAGRAM

The P85/d-water phase diagram has been mapped out (Mortensen, 1996) using the SANS 

technique (among other techniques). The main phases (unimers, spherical micelles, 

cylindrical micelles and lamellar phase) can be observed at low P85 weight fraction. Other 

phases (ordered spherical micelles, hexagonal phase and a disordered phase) are not 

discussed here. 
441

Figure 15: The P85/d-water phase diagram. 

8. TYPICAL SURFACTANTS

P85 was used as a typical nonionic surfactant. A class of nonionic surfactants is denoted CiEj. 

They are comprised of a hydrophobic alkyl chain containing i carbons H(CH2)i- and a 

hydrophilic ethoxy group containing j alkylpolyglycol ethers –(OCH2CH2)jH. 

Ionic surfactants contain a charged headgroup and a hydrophobic tail. Model ionic 

surfactants include cetyletrimethylammonium bromide (CTAB), sodium dodecyl sulfate 

(SDS), bis(2-ethylhexyl) sodium sulfosuccinate (AOT) also called diocyl sulfosuccinate, 

sodium salt, etc. The phase diagram for the water/oil/ionic surfactant case is similar (but not 

identical) to the nonionic surfactant case. 

P85 in d-water

P85 Fraction (%)

PPO - Lamellar

Hexagonal

Cylindrical 

Micelles

Spherical

Micelles

Cubic BCC

Unimers

PEO - and

PPO - Lamellar

0 10 20 30 40 50 60 70 80 90 100

20

40

60

80

0

Lamellar 

Micelles
442

A table summarizes the chemical formulas and scattering length densities for some common 

surfactants. Deuterated water and a typical deuterated oil (d-decane) have also been included. 

Note that some of the densities have been estimated. 

Table 1: Typical ionic surfactants and their scattering length densities.

Surfactant Chemical Formula Density 

(g/cm3)

Scattering Length 

Density (Å-2)

CTAB CH3(CH2)15N(CH3)3Br-Na+0.9 -1.54*10-7

SDS H(CH2)12OSO3

-Na+

0.9 3.72*10-7

AOT CH3(CH2)3CH(C2H5)CH2O2CCH2CH(SO3

-

Na+)CO2CH2CH(C2H5)(CH2)3CH3

0.9 5.08*10-7

d-water D2O 1.11 6.39*10-6

d-decane CD3(CD2)8CD3 0.73 5.7*10-6

9. SELF-ASSEMBLING SYSTEMS

A slightly more general category than “complex fluids” could be referred to as “selfassembling systems”. These include a large class of molecular systems that do not stay 

dissolved. The following couple of chapters cover “self-assembling” systems. Two nonionic 

systems will be considered: crystalline lamellae in solution and spherical micelle-forming 

Pluronic. An ionic system forming wormlike micelles will also be included. 

Soft materials include gels and networks that trap solvent and form regular amorphous 

structures. Strong driving forces such as hydrophobic/hydrophilic or Coulombic interactions 

and other specific interactions (such as hydrogen bonding or dipolar interactions) form selfassembling systems.

Various salts are added to micellar systems in order to control solubility. For example, 

lyotropic salts decrease the mutual solubility of water and amphiphiles (they increase the 

closed loop immiscibility island) whereas hydrotropic salts increase solubility (they shrink 

the closed loop). 
443

Micellar systems are part of main-stream chemical-engineering. They find applications in 

detergents and cosmetics formulation, drug design and delivery, pharmaceuticals, oil 

recovery and treatment and in the food industry. 

REFERENCES

B. Hammouda, “A New Guinier-Porod Model”, J. Appl. Cryst. 43, 716-719 (2010).

M. Kahlweit and R. Strey, “Phase Behavior of Ternary Systems of the Type H2O-OilNonionic Amphiphile (Microemulsions)”, Angewandte Chemie International Edition in 

English 24, 654–668 (1985). 

K. Mortensen, “Structural Studies of Aqueous Solutions of PEO-PPO-PEO Triblock 

Copolymers: Their Micellar Aggregates and Mesophases; A SANS Study” J. Phys.; 

Condensed Matter 8, A13 (1996).

http://www.ncnr.nist.gov/resources/sldcalc.html contains a scattering length density 

calculator. 

QUESTIONS

1. What is the difference between an LCST and a UCST phase separation?

2. What is a miscibility gap? How about an immiscibility island?

3. What is a tie line?

4. Define the critical point for a binary mixture. 

5. What name replaces the critical point for a ternary mixture?

6. Define the 2-phase region for a ternary water/oil/surfactant system. 

7. Describe the 1-phase for binary and for ternary mixtures. Contrast the main difference. 

8. How many types of surfactant are there? Name them. 

9. What is referred to as the “fish” phase diagram?

10. What is the “prism” phase diagram?

11. What are the two main criteria for the formation of a 3-phase region?

12. In which part of the phase diagram is the SANS technique most effective for a ternary 

system? Why?

ANSWERS

1. Phase separation occurs upon cooling for a UCST system and upon heating for an LCST 

system. 

2. A miscibility gap is a miscibility region between a UCST at low temperature and an LCST 

at high temperature. A closed loop immiscibility island is a phase separation region between 

an LCST at low temperature and a UCST at high temperature. 
444

3. A tie line is the line in the temperature-concentration phase diagram along which phase 

separation proceeds. The final product is a 2-phase mixture. 

4. The critical point for a binary mixture (say water and surfactant) is the maximum point on 

the binodal line (also called coexistence curve) in the temperature-concentration phase 

diagram. 

5. The plait point is used to refer to the critical point for a ternary mixture. 

6. The 2-phase for a ternary system consists of a mixture of two phases (one rich in water and 

one rich in oil). 

7. The 1-phase region for binary systems consists of a homogeneously mixed phase (i.e., 

mixed at the molecular level). The micelle-formation phase (m-phase) is sometime referred 

to as the 1-phase for ternary systems. This m-phase is mixed at the micrometer (so-called 

optical) scale but shows micellar structure at the nanometer (SANS) scale. 

8. There are two types of surfactants: nonionic and ionic (i.e., charged). 

9. The “fish” phase diagram maps out the 2-phase, the 3-phase and the m-phase regions in a 

temperature/surfactant concentration plot. 

10. The “prism” phase diagram is the extension of the ABC triangle phase diagram to 3D by 

adding the temperature variable (out of the page). A prism shape is obtained. 

11. A 3-phase region forms when the water-surfactant binary phase diagram contains a 

closed loop immiscibility island and the surfactant is a strong amphiphile. 

12. The SANS technique is most effective in the micelle-formation phase since micelles are 

of nanometer sizes. The 2-phase and the 3-phase regions are characterized by huge 

(micrometer scale or larger) droplets. 
445

Chapter 43 - SANS FROM CRYSTALLINE LAMELLAE 

Polymers with regular structures can crystallize whether in solution or in the melt state. 

Solution crystallization forms regular lamellar structures and is the focus here. Crystalline 

lamellae qualify for inclusion in this part on “self-assembling systems”. 

1. A CRYSTALLINE POLYMER SOLUTION 

Scattering from poly(ethylene oxide) (PEO) in d-ethanol (deuterated ethanol) solutions 

shows a strong low-Q SANS signal at low temperatures (below 40 oC). A low-Q, SANS 

signal could be due to many effects including aggregation, clustering, phase separation, 

crystallization or just undesired bubbles in the sample. 

10-4

10-3

10-2

10-1

100

101

102

103

104

0.01 0.1

4 % PEO/d-ethanol, MW = 50,000 g/mol 

23 oC 

35 oC 

45 oC 

 Scattering Intensity (cm-1) 

Scattering Variable Q (Å-1)

Figure 1: SANS from 4 % PEO/d-ethanol (mass fraction) measured at temperatures below 

and above the crystal melting temperature. The incoherent background component has been 

subtracted. 
446

10-6

0.0001

0.01

1

100

104

0.01 0.1

 PEO/d-Ethanol, Mw = 90,000 g/mol 

20 %

15 %

10 %

7 %

4 %

2 %

1 %

0.5 %

Scattered Intensity (cm-1

)

Scattering Variable Q (Å-1)

Figure 2: SANS data from PEO/d-ethanol for the various mass fractions measured. 

Measurements were taken at 23 oC, i.e., well into the crystalline region. The incoherent 

background component has been subtracted. 

In order to discriminate among the various effects that could cause the strong low-Q signal, a 

couple of standard characterization methods are used. These methods can detect 

crystallization. The first method is Differential Scanning Calorimetry (DSC) which clearly 

shows a crystal melting process upon heating and a crystallization process upon cooling. 
447

-30

-20

-10

0

10

20

30

20 30 40 50 60 70

 4% PEO/d-ethanol , MW = 90,000 g/mol 

Heating 

Cooling 

 Heat Flow (mW/g) 

Temperature (oC)

Figure 3: Differential Scanning Calorimetry (DSC) measurements from 4 % PEO/d-ethanol 

showing the crystalline nature. The heating and cooling curves show the effect of melting 

and crystallization. 

The second good monitor of sample crystallization is Wide-Angle X-ray Scattering (WAXS). 

This method clearly shows crystalline peaks even at low PEO mass fraction. 
448

102

103

0 10 20 30 40

PEO/d-ethanol, MW = 50,000 g/mol 

4 % PEO/d-ethanol 

10 % PEO/d-ethanol 

Scattering Intensity (counts)

2  (degrees)

Figure 4: Wide-Angle X-ray Scattering (WAXS) spectra from PEO/ethanol showing 

crystallinity in the sample. Note that most of the spectrum consists of amorphous halos but 

two crystalline peaks are observed. The second spectrum was shifted upward. 

Another method for determining crystal melting temperatures (while heating) and 

crystallization temperatures (while cooling) consists in precise density measurements. 
449

0.86

0.87

0.88

0.89

0.9

0.91

0.92

10 20 30 40 50 60 70

4% PEO/d-ethanol 

Heating

Cooling

Density (g/cm

3

)

Temperature (oC)

Figure 5: Density measurements for 4 % PEO/d-ethanol. The heating and cooling cycles are 

shown. Arrows show breaks in the data trend corresponding to melting and crystallization 

transitions. 

2. CLUES ABOUT THE PEO/D-ETHANOL SYSTEM

PEO crystallizes in ethanol even at low mass fractions. Minutes after mixing PEO and 

ethanol, the solution turns white and gel-like; it does not flow when turned upside-down. 

When probed using a needle, the structure feels sponge-like. This structure could be referred 

to as a lamellar sponge. 

In order to analyze the measured SANS data, the following model in used. 
450

3. FORM FACTOR FOR A SINGLE LAMELLA

Figure 6: Coordinates parametrization of a single lamella. 

The form factor amplitude F(Q,) for a single-lamella has two contributions: one for the Q 

component parallel to the z-axis and one in the horizontal plane. 

F(Q, ) F (Q, )F (Q, )  

z   

(1)

Q L / 2

sin(Q L / 2)

dz exp[ iQ z]

L

1

F (Q, )

c

c

L / 2

c L / 2

z

c

c





     



(2)

        



 





R

0

2

0

2

2

d d exp[ iQ 1 cos( ) ]

R

1

F (Q, ) . (3)

Here  = cos() and  is the inclination angle. After manipulations described elsewhere, one 

obtains:

Q 1 R

2J (Q 1 R)

F (Q, )

2

2

1

 

 

   . (4)

J1 is the cylindrical Bessel function. The final result for the form factor amplitude for a single 

lamella is:





































 

Q 1 R

2J (Q 1 R)

Q L / 2

sin(Q L / 2)

F(Q, )

2

2

1

c

c

. (5)

The form factor for a single isolated lamella is therefore given by the following orientational 

average:

x

y

z

R

Q

Qz



 Lc
451

2

1

1

d | F(Q, ) |

2

1

P(Q)





   . (6)

Lamellae form in stacks. Here, the lamellae are not isolated so that the orientational 

averaging is not performed until the inter-lamellae stack structure factor is included. 

4. INTER-LAMELLAE STRUCTURE FACTOR

Consider a stack of N lamellae consisting of alternating crystalline and amorphous regions. L 

is the inter-lamellar distance (also called long period), LC is the lamella thickness and R is the 

radius (Richter et al, 1997; Ho et al, 2006). 

Figure 7: Stack of multiple lamellae. 

Consider the following inter-lamellae Gaussian distribution function:



















 



2

L

2

2

L

k

2k

(z kL)

exp

2 k

1 W (z,L) . (7)

Note that the variance kL

2

gets larger and the Gaussian peak height gets smaller with 

increasing lamellar order k within a stack. 

The structure factor for a stack of N lamellae is given by:

z

x

y

R

L

Lc

Crystalline

Amorphous

Tie chain
452

     



N

i,j 0

I 2

exp[ iQ z]

N

1

S (Q, )

(8)

    





N

i,j 0

2 |i j| dz exp[ iQ z]W (z,L)

L

1

N

1

.

Here Qz = Q is the projection along the vertical (z) axis and the “I” subscript on the 

structure factor stands for “inter-lamellae”. The z integration is readily performed:















   

    

















 

 



  

 



2

Q | i j|

exp[ iQ | i j| L]exp

2 | i j|

(z | i j| L)

exp

2 | i j|

exp[ iQ z]

dz

L

1

2

L

2 2

2

L

2

2

L

(9)

Use the following summation identity:

   

   

N

k 1

N

i,j 1

F(| i j|) N 2 (N k)F(k). (10)

to obtain:















  

       

 2

Q k

(N k)exp[ iQ kL]exp

N

2

N

1

S (Q, )

2

L

2 2 N

k 1

I 2

. (11)

This result is general and applies for a finite stack of lamellae. 

Note that the hypothetical (unrealistic) case where L = 0 yields the following familiar 

function:











   

   I 2 A

exp( A) 1 A

2

N

1

S (Q, ) . (12)

This is the Debye function encountered when calculating the form factor for a Gaussian coil 

and the variable is 

A Q 2

2

L

2 2

  

. These two widely different systems involve the same 

function by mere coincidence. 

The infinite stack case is obtained by taking the 

N  

limit. The following result is 

obtained:

exp(Q / 2) exp( Q / 2) 2cos(Q L)

exp(Q / 2) exp( Q / 2)

N

1

S (Q, )

2

L

2 2 2

L

2 2

2

L

2 2 2

L

2 2

I

       

     

  

(13)
453

cosh(Q / 2) cos(Q L)

sinh(Q / 2)

N

1

S (Q, )

2

L

2 2

2

L

2 2

I

   

 

   .

This is a more compact result but applies only for stacks with infinite number of lamellae. 

5. THE SCATTERING FACTOR

Putting the single-lamella form factor and the inter-lamella structure factor together gives the 

following scattering factor:

d P(Q, )S (Q, )

2

1

I

1

1

   



(14)

S (Q, )

Q 1 R

2J (Q 1 R)

Q L / 2

sin(Q L / 2)

d

2

1

I

2

2

2

1

2

c

c

1

1



















 

 

















 



.

This model assumes uniform crystalline density within each lamella and neglects completely 

scattering from the amorphous phase between the lamellae. Note that in writing down this 

scattering factor, we did not worry too much about normalization factors. These factors are 

included in the next section where the scattering cross section is written down. 

6. THE STACKED LAMELLAE MODEL

Consider a scattering system consisting of a sponge-like structure where the lamellae trap 

solvent. A few lamellae packed into stacks form the partitions. Each stack contains a number 

of crystalline lamellae and amorphous regions in-between the lamellae. Solvent dissolves the 

amorphous regions and fills the pockets of the sponge-like structure. Note that the extent of 

the stack (its radius) is not well-defined. It is large and could be thought of as the average 

distance between bifurcation points. 
454

Figure 8: The stacked lamellae model consists of alternating amorphous and crystalline 

regions that surround solvent pockets. 

The following parameters are used in the model. 

LA: thickness of the amorphous region

LB: thickness of the crystalline region

L: lamellar spacing (L = LA+LB)

N: number of lamellae per stack

L: standard deviation of lamellar spacing

VA = R

2LA: volume of one amorphous region 

VB = R

2LB: volume of one crystalline region

R: average radius of the lamellar stack

VA+B = R

2L: volume of one crystalline and one amorphous regions

A = bA/vA: scattering length density of the amorphous region (region A)

B = bB/vB: scattering length density of the crystalline region (region B)

C = bC/vC: scattering length density of the solvent region (region C)

NS/V: lamellar stacks number density. 

The orientation-dependent cross section is given by:

V F (Q, ) V F (Q, ) S (Q, )

v

b

v

b

V F (Q, )

v

b

v

b

d V

d (Q, )

I

2

A B AB A A

C

C

B

B

A A

C

C

A

A

S

S

 











   















   















 













 















 



(15)

V F (Q, ) S (Q, )

v

b

v

b

V F (Q, )

v

b

v

b

d V

d (Q, )

I

2

A B AB

C

C

B

B

A A

B

B

A

A

S

S 

















 















   















 













 















 



.

z

Region B is crystalline

Region A is amorphous

Q

 = cos-1(

Region C is solvent

Q

z

z

z

lamellar stack
455

Note that the number density of the lamellar stacks NS/V has been expressed in terms of the 

stacks volume fraction S and stack volume VS as NS/V = S/VS. 

The form factors are:

















 

 

















 

Q 1 R

2J (Q 1 R)

Q L / 2

sin(Q L / 2)

F (Q, )

2

2

1

A

A

A

(16)

















 

 

















 

Q 1 R

2J (Q 1 R)

Q L/ 2

sin(Q L/ 2)

F (Q, )

2

2

1

AB

.

The inter-lamellae structure factor for each stack is given by:











  

      

 2

Q k

(N k)exp[iQ kL]exp

N

2

N

1

S (Q, )

2

L

N 2 2

k 1

I 2

. (17)

The orientationally averaged cross section is obtained as:



 

 









d

d (Q, )

d

2

1

d

d (Q)

1

1

. (18)

The scattering intensity consists of the following contributions:

B

d

d (Q)

Q

A

I(Q)

n







  . (19)

The lamellar stacks represent only the crystalline-amorphous regions forming the walls of the 

sponge-like structure. Scattering comprises also scattering from the sponge-like structure 

itself. A term A/Qnterm has been added to represent the lamellar non-stack scattering 

component. This component has contributions from the clustering network of the sponge-like 

structure (mass fractal network and surface fractal lamellae). B is a Q-independent (mostly 

incoherent) scattering background. 

7. MODEL FITTING

The stacked lamellae model is smeared with the instrumental resolution function and used to 

fit SANS data from PEO/d-ethanol samples in the crystalline region. 
456

0.001

0.01

0.1

1

10

100

1000

104

105

0.01 0.1

 15 % PEO/d-Ethanol, Mw = 90,000 g/mol 

Model Fit

SANS Data

Scattered Intensity (cm-1

)

Scattering Variable Q (Å-1) 

Figure 9: Comparison of the SANS data and the stacked lamellae smeared model for the 15 

% PEO/d-ethanol sample at 23 oC. The model fits the data fairly well except for a region in 

the middle of the window. 

Results of the fit follow. 

Lamellar stacks volume fraction S = 0.36

Lamellae radius R = 10,757 Å

Thickness of the amorphous region LA = 3 Å

Lamellar thickness LB = 50 Å

Scattering length density of the amorphous region A = 4.15*10-6 Å

-2

Scattering length density of the crystalline region B = 7.67*10-7 Å

-2

Scattering length density of the d-ethanol solvent region S = 6.07*10-6 Å

-2

Number of lamellae per stack NL = 6.86

Standard deviation of the inter-lamellar distance L= 0.56 Å

Clustering scaling factor A = 282*10-6cm-1

Clustering Porod exponent n = 2.8

Constant (incoherent) background level B = 0.004 cm-1. 

The solvent scattering length density S was fixed. All other 11 fitting parameters were 

varied. With so many varying parameters, it is difficult to find a unique solution. The one 

scattering 

from flat 

lamellae

interlamellar 

peak
457

presented here gives an idea of the magnitude of the various parameters. Constraints such as 

ordering the various scattering length densities as EO < B < A < dEth help the fitting.

0.0001

0.01

1

100

104

0.01 0.1

 15 % PEO/d-Ethanol, Mw = 90,000 g/mol 

I(Q) 

B

A/Qn

d(Q)/d

Scattered Intensity (cm-1

)

 Scattering Variable Q (Å-1) 

Figure 10: Breakdown of the various terms forming the fitting model functional form. Note 

that 

I(Q) d (Q) d A/Q B

n

     . 

8. THE MATERIAL BALANCE EQUATIONS

Consider a solution consisting of lamellar stacks comprising alternating amorphous and 

crystalline regions. We assume that D2O exists in the amorphous and crystalline regions; i.e., 

that there are yA d-ethanol molecules per ethylene oxide (EO) monomer in the amorphous 

region (region A) and yB d-ethanol molecules per EO monomer in the crystalline region 

(region B). Note that most of the solvent exists in the sponge-like pockets outside of the 

lamellar stacks. Define Nagg as the aggregation number, i.e., the number of PEO 

macromolecules per lamellar stack and note that there are n EO monomers per 

macromolecule. These are distributed as nA = nfA monomers in region A and nB = nfB in 

region B. 

The fitting parameters are LA, LB, A, B, R, NL, and L. Other known quantities are the 

specific volumes 

A

vEO

(EO in the amorphous region), 

B

vEO

(EO in the crystalline region) and 
458

vdEth and the scattering lengths 

bEO

and bdEth. The unknown parameters are yA, yB, fA, and 

Nag. Note that fB = 1-fA.

The material balance equations are:

(1) 

  dEth A

A

L ag A EO

2

LAR N  N f n v  v y

(20)

(2) 

  dEth B

B

L ag B EO

2

LBR N  N f n v  v y

(3) 

 

L

2

A

ag A EO dEth A

A

4 L R N

N f n b b y





 

(4) 

 

L

2

B

ag B EO dEth B

B

4 L R N

N f n b b y





  .

These four linear equations can be solved to obtain:

dEth A dEth

EO

A

A EO

A

b v

v b

y

 

 



(21)

dEth B dEth

EO

B

B EO

B

b v

v b

y

 

 



 

   

B

A

B dEth

B

A dEth EO

A

EO

B

A

B dEth

B

EO

A

L

L

v y v v y v

L

L

v y v

f

  





B A

f 1 f

  A dEth

A

A EO

A L

2

ag nf v y v

4 R L N

N





 .

We have transformed the four fitting parameters LA, LB, A and B into four meaningful 

parameters yA, yB, fA and Nag. This set of solutions is unique. Note that only Nag depends on 

the ill-defined stack radius R. 

9. NUMERICAL APPLICATION

Consider the PEO/d-ethanol system described earlier. Some of the known parameters follow:
459

v 42.82 cm / mol A 3

EO 

(23)

v 38.93cm /mol B 3

EO 

v 58.56 cm / mol 3

dEth 

b 4.14*10 cm 13

EO





b 59.12*10 cm 13

dEth



 . 

Note that the amorphous region’s specific volume has been taken to be 10 % higher than that 

for the crystalline region. Using these parameters along with the following fitting parameters:

S = 0.36 (24)

R = 10,757 Å

LA = 3 Å

LB = 50 Å

A = 4.15*10-6 Å

-2

B = 7.67*10-7 Å

-2

S = 6.07*10-6 Å

-2

NL = 6.86

L= 0.56 Å.

The material balance equations yield:

yA = 3.7 (25)

yB = 0.02

fA = 0.01

fB = 0.99

Nag = 926,000

Fit results yield 3.7 solvent molecules per EO monomer in the amorphous region and no 

solvent in the crystalline region. Moreover, most of the EO monomers are found in the 

crystalline regions. 

Figure 11: Schematic representation of the sponge-like structure. The sponge walls are 

formed of lamellar stacks that trap pockets of solvent. 
460

In order to acquire a “picture” of the sponge-like structure in direct space, TEM was 

performed. A micrograph is included.

Figure 12: Transmission Electron Micrograph (TEM) of a 4 % PEO/ethanol crystalline 

sample. The sample was stained. The sponge-like crystalline structure traps pockets of 

solvent. 
461

Figure 13: Electron diffraction micrograph from the same 4 % PEO/ethanol semicrystalline 

sample. The diffraction spectrum shows strong low-Q scattering and a series of peaks 

forming a ring at Q = 0.05 Å-1. 

The exercise described here gives an idea of what it takes to understand some SANS data: 

clues about the sample, model fitting and a great deal of common sense. 

REFERENCES

D. Richter, D. Schneiders, M. Monkenbush, L. Willner, L.J. Fetters, J.S. Huang, M. Lin, K. 

Mortensen and B. Farago, “Polymer Aggregates with Crystalline Cores: The System 

Polyethylene-Poly(ethylenepropylene)”, Macromolecules 30, 1053-1068 (1997). 

D.L Ho, B. Hammouda, S.R. Kline and W-R Chen, “Unusual Phase Behavior in Mixtures of 

Poly(ethylene oxide) and Ethyl Alcohol”, J. Polym. Sci., Polym. Phys. Ed. 55, 557-564 

(2006)
462

QUESTIONS

1. Crystalline lamellar morphology is formed in what conditions?

2. What are the main pieces used to work out the scattering from a lamellar system?

3. Why is the scattering from crystalline lamellae characterized by SANS oscillations at highQ?

4. Why is the inter-lamellae scattering (also called long period) characterized by a Bragg 

peak?

5. Lamellar growth occurs either following the “adjacent” or the “random switchboard” reentry. Which occurs in solution crystallization and which occurs in melt crystallization?

6. Are the crystallization and the crystal melting temperatures the same? Which is lower? 

Why?

ANSWERS

1. Crystalline polymers form lamellar morphology in solution as well as in the melt state. In 

the melt state, however, lamellae organize into spherulitic structures. 

2. Scattering from a lamellar system is calculated using the form factor for a lamella and the 

inter-lamellar structure factor. 

3. SANS scattering from crystalline lamellae is characterized by oscillations at high-Q 

because of the sharp (highly monodisperse) lamellar size. 

4. The inter-lamellae scattering is characterized by a Bragg peak because of the well-defined 

characteristic d-spacing between lamellae. 

5. Adjacent re-entry occurs in solution crystallization and random switchboard occurs in melt 

crystallization. This is due to the slower crystalline growth kinetics in solution crystallization. 

6. Crystallization is obtained through cooling whereas crystal melting happens though 

heating. For this, the crystallization temperature is lower than the melting temperature due to 

the “hysteresis” effect. This effect is seen on the DSC spectra included earlier. 
463

Chapter 44 - SANS FROM A PLURONIC

1. WHAT ARE PLURONICS?

Poly(ethylene oxide) (referred to as PEO) is the simplest water soluble polymer. Its structure 

–CH2CH2O- consists of a hydrophobic ethylene –CH2CH2– group and a hydrophilic –O–

oxygen. Its neighbors in the homologous series poly(methylene oxide) (or PMO) and 

poly(propylene oxide) (or PPO) do not dissolve in water at ambient temperature. This 

behavior can be traced to the just-right balance between hydrophobic and hydrophilic 

interactions in PEO. For instance, the oxygen-oxygen inter-distance of 4.7 Å in PEO 

corresponds to the oxygen-oxygen next-nearest neighbor inter-distance in the structure of 

pure water. 

Pluronics are triblock copolymers composed of one PPO block connected to two PEO blocks. 

The PEO blocks dissolve well in aqueous media because they are mostly hydrophilic while 

the PPO blocks do not dissolve because they are mostly hydrophobic at ambient temperature. 

This amphiphilic nature of Pluronics molecules makes them form micelles at ambient 

temperature. At low temperatures, this balance does not hold and both PEO and PPO blocks 

dissolve in water thereby inhibiting micelle formation. The Critical Micelle formation 

Temperature (CMT) is a temperature at which micelles form. The CMC is the Critical 

Micelle formation Concentration. CMT and CMC vary depending on the block composition 

of the various Pluronics. 

Pluronics are commercially available materials used in the cosmetics and pharmaceutical 

industries. The P85 Pluronic considered here consists of 26 EO monomers in each of the 

outside blocks and 40 PO monomers in the middle block. P85 is referred to as 

EO26PO40EO26. The molecular weight of P85 is around 4,600 g/mol (Slawecki et al, 1998).

2. SANS FROM P85 PLURONIC

SANS data were taken from 10 % P85 in d-water at various temperatures (from 20 oC to 60 

oC). Two instrument configurations (one low-Q and one high-Q) were used. Micelles are 

well formed by 30 oC. When micelles are formed, the scattering is characterized by two 

features: a peak characteristic of inter-micelles interactions, and decay at high Q 

characterizing the tail of the single-particle form factor. In-between these two features, the 

hint of a second peak (shoulder around Q = 0.15 Å-1) can be observed. This shoulder is also 

characteristic of the single particle form factor (oscillation of the spherical Bessel function) 

and is affected by polydispersity effects. 
464

0.1

1

10

100

0.01 0.1

Pluronics/d-water 

60 oC

50 oC

40 oC

30 oC

20 oc Scattered Intensity (cm-1

)

Scattering Vector Q (Å-1)

Figure 1: SANS data from the 10 % P85 in D2O at various temperatures. Data statistics are 

different for the two instrument configurations. The overlap region for the two configurations 

is between 0.03 Å -1and 0.04 Å -1. 

3. THE CORE-SHELL PARTICLE MODEL

The P85 Pluronic forms micelles above the CMT and CMC. The simple core-shell model is 

used to analyze the SANS data from P85 in d-water (Kline-Hammouda, 2004). This model is 

reviewed here. Consider three regions: a core (region A) of radius RA, a shell (region B) of 

thickness RB-RA and the solvent (region C). 
465

Figure 2: Schematic representation of the P85 micelle as a core-shell particle. 

The following parameters are defined:

N: number of core-shell particles in the solution.

A: scattering length density for region A.

B: scattering length density for region B.

C: scattering length density for region C.

vA: specific volume in region A (= density/molar mass).

vB: specific volume in region B (= density/molar mass).

VA: volume of region A. VA = (4/3)RA

3

. 

VA+B: volume of regions A and B. VA+B = (4/3)RB

3

. 

V: total volume of the solution.

At first, assume that the particles do not interact, i.e., consider the infinite dilution case. 

Assuming uniform densities in the core and shell regions, the macroscopic scattering cross 

section is given as the particle number density multiplied by the square of the single-particle 

form factor. 

   

2

A

1 A

A

B

1 B

B C A B

A

1 A

A C A

I D QR

3j (QR )

V

QR

3j (QR )

V

QR

3j (QR )

V

V

N

d

d (Q)





























         

















(1)

The VD subscript stands for infinite dilution. The first part is for the core contribution 

(integration between 0 and RA) with the right scattering length density difference and the 

second part is for the shell contribution (integration between RA and RB). The single-particle 

form factor for a sphere (of radius RA) is expressed in terms of the familiar spherical Bessel 

function 

RA

A RB

B
466

















 

A

A

2

A

A

1 A QR

cos(QR )

(QR )

sin(QR )

j (QR ) . (2)

Another form for the same cross section is:

   

2

B

1 B

B C A B

A

1 A

A B A

I D QR

3j (QR )

V

QR

3j (QR )

V

V

N

d

d (Q)





























        

















. 

(3)

These forms reproduce the limit of the scattering cross section for uniform density spheres 

(of radius RA) by assuming either B = C or RB = RA. 

4. CONCENTRATION EFFECTS

When the particle concentration is finite (i.e., non zero), the scattering cross section contains 

contributions from the inter-particle structure factor SI(Q). The simplest analytical form for 

expressing SI(Q) is through the Percus-Yevick approximation for spheres interacting through 

a hard sphere potential. The Percus-Yevick approximation was introduced as a “closure” 

relation to the Ornstein-Zernike equation. Within the Percus-Yevick approximation, SI(Q) for 

a finite concentration of spheres (of diameter D) is expressed as:

1 NC(Q)

1

S (Q)

I



 . (4)

NC(Q)

is the Fourier transform of the direct correlation function given by:

     

 

         

   

















   

  









 

    4

2

1 3 2

QD

QD cos QD 2 QD sin QD 2cos QD 2

6

QD

sin QD QD cos QD NC(Q) 24                  

  

















      

  6

4 3 2

2

QD

QD cos QD 4 QD sin QD 12 QD cos QD 24 QD sin QD 24cos QD 24

2

(5)

The parameters 1 and 2 are given by:

 

 

4

2

1

1

1 2

 

 

 

(6)

 

 

4

2

2

1

1 2

 

  

  .
467

 is the packing fraction which can be expressed in terms of the particle number density 

(N/V) and particle radius R (R = D/2) and is defined as

























 

3 R

3

4π

V

N

. Note that in the 

core-shell particle model R = RB.

The scattering cross section for a concentrated solution of hard spheres is obtained by 

multiplying the infinite dilution result by SI(Q): 

S (Q)

d

d (Q)

d

d (Q)

I

ID























. (7)

The single-particle form factor and the inter-particle structure factor are the main pieces of 

the scattering cross section for the P85 micelles. 

5. FIT RESULTS

Fits of the P85 SANS data were performed using the non-dilute solution of core-shell 

particles. Results for the 10 % P85 in D2O are presented here for the 40 oC temperature case 

where the micelles are well formed. 

Packing volume fraction  = 0.248 (8)

Core radius RA = 43.96 Å. 

Polydispersity parameter A/RA = 0.16

Shell outer radius RB = 72.91 Å. 

A

= 7.563*10-7 Å

-2

B

= 5.940*10-6 Å

-2

C

= 6.40*10-6 Fixed.

These fit results are used to obtain detailed information about the P85 micelles using the 

material balance equations. 
468

0.1

1

10

100

0.01 0.1

10% P85 Pluronic in D2O at 40 oC

SANS Data

Model Fit

Scattered Intensity (cm-1

)

Scattering Vector Q (Å-1)

Figure 3: SANS data and model fit for the 10 % P85 in D2O sample at 40 oC. The two curves 

are indistinguishable. 

6. MATERIAL BALANCE EQUATIONS

Consider a model whereby the micelle core contains PO blocks and a fraction f of the EO 

blocks and the shell contains the remaining fraction (1-f) of the EO blocks. Moreover assume 

that D2O exists in the core and in the shell to hydrate the EO blocks. Assume that there are yA

D2O molecules per EO monomer in the core (region A) and yB D2O molecules per EO 

monomer in the shell (region B). Define Nagg as the aggregation number, i.e., the number of 

P85 molecules per micelle and recall that there are 40 PO monomers per block and 26*2 = 52 

EO monomers per macromolecule.

The material balance equations are:

(1) 

3

4

RA

3 = Nag.[40.vPO + 52.f.vEO + 52.f.vD2O.yA] (9)

(2) 

3

4

(RB

3

- RA

3

) = Nag.[52.(1-f).vEO + 52.(1-f).vD2O.yB]
469

(3) 

3

A

a g P O EO D2O A

A

R

3

4

N [40b 52.b .f 52b .f.y ]



 

 

(4) 

R R )

3

4

N [52.b .(1 f) 52.b .(1 f).y ]

3

A

3

B

a g EO D2O B

B





  

  .

Specific volumes and scattering length densities have been defined for EO, PO and D2O. 

These four linear equations can be solved to obtain:

D2O D2O B

EO EO B

B

b v

b v

y

  

 



(10)

f

A Bf

yA





where:

D2O

3 PO

A

3

B

3

A

EO B D2O

52v

40v

R R

R

[52v 52y v ]

A







 (11)

D2O

3 EO

A

3

B

3

A

EO B D2O

v

v

R R

R

[ v y v ]

B





 



D F

E C

f







And:

PO D2O A C  [40v  52.A.v ]

(12)

EO D2O A D  [52v  52.B.v ]

E [40b 52.A.b ]  PO  D2O

F [52b 51.B.b ]  EO  D2O

The last parameter is given by:

PO EO D2O A

3

A

ag 40v 52v .f 52v .f.y

R

3

4

N

 



 .
470

We have transformed the four fitting parameters RA, RB, A and B into four meaningful 

parameters Nag, f, yA and yB. This set of solutions is unique.

7. RESULTS FOR THE 10 % P85 IN D2O AT 40 oC

The following results are obtained for the 10 % P85 in D2O at 40 oC. 

Table 1: Scattering lengths and specific volumes for PO, EO and D2O. 

Density

g/cm3

mw

g/mol

Molar volume

cm3/mol

Scattering Lengths 

cm

Scattering Length 

Densities (Å-2)

PO 1.004 C3H6O 58 vPO = 57.77 bPO = 3.307*10-13 PO = 3.44*10-7

EO 1.127 C2H4O 44 vEO = 39.04 bEO = 4.139*10-13 EO = 6.38*10-7 

D2O 1.11 D2O 20 vD2O = 18.02 bD2O = 19.145*10-13 D2O = 6.39*10-6

Using the table of scattering lengths and specific volumes for the various components, and 

taking care of expressing everything consistently in units of Å or cm, the following results 

are obtained. 

f = 0.53 (53 % of the EO monomers are in the core).

yA = 0.39 (there is less than one D2O molecule per EO monomer in the core).

yB = 26.44 (there are 26.44 D2O molecules per EO monomer in the shell). 

Nagg = 59.80 (there are 59.80 P85 molecules per micelle). 

The finding that 53 % of the EO monomers can be found in the micelle core is surprising. 

Not much D2O makes it to the core region as expected even with the presence of a fraction of 

the EO monomers. The micellar shell region is swollen by quite a bit of D2O as expected. 

Note that in this simple model we have assumed that the various specific volumes are 

constant with temperature. 

The micelles number density 

N V

(i.e., number of micelles per cm3) can be obtained from 

the packing volume fraction  as:











 



 











3

RB

3

V 4

N

. (13)

The estimated P85 volume fraction can be obtained as:
471

P85 = 













V

N

Nagg.(40.vPO+52.vEO). (14)

In the case considered here, one obtains 

N V

= 1.53*1017 cm-3and P85 = 0.066 which is 

lower than the real mixing fraction Mix = 0.1. 

With a rather simple (core-shell) model, one can obtain quite a bit of information from the 

SANS data. The purpose of this exercise is not to prove that the core-shell model used is 

correct, but to show an example of the mass balance equations that relate the fitted 

parameters to more meaningful properties. 

Figure 4: Schematic representation of a micelle with sizes, scattering length densities and 

contents. 

RB = 72.91 Å

v A

b













= 7.563*10-7

Å

-2

v B

b













= 5.940*10-6

Å

-2

v C

b













= 6.400*10-6

Å

-2

core region A

shell region B

solvent region C

In the core:

2,392 PO monomers

1,648 EO monomers

1,213 D2O molecules

In the shell:

1,462 EO monomers

38,642 D2O molecules

RA = 43.96 Å
472

8. POLYDISPERSITY EFFECTS

The polydispersity parameter obtained from the fit to the SANS data for the 10 % P85 

sample at 40 oC was A/RA = 0.16, where A is the standard deviation of the core radius size 

distribution. This distribution was assumed to correspond to the Schulz distribution. 

 

R Γ(z 1)

R

R

exp (z 1)

R

R

f(R) z 1

A

A

z

A

z 1















 

















 



. (15)

RA is the average value for the peaked distribution and the standard deviation A is related to 

z as:

(z 1)

1

R A

A







. (16)

The integration over the Schulz distribution is performed over the macroscopic cross section 

for the core-shell particle model. 

polydisperse 0 monodisperse

d

d (Q)

dR f(R)

d

d (Q)





















 















. (17)

This integration can be performed numerically and is available analytically. Due to the 

lengthy expression, the analytical form is not reproduced here. 
473

0.1

1

10

100

0.01 0.1

10% P85 Pluronic in D2O at 40 oC



A

/RA  = 30%



A

/RA  = 16%



A

/RA  = 1%

Scattered Intensity (cm-1

)

Scattering Vector Q (Å-1)

Figure 5: Simulated data corresponding to parameters for the 10 % P85 sample at 40 oC. The 

polydispersity parameter has been varied to see its effect. All other parameters were kept the 

same. The effect of polydispersity is seen to broaden peaked features. 

Note that the particle core volume averaged over the polydisperse size distribution is given 

by:

2

3

A

3

A

(z 1)

(z 2)(z 3)

R

3

4

R

3

4

V



  







In practice the average volume (4RA

3

/3) is often used for low polydispersity.

9. TEMPERATURE EFFECTS

SANS data were taken from the 10 % P85 in D2O at various temperatures going from the 

partly dissolved copolymer phase at 20 oC to the fully formed micelles at temperatures of 

30oC and above. The above described model was applied to the SANS data for 40 oC, 45 oC, 

50 oC, 55 oC, and 60 oC. For simplicity, the specific molar volumes were assumed to be 

independent of temperature. The results are presented in a series of figures. 
474

When temperature is increased, both RA and RB increase while the micelle packing volume 

fraction decreases. This is due to the fact that the aggregation number increases because the 

driving force for micelle formation gets stronger. As this happens, D2O gets squeezed out of 

the shell region. As temperature increases, the micelles packing volume fraction decreases 

because there are fewer larger micelles. 

50

60

70

80

90

100

35 40 45 50 55 60 65

10% P85 Pluronic in D2O

Fit Result

P85 Aggregation Number

Temperature (oC)

Y = M0 + M1*X

M0 -10.112

M1 1.7522

R 0.99974

Figure 6: Variation of the aggregation number with temperature. 
475

19

20

21

22

23

24

25

26

27

35 40 45 50 55 60 65

10% P85 Pluronic in D2O

Fit Result

Number of D2O Molecules 

per PEO Monomer 

Temperature (oC)

Y = M0 + M1*x + ... M8*x8 + M9*x9

M0 66.042

M1 -1.4325

M2 0.010919

R 0.99129

Figure 7: Variation of the number of D2O molecules per EO monomer in the shell region 

with increasing temperature. 
476

1 1017

1.1 1017

1.2 1017

1.3 1017

1.4 1017

1.5 1017

1.6 1017

35 40 45 50 55 60 65

10% P85 Pluronic in D2O

Fit Result

Number of P85 Micelles per Unit Volume

Temperature (oC)

Y = M0 + M1*x + ... M8*x8 + M9*x9

M0 2.9167e+17

M1 -4.2162e+15

M2 1.8671e+13

R 0.99923

Figure 8: Variation of the micelles number density with temperature. 

10. DISCUSSION

P85 is a typical nonionic micellar system comprising hydrophilic EO blocks and hydrophobic 

PO blocks. SANS data from P85 micelles were fitted to a core-shell model in order to obtain 

core sizes and shell thicknesses, scattering length densities of these two regions, fitted 

volume fractions and core polydispersity characteristics. Using a materials balance equation, 

useful information such as the fraction of EO blocks in the core, the aggregation number, and 

the number of D2O molecules per EO monomer in the two regions. The purpose of this last 

exercise was not to prove the correctness of the core-shell model but to show how useful 

material balance equations could be. 

REFERENCES

T. Slawecki, C. Glinka and B. Hammouda, “Shear-Induced Micellar Crystal Structures in an 

Aqueous Triblock Copolymer Solution”, Phys Rev E 58, R4084-4086 (1998).
477

S. Kline and B. Hammouda “Free Chains and Micelles of a Triblock Copolymer in Water”, 

NCNR Summer School (2004).

QUESTIONS

1. Do Pluronics form ionic or nonionic micelles?

2. What is the CMC? How about the CMT?

3. Are micelles always spherical?

4. What is the Percus-Yevick approximation?

ANSWERS

1. Pluronics are not charged. They form nonionic micelles. 

2. The CMC is the Critical Micelles (formation) Concentration. The CMT is the Critical 

Micelles (formation) Temperature. These are conditions at which micelles form. 

3. No, micelles are not necessarily spherical. Micelles come in many shapes including 

elongated wormlike micelles. 

4. The Percus-Yevick approximation is a closure relation that allows the analytical solution 

of the Ornstein-Zernike equation for hard sphere interaction potential. 
478

Chapter 45 - SANS FROM IONIC MICELLES

Ionic micelles use surfactants with charged head groups. When mixed with hydrophilic and 

hydrophobic molecules, these self-assemble into micelles of various shapes. Micelles form in 

order to screen the hydrophobic groups and avoid their contact with water. Coulomb 

interactions contribute to micelle formation. A system that forms rodlike ionic micelles is 

described here. 

1. AN IONIC RODLIKE MICELLES SYSTEM

Consider the following ionic micellar system: cationic surfactant cetyltrimethyl ammonium 

4-vinylbenzoate (CTVB) in aqueous (d-water) solution (Kline, 1999; Kim et al, 2006). These 

form rodlike micelles. Free radical polymerization is performed on the VB groups in order to 

obtain polymerized micelles. After polymerization, negative charges (VB-) are on the outer 

surface and positive charges (CTA+) are on the inner surface of the rodlike polymerized 

micelles.

Figure 1: Schematic representation of the CTVB polymerized rodlike micelle. 

A set of four CTVB/d-water samples were prepared with different micelles fractions. These 

correspond to 0.25 %, 0.5 %, 1 % and 1.9 % CTVB mass fractions. SANS measurements 

were made at 25 oC. As the mass fraction increases, an inter-particle “interaction” peak is 

seen to develop. 

VBCTA+

(CO)OCH3(CH2)15N

+

(CH3)3
479

0.1

1

10

0.01 0.1

CTVB/d-water at 25 oC 

1.9 %

1 %

0.5 %

0.25 %

Scattering Intensity (cm-1

)

Scattering Variable Q (Å-1)

Figure 2: SANS data from CTVB/d-water for increasing concentration. 

2. SCATTERING MODEL

A scattering model consisting of a solution of interacting rodlike particles is used to fit the 

SANS data. The scattering intensity (cross section) is given by:

(Q) B

I

V P(Q)S

2 B

d

d (Q)

I(Q)     P 





 . (1)

Here 

2

is the contrast factor,  is the particles volume fraction, VP is the particle volume 

and B is a constant used to represent the Q-independent (mostly incoherent scattering) 

background. 

The form factor for a cylinder is given by the following orientational average:

2

1

1

d | F(Q, ) |

2

1

P(Q)





   . (2)
480





































 

Q 1 R

2J (Q 1 R)

Q L / 2

sin(Q L / 2)

F(Q, )

2

2

1

. (3)

 represents the rod orientation with respect to the scattering vector direction. R is the 

cylinder radius and L is its length. J1 is the cylindrical Bessel function. 

The structure factor for a solution of charged particles is obtained from the Ornstein-Zernike 

(OZ) equation solved with the Mean Spherical Approximation (MSA) closure relation. The 

MSA approach was used to account for the Coulomb interactions. Note that the MSA 

solution was originally introduced for spherical particles. Since there is no simple analytical 

approach that can model the structure factor for rodlike particles, the MSA is used here for 

lack of a better model. The structure factor is given by:

1 NC(K)

1

S (Q)

I





(4)

 

3

2

3 K

K

2

1 Kcos(K) 2sin(K)

K

2

B

K

A sin(K) Kcos(K)

24

NC(K)













  





















3

3 2 2 4

2K

Kcos(K)

K

24

K

12 sin(K) 1

K

6

4 1

K

24 A 























   











  



 

 

2 2 K K k

C k cosh(k)sin(K) Ksinh(k)cos(K)







  

 

2 2 K K k

F k sinh(k)sin(K) K cosh(k)cos(K) 1



 



    

 

2 2 2 K K k

exp k k sin(K) Kcos(K)

K

F cos(K) 1



  





 .

Here K = QD is the reduced scattering variable and D is the rodlike micelle diameter. 

Note that it is difficult to model overlapping rods since these could form liquid crystalline 

(such as nematic or smectic) phases. Only the isotropic phase (obtained for a low 

concentration of rods) can be modeled by the MSA approach and is of interest here. 
481

Figure 3: Schematic representation of the main liquid crystal phases for rodlike particles. 

These are the isotropic, nematic and smectic phases obtained when the rodlike particle 

concentration increases. 

3. FITS OF THE SANS DATA

Fits of the model to the SANS data are performed. Results for the 1 % mass fraction sample 

are included here. The contrast factor for the CTVB/d-water mixture was fixed as well as the 

dielectric constant  and the sample temperature T. 

 = d-water -CTVB= 6.39*10-6- 0.35*10-6 = 6.04*10-6 Å

-2

(5)

 = 77.94 

T = 298 K.

The remaining fitting parameters were varied and found to be: 

 = 0.01 (6)

R = 20.9 Å

L = 184 Å

zm = 0.06

B = 0.074 cm-1. 

In order to appreciate the contributions from the form factor and the structure factor terms, 

SANS data are compared to the results of the model fits in the dilute limit (i.e., when 

SI(Q)=1). The structure factor SI(Q) is also plotted. The effect of inter-particle interactions is 

small but finite. 

Isotropic Phase Nematic Phase Smectic Phase
482

0.1

1

10

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0.01 0.1

1 % CTVB/d-water at 25 oC 

SANS Data

Dilute Limit

S

I

(Q) 

Scattering Intensity (cm-1

)

S

I(Q)

Scattering Variable Q (Å-1)

Figure 4: Comparison of the SANS data to the model fit in the dilute limit. The structure 

factor SI(Q) is also plotted. 

The apparent radius of gyration for the rodlike cylinders of radius R and length L is 

calculated using:

12

L

2

R

R

2 2

g

  . (7)

Results are plotted for increasing CTVB mass fraction. A polynomial fit is performed and 

yields:

        

2

1 2 0g app

Rg R m m

(8)

This is sometime referred as a “virial expansion”. The “real” radius of gyration is obtained at 

the infinite dilution limit as:

R  124 Å

0

g

 . (9)
483

Note that the first dominant correction term in the expansion is negative. In the infinite 

dilution (subscript ID) limit, P(Q) decreases (and therefore Rg decreases) with increasing 

micelles fraction.

V P(Q)

B

d

d (Q)

P

ID 2

 



 















. (10)

40

60

80

100

120

0 0.005 0.01 0.015 0.02

CTVB/d-water at 25 oC 

Data and Polynomial Fit 

Apparent Radius of Gyration (Å) 

 CTVB Fraction

Y = M0 + M1*x + ... M8*x8 + M9*x9

M0 123.46

M1 -8638.9

M2 2.2172e+5

R 0.98079

Figure 5: Variation of the apparent radius of gyration for increasing CTVB fraction. 

Temperature was varied for the 1 % CTVB/d-water sample. Rodlike particle dimensions (R 

and L) were obtained from the fits. Since the micelles are polymerized, there is very weak (to 

non-existent) temperature dependence of the radius R but noticeable temperature dependence 

of the rod length L. 
484

180

185

190

195

200

205

210

20 30 40 50 60 70 80 90

1 % CTVB/d-water 

Model Fit 

Length L (Å)

Temperature (oC)

Y = M0 + M1*x + ... M8*x8 + M9*x9

M0 172.26

M1 0.52429

M2 -0.0011226

R 0.99823

Figure 6: Temperature dependence of the rodlike particles length for increasing sample 

temperature. 

In order to perform the fits to the SANS data when sample temperature was varied, 

temperature dependence of the dielectric constant for d-water was required.  is seen to 

decrease with temperature as tabulated (CRC Handbook, 1984). 
485

55

60

65

70

75

80

85

90

0 20 40 60 80

Published Data 

Dielectric Const for D2O

Temperature (o C)

Y = M0 + M1*x + ... M8*x8 + M9*x9

M0 87.679

M1 -0.4249

M2 0.0013677

M3 -3.6439e-6

R 0.99993

Figure 7: Temperature dependence of the dielectric constant for d-water. 

4. DISCUSSION

SANS data from the CTVB/d-water ionic polymerized micelle system are characterized by 

varying features when the CTVB fraction increases. For low volume fractions, the MSA 

model developed for particulate systems with Coulomb interactions applies. A model 

consisting of the form factor for rodlike particles and the MSA structure factor was used to fit 

the SANS data. Fit results included rod dimensions (rod radius and length) in each case along 

with the macroion charge. This charge was found to be very small pointing to almost neutral 

rodlike particles. 

REFERENCES

R.C. Weast, Editor-in-Chief, “CRC Handbook of Chemistry and Physics”, 65th Edition, Page 

E57 (1884). 

S.R. Kline, “Polymerization of Rodlike Micelles”, Langmuir 15, 1726-1732 (1999)
486

T-H Kim, S-M Choi and S.R. Kline, “Polymerized Rodlike Nanoparticles with Controlled 

Surface Charge Density”, Langmuir 22, 2844-2850, (2006)

QUESTIONS

1. What is the form factor for an infinitely thin rod of length L?

2. Name two possible closure relations used to solve the Ornstein-Zernike equation for 

particulate systems?

3. What is the Debye-Huckel screening length? Define it for a neutral solution of macroions 

of charge zme and electrons. 

4. What is the CMC?

5. What is the radius of gyration for a cylindrical rod of radius R and length L?

ANSWERS

1. The form factor for an infinitely thin rod of length L is given by: 

2 1

1 Q L/ 2

sin(Q L/ 2)

d

2

1

P(Q)

















  



. 

2. Two possible closure relations used to solve the Ornstein-Zernike equation for particulate 

systems are: the Percus-Yevick and the Mean Spherical Approximation. 

3. The Debye-Huckel screening length is the distance beyond which Coulomb interactions 

die out (are screened). The Debye-Huckel screening parameter (inverse length) is given by: 

z N

k T

e

m

B

2

2

 

where e is the electron charge, zme is the macroion charge, 

N

is the 

macroion number density (number per unit volume) and kBT is the sample temperature in 

absolute units. 

4. The Critical Micelle Concentration (CMC) is the surfactant concentration for which 

micelles form. 

5. The radius of gyration for a cylindrical rod of radius R and length L is given by: 

12

L

2

R

R

2 2

g

  .
487

Chapter 46 – OTHER SANS FROM COMPLEX FLUIDS PROJECTS

Complex fluids or self-assembling systems represent a large fraction of SANS projects. A 

few of these projects of interest to this author are included here. 

1. ENCAPSULATION OF IBUPROFEN DRUG IN PLURONIC MICELLES

Pluronic P103, P104 and P105 are triblock copolymers PEO-PPO-PEO with the same PPO 

block but with increasing PEO blocks [1]. The aggregation of the hydrophobic central PPO 

block prompts the formation of micelles. The hydrophilic PEO blocks form a dissolved shell 

outside of the micelles. Added Ibuprofen drug resides in the hydrophobic (core) region of the 

micelles. A series of samples were measured corresponding to a range of added Ibuprofen 

and measurement temperatures with 5 % Pluronic fraction. 

The uniform core-grafted chains model was used to fit SANS data. Increasing the Ibuprofen 

content or the temperature favors micelle formation as evidenced by the increase in the 

aggregation number and micelle core radius. The critical micelle temperature is also reduced 

when the Ibuprofen content is increased. This is accompanied by a decrease in the volume 

fraction of the d-water solvent in the core. 

Figure 1: SANS from 5 wt %/vol Pluronic P105 in D2O with increasing quantities of 

ibuprofen, equilibrated at 293 K: (O) 0.00 wt %/vol ibuprofen, (3) 0.25 wt%/vol ibuprofen, 

(0) 0.50wt%/ vol ibuprofen, and (]) 0.75 wt%/vol ibuprofen. Solid lines are fits to the 

Pedersen model for Pluronic triblock copolymers.
488

At high temperature, micelles become elongated pointing to a structural phase transition 

from globular (spherical) to cylindrical micelles. 

Figure 9. SANSfrom5wt%/vol Pluronic P104 inD2O, at a range of temperatures. Solid lines 

are fits to the Pedersen model for Pluronic triblock copolymers. Data have been shifted 

vertically to see the clarity of fits, and the inset shows the data (in the same order as the main 

graph) without shifting.

Adding ibuprofen not only decreases the critical micelle formation condition but also 

modifies the micellar shape to ellipsoidal as evidenced by the appearance of an additional 

hump in the scattering data. 
489

Figure 10. SANS from 5 wt %/vol Pluronic P104 and 0.5 wt %/ vol ibuprofen inD2O, at a 

range of temperatures. Solid lines are fits to the Pedersen model for Pluronic triblock 

copolymers. Data have been shifted vertically to see the clarity of fits, and the inset shows

the data (in the same order as the main graph) without shifting.

SELF-ASSEMBLY OF PLURONIC TRIBLOCK COPOLYMER 17R4 

Pluronic copolymer 17R4 PPO-PEO-PPO contains the hydrophobic PPO blocks at the end 

and the hydrophilic PEO one in the middle [2]. PPO blocks aggregate together to form 

micelles with the PEO blocks looping out then back into the micelles. This copolymer 

presents a rich phase diagram comprising many regions. Region I (low temperature and 

copolymer fraction) in which the cloudy solution includes large clusters of PPO blocks 

forming hydrophobic domains (cores) bridged by hydrophilic PEO blocks. Region II is 

above the micelle formation line whereby flower-like structures coexist with unimers. 

Flower-like structures form since for the middle PEO block to remain dissolved, it has to 

loop back onto the hydrophobic PPO core. Region III sees complete phase separation with a 

top phase and a bottom phase separated by a meniscus. The concentrated upper phase 

contains high concentration of micelles forming a connected network. 
490

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5

Pluronic 17R4 in d-water

 Temperature (

oC) 

Pluronic Fraction 

Phase I

Phase II

Phase III

Figure 2. Phase diagram of Pluronic 17R4 in d-water.

A series of figures represents the trends observed in the data which consist in a low-Q 

clustering feature (power law model) and a high-Q micelles structure feature (either the 

correlation length model or the Guinier-Porod model). The correlation length model yielded 

a correlation length and a Porod exponent while the Guinier-Porod model yielded a radius of 

gyration and a high-Q Porod exponent along with a “dimensionality” parameter (equal to 

zero for globular objects and equal to one for cylindrical particles). 
491

0.1

1

10

100

1000

0.01 0.1

Pluronic 17R4 in d-water

24.9 oC 

15.5 oC 

 Scattering Intensity (cm-1) 

 Scattering Variable Q (Å-1) 

Figure 3. SANS data from sample with mass fraction of 0.23 in region I.
492

0.1

1

10

100

1000

0.01 0.1

Pluronic 17R4 in d-water

49.3 oC 

11.9 oC 

 Scattering Intensity (cm-1) 

 Scattering Variable Q (Å-1) 

Figure 4. SANS data from sample with mass fraction of 0.11 17R4 in region I (11.9 oC and 

30.8 °C) and in region III (49.3 °C). Scattering from unimers or unimers bridging between 

clusters (high Q) and from clusters (low Q) appear at all temperatures. At 49.3 oC, the peak

indicates that phase separation has occurred.
493

1

10

100

0.01 0.1

Pluronic 17R4 in d-water

29.8 oC 

34.5 oC 

 Scattering Intensity (cm-1) 

 Scattering Variable Q (Å-1) 

Figure 5. SANS data from  = 0.23 17R4 in D2O in region II and in Region I. In Region II 

scattering does not show the low-Q clustering feature.

Fit results along with the use of other characterization techniques (dynamic light scattering, 

microscopy and rheology) provided a basis for conclusions about what structures are 

involved in each of the three phases. 

Region I contains clustered unimers, region II contains flower-like structures while region III 

contains demixed phases containing a micelles-rich top phase and a micelles-poor bottom 

phase. The demixing is prompted by the lower critical solution temperature (LCST) 

demixing of PEO at high temperature. The top phase may contain flower-like structures. 

NANOSTRUCTURE OF SDS MICELLES IN WATER

A series of sodium dodecyl sulfate (SDS) solutions in d-water were investigated by SANS 

[3]. Samples with the following SDS mass fractions were measured: 0.1 %, 0.5 %, 1 %, 2 %, 

5 %, 10 %, and 20 %. Temperature was varied between 10 oC and 90 oC with 10 oC intervals. 

SANS data show a weak low-Q (long-range) feature and a dominant intermediate-Q (shorterrange) feature which is due to the micelle particles structure. The intermediate-Q peak and 

shoulder features observed in the SANS data are characteristic of anisotropic micelles such 

as ellipsoidal particles. These are seen to move to higher Q upon heating implying that 

particles get smaller with increasing temperature. The low-Q feature (observed at low SDS 
494

fractions) is likely due to clustering and characterizes water-soluble (especially ionic) 

systems. 

0.1

1

10

0.01 0.1

 5% SDS/d-Water 

11 oC 

30 oC 

49 oC 

68 oC 

87 oC 

Scattering Intensity (cm-1

)

Scattering Variable Q (Å-1)

Figure 6: SANS data for 5 % SDS mass fraction while varying temperature. The peak and 

shoulder features are characteristic of ellipsoidal micelles. 

Increasing the SDS mass fraction for fixed high temperature (68 oC) shows smooth shape 

change for the ellipsoidal micelles. Only samples that are above the critical micelle formation 

concentration (i.e., at or above 0.5 %) are included in this trend. The peak and shoulder 

features become more pronounced and move to higher Q, which means that the micelles 

become more ellipsoidal and their packing gets tighter.
495

0.1

1

10

0.01 0.1

 68 oC 

20% SDS

10% SDS

5% SDS

2% SDS

1% SDS

0.5% SDS

Scattering Intensity (cm-1

)

Scattering Variable Q (Å-1)

Figure 7: SANS data for varying SDS mass fraction and a fixed temperature of 68 oC.

The recurring clues characterizing the SANS data consist of two size scales observed on the 

intermediate-Q peak. This points to ellipsoidal shape micelles. A scattering model consisting 

of a solution of interacting ellipsoidal particles is used to fit the SANS data. The scattering 

cross section is expressed as:

V P(Q)S (Q)

2

ellipsoids d

d (Q)



   P I















. (1)

Here 

2

is the contrast factor,  is the particle volume fraction, VP is the particle volume, 

P(Q) is the single-particle form factor, and SI(Q) is the inter-particle structure factor. This 

model works best for spherical particles, and is used here for ellipsoidal particles that are not 

too distorted. 

The form factor represents an average over orientations of the anisotropic particles. Particles 

are assumed to be ellipsoidal with half axes Ra and Rb. For an oblate ellipsoid particle (with 

Rb>Ra), an effective radius Reis defined as 

R R (R R )

2

b

2

a

2 2

b

2

e

    .

The form factor amplitude is the same as the one for a sphere of radius Re: 
496 2

e

1 e

QR

3j (QR )

P(Q, ) 











  . (4)

Here 

j (QR ) 1 e

is the spherical Bessel function of order 1. The Mean Spherical

Approximation (MSA) is used to model the structure factor 

(Q)

I

S

. This model is known to 

be reliable when screened Coulomb interactions are present (such as for ionic micelles), and 

relies on the MSA closure relation to solve the Ornstein-Zernike equation. It should be 

mentioned that the approximate MSA model is often used since it relies on an analytical 

solution. Fits to this model yield effective sizes. 

The following model parameters are used:  is the dielectric constant, D is the micelle (also 

called macroion) effective diameter, 



is the Debye-Huckel inverse screening length, and 

zme is the electric charge on the micelle surface where e is the electron charge. 

The Debye-Huckel screening parameter (inverse length) squared is expressed as follows:















 







 

salt

salt

P

m

B

2

2

V V

z

k T

e

(5)



and 

salt

are the micelle particle and salt volume fractions, 

VP

and 

Vsalt

are the particle 

and salt molecule volumes, and kBT is the sample temperature in absolute units. 

The micelle volume fraction 



is expressed in terms of the number density 

N

and micelle 

volume 

V D 6

3

P  

as 

  NVP

. 

In order to perform fits to the SANS data when sample temperature was varied, tabulated 

temperature dependence of the dielectric constant for d-water is used (i.e., is fixed to help the 

fits). 

The model used to fit the SANS data consists of the sum of two functional forms: a low-Q 

power law function and the ellipsoidal micelles model:

B

d

d (Q)

Q

A

I(Q)

ellipsoids

n

 















  . (6)

n is a low-Q Porod exponent, 

 ellipsoids d(Q) d

was discussed above and B is a constant 

representing the Q-independent background mostly due to incoherent scattering from 

hydrogen.

Both ellipsoidal micelles half axes Ra and Rb decrease with increasing temperature. The 

value of Rb was systematically larger than Ra pointing to oblate (i.e., compressed) ellipsoidal 

micelles as expected.
497

12

14

16

18

20

22

24

0 20 40 60 8 0 100

1 % SDS/d-Water 

Rb

Ra Ellipsoid Half Axes (Å)

Temperature ( oC)

Figure 8: Variation of the ellipsoid micelles half axes with increasing temperature for the 1 % 

SDS sample. The lines going through the points are guides to the eye (smooth fitting). 

Fit results show that the charge on the micelles increases with SDS weight fraction as it 

should since the size of micelles increases with increasing SDS fraction. Micelle charges, 

however, decrease with increasing temperature since the micelle volume decreases with 

increasing temperature. This trend breaks down for the highest SDS mass fraction (20 %) 

sample. The same saturation trend at high SDS fraction was observed previously. Since the 

SDS molecule carries one electron charge on the ionized oxygen atom, the micelle charge 

scales with the micelle aggregation number. 
498

10

100

0.01 0.1

21 oC

40 oC

59 oC

78 oC

 Micellar Charge

SDS Mass Fraction

Figure 9 Variation of the micelle charge with increasing SDS mass fraction for various 

sample temperatures. 

Note that a materials balance equation approach can be used to obtain more information of 

the micelles content. 

REFERENCES

1. B. Foster, T. Cosgrove and B. Hammouda, “Pluronic Triblock Copolymer Systems and 

Their Interactions with Ibuprofen”, Langmuir 25, 6760-6766 (1009). 

2. B. Kumi, B. Hammouda and S. Greer, “Self-Assembly of the Triblock Copolymer 17R4 in 

D2O”, J. of Colloid and Interface Science 434, 201-207 (2014).

3. B. Hammouda, “Temperature Effect on the Nanostructure of SDS Micelles in Water”, J. of 

Research of NIST 118, 151-167 (2013).
499

Part J – SANS IN BIOLOGY

Chapter 47. Elements of Biology

47.1 Cell Biology

47.2 Lipids

47.3 DNA and RNA

47.4 Proteins

47.5 Polysaccharides

47.6 Cells and Organelles

47.7 Characterization Methods

47.8 Neutron Scattering Lengths

References

Questions

Answers

Chapter 48. SANS from Phospholipid Bilayers under Pressure

48.1 Phospholipids

48.2 Phospholipid Bilayers Phases

48.3 SANS Under Pressure

48.4 The Phase Diagram

48.5 Comments

References

Questions

Answers

Chapter 49: SANS from DNA

49.1 The DNA Double Helix

49.2 UV Absorption Spectroscopy 

49.3 Helix-to-Coil Transition in mixed Solvents

49.4 Helix-to-Coil Transition by SANS

49.5 A Helix-to-Coil Transition Model 

49.6 Application of the Model to UV Absorption Data

References

Questions

Answers

Chapter 50: SANS from a Protein Complex

50.1 Introduction

50.2 SAXS from the Protein Complex

50.3 SANS from the Protein Complex

50.4 Comments

References

Questions

Answers

Chapter 51: SANS in Biology Literature Review
500

52.1 1. DNA and RNA

52.2 Lipids and Membranes

52.3 Proteins

References
501

Chapter 47 – ELEMENTS OF BIOLOGY

Biology is the science of life. The word microbiology refers to the observation of biological 

systems in the optical (micrometer) size scale. Molecular biology is the science of biology at 

the molecular level. The new term nanobiology could refer to investigations in the nanometer 

scale. This is our focus. 

1. CELL BIOLOGY

Understanding the cell drives a great deal of biology research. Microbiology is moving from 

a mode driven by clinical research to new developments in bio-genetics. The human genone 

has been mapped out a few years ago. The 23 pairs of chromosomes contain DNA which 

codes for around 23,000 protein-coding genes involving some 3 billion base pairs. These 

control the synthesis of all proteins. The order of the nucleotides in DNA controls the order 

of amino acids in protein synthesis. Proteins perform most of the biological functions 

essential for life (Becker et al, 1999). 

Cells contain organelles which perform specific functions. One of these, the nucleus, 

contains the DNA double helix packed into the chromosomes. The same DNA is contained in 

every nucleus of every cell in the body. Cell division involves DNA replication whereby new 

copies are made. 

Cells are bounded by a membrane with a hydrophobic interior and a hydrophilic exterior. 

Lipids are surfactant-like molecules containing (hydrophobic) hydrocarbon tails and polar 

(hydrophilic) head groups; lipids form the cell membrane. Special proteins help in the 

transport of essential ingredients into and out of the cell. The passage of small molecules 

(such as CO2 and H2O) and of essential ionic groups also occurs.

The interplay of hydrophobic and hydrophilic interactions is a major diving force in the 

working of membranes, in the assembly of cellular structures as well as in the folding of 

macromolecules such as DNA and proteins. Proteins are folded into helical and sheet 

portions in their active form which is essential for specific functions. 

Polysaccharides perform storage and structural functions in cells. Starch and cellulose are 

typical plant polysaccharides. They are formed of repeating sugar glucose units. 

2. LIPIDS

The simplest components of a cell are the lipid molecules forming the membranes. These are 

amphiphile (surfactant) molecules with hydrophilic head groups and hydrophobic tails. 

Membranes surround the cell as well as the organelles inside the cell. Proteins form channels 

across the membrane and other structures in membranes. 
502

Figure 1: Schematic representation of a membrane.

Lipids are formed from a glycerol backbone and fatty acid side chains. These two undergo a 

condensation reaction that removes a water molecule. 

Figure 2: Reaction that forms lipids.

One specific lipid corresponding to a C17 hydrocarbon tail is included here. 

lipids

proteins

MEMBRANE

0 5 nm

Glycerol

 

 H

 H-C-OH

 H-C-OH

 H-C-OH

 H

HOC

Fatty acid
503

Figure 3: Example of a lipid molecule.

The fatty acids forming the lipid tails contain either C=C double bonds or saturated C-C 

single bonds. The double bonds produce kinks in the fatty acid chain. 

3. DNA AND RNA

DNA is the key component for the passage of genetic information between parent and 

offspring cells or organisms. It is composed of linear chains of nucleotides, whose sequence 

determines and regulates subsequent expression of proteins and this, in turn, all phenotype 

traits. RNA plays an intermediate role in reading the DNA code. Genetic information in the 

deoxyribonucleic acid (DNA) and the ribonucleic acid (RNA) molecules is identical except 

for a small difference on the sugar ring. DNA contains deoxyribose whereas RNA contains 

ribose. The DNA chain is synthesized through a condensation reaction as well. 

Figure 4: Chemical formula for the sugar ring in DNA and in RNA. 

Sugar

O

H

HO H

HOCH2 OH

H H

D-deoxyribose

H

DNA

O

H

HO OH

HOCH2 OH

H H

D-ribose

RNA H

Lipid

Glycerol

Tristearin

 

 H

 O-C-H

 O-C-H

 O-C-H

 H

 O

 C17H35 CStearic acid

 O

 C17H35 C-

 O

 C17H35 C-
504

Messenger RNA contains the genetic information (transcribed from DNA) that dictates the 

amino acid sequence during the synthesis of polypeptides. Transfer RNA brings the correct 

amino acid to the next site during this synthesis process on ribosomes. RNA is formed of 

single strands. 

Each DNA nucleotide is formed of a phosphate group, a five-carbon sugar, and an amine 

base. Different nucleotides contain different amine bases. Please note that the DNA chain 

runs either from the 5’ position to the 3’ position on the sugar or in the other direction 

(position 3’ to 5’).

Figure 5: Components of the DNA nucleotide. 

There are four amine bases in DNA: Adenine (A), Thymine (T), Guanine (G) and Cytosine 

(C). In RNA, Thymine (T) is replaced by Uracil (U). 

DNA

H

P

O

O

O

C

H

O

C

C

C C

HH H

H H

A

O

O

O H

P O C

H

O

C

C

C C

HH H

H H

C

O

Phosphate

Base

Sugar deoxyribose

Position 3’

Position 5’

Position 1’
505

Figure 6: Components of the DNA molecule. 

DNA of higher organisms folds into a double helix structure whereby the amine bases are 

stacked and hydrogen bonded. The pair A-T has two hydrogen bonds whereas the pair C-G 

has three hydrogen bonds. The two DNA chains forming the double helix run in opposite 

direction. Stacking of the amine bases and hydrogen bonding between them is the key 

driving force for the helix formation. Water hydrates the outside phosphate groups as well as 

produces hydrogen bonding between the base pairs. 

P O

O

O



O

Phosphate

 group

Sugar

Amine

bases

O

H

HO H

HOCH2 OH

H H

D-deoxyribose

Purines Pyrimidines

H

Adenine (A)

Guanine (G) Cytosine (C)

Thymine (T)

H H

N

N

N N

N

H

H

H

H

H

H H

H

N

N

N

N N

O

H

H

H

H

N

N

N O

H

H

H

O

N O

N

H3C

H

H

H

H

H

H

P

O

C

C C

O

C

N

N

H N O

O

HO

CH2

O



O

Nucleotide

H H

Phosphate 

group Sugar

Base

THE DNA MOLECULE
506

Figure 7: Amine base stacking and hydrogen bonding that form the DNA double helix.

X-ray diffraction helped in the determination of the precise structure of the DNA double 

helix. Information like the repeat distance per base pair (3.4 Å) and the helix pitch (34 Å) 

were determined.

3’ end

H

N

N

N

N N H

H

H

H

H

N N

O

CH

O 3

O

O

OO P O

O

H CH

2

O

H

H

H

N N H

H

H

H

N

N

N

N

O

O

OO P O

O

H CH

2

O

H2

C

O

O

O

P O

O

OO P O

O

H CH

2

O

O

O

O

H

H2

C

O

O

-

O

P

H2

C

O

O

-

O

P

Adenine Thymine

Cytosine Guanine

Thymine Adenine

Guanine Cytosine

O

OO P O

O

H CH

2

O

H2

C

O

O

-

O

P

O

H

5’ end 3’ end

5’ end

N

N

H
507

Figure 8: The DNA double helix. Structure for the dominant form of DNA (the B type) is 

schematically represented here. 

4. PROTEINS

Proteins are responsible for the enzymatic (or catalytic), structural and regulatory functions 

of the cell. Most of the cell processes are mediated by proteins. Proteins are formed of 

polypeptide chains which are coiled and folded in a precise way dictated by the amino acid 

sequence. Unfolding (denaturation) leads to loss of biological activity. Each polypeptide 

consists of a specific amino acid sequence. No two long amino acid sequences are alike. 

Peptides are synthesized through a condensation reaction of two amino acids to form the 

amide (also called peptide) bond. A water molecule is a byproduct. Energy is provided for 

this chemical reaction. RNA plays the roles of messenger (mRNA), adapter or transfer 

(tRNA), and linker or ribosomal (rRNA) in order to transcribe the DNA code into a correct 

amino acid sequence. 

THE DNA DOUBLE HELIX

A

A

A

A

A

A

A

T

T

T

T

G

G

G C

C

C

Pitch

34 Å

Repeat distance

per base pair=3.4 Å

Major groove

Minor groove

0 1 nm
508

Figure 9: A peptide sequence of two amino acids. 

There are 20 amino acids involved in protein synthesis. These consist of charged 

(hydrophilic) head groups and side chains which can be either polar (charged) or nonpolar 

(uncharged, hydrophobic). The hydrophobic groups tend to be buried in the middle of 

proteins out of contact with water.

Figure 10a: The group of nonpolar (hydrophobic) amino acids. 

OH

C

O

H

H2N

H

C

Glycine 

(Gly or G)

H2N C

O

CH3

H

C

OH

Alanine 

(Ala or A) Valine 

(Val or V)

Leucine 

(Leu or L)

Isoleucine 

(Ilu or I)

Nonpolar (Hydrophobic) Amino Acids

C

O

CH

H2N

H

C

OH

CH3CH3

C

O

H2N

H

C

OH CH

CH2

CH3

OH H3C

C

O

CH2

H2N

H

C

CH

CH3CH3

Methionine 

(Met or M)

C

O

H2N

H

C

OH CH2

CH2

CH3

S

Phenylalanine 

(Phe or F)

OH

C

O

CH2

H2N

H

C

OH

C

O

CH2

H2N

H

C

NH

Tryptophan 

(Trp or W)

C

O

H2C

HN

H

C

CH2

CH2

OH

Proline 

(Pro or P)

PEPTIDE SEQUENCE

Amino acid Amino acid

H2N-C-C-OH

O

R

H R’

H2N-C-C-OH

O

H

+ H2N-C-C

O

R

H

R’

N-C-C-OH

O

H

+ H2O

Peptide bond

N terminal end C terminal end

H
509

Figure 10b: The group of polar (hydrophilic) uncharged amino acids. 

Figure 10c: The group of polar (hydrophilic) charged amino acids. 

Polar (Hydrophilic) Charged Amino Acids

Aspatate 

(Asp or D)

C

O

C

H2N

H

C

OH

O

-O

CH2

Glutamate 

(Glu or E)

C

O

C

H2N

H

C

OH

O

-O

CH2

CH2

Acidic

Lysine 

(Lys or K)

C

O

H2N

H

C

OH CH2

CH2

NH3

+

CH2

CH2

H

C

O

H2N C

OH CH2

CH2

NH2

NH

CH2

C=NH2

+

Arginine 

(Arg or R)

OH

C

O

CH2

H2N

H

C

NH

NH+

Histidine 

(His or H)

Basic

Serine 

(Ser or S)

Polar (Hydrophilic) Uncharged Amino Acids

OH

C

O

H2N

H

C

CH2

OH

C

O

CH

H2N

H

C

OH

OH CH3

Threonine 

(Thr or T)

OH

C

O

H2N

H

C

CH2

SH

Cysteine 

(Cys or C)

OH

C

O

CH2

H2N

H

C

OH

Tyrosine 

(Tyr or Y)

Asparagine 

(Asn or N)

C

O

C

H2N

H

C

OH

H2N O

CH2

H

C

O

H2N C

OH CH2

CH2

C

H2N O

Glutamine 

(Gln or Q)
510

Molecular chaperones are needed in the folding of proteins in order to prevent incorrect 

molecular conformations. Proteins comprise four hierarchical self-assembling structures. 

These are the primary, secondary, tertiary and quaternary structures. 

The primary structure refers to the amino acid sequence along the peptide chains. mRNA, 

tRNA and rRNA play the roles of transcribing, transferring and linking the amino acid 

sequence. The secondary folding structure is determined by the amino acid sequence and 

happens spontaneously under favorable environmental conditions. Folding occurs in the form 

of alpha helix and beta sheet while random coil corresponds to no folding. The alpha helix 

contains 3.6 amino acids per helical turn bridging the peptide bond of every fourth amino 

acid. For example, leucine forms alpha helices. The beta sheet is a planar structure involving 

–CO to HN– hydrogen bonds between two polypeptides. For example, valine forms beta 

sheets. Complex proteins require chaperone assistance in folding. 

The tertiary structure involves interaction between the R groups of amino acids. Electrostatic, 

hydrophobic and hydrogen bonding interactions as well as S-S covalent bonds contribute to 

the structure formation. Polypeptides are folded, coiled and twisted into the desired protein’s 

native configuration. A domain of 50 to 350 amino acids is required to fulfill a specific 

function. Some proteins consist of multiple functions played by multiple domains. The 

quaternary structure is made of two or more chains. Each chain can have two or more 

domains. 

Information included in the DNA gene sequence (A, G, T, C bases) is used to set the amino 

acid sequence in protein synthesis. For example, an AAG sequence in a DNA strand (running 

from 5’ to 3’ positions) transcribes to an AAG sequence in the mRNA and contributes to the 

addition of Lys (lysine) amino acid in the polypeptide synthesis. The beginning (start) and 

end (stop) of a gene sequence are also included in the DNA code. 

Beta sheet

Alpha helix

Random coil

Ribonuclease Protein

0 5 nm
511

Figure 11: Schematic representation of Ribonuclease which is a globular protein containing 

beta sheet, alpha helix and random coil secondary structures. Ribonuclease is an enzyme that 

catalyzes the hydrolysis of RNA into smaller components. 

5. POLYSACCHARIDES

Polysaccharides are polymers formed of monosaccharides (Greek for “single sugar”) joined 

together by glycosidic bonds. There are two categories of sugars: the aldosugars and the 

ketosugars. 

Figure 12: The two categories of sugar molecules

The most common form of aldosugar is the D-glucose (C6H12O6) which occurs in linear and 

cyclic forms. The general formula CnH2nOn is characteristic of sugars. These are also referred 

to as carbohydrates. 

Figure 13: The linear and cyclic forms of D-glucose. 

 O

 C-H

 H-C-OH

HO-C-H

 H-C-OH

 H-C-OH

 CH2-OH

Linear Form

 

 HC-OH

 H-C-OH

HO-C-H

 H-C-OH

 H-C

 CH2-OH

Cyclic Form

Glucose

O

H

CH2OH

O

OH

OH

H

OH

H

H H

HO

Cyclic Form

1

6

1

3

6

 

 CH2-OH

 C=O

 H-C-OH

HO-C-H

 CH2-OH

Aldosugar Ketosugar

 O

 H-C

HO-C-H

 H-C-OH

HO-C-H

 H2-C-OH
512

Two well-known polysaccharides in plant cells are starch and cellulose. Two well-known 

polysaccharides in animal cells are glycogen and chitin. Starch and glycogen contain alpha 

glycosidic bonds whereas cellulose and chitin contain beta glycosidic bonds. Polysaccharides 

are often branched and are represented by the chemical formula (C6H10O5)n with n being a 

large number (hundreds to thousands). 

Figure 14: Chemical formulas for two plant polysaccharides. 

Mammals do not possess enzymes that can hydrolyze the beta glycosidic bonds and cannot 

therefore digest cellulose. 

6. CELLS AND ORGANELLES

The basic unit in biology is the cell. Eukaryotes are characterized by well defined membranes 

around the organelles (mitochondria, chloroplasts, inner cell membrane systems) as well as 

around the nucleus. In contrast, bacteria have no nucleus or organelles and are termed 

prokaryotes. Eukaryotes evolved from prokaryotes. 

O

H

H

CH2OH

O HO

O

H

Starch 

O

H

H

CH2OH

O

H

H

CH2O

H O

O

H

Cellulose 

POLYSACCHARIDES

CH2OH

H

OH

OH

OH

OH

OH

OH

OH

OH

H

H

Beta

glycosidic

bond

Alpha

glycosidic

bond
513

Figure 15: Schematic representation of an animal cell showing the various organelles. 

A brief description of the various organelles is included here. These include the nucleus, the 

endoplasmic reticulum, the Golgi complex, the mitochondria, the lysosomes, the vacuoles 

and vesicles. 

The cell nucleus contains the chromosomes where DNA is tightly packed. The nucleus is 

bounded by a double membrane that keeps its contents separate from the cytoplasm. 

Membrane pores allow the free movement of small molecules and ions. The movement of 

larger molecules such as proteins is controlled by specialized carrier proteins (enzymes)

The endoplasmic reticulum (ER) consists of an interconnected network of tubular 

membranes and vesicles and is the site of protein synthesis and transport. The ER is also the 

site for the sequestration of calcium and the production and storage of glycogen and steroids. 

The Golgi complex serves in the processing and packaging of secretory proteins, in the 

synthesis of polysaccharides and in the processing of lipids. 

Golgi complex

mitochondrion

lysosome

ribosome

nucleus

endoplasmic

reticulum

plasma cell 

membrane

vacuole

cytoplasm

0 10 m
514

Mitochondria play an important role in the oxidation of sugars which are the main source of 

energy for the cell. It is the site for the generation of adenosine triphosphate (ATP) which is 

the energy currency within the cell. The cell localizes most of the enzymes in the 

mitochondrion. Some proteins are synthesized in the mitochondria. In addition to being the 

energy production center, mitochondria are also involved in cell signaling, cell differentiation 

as well as cell growth. Mitochondria have their own DNA which is independent of the DNA 

in the nucleus. 

Lysosomes contain digestive enzymes along with carbohydrates. The membrane surrounding 

the lysosome prevents the digestive enzymes from destroying the cell. Lysosomal proteins 

are made in the ER and Golgi apparatus. Peroxisomes play a role in breaking down fatty 

acids. 

Vacuoles are used for temporary storage or transport. They remove and export unwanted 

substances, maintain the proper internal cell hydrostatic pressure and pH and enable cell 

flexibility. Vacuoles are important components of plant cells. 

The cell contains many other components. Vesicles are used for storage. Plant cells contain 

chloroplasts where photosynthesis takes place. The cytoplasm contains tubules and filaments 

for flexibility and mobility of the cell. Chaperones help in the correct assembly of proteins. 

Enzymes are synthesized in the rough ER, packaged in the Golgi complex and released by 

vesicle transport.

7. CHARACTERIZATION METHODS

There are many characterization methods used in biology. Some of the routine techniques 

include: mass spectroscopy to measure molecular weights, circular dichroism (CD) to 

distinguish between coil, beta sheet and alpha helix configurations, Cryo-Transmission 

Electron Microscopy (TEM) to visualize down to the molecular level, Vis-UV absorption 

spectroscopy to detect the existence of DNA or proteins in the sample, Electrophoresis to 

determine fragment sizes, etc. For example, UV absorption spectroscopy is sensitive to 

bonding in the amine bases in DNA (260 nm line) and to the existence of Tryptophan amino 

acid in proteins (280 nm line). Wide-angle x-ray diffraction is used to determine crystalline 

structures. When crystalline bio-material cannot be obtained, small-angle x-ray scattering 

(SAXS) and small-angle neutron scattering (SANS) are used in order to determine 

amorphous structures. Dynamic Light Scattering (DLS) is also used to estimate particle sizes. 

8. NEUTRON SCATTERING LENGTHS

Neutron scattering works best when deuteration is possible. Deuteration is achieved either by 

using a deuterated solvent or by deuterium labeling the macromolecules. Due to the complex 

“natural” processes involved, deuterium labeling is not easy to achieve in biology. Synthesis 

using deuterated amino acids is achieved for short polypeptide sequences of less than 100 

amino acids. That process is complex and time consuming even with the use of automated 
515

equipment. Another more involved route for obtaining deuterated bio-macromolecules is to 

grow organisms in a d-water culture then separate the deuterated proteins. 

Water is the solvent of choice in biology research since it is the major component in the cell. 

The dialysis process allows the exchange of H by D and vice versa. This helps enhance the 

neutron contrast and helps deuterium exchange on the macromolecules. C-H bonds do not 

exchange but O-H and N-H bonds exchange into O-D and N-D. 

The tabulated values for the scattering lengths of the light chemical elements found in 

proteins and DNA are included:

bH = -3.739*10-13 cm, bD = 6.671 *10-13 cm, bC = 6.646 *10-13 cm

bO = 5.803 *10-13 cm, bP = 5.130 *10-13 cm, bS = 2.847 *10-13 cm.

The scattering lengths, densities and molecular volumes for water and d-water are:

bH2

O = -1.675*10-13 cm, bD2O = 19.145*10-13 cm

dH2

O = 1 g/cm3

, dD2

O = 1.11 g/cm3

vH2

O = 29.9 Å

3

, vD2

O = 29.9 Å

3

The scattering length densities are given by:

7 -2

H O

H O

H O 5.702*10 Å

v

b

2

2

2



    , 

6 -2

D O

D O

D O 6.403*10 Å

v

b

2

2

2



  

A table summarizes the chemical formulas, the scattering lengths and scattering length 

densities for the various amino acids (Jacrot, 1976). Four cases are considered: (1) the case of 

hydrogenated molecules (amino acids or nucleotides), (2) the case of hydrogenated 

molecules with H/D exchange, (3) the case of deuterated molecules, and (4) the case of 

deuterated molecules with D/H exchange. The same information is included for the DNA and 

the RNA nucleotides These numbers are estimates for thermal neutron scattering and will 

vary with neutron wavelength and solution conditions (such as pH, etc). 

Table 1: Scattering lengths for amino acids and deuterated amino acids after H/D or D/H 

exchange. Scattering lengths for nucleotides and deuterated nucleotides are also included. 

Name H/D Content Formula

Scattering 

Length

(10-12 cm)

Density

(g/cm3)

Scattering 

Length

Density

(10-6 Å

-2

)

Amino Acids

Hydrogenated C2NOH3 1.72 1.43 2.60
516

Glycine (Gly 

or G)

H/D ExchangeC

2NOH2

D 2.77 1.45 4.16

DeuteratedC

2NOD3 4.85 1.50 7.30

D/H ExchangeC

2NOHD2 3.81 1.48 5.73

Alanine (Ala 

or A)

HydrogenatedC

3NOH5 1.64 1.29 1.79

H/D ExchangeC

3NOH4

D 2.68 1.31 2.93

DeuteratedC

3NOD5 6.85 1.38 7.48

D/H ExchangeC

3NOHD4 5.80 1.36 6.34

Valine (Val 

or V)

HydrogenatedC

5NOH9 1.47 1.16 1.04

H/D ExchangeC

5NOH8

D 2.52 1.17 1.78

DeuteratedC

5NOD9 10.84 1.27 7.65

D/H ExchangeC

5NOHD8 9.80 1.25 6.92

Leucine (Leu 

or L)

HydrogenatedC

6NOH11 1.39 1.12 8.28

H/D ExchangeC

6NOH10

D 2.43 1.13 1.45

DeuteratedC

6NOD11 12.84 1.23 7.65

D/H ExchangeC

6NOHD10 11.80 1.22 7.03

Isoleucine 

(Ilu or I)

HydrogenatedC

6NOH11 1.39 1.11 8.24

H/D ExchangeC

6NOH10

D 2.43 1.12 1.44

DeuteratedC

6NOD11 12.84 1.22 7.61

D/H ExchangeC

6NOHD10 11.80 1.21 6.99

Phenylalanine 

(Phe or F)

HydrogenatedC

9NOH9 4.13 1.20 2.03

H/D ExchangeC

9NOH8

D 5.17 1.21 2.54

DeuteratedC

9NOD9 13.50 1.27 6.64

D/H ExchangeC

9NOHD8 12.46 1.27 6.13

Tyrosine (Tyr

or Y)

HydrogenatedC

9NO2

H

9 4.71 1.33 2.31

H/D ExchangeC

9NO2

H

7

D

2 6.79 1.35 3.34

DeuteratedC

9NO2

D

9 14.08 1.40 6.92

D/H ExchangeC

9NO2

H

2

D

7 12.00 1.39 5.89

Tryptophan 

(Trp or W)

HydrogenatedC11N

2OH10 6.02 1.30 2.54

H/D ExchangeC11N

2OH8

D

2 8.11 1.32 3.41

DeuteratedC11N

2OD10 16.43 1.37 6.92

D/H ExchangeC11N

2OH2

D

8 14.35 1.36 6.04

Aspartate 

(Asp or D)

HydrogenatedC

4NO3

H

4 3.84 1.67 3.38

H/D ExchangeC

4NO3

H

3

D 4.88 1.68 4.30

DeuteratedC

4NO3

D

4 8.00 1.73 7.05

D/H ExchangeC

4NO3HD3 6.96 1.71 6.13

HydrogenatedC

5NO3

H

6 3.76 1.51 2.67
517

Glutamate 

(Glu or E)

H/D ExchangeC

5NO3

H

5

D 4.80 1.52 3.41

DeuteratedC

5NO3

D

6 10.00 1.58 7.11

D/H ExchangeC

5NO3HD5 8.96 1.57 6.37

Serine (Ser or 

S)

HydrogenatedC

3NO2

H

5 2.22 1.46 2.24

H/D ExchangeC

3NO2

H

3

D

2 4.30 1.49 4.34

DeuteratedC

3NO2

D

5 7.43 1.54 7.49

D/H ExchangeC

3NO2

H

2

D

3 5.34 1.51 5.39

Threonine 

(Thr or T)

HydrogenatedC

4NO2

H

7 2.14 1.37 1.75

H/D ExchangeC

4NO2

H

5

D

2 4.22 1.40 3.46

DeuteratedC

4NO2

D

7 9.43 1.47 7.72

D/H ExchangeC

4NO2

H

2

D

5 7.34 1.44 6.01

Asparagine 

(Asn or N)

HydrogenatedC

4

N

2

O

2

H

6 3.45 1.40 2.55

H/D ExchangeC

4

N

2

O

2

H

3

D

3 6.57 1.44 4.86

DeuteratedC

4

N

2

O

2

D

6 9.69 1.47 7.17

D/H ExchangeC

4

N

2

O

2

H

3

D

3 6.57 1.44 4.86

Glutamine 

(Gln or Q)

HydrogenatedC

5

N

2

O

2

H

8 3.36 1.32 2.09

H/D ExchangeC

5

N

2

O

2

H

5

D

3 6.49 1.35 4.03

DeuteratedC

5

N

2

O

2

D

8 11.69 1.40 7.26

D/H ExchangeC

5

N

2

O

2

H

3

D

5 8.57 1.37 5.32

Lysine (Lys 

or K)

HydrogenatedC

6

N

2OH13 1.58 1.22 8.96

H/D ExchangeC

6

N

2OH9

D

4 5.74 1.25 3.26

DeuteratedC

6

N

2OD13 15.11 1.34 8.58

D/H ExchangeC

6

N

2OH4

D

9 10.95 1.30 6.21

Arginine 

(Arg or R)

HydrogenatedC

6

N

4OH13 3.45 1.44 1.91

H/D ExchangeC

6

N

4OH8

D

5

9.70 1.50 5.36

DeuteratedC

6

N

4OD13 16.984 1.56 9.39

D/H ExchangeC

6

N

4OH5

D

8 10.74 1.51 5.94

Histidine (His 

or H)

HydrogenatedC

6

N

3OH7 4.96 1.36 2.84

H/D ExchangeC

6

N

3OH5

D

2 6.84 1.38 4.09

DeuteratedC

6

N

3OD7 12.05 1.43 7.20

D/H ExchangeC

6

N

3OH2

D

5 9.96 1.41 5.96

Methionine 

(Met or M)

HydrogenatedC

5NOSH9 1.76 1.28 1.03

H/D ExchangeC

5NOSH8

D 2.80 1.29 1.64

DeuteratedC

5NOSD9 11.13 1.36 6.52

D/H ExchangeC

5NOSHD8 10.09 1.35 5.91

HydrogenatedC

3NOSH5 1.93 1.62 1.82
518

Cysteine (Cys 

or C)

H/D Exchange C3NOSH3D2 4.00 1.65 3.79

Deuterated C3NOSD5 7.13 1.70 6.75

D/H Exchange C3NOSH2D3 5.05 1.67 4.78

Proline (Pro 

or P)

Hydrogenated C5NOH7 2.22 1.25 1.72

H/D Exchange C5NOH7 2.22 1.25 1.72

Deuterated C5NOD7 9.51 1.34 7.36

D/H Exchange C5NOD7 9.51 1.34 7.35

DNA Nucleotides

Adenine

Hydrogenated PN5O5C10H11 10.63

H/D Exchange PN5O5C10H9D2 12.71

Deuterated PN5O5C10D11 22.08

D/H Exchange PN5O5C10H2D9 20.00

Guanine

Hydrogenated PN5O6C10H11 11.21

H/D Exchange PN5O6C10H8D3 14.33

Deuterated PN5O6C10D11 22.66

D/H Exchange PN5O6C10H3D8 19.54

Cytosine

Hydrogenated PN3O6C9H11 8.67

H/D Exchange PN3O6C9H9D2 10.75

Deuterated PN3O6C9D11 20.12

D/H Exchange PN3O6C9H2D9 18.04

Thymine

Hydrogenated PN2O7C10H12 8.61

H/D Exchange PN2O7C10H11D1 9.65

Deuterated PN2O7C10D12 21.10

D/H Exchange PN2O7C10H1D11 20.06

RNA Nucleotides

Adenine

Hydrogenated PN5O6C10H11 11.21

H/D Exchange PN5O6C10H8D3 14.33

Deuterated PN5O6C10D11 22.66

D/H Exchange PN5O6C10H3D8 19.54

Hydrogenated PN5O7C10H11 11.79

H/D Exchange PN5O7C10H7D4 15.95
519

Guanine Deuterated PN5O7C10D11 23.24

D/H Exchange PN5O7C10H4D7 19.08

Cytosine

Hydrogenated PN3O7C9H11 9.25

H/D Exchange PN3O7C9H8D3 12.37

Deuterated PN3O7C9D11 20.70

D/H Exchange PN3O7C9H3D8 17.58

Uracil

Hydrogenated PN2O8C9H10 9.27

H/D Exchange PN2O8C9H8D2 11.35

Deuterated PN2O8C9D10 19.68

D/H Exchange PN2O8C9H2D8 17.60

Note that the density of nucleotides is estimated to be between 1.73 g/cm3and 1.78 g/cm3. 

This gives scattering length densities for hydrogenated nucleotides between 3.18*10-6 Å

-2 and 

3.66*10-6 Å

-2

. 

A figure summarizes the average scattering length densities ( = b/v where b is the scattering 

length and v is the molecular volume) values for hydrogenated proteins and DNA without 

H/D exchange (left axis) and with H/D exchange (right axis). Average values for deuterated 

proteins and deuterated DNA without and with D/H exchange are also included (Jacrot, 

1976). 

Note that the neutron contrast 

2

is defined as the difference in scattering length densities 

(squared) between the macromolecules (proteins or DNA) and the solvent (water). The 

average scattering length density for hydrogenated proteins is 1.8*10-6 Å

-2

(left axis) whereas 

after H/D exchange (in D2O) it is 3*10-6 Å

-2

(right axis). For example, proteins are contrast 

matched in mixtures of 40 % D2O and 60 % H2O, DNA is contrast matched for 65 % D2O 

and 35 % H2O, lipids are contrast matched for 15 % D2O and 85 % H2O and polysaccharides 

are contrast matched for 30 % D2O and 70 % H2O. 
520

0

2

4

6

8

0 0.2 0.4 0.6 0.8 1 Scattering Length Density (10-6 Å-2

)

d-Water Fraction

d-DNA

d-proteins

DNA

proteins

water

lipids

d-lipids

H2O D2O 

d-polysaccharides

polysaccharides

Figure 16: Average scattering length densities for DNA, proteins, lipids and polysaccharides 

as well as deuterated DNA, deuterated proteins, deuterated lipids and deuterated 

polysaccharides following H/D exchange in H2O (left) or D2O (right). Arrows mark the 

D2O/H2O contrast match conditions. 

REFERENCES

W.M. Becker, L.J., Kleinsmith and J. Hardin, “The World of the Cell”, Benjamin/Cummings 

Publishing (1999)

B. Jacrot, “The Study of Biological Structures by Neutron Scattering from Solution”, Rep. 

Prog. Phys. 39, 911-953 (1976)

QUESTIONS

1. What are biological membranes made out of?

2. What is a condensation reaction? Where do condensation reactions take place?

3. What are the main components of DNA? How many amine bases are there? Name them. 

4. What is the difference between DNA and RNA? 

5. What are the main conformations of proteins?
521

6. How many amino acids are used in the synthesis of proteins?

7. How is the genetic DNA code used for the synthesis of proteins?

8. What is the prominent component of polysaccharides? Name a couple of polysaccharides. 

9. Which cell organelle is the center of energy production?

10. How is dialysis used to enhance the neutron contrast?

ANSWERS

1. Biological membranes (those found in living organisms) contain lipids, but also proteins 

and their glycosylated derivatives. Lipids consist of hydrophilic heads and hydrophobic tails. 

Lipids are produced through the condensation reaction of glycerol and a fatty acid group. 

2. A chemical reaction is referred to as a “condensation reaction” when it produces a water 

molecule. Condensation reactions occur during the synthesis of lipids, of DNA and of 

proteins. 

3. DNA is formed of phosphate groups, sugars and amine bases. There are four amine bases. 

These are Adenine (A), Thymine (T), Guanine (G) and Cytosine (C). 

4. The main difference between DNA and RNA is in the sugar group. DNA contains 

deoxyribose whereas RNA contains ribose. Moreover, in RNA, Thymine is replaced by 

Uracil (U). 

5. Protein conformations are the alpha helix, the beta sheet and the random coil. 

6. There are 20 amino acids involved in the synthesis of proteins. 

7. The sequence of three nucleotides codes for the addition of a specific amino acid in the 

synthesis of a peptide sequence. 

8. Polysaccharides are made through the polymerization of sugars. Starch and cellulose are 

plant polysaccharides. Glycogen and chitin are animal polysaccharides. 

9. The mitochondrion is the center of energy production. Energy is produced through the 

oxidation mainly of sugars. 

10. A dialysis bag (containing the macromolecules) is used inside a container full of d-water 

in order to enhance deuterium exchange. This results in partially deuterated macromolecules 

(DNA or proteins) by deuterium exchange on O and N atoms. Dialysis is sometime 

performed more than once in order to enhance the deuteration level. 
522

Chapter 48 - SANS FROM PHOSPHOLIPID BILAYERS UNDER PRESSURE

Membranes are essential biological components for the integrity of cells and cell 

constituents. They create a barrier between the outside and the inside and regulate the input 

and output processes. Cells membranes are composed of surfactant bilayers with hydrophilic 

head groups and hydrophobic tails. Phospholipid molecules are typical amphiphile 

surfactants that form biological membranes. 

1. PHOSPHOLIPIDS

Phospholipids form a chemical homologous series with increasing hydrocarbon tail. 

Dimyristoylphosphatidylcholine (DMPC) has a C14 carbon tail, dipalmitoyl 

phosphatidylcholine (DPPC) has a C16 carbon tail, distearoylphosphatidylcholine (DSPC) 

has a C18 carbon tail, and diactylphosphatidylcholine (DAPC) has a C20 acyl hydrocarbon 

tail, etc. These are membrane lipids that contain two saturated symmetric hydrocarbon 

chains. The choline group is charged and acts as the surfactant head group which interfaces 

with water. The acyl fatty acid groups form the hydrophobic bulk of the lipid layers. 

Figure 1: The various groups that form DAPC.

2. PHOSPHOLIPID BILAYERS PHASES

Phospholipids form smectic phase bilayers in the presence of water. The bilayers are formed 

of alternating hydrophilic and hydrophobic layers. Many phases have been identified using 

C

C18H36

CH3

O

O

H2C CH

C

C18H36

CH3

O

O

CH2

O

O P O

-

O CH2 CH2 N

+

(CH3)3

choline

glycerol group

acyl fatty acid

phosphate group

Diacyl-phosphatidylcholine (DAPC)
523

optical and diffraction methods. Some of these phases are: the liquid crystalline phase (L) 

obtained at high temperature, the ripple gel phase (P) obtained at intermediate temperature, 

the lamellar gel phase (L) obtained at low temperature, and the lamellar crystalline phase 

(Lc) obtained at even lower temperature. Note that similar primed phases such as L’ and P’

correspond to tilted hydrocarbon tails forming the structures. A variation of the L phase is 

the interdigitated phase (LI). 

Figure 2: The main phospholipid bilayer phases. 

The interdigitated phase is induced by a number of factors including (1) small amphiphilic 

(cosurfactant) molecules such as ethanol, ethylene glycol, etc and (2) pressure. The effect of 

hydrostatic pressure is described here when used in conjunction with small-angle neutron 

scattering (SANS). 

P

L

Lamellar gel phase Ripple phase

Liquid crystalline phase

Lc

Lamellar crystalline phase

L

Interdigitated phase

L
524

3. SANS UNDER PRESSURE

The SANS technique was used to investigate the transitions between the various bilayer 

phases in DAPC (characterized by C20) and with in-situ pressure and temperature control 

(Worcester-Hammouda, 1997). The sample contained about 2 % DAPC (mass fraction) in dwater and was formed of multi-lamellar vesicles. Pressure was varied up to 1 kbar (14.7 kpsi) 

and temperature was varied from 50 oC to 85 oC. A figure shows the case corresponding to 

70 oC. The liquid crystalline phase L, the ripple gel phase P and the interdigitated phase LI

are clearly observed. They are identified by their widely different structural features. 

10

100

1000

0 0.05 0.1 0.15

DAPC at 70 oC 

200 psi

10 kpsi

15 kspi

Scattered Intensity (cm-1

)

Scattering Variable (Å-1)

Ripple Gel Phase

Liquid Crystalline Phase 

Interdigitated Phase 

Figure 3: SANS from DAPC phospholipid bilayers at 70 oC. Increasing pressure induces 

structural phase transitions. 

The following peak positions are obtained for the P and LI spectra Q0 = 0.085 Å-1and Q0 = 

0.114 Å-1respectively. These correspond to inter-layer d-spacings (= 2/Q0) of 73.57 Å and 

54.92 Å for the P and LI phases respectively at 70 oC. 

4. THE PHASE DIAGRAM
525

Using this method of identifying the various phases by their SANS diffraction pattern, an 

entire phase diagram has been constructed by varying pressure and temperature. The 

interdigitated phase forms only at high pressure. 

Formation of the interdigitated phase by hydrostatic pressure is largely driven by the packing 

of acyl chains. This is otherwise difficult to achieve because of the large cross-sectional areas 

of the polar head groups. The curved phase boundary for interdigitation demonstrates 

complex properties for this packing. 

50

55

60

65

70

75

80

85

0 20 40 60 80 100

DAPC (C20) Phospholipid Bilayers

Temperature (

oC)

Pressure (MPa)

L



 liquid 

crystalline phase

P



 ripple gel phase

L



 lamellar gel phase

L

 

 interdigitated 

gel phase

Figure 4: The temperature-pressure phase diagram for DAPC. Note that 14.7 kpsi = 1 kbar = 

100 MPa. 

Similar temperature-pressure phase diagrams have been mapped out for other 

phospholipid/d-water systems (for example DPPC and DSPC). 

5. COMMENTS

This exercise shows once again that the SANS technique is a useful tool for resolving 

structures and mapping out phase boundaries. It also demonstrates the advantage of SANS 

measurements with in-situ pressure. 
526

The field of membrane research has benefited from SANS as well as Neutron Reflectometry 

(NR). Just like SANS, NR can probe in-plane structures through the so-called off-specular 

scattering component. 

REFERENCES

D. Worcester and B. Hammouda, “Interdigitated Hydrocarbon Chains in C20 and C22 

Phosphatidylcholines Induced by Hydrostatic Pressure”, Physica B: Condensed Matter 241-

243, 1175-1177 (1997). 

QUESTIONS

1. What are the major components of a biological membrane?

2. What is a surfactant?

3. What is the difference between DMPC, DPPC, DSPC and DAPC?

4. What is a smectic phase? How about a nematic phase? How about an isotopic phase? 

ANSWERS

1. A biological membrane is formed of phospholipid bilayers. 

2. Phospholipids are surfactant molecules containing a hydrophilic headgroup and a 

hydrophobic tail. 

3. The difference between the various phospholipids in the series is the length of the 

hydrocarbon tail; DMPC has C14, DPPC has C16, DSPC has C18 and DAPC has C20.

4. A smectic phase is a liquid crystalline phase with both translational and orientational 

order. A nematic phase is characterized by translational order and orientational disorder. An 

isotopic phase has translational and orientational disorder. 
527

Chapter 49 - THE HELIX-TO-COIL TRANSITION IN DNA

1. THE DNA DOUBLE HELIX

Deoxyribonucleic acid (DNA) is the basic building template for life. It codes for the 

synthesis of proteins inside living cells by controlling the amino acid sequences that form the 

genes. DNA is formed of backbone phosphates, linked to deoxyribose sugars and side group 

amine bases. The charged phosphate groups are hydrophilic, the deoxyribose sugar groups 

are mostly hydrophobic, and the amine bases contain both hydrophobic and hydrophilic 

groups. DNA forms a helical structure which is stable because of the stacking of the amine 

bases and of the hydrogen bonding between them. The helical structure is effective at 

“hiding” the hydrophobic sugar groups from contact with water. The helix phase melts into 

disordered coils under various conditions including heating. 

Figure 1: Schematic representation of the DNA helix and coil conformations

DNA macromolecules form helical structures in their active form and melt to a random coil 

phase in their denatured form. The denaturation transition consists in a helix-to-coil transition 

that can be promoted either using denaturing agents or through heating. Here, the simpler 

heating route is discussed using two characterization methods: the UV absorption 

spectroscopy and SANS. 

Helix Coil
528

2. UV ABSORPTION SPECTROSCOPY

Ultra-violet (UV) light absorption spectroscopy is sensitive to the stacking of -bonded 

groups such as the amine bases in DNA. It is an effective method to monitor the helix-to-coil 

transition. 

0

0.5

1

1.5

2

2.5

3

200 220 240 260 280 300 320

UV Absorption Spectroscopy 

Measured 

UV Absorbance

UV Wavelength (nm) 

Figure 2: Typical UV absorption spectrum from DNA showing a characteristic peak at a 

wavelength of 260 nm. 

The monitoring of the intensity at the peak position (260 nm) with increasing temperature 

yields a sigmoid function centered at the helix-to-coil transition temperature. The helix-tocoil transition is mediated by the un-stacking of the amine bases and the breaking of 

hydrogen bonds between these bases. 

A sample containing 4 % salmon DNA (molecular weight of 1.3 *106g/mol) in water is 

characterized by a helix-to-coil transition temperature of 94 oC. Since this temperature is too 

close to the boiling temperature for water (100 oC), another solvent was considered as well. 

A 4 % DNA in ethylene glycol is characterized by a more convenient helix-to-coil transition 

temperature of 38 oC. The characteristic sigmoid shape function was obtained in each case. 

The inflection point corresponds to the helix-to-coil melting temperature. In order to avoid 

saturation of the UV absorbance, 50 m thin samples were measured. 
529

Since the same DNA samples were investigated by UV absorption spectroscopy and SANS, 

deuterated solvents were used with both characterization methods. Moreover a 100 mM NaCl 

salt content was added throughout in order to screen the charges on the DNA phosphate 

groups. 

1.8

2

2.2

2.4

2.6

2.8

20 30 40 50 60 70 80 90 100

4 % DNA in d-ethylene glycol 

and in d-water, 100 mM NaCl

4% DNA/d-ethylene glycol

4% DNA/d-water

260 nm Absorption Peak Height

Temperature (oC)

Transition Temperatures

Figure 3: Variation of the UV absorption spectroscopy peak intensities at 260 nm with 

increasing temperature for 4 % DNA in d-water or in d-ethylene glycol. The helix-to-coil 

transitions can be observed clearly. 

3. HELIX-TO-COIL TRANSITION IN MIXED SOLVENTS

The UV absorption spectroscopy provides an effective way to monitor the helix-to-coil 

transition in d-water/d-ethylene glycol mixed solvents. The same 4 % salmon DNA weight 

fraction and 100 mM NaCl salt content were used. 
530

30

40

50

60

70

80

90

100

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

4 % DNA in d-water/d-ethylene glycol mixtures, 

100 mM NaCl

Second Temperature

Transition Temperature

First Temperature

Transition Temperatures (

oC)

d-Ethylene Glycol Fraction

DNA/d-water

DNA/d-ethylene glycol

Figure 4: Variation of the helix-to-coil transition temperature for 4 % DNA in mixed dwater/d-ethylene glycol mixed solvents. 

The monotonic linear variation is attributed to the fact that the transition was approached 

from the helix side whereby solvents mix randomly (ideal solvent mixing behavior). The fact 

that the melting temperature decreases with d-ethylene glycol fraction points to the 

conclusion that the hydrophobic groups CD2 in d-ethylene glycol play an important role in 

the melting transition. They help solvent molecules cross the hydrophobic zone of the 

deoxyribose sugar groups thereby loosening the helical structure. This argument helps 

understand the micellar nature of the DNA macromolecules in terms of a hydrophobic sugar 

region and hydrophilic phosphate and amine base regions. The ability of the solvent to cross 

the hydrophobic region controls the stability of the helix phase. 
531

Figure 5: Simple representation of a cross section of the DNA macromolecule. 

4. THE HELIX-TO-COIL TRANSITION BY SANS

The SANS technique is effective at determining macromolecular structures. A series of 

measurements were performed from a 4 % DNA/d-ethylene glycol/100 mM NaCl sample at 

temperatures ranging from 10 oC to 80 oC (at 5 oC intervals). A figure shows a typical SANS 

spectrum at two temperatures; one below (25 oC) and the other one above (50 oC) the helixto-coil transition temperature. This temperature is known to be 38 oC based on US absorption 

measurements (Hammouda-Worcester, 2006). 

hydrophilic regions hydrophobic region

PHOSPHATES

SUGARS

AMINE BASES
532

0.1

1

10

0.01 0.1

4 % DNA in d-ethylene glycol, 

100 mM NaCl

25 oC

50 oC

Scattering Intensity (cm-1

)

Scattering Variable Q (Å-1)

Figure 6: SANS from a 4 % mass fraction DNA/d-ethylene glycol/100 mM NaCl sample 

measured at temperatures below (25 oC) and above (50 oC) the helix-to-coil melting 

temperature.

The high-Q signal is distinctively different in the two cases. The data show an abrupt 

decrease in the high-Q intensity for the helix phase but a gradual decrease for the coil phase. 

The SANS data were fit to the following empirical functional form that reproduces the main 

data features:

 

B

1 Q

C

Q

A

I(Q)

n m



 

  . (1)

The term A/Qnrepresents the low-Q clustering (network) scattering part and the term 

C/[1+(Q)

m] represents the high-Q solvation part. B represents a Q-independent (mostly 

incoherent) background. The low-Q part represents scattering from a large gel network 

structure. It does not change much across the melting transition. Our focus here is on the 

high-Q signal exclusively. 

A figure shows the variation of the “solvation intensity” (the fitted quantity C) for increasing 

temperature. The intensity drop between 25 oC and 40 oC characterizes the helix melting 
533

transition. Lowering temperature shows that this transition is weakly reversible with 

substantial hysteresis. Further temperature increase beyond the melting transition increases 

the solvation intensity. This result is typical of water-soluble polymers which are 

characterized by a Lower Critical Solution Temperature (LCST). 

0.135

0.14

0.145

0.15

0.155

0.16

0.165

0.17

0 20 40 60 80

4 % DNA in d-ethylene glycol, 

100 mM NaCl

Solvation Intensity (cm-1

)

Temperature (oC)

helix

coil

transition

Figure 7: Variation of the SANS solvation intensity (the quantity C in the empirical model) 

for increasing temperature. For temperatures beyond the melting transition, the solvation 

intensity increases. 

The correlation length  also varies across the melting transition. This correlation length 

represents a weighted-average inter-distance between the hydrogen-containing (sugar-amine 

base) groups. It is around 8.5 Å in the helix phase and increases to 12.3 Å in the coil phase. 

In the helix phase the sugar-amine base groups are closer together than in the coil phase. This 

increase in  is due to the opening of the tight helical structure into a loser coil configuration. 

This correlation length is not a measure of the DNA radius. Raising the temperature further 

in the coil phase increases the correlation length even further; this is a familiar trend for 

LCST systems. 
534

8

9

10

11

12

13

14

15

0 20 40 60 80

4% DNA in d-ethylene glycol, 

100 mM NaCl

Correlation Length (Å)

Temperature (oC)

helix

coil

transition

Figure 8: Variation of the correlation length (the quantity  in the empirical model) for 

increasing temperature. After melting, DNA coils swell with further temperature increase. 

Finally the high-Q Porod exponent m is seen to vary between values around 3.7 in the helix 

phase to values close to 1.7 in the coil phase. DNA helices are appearing like cylinders with 

fairly smooth surfaces (Porod exponents close to 4) and DNA coils behave like polymer 

chains in good solvent conditions or in a fully swollen chain configuration (Porod exponent 

of 5/3 = 1.67). 
535

1.5

2

2.5

3

3.5

4

0 20 40 60 80

4 % DNA in d-ethylene glycol, 

100 mM NaCl

Porod Exponent 

Temperature (oC)

helix

coil

transition

Figure 9: Variation of the high-Q Porod exponent m for increasing temperature. This 

exponent varies from 3.7 (close to 4 for cylinder) to 1.7 (swollen coil).

It is noted that the rod-like nature of DNA (Porod exponent around 1) has not been seen due 

to the clustering signal overwhelming the low-Q scattering. It is also noted that once the 

melting transition has taken place, DNA coils behave like typical water-soluble synthetic 

polymer chains. 

5. A HELIX-TO-COIL TRANSITION MODEL

Helix-to-coil transition models have been published by many authors including Zimm 

(Zimm, 1959). The formulation from Flory’s book (Flory, 1969) will be followed closely 

here. 

Consider a macromolecule consisting of N units (think residues) comprising  helical 

sequences. There is a total of NH helical units and NC = N-NH coil units. Define the partition 

function for the melting of one helical unit as s = exp(Hm/RT) where Hm is the enthalpy 
536

needed, R is the molar gas constant (related to the Boltzman constant kB through Avogadro’s 

number NAv as R = kBNAv) and T is the temperature in absolute units. Assume that it takes no 

enthalpy to form a coil so that the partition function for a coil unit is equal to 1. Define the 

partition function for the removal of one helical sequence as . The partition function for the 

helix-to-coil melting process is:

 Z    s

H 

N

configs

. (2)

The product 



is taken over all helical units NH and all helical sequences  and the 

summation 



is taken over all possible configurations (i.e., over all possible unit 

arrangements to form the macromolecule with N units). 

Figure 10: Schematic representation of the helical and coil sequences. 

The partition function can be expressed in matrix notation as:

Z J *U J.

N



(3)

With:

J*  1 0 









 



1 s

1 s

U 













1

1

J

.(4)

J* means that the macromolecule starts with a coil unit and J means that it finishes with 

either a coil or a helical unit. In order to perform the UNproduct, the configuration matrix U 

helical sequence

coil sequence

helical unit
537

is diagonalized into the form 

1 U A A

 

where  is a diagonal matrix. UNsimplifies as 

N N 1 U A A

 

. The eigenvalues 1 and 2 (diagonal elements of matrix  are given by:

2

(1 s) (1 s) 4 s

2

1

    

 

2

(1 s) (1 s) 4 s

2

2

    

  . (5)

The partition function can then be summed up to become:

N 1

2

1 2

N 1 1

1

1 2

2

( )

( 1)

( )

(1 )

Z

 



  

 

 

  

 

 . (6)

The fraction of units in the helical state is given by:

ln(s)

ln(Z)

N

1

pH





 . (7)

The fraction of units in the coil state is pC = 1-pH. In the case of long macromolecules 

(N>>1/2



), pH simplifies to:

( )

( 1)

ln(s)

ln( )

p

1 2

1 1

H

  

 





 

 . (8)

The relative number of helical sequences is given by:

( )

( 1)(1 )

ln( )

ln( )

p

1 1 2

1 1 2

   

   



 

 



 . (9)

The average number of helical units per helical sequence yH is given by the ratio:

p (1 )

p

y

2

H 1

H

 



 



. (10)

In the notation used here, the total number of helical units is NH = N pH and the number of 

helical sequences is  = N p. 

The meaning of the various parameters is discussed here. First what is the meaning of 

parameter s? The helix-to-coil transition is driven by heating of the sample, i.e., by the 

enthalpy needed to melt one unit Hm. From the definition of s, one can express the deviation 

from the melting temperature Tm as:

(s 1)

H

RTT T T

m

m

m 



  . (11)
538

This simple relation is obtained by expanding the exponential in the definition of s. The 

dimensionless variable s can be used instead of T. How to understand the meaning of 

parameter ? Note that at s = 1 (middle of the helix-to-coil melting transition corresponding 

to temperature Tm) the preceding results simplify to:

pH = 

2

1

, p 

2



, yH 





1

. (12)

Right at the transition point (i.e., at T = Tm),  = N



/2 represents the number of helical 

sequences. One can think of  as a helical sequence “nucleation” parameter. The helix-tocoil transition happens through either a few or many helix-to-coil sequences (think 

“nucleation centers”) depending on the temperature conditions. This transition is similar to 

the melting transition of crystalline materials that happens through nucleation centers. 

Note that this simple model applies to the simplest form of helix-to-coil transition involving 

single-strand helical structures and “isolated” macromolecules. This would apply to very 

dilute polypeptide sequences. The case of double-strand helical structures (as in DNA) would 

involve larger configuration matrices U with different enthalpies for the various processes 

(hydrogen bonding and base stacking for AT or GC pairs). The results described here are 

approximate but still give useful predictions for the DNA case. They are applied to some of 

our UV absorption data. 

6. APPLICATION OF THE MODEL TO UV ABSORPTION DATA

Consider the helix-to-coil transition UV absorption data obtained for 4 % DNA/100 mM 

NaCl in d-ethylene glycol. In order to apply the simple model described above, two rescaling 

steps of the UV data have are performed: (1) modification of the horizontal temperature axis 

into the variable s axis using the relationship between T and s given in the previous section 

and (2) rescaling of the UV data vertical axis to a variation between 0 and 1. Moreover the 

values R = 1.989 cal/mol.K and 1 cal = 4.18 J are used. The model described here is not 

sensitive enough to let both Hm and  float. The melting temperature (T= Tm = 38oC) 

corresponds to s = 1. A reasonable value for the enthalpy of melting Hm is taken to be Hm 

= -6 kcal/mole. The data are plotted along with model best fit for the predicted pC using this 

value and the best fit value of  = 0.0037. This simple model reproduces the sigmoid shape 

of the UV absorption data well. 
539

0

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

rescaled UV absorption data

model prediction for =0.0037

UV Absorption Data and 

p

C=1-p

H model predictions

s variable

Figure 11: The UV absorption data across the helix-to-coil transition is compared to model 

prediction for pC = 1-pH with the best fit parameter   . Hm = -6 kcal/mole has been 

used. 

The relative number of helical sequences p increases, reaches a maximum at s=1 (or T=Tm) 

then decreases. Note that at the melting transition (i.e., for T=Tm), NH = N/2 and

p = 0.029 which means that there are 29 helical sequences per 1000 base units. 

REFERENCES

B.H. Zimm, and J.K. Bragg, “Theory of the Phase Transition between Helix and Random 

Coil in Polypeptide Chains”, J. Chem. Phys. 31, 526-535 (1959).

P.J. Flory, “Statistical Mechanics of Chain Molecules”, Interscience Publishers (1969). Look 

up mostly Chapter VII.

B. Hammouda and D.L. Worcester, “The Denaturation Transition of DNA in Mixed 

Solvents”, Biophysical Journal 91, 2237-2242 (2006).
540

QUESTIONS

1. DNA is formed of what units?

2. What drives the formation of the helical structure of DNA?

3. What is the analytical measurement method of choice for observing the helix-to-coil 

transition?

4. What is the typical helix-to-coil transition temperature for DNA/water? How about for 

DNA/ethylene glycol?

5. What is the SANS Porod exponent for the helical structure? How about for the coil 

structure? What do these exponents mean?

6. What is the activation enthalpy for the melting of a helical unit?

ANSWERS

1. DNA is formed of nucleotides.

2. The helical structure of DNA is driven by the stacking of the amine bases and the 

hydrogen-bonding between them.

3. UV absorption spectroscopy is the analytical measurement method of choice for observing 

the helix-to-coil transition. 

4. A typical helix-to-coil transition temperature of 94 oC characterizes DNA/water. That 

temperature is around 38 oC for DNA/ethylene glycol. 

5. A SANS Porod exponent close to 4 characterizes the helical structure and an exponent 

close to 5/3 characterizes the coil structure. A Porod exponent of 4 is for a cylinder with 

smooth surface whereas an exponent of 5/3 is for a fully swollen coil. 

6. The melting of a helical unit is characterized by an activation enthalpy of -6 kcal/mol. 
541

Chapter 50 - SANS FROM A PROTEIN COMPLEX

1. INTRODUCTION

Proteins perform the basic tasks essential for life. Enzymes are proteins that catalyze 

reactions and perform specialized cell functions. Kinases are a large class of enzymes that 

add phosphate groups to proteins and other biomolecules. In bacteria, kinases coordinate 

cellular responses to external signals. The function of histidine kinases is to transfer a 

phosphate group from an ATP molecule to a specific amino acid (histidine) site on the 

kinase. This transfer mechanism is referred to as phosphorylation. The phosphate is further 

transferred from the histidine site to another protein.

A histidine kinase named KinA is essential in the sporulation mechanism of bacteria. 

Sporulation is the formation of spores which are capable of surviving dormant for a long time 

before reproducing again when external conditions become more favorable. KinA functions 

in concert with a response regulator named Sda which can halt sporulation when DNA 

damage is detected. Sda binds onto KinA to stop its autokinase activity thereby stopping its 

sporulation function by halting the phosphate transfer mechanism. 

Histidine kinases are comprised of two domains: (1) a “sensor” domain which recognizes the 

sporulation signal and (2) an “autokinase” domain which performs the phosphotransfer 

function. 

Figure 1: Representation of the two domains that form KinA. The 606 amino acid sequence 

is marked. The histidine used in the phosphate transfer mechanism is located in position 405. 

The crystalline structures of KinA and of its response regulator Sda are known from x-ray 

crystallography. The structure of the KinA/Sda complex, however, is not known. The 

autokinase domain is divided into two sub-domains: two catalytic and ATP binding parts 

named the CA dimer and a histidine phosphotransfer part named the DHp stalk. 

0 200 400 600

1 2

sensor domain autokinase domain

DHp CA

His
542

Figure 2: Representation of the dimeric KinA protein comprising the two CA domains and 

the DHp stalk. One of the two histidine amino acids used in the phosphate transfer 

mechanism is shown in green. The other one is behind. The response regulator Sda is also 

included on the left side. 

KinA is comprised of 606 amino acid residues. The first 383 constitute the sensor domain 

with three PAS regions and the next 223 constitute the autokinase domain with residues 383-

465 for the DHp region and residues 456-606 for the CA dimer region. The amino acid 

sequence for KinA has been only partly resolved. 

Figure 3: Spelling out of the amino acid residue sequence (383-606) showing two helices 1 

and 2 located in the DHp stalk. The histidine used in the phosphate transfer mechanism is 

shown at position 405. Dash marks correspond to the sequences that have not been resolved. 

sensor domain

CA 

domain

DHp stalk

CA 

domain

ATP

phosphate

group

SpoA

Histidine

on KinA

KinA

Sda

TEELMLKSEKLS-------IAGQLAAGIAHEIRNPLTAIKGFLQ-----LMKPTMEGNEHYFDIVFSELSRIELILSELLMLAKPQQNAV

---KEYLNLKKLI-GEVSALLE—TQANLNGIFIRTSYEKDSIYINGDQNQLKQVFINLIKNAV----ESMPDGGTVDIIITEDEHSVHV

1 2

TVKDEGEGIPEKVLNRIGEPF------LTTKEKGTGLGLMVTFNIIENHQGVIHVDSHPEKGTAFKISFPKK

His
543

The DHp stalk is formed of a bundle of four helices. The two histidine phosphorylation 

active sites are located halfway along the stalk (helices 1 and ’1). When the sensor domain 

receives the signal to sporulate, each of the two CA domains transfers a phosphate group 

from an adenosine triphosphate (ATP) molecule to one of the two histidine active sites. This 

is performed through a hinge-like motion of the CA domains; these domains pivot to transfer 

the phosphate groups. 

The response regulator Sda halts (inhibits) the sporulation mechanism performed by KinA 

when necessary. When Sda binds onto KinA, the phosphotransfer mechanism performed by 

the CA domains is stopped. The Sda binding region on KinA has been identified as located 

toward the lower part of the DHp stalk with no direct interaction with the CA domains. 

The purpose of the investigations reported here is to understand the positioning of the KinA 

and Sda parts of the protein complex and to get insight into the sporulation inhibition 

mechanism (Whitten et al, 2007). This is performed using the SAXS and the SANS 

techniques. 

2. SAXS FROM THE PROTEIN COMPLEX

The KinA and the Sda parts of the protein complex have been characterized by standard 

methods. The molecular weights have been determined using mass spectroscopy and size 

exclusion chromatography (SEC). These showed the dimer nature of Sda. Kinase assay 

identified the Sda surface involved in binding the KinA protein. Characterization was 

performed both on the protein complex and on its individual components. 

SAXS was performed on KinA and Sda alone and on the KinA-Sda complex. Two protein 

complex concentrations in the dilute regime in water were measured. 
544

0.01

0.1

1

10

100

1000

104

0 0.05 0.1 0.15 0.2 0.25 0.3

SAXS from Protein Complex 

KinA-Sda 

KinA 

Sda 

Scattered Intensity 

Q (Å-1) 

Figure 4: SAXS from dilute solutions of KinA and Sda alone and of the KinA-Sda complex. 

Curves have been shifted arbitrarily upward to avoid overlap. 

The pair-distance probability distribution function 

P(r)



is the inverse Fourier transform of 

the scattering form factor P(Q). 

 















    



0

2

P(r)

Qr

sin(Qr)

P(Q) dr exp( iQ.r)P(r) dr4 r

  

 

. (1)

The distance distribution function 4

r P(r)

2 

(also referred to as the pair correlation function) 

was obtained and plotted. It gives an estimate of the average size of the KinA, Sda and 

KinA/Sda complex (peak position) and goes to zero at the particle edge (at Dmax). 
545

0

0.1

0.2

0.3

0.4

0 20 40 60 80 100

SAXS from Protein Complex 

KinA 

Sda*10 

KinA-Sda 

 4 r

2 P(r)

 r (Å) 

Figure 5: Distance distribution functions obtained from SAXS data from the individual 

protein and inhibitor and from the protein complex. Scattering from Sda was scaled up (*10). 

The radius of gyration is obtained as the second moment of 

P(r)



:

  

2 2

g R r

 

 







0

2

0

2 2

dr4 r P(r)

dr4 r r P(r)

. (2)

Rgs for KinA and Sda were obtained from the Guinier analysis and from the second moment 

of P(r) analysis and are summarized in a table. Sizes obtained for the KinA/Sda complex 

showed a compaction of KinA after Sda binding. 

Table 1: Size parameters derived from the SAXS Guinier and P(r) data analyses.

Sample Concentration

(mg/ml)

Guinier Rg

(Å)

Rg from P(r)

(Å)

Dmax from P(r) 

(Å)

KinA 3.7 29.3 29.6 95

Sda 5.2 15.3 15.4 52
546

KinA/Sda 4.4 29.2 29.1 80

KinA/Sda 3.7 29.4 29.1 80

3. SANS FROM THE PROTEIN COMPLEX

SANS measurements were performed with deuterated Sda (d-Sda) complexed with nondeuterated KinA. Both KinA and Sda were overexpressed using E. coli cultures. D-Sda was 

obtained using a culture in d-water. Two dilute solution concentrations (3.5 mg/ml protein in 

50 mM NaCl and 11.9 mg/ml protein in 200 mM NaCl) were measured. A contrast variation 

series was performed in each case using mixtures of water and d-water. 

Radius of Gyration Analysis

Here also, radii of gyration were obtained for KinA, d-Sda and the KinA/d-Sda complex. In 

order to estimate the radius of gyration of the complex in terms of the individual radii of 

gyration, the following argument is used. Consider a compound object formed of two 

particles: particle 1 (for d-Sda) of volume V1 and particle 2 (for KinA) of volume V2

separated by a distance D (between their centers of mass) and define 1 and 2 for their 

excess scattering length densities (1 = 1-0 and 2 = 2-0 where index 0 refers to the 

solvent). 

Figure 6: Schematic representation of a compound object formed of two distinct particles. 

The inter-distance between the centers of mass is D. 

According to the parallel axis theorem, the radius of gyration of the compound object is 

expressed in terms of the individual radii of gyration Rg1 and Rg2 as follows:

D

Particle 2

Particle 1

KinA

Sda
547

2

1 1 2 2

2 2

1 1 2 2

2 1 1

g2

1 1 2 2

2 2 2

g1

1 1 2 2

2 1 1

g D

V V

V

V V

V

R

V V

V

R

V V

V

R 















  



















  





  





  





(3)

After a few straightforward manipulations, this can be put into the following Stuhrmann 

relation (Ibel-Stuhrmann, 1975):

2

2

gm

2

Rg R











  . (4)

This is expressed in terms of the average neutron contrast 



defined as:

1 2

1 1 2 2

V V

V V



  

  . (5)

The various parameters are defined as follows:

2

1 2

2

1 2

2 1

g2

1 2

2 2

g1

1 2

2 1

g m D

V V

V

V V

V

R

V V

V

R

V V

V

R 













































 





































  



































    

2

2

1 2

2

1

2

2 2

g2

2

g1

1 2

2

1 2

1

1 2 D

(V V )

V V

R R

V V

V

V V

V

 

2

2

1 2

2

2

1 2

2 1

1 2 D

V V

V

V V

V





































     . (6)

Variation of the measured radius of gyration (squared) with 

1/ 

yields three parameters 

(

2 Rgm

,  and  or 

Rg1

, Rg2and D). The various points were obtained from the contrast 

variation series. Rg

2

values were obtained from Guinier plots. 
548

100

200

300

400

500

600

700

800

900

-1.5 -1 -0.5 0 0.5 1 1.5 2

SANS from Protein Complex 

Parabola Fit 

SANS Data Rg

2 (Å

2

)

(106 Å2)

Figure 7: Stuhrmann plot for the KinA/d-Sda complex. 

The positive sign for the coefficient  reveals that the higher contrast component (d-Sda) lies 

towards the periphery of the complex. The peak position (maximum) corresponds to the 

condition 

   2

. Since  > 0, the condition  > 0 implies: 

 

2

g1

2

g2

2

2

1 2

2

1

2

2 D R R

(V V )

V V

 





















. (7)

Moreover, it was found that the distance between the centers of mass is D = 27.0 Å along 

with Rg1 = 25.3 Å for d-Sda and Rg2 = 25.4 Å for KinA. 

Pair Correlation Function Analysis 

The SANS intensity for the contrast variation series from the KinA/d-Sda complex can be 

modeled as:

I(Q) I (Q) I (Q) 2I (Q)

2 2 1 2 1 2

2

1 1 2

2

 1      . (8)
549

I11(Q) is the scattering from particle 1, I22(Q) is the scattering from particle 2 and I12(Q) is the 

cross term representing correlations between a pair of scatterers belonging to the two 

particles. 

-0.5

0

0.5

1

1.5

2

2.5

3

0 0.05 0.1 0.15 0.2 0.25 0.3

SANS from Protein Complex 

I

22

(Q), KinA 

I

11

(Q), d-Sda 

2I

12

(Q), KinA/d-Sda 

Scattered Intensity 

Q (Å-1) 

Figure 8: Single-particle and inter-particle form factors I11(Q), I22(Q) and I12(Q) obtained 

from SANS measurement. Scattered intensity is in arbitrary units. 

The inverse Fourier transform for the self and cross terms has been obtained and plotted. 
550

-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 20 40 60 8 0

SANS from Protein Complex 

P

22

(r), KinA 

P

11

(r), d-Sda*10 

2P

12

(r), KinA/d-Sda 

4 r

2 P(r)

r (Å) 

Figure 9: Single-particle and inter-particle pair correlation functions P11(r), P22(r) and P12(r) 

obtained as the inverse Fourier transform of the form factors. 

The pair distribution function for the d-Sda component is characterized by two peaks. 

Although d-Sda forms dimers when it is alone in solution, it binds to KinA as two individual 

distinct monomers. The two peaks indicate that centers of mass of the two d-Sda monomers 

are separated by 45 Å. The KinA and the d-Sda molecules themselves are separated by 27 Å. 

It is noted that the KinA molecule appears more compact when d-Sda is bound to it. Size 

parameters obtained from SANS data analyses are summarized in a table. 

Table 2: Size parameters derived from the SANS Guinier and P(r) analyses. 

Sample Concentration

(mg/ml)

% H2O Guinier Rg

(Å)

P(r) Rg

(Å)

P(r) Dmax

(Å)

KinA/d-Sda 3.7 0

10

20

80

90

100

28.9

28.3

28.0

22.5

27.1

25.7

28.3

28.5

27.9

22.7

24.4

25.1

80

80

80

80

80

80

KinA/d-Sda 11.9 0 28.3 28.9 80
551

10

20

80

90

100

28.2

29.1

22.5

24.3

25.3

28.3

28.2

23.0

24.2

25.1

80

80

70

75

75

KinA/d-Sda 26.6 40 26.1 23.4 70

The cross term P12(r) (inverse Fourier transform of I12(Q)) is characterized by one peak. This 

means that the two d-Sda monomers lie approximately equidistant from the two CA catalytic 

domains and must be located on opposite sides of the lower part of the DHp stalk. Even 

though the d-Sda monomers are nowhere near the CA domains, they affect their 

phosphorylation function. This must be performed through a remote (called allosteric) 

control via the four helix bundle. This bundle is probably disturbing the hinge motion of the 

CA domains necessary for the transfer of a phosphate group from an ATP molecule to the 

histidine site in the middle of the DHp stalk. The d-Sda inhibitor is therefore not acting as a 

molecular barrier to the autokinase motion of the CA domains. 

Figure 10: Computer rendering of the KinA/Sda protein complex consistent with the 

scattering results obtained. The second histidine and the second Sda monomer located on the 

back are not shown. 

DHp Stalk 

on KinA

Sda

CA Domain 

on KinA

CA Domain 

on KinA

Histidine

on KinA
552

4. COMMENTS

The structure of amorphous protein complexes are hard to resolve since no single-crystal can 

be obtained and therefore no diffraction data can be obtained. Small-angle scattering helps 

resolve the main structural characteristics of such protein complexes. The SANS technique 

when used with contrast variation and deuterated macromolecules can map out sizes and 

inter-particle distances between the various components forming the complex. Information 

obtained from scattering methods and from other techniques helps in the understanding of the 

mechanisms involved. 

A histidine kinase KinA and its inhibitor Sda have been investigated using SAXS and SANS. 

This protein complex is relevant to the sporulation mechanism in bacteria. Sporulation 

happens through the transfer of a phosphate group from an ATP molecule to a specific 

histidine site on the KinA stalk. This is performed by the hinge-like motion of the two CA 

domains of KinA. The phosphate is further transferred from the histidine site to a sporulation 

protein (SpoA) that docks onto KinA. The inhibition of the sporulation mechanism is 

performed when protein Sda forms a complex with KinA. Even though Sda is located at the 

bottom of the DHp stalk, it stops the sporulation mechanism of the KinA CA domains 

allosterically (i.e., remotely) through conformational changes in the four helix bundle of the 

DHp stalk. 

REFERENCES

A.E. Whitten, D.A. Jacques, B. Hammouda, T. Hanley, G.F. King, J.M. Guss, J. Trewhella, 

D.B. Langley, “The Structure of the Sda-KinA Complex Suggests an Allosteric Mechanism 

of Histidine Kinase Inhibition”, Journal of Molecular Biology 368, 407-420 (2007)

K. Ibel, H.B. Stuhrmann, “Comparison of Neutron and X-ray Scattering of Dilute 

Myoglobin Solutions”, Journal of Molecular Biology 93, 255-265 (1975)

QUESTIONS

1. What is a histidine?

2. What is a kinase?

3. What is an inhibitor? Name a kinase protein and its inhibitor. 

4. What is sporulation?

5. What is the difference between a spore and a seed?

6. What is measured through the Guinier plot?

7. What is the pair correlation function 

P(r)



? How about the size distribution function? 

8. State the so-called Stuhrmann relation used to analyze SANS data. 

9. What is the parallel axis theorem for the radius of gyration? What else is it used for? 

10. How are deuterated proteins obtained?
553

ANSWERS

1. Histidine is one of the 20 amino acid residues. These are the building blocks for proteins. 

2. A kinase is a protein that performs the function of phosphorylation. Phosphorylation is the 

transfer of a phosphate group from an ATP molecule to an active site on the kinase or other 

molecules. 

3. An inhibitor is a protein that stops (inhibits) a specific function. KinA is a histidine kinase 

used in the sporulation function and Sda is its inhibitor; i.e., it stops that function. 

4. Sporulation consists in the formation of spores which package the genetic material of a 

species. Spores are capable of surviving for a long time before reproducing when external 

conditions become more favorable. 

5. Spores contain the genetic material for reproduction only. Seeds contain the genetic 

material as well as food for initial growth. 

6. The Guinier plot measures the radius of gyration which is a measure of the particle density 

distribution around the center-of-mass. 

7. The pair correlation function 

P(r)



is the inverse Fourier transform of the scattering 

intensity (form factor). It is the probability of finding a scatterer at position 

r



given that there 

is a scatterer at the origin. The size distribution function is defined as 

4 r P(r)

2 

 . 

8. The Stuhrmann relation is used to analyze Rg data when a contrast variation series is 

measured. It related Rg

2

to the average excess scattering length density 



as follows: 

2

2

gm

2

Rg R











 

. Rgm

2

,  and  are related to structural parameters. 

9. The parallel axis theorem relates the radius of gyration of a compound particle to the radii 

of gyration of the individual components. The parallel axis theorem is used in cases where 

the second moment is used. The moment of inertia for compound particles is also calculated 

that way. 

10. Deuterated proteins are obtained from E. coli cultures grown (overexpressed) in d-water. 
554

Chapter 51 – OTHER SANS IN BIOLOGY PROJECTS

Biology is the third major SANS research area. It has been growing steadily. A few projects 

where this author was involved are included here. 

1. POLY(ETHYELENE GLYCOL)-LYSOZYME CONJUGATE

PEGlated proteins have PEG chains attached to the protein. This is referred to as the PEGlysozyme conjugate. The goal of this project (Pai et al, 2011) was to figure out whether the 

PEG chains were wrapping around the protein (shroud model) or assuming a separated 

configuration consisting of PEG random coil on one side and the protein on the other 

(dumbbell model). 

Note that poly(ethylene glycol) and poly(ethylene oxide) have the same monomer (-

CH2CH2O-) but have different terminal groups. Poly(ethylene glycol) or PEG has –OH 

groups at both ends while poly(ethylene oxide) or PEO has an –OH group at one end and a –

CH3 group at the other.

Chicken egg lysozyme was covalently bonded to PEG chains (Mw = 20,000 g/mol). SANS 

investigations were conducted with just lysozyme or attached to PEG-lysozyme conjugate in 

D2O in order to estimate their respective sizes. Measured volume fractions were around 0.1 

%. Then deuterated PEG (d-PEG) was grafted onto lysozyme and SANS investigations were 

performed from d-PEG-lysozyme in 46 % D2O/54 % H2O in order to contrast match the 

lysozyme and enhance the contrast of d-PEG thereby focusing on its conformation. 

Standard data analysis methods were used including the inverse Fourier transform method, 

the Guinier plot method and fits to standard models. In particular, the core-shell model and 

the form factor for Gaussian polymer coils (the Debye function) were used to discriminate 

between the shroud model (polymer shrouding the protein and forming a core-shell structure) 

and the dumbbell model whereby the scattering would be dominated by the PEG Gaussian 

coils when lysozyme is contrast matched. 

The inverse Fourier transform method yields a pair distribution function. The peak position 

indicates the size of the scattering object while the value of rmax (value for which the intensity 

falls back to zero) yields the maximum extension. 
555

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250

PEG-lysozyme 

lysozyme

Rescaled r

2 P(r) 

Size r (Å) 

Figure 1: Pair-distance distribution functions P(r) for lysozyme, and PEG-lysozyme in 

deuterated water (D2O). Maximum extensions of 34.4 Å for lysozyme and 212 Å for PEGlysozyme were obtained. 

SANS data from the d-PEG-lysozyme sample where lysozyme is contrast matched did not fit 

the core-shell model well, but fitted to the random coil model were good. 
556

Figure 2: Scattering data for deuterated PEG-lysozyme in 45% d-water and fitted curve to the 

form factor for random polymer coil (Debye function). Fit result yields a radius of gyration 

of Rg = 51.4 Å. 

Based on the SANS measurements and data analysis, it was concluded that the dumbbell 

model is more realistic than the shroud model to describe the PEG-lysozyme conjugate. 

2. CHIRALITY IN PEPTIDE BIOMATERIALS

This project investigates the application of chirality as a tool to tune the rate of gelation as 

well as the final mechanical properties of biomaterials (Taraban et al, 2012). The origin of 

homochirality and its effects in the biological world continuously stimulates much interest. 

Using a pair of oppositely charged peptides that co-assemble into hydrogels, we have 

systematically investigated the effect of chirality on the mechanical properties of these 

hydrogels. The results highlight the possible role of biohomochirality in the evolution and/or 

natural selection of biomaterials. A facet of such research is described here. 

Chiral substances can rotate the polarization plane of linearly polarized light. This is referred 

to as the Faraday rotation. Rotation could be to the left (L-type) or the right (D-type). In the 

biological world, almost all biological substances are homochiral; proteins are of L-type 

while sugars are of D-type. Peptides make up proteins that consist of amino acid sequences. 

Natural peptides are of L-type whereas man-made ones can be synthesized to be of the Ltype or D-type. Series of such peptides containing 11 amino acids were synthesized in the 

lab. These were mixed to make either L/D- or D/L- heterochiral biogels or L/L- or D/Dhomochiral ones. The formed peptide biogels were characterized using mechanical testing 

(shear response using a rheometer) as well as small-angle neutron scattering (SANS). It was 

found that homochirality confers mechanical advantage, resulting in higher elastic modulus 
557

and strain yield value. Yet, heterochirality confers kinetic advantage resulting in faster 

gelation. 

Figure 3: SANS data for the homochiral gels and the heterochiral ones. 

Inverse Fourier transform of SANS data combined with shape reconstruction methods 

(simulation) have given valuable insight into the fibrilar structure. Homochiral fibers are 

found to be thicker and denser. 
558

Figure 4: Inverse Fourier transform of SANS data. The rescaled pair distance-distributions 

r

2P(r) for homochiral gels are at the top while for those for heterochiral ones are at the 

bottom. Peak positions denote characteristic sizes. 

In order to estimate the fibrilar structure of these peptide hydrogels, an ab-initio shape 

reconstruction method (simulation) was used and compared to the scattering data. Since 

SANS data are characterized by Porod-type power law scattering (and no Guinier region was 

observed at low-Q), only cross fiber sectional size information was obtained (but no overall 

fiber sizes). Note the cross sectional sizes obtained from the ab-initio calculation agree with 

those obtained from the inverse Fourier transform method. 
559

Figure 5: Results of the shape reconstruction method showing the fibril cross sections and the 

fibrilar structure. These are plausible (not unique) solutions. 

Mechanical testing using a Couette-type rheometer shows that homochiral gels are stronger 

over long time scales as compared to heterochiral ones. 
560

Figure 6: Results of monitoring of the gelation process using mechanical testing (shear 

response) over time.

3. POLYSACCHARIDE NETWORKS

Polysaccharides are formed of long chains of carbohydrate biopolymers. Chitosan and 

alginate are polysaccharides that are effective structural biomaterials. When used together, 

these form fibers that are used in tissue engineering. In this project, networks made of 

chitosan and alginate have been investigated as potential engineering scaffolds (Hyland et al, 

2011). Different mixtures of these components yield different bulk material properties such 

as the elastic response measured using a rheometer. Moreover, additives like chondroitin or 

calcium chloride were added to modify the properties of these polysaccharide networks. 

A number of networks of various compositions were prepared, subjected to mechanical 

testing and measured by electron microscopy (SEM) and SANS to investigate the 

nanostructure such as porosity and the fiber like structure. SANS data are characterized by 

power law scattering with no Guinier region. Higher intensities correspond to larger and 

denser structures.
561

Figure 7: SANS data from biopolymer networks of various compositions. The color coding is 

as follows: cyan for chitosan, black for alginate, orange for chitosan + alginate, violet for 

chitosan + alginate + calcium chloride, green for chitosan + alginate + chondroitin. 

Inverse Fourier transform yields pair distance distributions for the various network 

compositions. For instance, it was found that the addition of chondroitin increased the fiber 

cross section (thickness). Moreover, the addition of calcium chloride led to the contraction of 

the alginate component in the network. 

Figure 8: Distance distribution functions obtained from the inverse Fourier transform 

method. The color coding as the same as before. 

Formation of the fiber structure is accompanied by increase in the SANS intensity. Chitosan 

and alginate interact to form the polysaccharide network. Calcium chloride shields 

electrostatic interactions thereby causing fiber network contraction while the addition of 

chondroitin increases the fiber thickness. Adding these modifiers improved the network 

stiffness and tensile strength. 

4. NIH DATABASES

A number of biology databases are available online at the National Institutes of Health (NIH) 

web site at http://www.ncbi.nlm.nih.gov/Database/index.html. These cover information on 
562

nucleotides, proteins, biological structures, taxonomy (science of classification), genomics, 

gene expression, and chemical structures. 

REFERENCES

S. Pai, B. Hammouda, K. Hong, D. Pozzo, T. Przybycien and R.D. Tilton, “The 

Conformation of the Poly(ethylene glycol) Chain in Mono-PEGylated Lysozyme and MonoPEGylated Human Growth Hormone”, Bioconjugate Chemistry 22, 2317-2323 (2011).

M.B. Taraban, Y. Feng, B. Hammouda, L. Hyland, and Y.B. Yu, “Chirality-Mediated 

Mechanical and Structural Properties of Oligopeptide Hydrogels”, Chemistry of Materials 

24, 2299-2310 (2012).

L. Hyland, M.B. Taraban, B. Hammouda and Y.B. Yu, “Mutually Reinforced 

Multicomponent Polysaccharide Networks”, Biopolymers 95, 840-851 (2011).
563

Part K – OTHER SANS TOPICS

Chapter 52. SANS from Polymer Blends under Pressure

52.1 Introduction

52.2 The DPS/PVME Polymer Blend under Pressure

52.3 Compressible Polymer Blend Model

52.4 A Polyolefin Polymer Blend under Pressure

52.5 The DPS/PBMA Polymer Blend under Pressure

52.6 Summary and Discussion

References

Questions

Answers

Chapter 53: SANS under Shear

53.1 Sheared Discotic Liquid Crystal Micelles

53.2 Sheared Copolymer Lamellae

53.3 Pluronics under Shear

53.4 Mixed Copolymer Morphologies under Shear

53.5 Comments

References

Questions

Answers

Chapter 54. Solvation with Mixed Solvents

54.1 Solvation of Macromolecules

54.2 Poly(Ethylene Oxide) in Water/Ethanol Solvent Mixtures

54.3 PEO/d-Water/d-Methanol Ternary Mixtures

54.4 DNA Solvation in Mixed Solvents

References

Questions

Answers

Chapter 55. Clustering in Macromolecular Media

55.1 Introduction

55.2 Chain End Clustering

55.3 Clustering Due to Monomer Sticking Interactions

References

Questions

Answers

Chapter 56: SANS from Polymeric Materials

56.1 Materials and Method

56.2 Macromolecular Orientation with Hot Stretching

56.3 Hot Stretching Revisited

56.4 Macromolecular Orientation with Injection Molding

56.5 Macromolecular Orientation in Shear Band Deformations 
564

References

Questions

Answers

Chapter 57: Neutron Scattering with Spin Polarization

57.1 The Spin Dependent Elastic Neutron Scattering Cross Section

57.2 Separate-Out the Coherent and Incoherent Cross Sections

57.3 Sum of the Spin-Dependent Cross Sections

57.4 Neutron Spin Polarization Factor

57.5 Nuclear Spin Polarization

57.6 Supermirror Neutron Spin Polarizer

57.7 He-3 Gas Neutron Spin Analyzer

References

Questions

Answers

Chapter 58: Other SANS Topics Literature Review

58.1 Instrumentation

58.2 Magnetism

58.3 Carbon Nanotubes

58.4 Ceramics Materials

58.5 Fuel Cells Materials

58.6 Geological Materials

58.7 Dielectric Materials

58.8 Metals

58.9 Nanoscale Materials

References
565

Chapter 52 – SANS FROM POLYMER BLENDS UNDER PRESSURE 

1. INTRODUCTION

Pressure is another parameter relevant to the understanding of the phase separation and 

thermodynamic behavior of polymer blends. Polymer blend mixtures phase separate either 

upon cooling and are characterized by an Upper Critical Spinodal Temperature (UCST) 

behavior or upon heating in which case they are characterized by a Lower Critical Spinodal 

Temperature (LCST) behavior. The UCST behavior is driven by enthalpic interactions 

between monomers whereas it is argued that the LCST behavior is due to “free volume”. 

Free volume is related to the packing ability of monomers and is related to many factors 

among which polymer chemistry (side group bulkiness). 

The mean field Flory-Huggins theory is a broadly used model that describes polymer blend 

miscibility. The chi parameter has two main contributions, one entropic and one enthalpic in 

nature. This theory is the basis for the Random Phase Approximation (RPA) approach used 

to model the scattering from homogeneous polymer blends. The commonly used RPA 

equations assume incompressible blends. Extension of the RPA equations to describe 

compressible blends will be discussed. 

The goal here is to investigate the effect of hydrostatic pressure on the UCST and LCST 

phase behaviors and to account for equation-of-state (i.e., compressibility) effects using the 

SANS technique with in-situ pressure. Pressure can be controlled faster than temperature and 

is therefore more effective for thermodynamic studies. Pressure is also an important factor in 

polymer processing. 

The in-situ pressure cell consists mainly of two sapphire windows separated by a fixed gap 

(typically 1 mm) in which a polymer wafer is confined inside an o-ring. Pressure is applied to 

the sample through the o-ring so that the pressurizing fluid never gets in contact with the 

sample. In-situ pressure can be varied typically up to a couple of kilobars (note that 1 bar = 1 

atm = 760 mm Hg = 14.7 psi = 100 kPa) and temperature can be varied up to 160 oC. This 

gives a wide window in parameter space.

2. THE DPS/PVME POLYMER BLEND UNDER PRESSURE

SANS with in-situ pressure has been applied to a series of deuterated polystyrene (dPS) and 

polyvinyl methyl ether (PVME) blends with various compositions (Hammouda-Bauer, 

1995). Molecular weights were Mw = 188,000 g/mol (Mw/Mn = 1.02) for dPS and Mw = 

201,000 g/mol (Mw/Mn = 1.49) for PVME. The dPS volume fractions in dPS/PVME were 

chosen as 10 %, 30 % and 50 % respectively. It is known that the dPS/PVME blend is 

characterized by an LCST behavior with a minimum (critical point) at around 20 % dPS. The 

effect of pressure is to raise the LCST line since its effect is to damp out composition 

fluctuations. This is manifested as a lowering of the scattering intensity. In order to monitor 

composition fluctuations, the forward scattering intensity I(0) is obtained using a Zimm plot 
566

(plot of I-1(Q) vs Q

2

). It was found that pressure favors mixing for the dPS/PVME blend in 

most cases. 

0

100

200

300

400

500

0 20 40 60 80

 10%/90% dPS/PVME Polymer Blend 

60 oC

80 oC

100 oC

110 o I(0)C

 Pressure (MPa) 

Figure 1: Variation of the forward scattering intensity I(0) with pressure for the 10 %/90 % 

dPS/PVME sample at various temperatures. A line through the points is included as a guide 

to the eye. 

A pressure-induced increase of the spinodal line with pressure (evidenced by a decrease of 

the scattering intensity I(0)) of as much as 30 oC/kbar was observed. 

Since pressure can be changed fast, spinodal decomposition can be controlled at will even for 

this strongly interacting blend. Following pressure change strategies, one could get in and out 

of the spinodal phase demixing region. This is observed as the formation of a spinodal ring 

which quickly disappears under the beamstop. 

As a first step in understanding the SANS data, recall the incompressible RPA equation for 

polymer blends:

0

12

1 1 1 1 2 2 2 2

2

1 2

v

2

n v P (Q)

1

n v P (Q)

1

d (Q) d

( ) 











 

  

. (1)
567

Here, the standard notation has been used for the degrees of polymerization (n1 and n2), 

volume fractions 1 and 2, monomer volumes v1 and v2, Debye functions P1(Q) and P2(Q), 

scattering length densities 1 and 2 and chi parameter 12/v0. 

The Flory-Huggins interaction parameter has two contributions: one entropic and one 

enthalpic in nature, 

T

D

12  C 

. Our measurements showed that both contributions increase 

with pressure. This interaction parameter is characterized by a composition-dependence. Our 

measurements also showed that this dependence is not due to compressibility effects. 

The form factors can be expanded at low-Q as 

3

Q R

1

P (Q)

1

2

g1

2

1

 

. This changes the 

incompressible RPA equation to the low-Q expansion as:

2

2

g2

2 2 2

2

g1

0 1 1 1

12

1 1 1 2 2 2

2

1 2 Q

3

R

n v

1

3

R

n v

1

v

2

n v

1

n v

1

d (Q) d

( )





































 

  

. (2)

The Zimm plot slope B in I-1(Q) = I-1(0)+BQ2 is proportional to the radii of gyration. 
568

32

34

36

38

40

0 20 40 60 80

30%/70% dPS/PVME Polymer Blend 

130 oC

120 oC

110 oC

100 oC

60 oC 

Zimm Plot Slope 

Pressure (MPa)

Figure 2: Variation of the Zimm plot slope B (obtained from I-1(Q) = I-1(0)+BQ2) with 

pressure for the 30 %/70 % dPS/PVME sample measured at various temperatures. 

This Zimm plot slope is seen to decrease with pressure. There is no reason for pressure to 

affect the radii of gyration. This artifact is due to the incompressible assumption in the RPA 

model. A compressible RPA model is described next. 

3. COMPRESSIBLE POLYMER BLEND MODEL

In order to include compressibility effects, an equation-of-state is used. It describes density 

variation with temperature and pressure (using so-called PVT measurements). Various 

equation-of-state models are available to describe free volume effects in polymers (1) cell 

models associate a free volume component as part of the monomer volume, (2) lattice-fluid 

models assume free volume as a separate component, and (3) hole models assume a 

combination of the above two features. Here a simple lattice-fluid model is used (SanchezBidkar, 1995).

Consider the “mixing” polymer volume fractions for the two polymer components 1 and 2

defined previously such that 1+2 = 1. Introduce a free volume fraction f0 and new fractions 

f1 = 1(1-f0) and f2 = 2(1-f0) such that f1+f2+f0 =1. f0 is not directly measurable but can be 
569

estimated through PVT (density) measurements. Define monomer “hard-core” volumes v1* 

and v2*, the statistical segment lengths a1 and a2, neutron scattering length densities 1 and 2

and degrees of polymerization n1 and n2. 

The lattice-fluid equation-of-state uses a characteristic energy density (internal pressure) P* 

and characteristic temperature T* for each of the components. These are tabulated quantities 

for each polymer. The lattice-fluid equation-of-state reads:

  0

T*

T

ln(f ) 1 f

P*

P

(1 f )

0 0

2

 0      . (3)

The term 

2

2

1

1

n n n

1 







appearing in the original model has been neglected for typical (large) 

degree of polymerization (n>>1). 

Mixing rules are used to connect the two polymer components:

P* P * 2 P * P2*

2

1 1 2 12 2

2

 1    

(4)

k T *

P *

k T *

P *

v

1

k T*

P*

B 2

2 2

B 1

1 1

B 0







  .

These equations describe the compressibility part of the blend mixture. 

The compressible binary mixture can be assumed to be a ternary incompressible mixture 

where the third component consists of “holes” (think free volume). The Gibbs free energy 

density for this ternary mixture (polymers 1 and 2 and holes as the third component) has the 

usual entropic part (three terms) and the entropic part (3 binary interactions). The scattering 

intensity is obtained using a three component RPA approach. The ternary RPA equations 

have been discussed previously. They are repeated here for convenience. The SANS cross 

section is given by the following set of equations (Hammouda-Benmouna, 1995):

S (Q) S (Q) 2 S (Q)

d

d (Q)

2 2 1 2 1 2

2

1 1 2

2

 1     





. (5)

 

1 v S (Q)1 v S (Q) v S (Q)S (Q)

S (Q) 1 v S (Q)

S (Q)

0

2 2

0

1 1

2

1 2

0

2 2 2 2

0

1 1 1 1

0

2 2 2 2

0

1 1

1 1

  





 

1 v S (Q)1 v S (Q) v S (Q)S (Q)

S (Q) 1 v S (Q)

S (Q)

0

2 2

0

1 1

2

1 2

0

2 2 2 2

0

1 1 1 1

0

1 1 1 1

0

2 2

2 2

  





1 v S (Q)1 v S (Q) v S (Q)S (Q)

S (Q)v S (Q)

S (Q)

0

2 2

0

1 1

2

1 2

0

2 2 2 2

0

1 1 1 1

0

1 2 2 2

0

1 1

1 2

  





S (Q) n f v *P (Q) 1 1 1 1

0

11 
570

S (Q) n f v *P (Q) 2 2 2 2

0

22 

0 0

0

S00  f v

11

B

1

0

00

11 C

k T

P *

2

S

1

v   

22

B

2

0

00

22 C

k T

P *

2

S

1

v   

12

B

12

0

00

12 C

k T

P *

2

S

1

v   

 





















 

  

k T *

P *

k T *

P *

(1 f )

ln(f ) (1 f )

C 2

B 2

2

B 1

1

2

0

0 0

1 1 2

 





















 

 

k T *

P *

k T *

P *

(1 f )

ln(f ) (1 f )

C 2

B 2

2

B 1

1

2

0

0 0

2 2 1

 





















 

   

k T *

P *

k T *

P *

(1 f )

ln(f ) (1 f )

C ( )

B 2

2

B 1

1

2

0

0 0

1 2 1 2

.

This formalism is tied up using the following bridging relation between the energy densities 

Pij* and the monomer-monomer interaction parameters Wij:

2v *

W

P *

ij

ij   . (6)

Here 

v*  v1* v2*

is a reference volume and P11* = P1* is understood. 

The lattice-fluid equation-of-state and the compressible RPA equations are solved selfconsistently (i.e., iteratively till convergence is obtained). The iterative process is carried out 

using an initial guess: 

P12*  P1*.P2* . 
571

Figure 3: Self-consistent approach to calculate the scattering intensity for compressible 

polymer blends using the lattice-fluid equation-of-state and the compressible RPA equations. 

This formalism is applied here to the dPS/PVME blend under pressure. Tabulated values for 

dPS are P1* = 355 MPa and T1* = 731 K and for PVME they are P2* = 353 MPa and T2* = 

657 K. Some results for the free volume fraction for increasing temperature and pressure 

follow. 

Input: 1, 2, v1*, v2*, 

P1*, P2*, T1*, T2*

Initial Guess: 

P12*  P1*.P2*

P12*=

Use Mixing Rule: 

P*=1

2P1*+212P12*+22P2*

P*/kBT*=1P1*/kBT1*+ 2P2*/kBT2*

Solve Lattice Fluid Equation of State:

(1-f0)

2 + P/P* + [ln(f0)+1-f0]T/T* = 0

Obtain Free Volume: f0

Fit RPA Equations to SANS Data:

S (Q) S (Q) 2 S (Q)

d

d (Q)

2 2 1 2 1 2

2

1 1 2

2

 1     





Using expressions for S11(Q), S22(Q) and S12(Q)

in terms of v11, v22, v12 and C11, C22, C12, v0, f0

Obtain: P12*

Final f0 and P12*

Input: 1,2, a1, a2

Iterate
572

0.02

0.04

0.06

0.08

0.1

0.12

20 40 60 80 100 120 140

10%/90% dPS/PVME Polymer Blend 

0.1 MPa

27.2 MPa

54.4 MPa

81.6 MPa

Free Volume Fraction 

Temperature (oC)

Figure 4: Variation of the free volume fraction f0 for increasing temperature for the 10 %/90 

% dPS/PVME polymer blend. 
573

0.02

0.04

0.06

0.08

0.1

0.12

0 20 40 60 80

10%/90% dPS/PVME Polymer Blend

60 oC

100 oC

120 oC

Free Volume Fraction

Pressure (MPa)

Figure 5: Variation of the free volume fraction f0 for increasing pressure for the 10 %/90 % 

dPS/PVME polymer blend. 

The free volume fraction f0 is seen to increase with temperature and decrease with pressure 

as it should. 

The inter-monomer interaction energy density P12* is plotted next with and without free 

volume (i.e., with f0 = 0). This variation is seen to have weak variation for increasing 

pressure and to decrease consistently with increasing temperature. These results are 

reasonable since increasing temperature moves the blend closer to phase separation. 
574

354.2

354.3

354.3

354.4

354.4

354.5

354.5

0 20 40 60 80

50%/50% dPS/PVME Polymer Blend

100 oC

100 oC, f

0

 = 0 P12

*

Pressure (MPa)

Figure 6: Variation of the inter-component interaction energy density P12* for increasing 

pressure for the 50 %/50 % dPS/PVME. 
575

354.1

354.2

354.3

354.4

354.5

354.6

354.7

60 80 100 120 140

50%/50% dPS/PVME Polymer Blend

81.6 MPa

81.6 MPa, f

0

 = 0

P

12

*

Temperature (oC)

Figure 7: Variation of the inter-component interaction energy density P12* for increasing 

temperature for the 50 %/50 % dPS/PVME. 

The compressible RPA model used here to fit the SANS data is highly nonlinear and yields 

wide variations in intensity for slight variation in interaction energy density P12*. P12* was 

found to depend on pressure (weakly) and on temperature (linearly). 

4. A POLYOLEFIN POLYMER BLEND UNDER PRESSURE

Model polyolefins of molecular weight around 200,000 g/mol consisting of deuterated 

polymethyl butylene (dPMB) and polyethyl butylene (PEB) were blended and investigated 

under pressure. A PMB-PEB diblock copolymer was added in order to adjust the phase 

diagram to a convenient temperature range. The degrees of polymerization were 4260 for 

dPMB, 3350 for PEB and 3740 for the PMB-PEB diblock. The relative fraction of PEB 

monomers in the PMB-PEB diblock was 0.33. The dPMB/PEB relative volume fraction was 

also 0.33. 

The SANS technique was used with in-situ pressure in order to investigate the effect of 

pressure on the spinodal and binodal temperatures for this polyolefin blend (Hammouda et al, 

1997). The spinodal temperature Ts is obtained from an extrapolation of the plot of I-1(0) vs

T

-1 where I(0) is the forward scattering intensity obtained from a Zimm plot and T is the 
576

absolute temperature (in K). The binodal temperature Tb on the other hand is obtained when 

I

-1

(0) becomes negative. Note that the region between Tb and Ts is the nucleation and growth 

region. When pressure is increased, both Ts and Tb are seen to increase. In other words, 

pressure favors phase separation (demixing). 

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

dPMB/PEB Polyolefin Blend

Binodal Temperature T

b

Spinodal Temperature T

s Transition Temperatures (

oC)

Pressure (kBar)

mixed phase region

nucleation and 

growth region

spinodal region

Figure 8: Variation of the binodal and spinodal transition temperatures with pressure for the 

dPMB/PEB polyolefin blend. The nucleation and growth and the spinodal regions constitute 

the so-called phase separation region. 

A simple description of compressibility effects through a pressure-dependent Flory-Huggins 

interaction parameter 

12  E  FP

showed that E (related to the internal energy change U 

upon mixing) and F (related to the volume change V upon mixing) are proportional. This 

means that the volume increase upon mixing is a linear response of the repulsive interactions 

between monomers (Lefebvre et al, 1999). 

Here also, varying pressure gives a fast-response way for moving from the mixed phase 

region to the phase separated region of the phase diagram. This allows the monitoring of the 

phase separation kinetics during phase separation. The reverse phase-mixing kinetics have 

also been investigated (Hammouda et al, 1997). 
577

5. THE DPS/PBMA POLYMER BLEND UNDER PRESSURE

Another polymer blend was investigated using in-situ pressure. It consisted of dPS blended 

with poly-n-butyl methacrylate noted PBMA for short (Hammouda-Bauer, 1995). The 

molecular weights were Mw = 10,000 g/mol (Mw/Mn = 1.03) for dPS and Mw = 34,000 g/mol 

(Mw/Mn = 1.96) for PBMA. This blend is characterized by a miscibility gap. In-situ pressure 

affects both the UCST and the LCST branches of the phase diagram. This is seen as a 

systematic decrease of the scattering intensity for all temperatures. Here also, pressure 

dampens composition fluctuations and widens the miscibility gap. 

3.8

4

4.2

4.4

4.6

4.8

5

20 40 60 80 100 120 140 160

10%/90% dPS/PBMA Polymer Blend

100 kPa

27.2 MPa

54.4 MPa

81.6 MPa

I(0)

Temperature (oC)

Figure 9: Variation of the forward scattered intensity I(0) with temperature for the 10 %/90 

% dPS/PBMA sample for various pressures. The upturn variation is characteristic of a 

miscibility gap. The lines are parabolic fits included to better visualize the trends. 

Pressure seems to affect the high temperature branch (LCST) more than the low temperature 

branch (UCST). It also shifts the miscibility gap to a slightly higher temperature. 
578

6. SUMMARY AND DISCUSSION

Pressure affects both the UCST and the LCST branches of the phase diagram in polymer 

blends. Based on the blends considered here, a few conclusions can be summarized. Pressure 

can raise the LCST which is driven by free volume. It can either raise or lower the UCST 

(driven by monomer-monomer interactions). It should be noted that in other instances (not 

described here), pressure can lower the LCST such as in the case of solutions of 

poly(ethylene oxide) in deuterated water (PEO/d-water). In that case, the LCST is driven by 

specific interactions (hydrogen bonding) that soften under pressure. These points are 

summarized elsewhere (Hammouda, 2001). 

Figure 10: Pressure effects on the thermodynamics of phase-mixing for three polymer blends. 

The phase separation lines can either increase or decrease with pressure. This can be 

understood in terms of the Clausius-Clapeyron equation: 

H

V

dP

dT





 . (7)

Here dT/dP is the derivative of the temperature variation with pressure along the phase 

separation curve and V and H are the volume change and enthalpy change upon phasemixing. For a UCST system, mixing happens upon heating (H>0). If the volume change 

upon mixing is positive V>0, then pressure increases the UCST line (as in the case of 

dPMB/PEB). If on the other hand V<0, then pressure decreases the UCST line (as in the 

case of dPS/PBMA). For an LCST system, mixing happens upon cooling (H<0). Similarly, 

if V<0 then dP/dT>0 and pressure increases the LCST line (as in the cases of dPS/PVME 

mixed phase

phase

separated

composition

T

e

m

p

dPS/PVME

pressure

H<0, V<0

phase

separated

mixed phase

composition

T

e

m

p

dPMB/PEB

pressure

H>0, V>0

phase

separated

phase

separated

mixed phase

composition

T

e

m

p

dPS/PBMA

pressure

pressure

H>0, V<0

H<0, V<0
579

and dPS/PBMA). This is the case of LCST driven by free volume (which decreases upon 

cooling). If on the other hand V>0, then dP/dT<0 and pressure decreases the LCST line as 

in the case of PEO/d-water (not shown here). This is the case of LCST due to hydrogen 

bonding (which causes the volume to increase upon cooling). 

REFERENCES

B. Hammouda and B. Bauer, “Compressibility of Two Polymer Blend Mixtures”, 

Macromolecules 28, 4505-4508 (1995)

U. Bidkar, and I. Sanchez, “Neutron Scattering from Compressible Polymer Blends: A 

Framework for Experimental Analysis and Interpretation of Interaction Parameters” 

Macromolecules 28, 3963-3972 (1995)

B. Hammouda and M. Benmouna, “Neutron Scattering from Polymer Blends Under 

Pressure”, J. Polym. Sci., Polym. Phys. Ed. 33, 2359-2364 (1995)

B. Hammouda, N.P. Balsara and A.A. Lefebvre, “SANS from Pressurized 

Polyethylebutylene/Polymethylbutylene Blends”, Macromolecules 30, 5572-5574 (1997)

A. Lefebvre, J. Lee, N. Balsara, B. Hammouda, and R. Krishnamoorti “Relationship Between 

Internal Energy and Volume Change on Mixing in a Simple Polymer Blend”, 

Macromolecules 32, 5460-5462 (1999)

B. Hammouda, “SANS from Polymers under Pressure: a Short Review”, Neutron News 12, 

20-24 (2001)

QUESTIONS 

1. How is pressure applied to the sample in an in-situ pressure cell?

2. Describe the LCST and UCST phase behaviors. 

3. Describe a miscibility gap. 

4. What is meant by “free volume” in polymer blends? What causes free volume?

5. What is the effect of pressure on an LCST system driven by free volume?

6. Does the UCST line increase or decrease with increasing pressure? Why?

7. What is an equation-of-state? Name an equation-of-state used for polymers. 

8. State the Clausius-Clapeyron equation. Define the various terms. 
580

ANSWERS

1. The in-situ pressure cell uses an o-ring confined between two sapphire windows. The 

sample itself is melt-pressed into the right volume and confined in the o-ring. The 

pressurizing fluid compresses the o-ring thereby pressurizing the sample. 

2. Phase separation occurs through heating in a Lower Critical Spinodal Temperature (LCST) 

system whereas it occurs through cooling in a UCST system. 

3. Phase separation occurs both though heating and cooling when a polymer blend is 

characterized by a miscibility gap with an LCST at high temperature and a UCST at low 

temperature. 

4. Free volume means the less-than-perfect packing of monomers in the blend. Free volume 

decreases during densification. Free volume can be caused by bulky side groups that are hard 

to pack tightly. 

5. Free volume is squeezed out when pressure is applied. This raises the LCST with 

increasing pressure. 

6. The UCST could either increase or decrease with increasing pressure. This is due to the 

possibility of a positive or negative volume change upon mixing in blends. 

7. An equation-of-state describes the variation of density with temperature and pressure. The 

lattice-fluid equation-of-state is used to describe polymers. 

8. The Clausius-Clapeyron equation is stated as: 

H

V

dP

dT







where dT/dP is the derivative of 

the temperature variation with pressure along the phase separation curve and V and H are 

the volume change and enthalpy change upon phase-mixing.
581

Chapter 53 – SANS UNDER SHEAR

SANS is useful for investigating structures under shear. The orientation of layered structures 

can be monitored by SANS using shear cells. Here, a couple of projects are described using 

in-situ shear cells. In-situ shear cells include the Couette type and the plate-plate geometry 

type. 

1. SHEARED DISCOTIC LIQUID CRYSTAL MICELLES

A lyotropic mixture of cesium-perfluoro-octanoate (CsPFO) in water (55 % mass fraction) 

was investigated using a Couette shear cell (Mang et al, 1994; Hammouda et al, 1995). This 

mixture forms discotic liquid crystal micelles with a characteristic smectic-to-nematic 

transition temperature of 47 oC and a nematic-to-isotropic transition temperature of 52 oC. 

The in-situ Couette shear cell consists of a cylindrical stator and rotor pair made out of quartz 

and separated by a 0.5 mm gap. The rotor diameter is 6 cm so that the sheared sample 

volume is around 12 ml. Its computer controlled rotation can be steady (for simple shear) or 

reciprocating (for oscillatory shear). Shear can be controlled up to a shear rate (or shear 

frequency) of 6000 Hz (note that 1 Hz corresponds to 1 rotation per second). Sample heating 

was performed using circulating fluid through the stator. The shear cell has two measurement 

configurations: one radial and one tangential. The radial configuration uses standard beam 

geometry with a 1.27 cm circular sample aperture diameter whereas the tangential 

configuration uses a vertical beam defining slit 0.5 mm in width. The oscillatory mode of

operation uses 200 % strain, i.e., it oscillates by about 1 mm in each direction. 

The SANS spectrum for the discotic micelles is characterized by two peaks: one at Q = 0.113 

Å

-1

and one at Q = 0.146 Å-1. The first one represents the center-to-center inter-distance for 

micelles that lie planar in an edge-to-edge configuration while the second one corresponds to 

the face-to-face inter-distance for stacked disks. SANS measurements with the Couette shear 

cell were performed in the smectic phase at 45 oC, in the nematic phase at 49.7 oC and in the 

isotropic phase at 54 oC. Monitoring of the two peaks in the radial and tangential 

configurations gave clues as to the orientation of the discotic structures.
582

0

2000

4000

6000

8000

1 104

1.2 104

1.4 104

0 0.05 0.1 0.15 0.2

Vertical

Horizontal

Scattering Intensity 

Q (A-1)

Figure 1: The two SANS peaks characterizing the discotic liquid crystal micelles structure. 

Vertical and horizontal sector cuts through the anisotropic SANS data are shown. 

The sheared discotic micelles orient mostly parallel to the moving shear cell (rotor) walls. 

This is referred to as the “C alignment” type. 
583

Figure 2: The three types of possible alignments as viewed using the radial and tangential 

configurations. 

Flipping of the discotic structures occurred from the C type alignment observed in the 

nematic phase to the A type alignment observed in the smectic phase. The flipping transition 

has been observed with oscillatory shear either by varying temperature or shear rate. The 

flipping was not complete so that mixtures of A and C alignment types were often mixed. 

A Alignment Type

B Alignment Type

C Alignment Type

Top View of Shear Cell Radial Beam Tangential Beam

Shear

Shear 

gradient
584

Figure 3: Iso-intensity contour plots for an oscillatory shear rate of 5835 s-1at a temperature 

of 49.7 oC (i.e., in the nematic phase) for radial (top) and tangential (bottom) beam 

geometries. The top contour plot shows an inner scattering ring and two outer scattering 

peaks in the vertical direction pointing to the A alignment type. The bottom contour plot 

shows four outer scattering peaks in the vertical and horizontal directions pointing to a 

mixture of A and C alignment types. 

When oscillatory shear is used, shear-induced shifts of the phase transition temperatures were 

observed. These are interpreted as shear-induced damping of critical fluctuations that become 

stronger close to phase boundary lines. Moreover, interesting competing “bulk” and “wall” 

effects have also been observed. These effects were seen by changing the shear cell sample 

gap (from 0.5 mm to 1 mm). 

2. SHEARED COPOLYMER LAMELLAE

Diblock copolymers form lamellar, cylindrical and spherical morphologies. Lamellar 

morphologies are amenable to investigations under shear. A polystyrene-polyisoprene (SI) 

diblock copolymer in concentrated DOP solution was investigated under Couette shear 

(Balsara-Hammouda, 1994; Balsara et al, 1994). The diblock molecular weights were Mw = 

Inner ring

Outer peak

RADIAL

Outer peaks

A alignment type

TANGENTIAL

A alignment type

C alignment type
585

11,000 g/mol for the styrene block and Mw = 17,000 g/mol for the isoprene block. This 

corresponds to a lamellar morphology. DOP solvent was added (65 % polymer fraction) in 

order to lower the order-disorder temperature (ODT) to an easily reachable value of 38 oC. 

Scattering from the SI diblock is characterized by a peak at Q = 0.032 Å-1. Monitoring of this 

peak in the vertical and horizontal directions with radial or tangential beam configurations 

provides helpful clues to determine the lamellar orientation in each case. 

The diblock thermal history was “erased” in each case by heating the sample above the ODT. 

The temperature was lowered before starting the shear. Some of the observations follow. 

Couette shear can orient the diblock lamellae even above the ODT as shown on an 

azimuthally averaged SANS data. This is due to a shear-induced shift of the phase boundary. 

700

800

900

1000

1100

1200

-100 -50 0 50 100 150 200 250 300

balsara_prl_1994_fig1_shear_rate_0hz

shear rate = 0 Hz

shear rate = 1.95 Hz

Scattered Intensity 

Azimuthal Angle (degrees)

Figure 4: Effect of shear on azimuthally averaged SANS profiles obtained at 43 oC, i.e., 

above the quiescent ODT. The zero degree orientation is along the vertical axis. 

The shear-induced orientation occurs above a critical shear rate 

c





. The lamellar orientation 

is quantified through an anisotropy ratio  (ratio of the intensities in the vertical and 

horizontal directions). This ratio is seen to follow a universal behavior when plotted vs the 

scaled Couette shear rate 

c



 / 

. Curves corresponding to different temperatures fall on a 

same master curve represented by 

 

0.19

c  ~ 

 / 

. This is reminiscent of the time-temperature 

superposition principle for polymer melts in the rubbery region. 
586

0.5

0.6

0.7

0.8

0.9

1

0.01 0.1 1 10

balsara_prl_1994_fig3_38p3c

38.3 oC 

42.6 oC

47.1 oC

51.7 oC

56.1 oC

60.4 oC

Anisotropy Ratio 

Scaled Shear Rate

Figure 5: Master curve for the dependence of the peak anisotropy 



with increasing scaled 

shear rate 

c



 / 

for various temperatures. 

A 3D map of the SANS data under shear with the radial and tangential beam geometries 

leads to the following conclusions. The lamellar morphology is not perfect and contains 

lamellar “crumples” or “ripples” that are seen in the radial geometry (vertical peaks). The 

much higher intensity horizontal peaks in the tangential geometry show that the lamellae 

themselves are oriented mostly parallel to the shear cell walls (C type alignment). 
587

Figure 6: SANS iso-intensity contour plots obtained at ambient temperature (25 oC) and for a 

simple shear rate of 0.2 s-1for the radial configuration (top) and tangential configuration 

(bottom). Note that the scattering volume is not uniform for the tangential configuration 

resulting in non-uniform peak heights. 

Oscillatory shear produces lamellar alignment of the C type. Steady shear on the other hand 

produces lamellar alignment of the C type at high shear rates and of the A type at low shear 

rates. This is the “flipping” transition discussed before. The kinetics of lamellar flipping are 

characterized by time scales taking up to 90 minutes (Wang et al, 1999). 

3. PLURONICS UNDER SHEAR

Pluronics are triblock copolymers of the type PEO-PPO-PEO containing poly(ethylene 

oxide) and poly(propylene oxide) blocks. They form micelles in water. P85 Pluronic forms 

spherical micelles in d-water in the mass fraction range of 25 % or 30 % (used here) and for 

temperatures above 20 oC. SANS investigations have been performed using SANS and the 

Couette shear cell. Interesting shear-induced texture of the packed spheres structure was 

observed (Slawecki et al, 1998). For example, a novel shear-induced structure with 2D 

hexagonal symmetry was observed. The unit cell of this “crystalline” structure was 

determined to be characterized by a = b=133 Å and c = 209 Å. This structure changes upon 

shear cessation. 

Velocity

Neutral

TANGENTIAL

RADIAL

Velocity gradient

Neutral

A alignment type

C alignment type
588

Tangential Beam

shear

shear gradient

Radial Beam

shear gradient

shear

0.5 mm

20 s-1

0 s-1

sample gap
589

Figure 7: SANS patterns from 25 % P85 in d-water under steady shear (20 s-1) and after shear 

cessation. The left and right sides are for the radial and tangential beam geometries 

respectively. The schematics on the top show the Couette shear cell flow geometry. 

4. MIXED COPOLYMER MORPHOLOGIES UNDER SHEAR

Individual lamellar and spherical copolymer morphologies have been investigated 

extensively under shear. A question comes to mind: what morphologies would be obtained 

when mixing samples with lamellar and spherical morphologies together? The answer to this 

question is addressed here. 

The plate-plate shear cell is well suited for SANS investigations of sheared copolymers. It 

consists of sandwiching the copolymer sample in-between a sliding plate and a fixed one. 

The shear rate (or frequency) and the strain (or travel distance) are controlled. Note the three 

characteristic directions: shear, shear gradient and neutral. 

Figure 8: The plate-plate shear cell used for copolymers. 

Asymmetric copolymer samples characterized by lamellar and spherical morphologies were 

homogeneously mixed and then investigated by SANS with in-situ plate-plate reciprocating 

shear. Two separate copolymers were measured (1) block copolymers of polystyrene and 

poly(ethylene-butene-1) (Krishnamoorti et al, 2000) and (2) block copolymers of polystyrene 

and polyisoprene (PS-PI) (Krishnamoorti et al, 2000). 

Shearing at various temperatures (below the ODT) helps orient the sample morphology 

yielding scattering peaks in the horizontal and vertical directions. Two geometries were used 

(1) with a circular neutron beam along the shear gradient direction or (2) with a (vertical) slitdefined neutron beam along the neutral direction. Note that lamellae are characterized by a 

series of reflections at Q*, 2Q*, 3Q* where Q* is the first reflection. Cylinders are 

Neutral 

Shear

Shear 

gradient

3 cm

1.5 cm

1 mm

Fixed plate

Sliding plate
590

characterized by Q*, 

3

Q*, 

4

Q* while spheres are characterized by Q*, 

2

Q*, 

3

Q*. 

Monitoring the first couple of peaks gives clues as to the sample structure. 

Figure 9: Shear aligned copolymer sample with cylindrical morphology characterized by a 

hexagonally close packed structure for cylinders aligned vertically with peaks at Q*, 

3Q*, 

etc. The neutron beam is parallel to the shear gradient direction. 

This technique yields a wide range or possible morphologies obtained by mixing lamellae 

and spheres. These include spherical, lamellar and cylindrical morphologies among others. 

Temperatures above the order-to-disorder temperature (ODT) correspond to the disordered 

phase whereas those below the ODT correspond to ordered phases. These SANS results were 

verified using rheology. Simple ideal mixing predictions are far from what was observed 

experimentally. Two copolymers characterized by different morphologies seem to follow 

non-ideal mixing behavior. 

Q*

3

Q

*
591

80

90

100

110

120

130

140

150

160

0 0.2 0.4 0.6 0.8 1 Temperature (

oC)

Copolymer Mixing Fraction

ideal mixing 

prediction

lamellae

spheres

cylinders

disordered lamellae 

ODT 

Figure 10: Different morphologies obtained by mixing PS-PI copolymer samples 

corresponding to lamellar and spherical morphologies at various fractions. The order-todisorder temperature is plotted. Non-ideal mixing behavior is observed.

5. COMMENTS

Our focus here was on demonstrating the richness of possibilities afforded by the use of 

SANS with in-situ shear. We have described the effect of Couette and plate-plate shear on 

liquid crystal micelles and copolymer systems. Shear is useful for the investigation of various 

morphologies including lamellar, cylindrical and spherical. It is noted that the projects 

described here made use of in-situ shear cells with no regard to stress measurements; these 

are not rheometers. Rheometers have recently been adapted for SANS geometry. 

REFERENCES

J. Mang, S. Kumar and B. Hammouda, “Discotic Micellar Nematic and Lamellar Phases 

under Shear Flow”, Europhysics Lett. 28, 489-494 (1994)

B. Hammouda, J. Mang and S. Kumar, “Shear-Induced Orientational Effects in Discotic 

Liquid Crystal Micelles”, PRE 51, 6282-6285 (1995)
592

N.P. Balsara and B. Hammouda, “Shear Effects on Solvated Block Copolymer Lamellae: PSPI in DOP”, PRL 72, 360-364 (1994)

N.P. Balsara, B. Hammouda, P.K. Kesani, S.V. Jonnalagadda and G.C. Straty, “In-Situ 

Small-Angle Neutron Scattering from a Block Copolymer Solution under Shear”, 

Macromolecules 27, 2566-2573 (1994)

H. Wang, M.C. Newstein, A. Krishnan, N.P. Balsara, B.A. Garetz, B. Hammouda and R. 

Krishnamoorti, “Ordering Kinetics and Alignment of Block Copolymer Lamellae under 

Shear Flow”, Macromolecules 32, 3695-3711 (1999)

T.M. Slawecki, C.J. Glinka and B. Hammouda, “Shear-Induced Micellar Crystal Structures 

in an Aqueous Triblock Copolymer Solution”, PRE 58, R4084-R4087 (1998)

R. Krishnamoorti, A.S. Silva, M.A. Modi and B. Hammouda, “SANS Study of a Cylinder-toSphere Order-to-Order Transition in Block Copolymers”, Macromolecules 33, 3803-3809 

(2000).

R. Krishnamoorti and B. Hammouda, “Mixed Copolymer Morphologies under Shear”, 

Unpublished Results (2000).

QUESTIONS

1. What is the difference between a shear rate and a shear frequency? How about a rotation 

frequency and an angular rotation frequency? 

2. What are the various types of shear cells? Which ones were used here? 

3. What is the most common orientation of layered (lamellar) structure under Couette shear?

4. Define the three main axes used with shear geometry. 

5. Define the three (called A, B, and C) types of shear alignment. 

6. Describe the “flipping” transition. What shear alignment types does it involve?

7. How much travel is required to shear a 0.5 mm thick sample to a strain of 200 %?

ANSWERS

1. The shear rate and shear frequency are the same thing. This is the number of rotations per 

second (given in units of Hz). The rotation frequency  (units of Hz) is related to the angular 

rotation frequency  (units of rad/s) as  = 2. 

2. Shear cells include the Couette type, the Poiseuille type, the plate-plate type and the coneplate type among others. The Couette and the plate-plate were used here. 

3. Layered (lamellar) structures tend to orient parallel to the moving shear cell walls under 

Couette flow. 

4. The three main axes used in shear geometry are: the shear direction, the shear gradient 

direction and the neutral (also called the vorticity) direction. 

5. The three types of shear alignment are described in Figure 2 in the text. 
593

6. The flipping “transition” corresponds to a transition from the C alignment type to the A 

alignment type. 

7. 1 mm of travel is required in order to shear a 0.5 mm thick sample to a strain of 200 %.
594

Chapter 54 - SOLVATION IN MIXED SOLVENTS

1. SOLVATION OF MACROMOLECULES

Solvent interaction with macromolecules determines the miscibility characteristics and chain 

conformations. Macromolecules dissolve in good solvents, precipitate in poor solvents and 

cluster or aggregate in marginal solvents. Clustering characterizes water-soluble 

macromolecules. SANS measures both the clustering component at low-Q as well as the 

solvation component at high-Q. A typical SANS spectrum is shown from 4 % PEO/d-water. 

0.1

1

0.01 0.1

4% PEO/d-water, Mw= 100,000 g/mol, T = 10 oC

Scattering Intensity (cm-1

)

Scattering Variable Q (Å-1)

Low-Q clustering

High-Q solvation

Figure 1: SANS data from 4 % PEO/d-water. 

The low-Q and high-Q features are separated through a fit to the following empirical 

functional form. 

 

B

1 Q

C

Q

A

I(Q)

n m



 

 

(1)

The first term (power law) describes Porod scattering from clusters and the second term 

describes scattering from solvated polymer chains. B represents a constant incoherent 

scattering background. The solvation part is of interest here; parameter C is referred to as the 
595

solvation intensity. Non-solvation would have been a better name for this parameter since C 

decreases when solvation gets better. 

2. POLY(ETHYLENE OXIDE) IN WATER/ETHANOL SOLVENT MIXTURES

PEO/d-water solution is characterized by an LCST phase separation behavior (it phaseseparates upon heating) whereas PEO/d-ethanol solution is characterized by a UCST phase 

separation behavior (it phase separate upon cooling). Note that PEO crystallizes in d-ethanol 

at low temperatures and that the UCST behavior is observed at high temperatures. Here, the 

solvation behavior of PEO in d-water/d-ethanol mixed solvent is described. 

When the d-water fraction is increased, the solvation intensity (parameter C) is characterized 

by a minimum. It decreases then increases. This is the signature of non-ideal mixing 

behavior. Solvent molecules are not randomly mixed around the polymer chain. SANS 

cannot resolve the local cage-like structure formed by the solvent molecules, but it shows 

that the mixed solvents are more effective solvating agents than any of the individual 

solvents. 

The solvation intensity is seen to decrease with increasing temperature at low d-water 

fractions (signature of UCST behavior) but increases with increasing temperature at high dwater fractions (signature of LCST behavior). 
596

0.5

1

1.5

2

2.5

0.01 0.1 1

4% PEO, Mw = 90,000 g/mol 

T=45 oC

T=55 oC

T=65 oC

Solvation Intensity (C) 

d-Water Fraction

Figure 2: Variation of the solvation intensity (parameter C) with increasing d-water fraction 

(in d-water/d-ethanol mixtures) for three temperatures. 

Variation of 1/C with 1/T yields an estimate of the spinodal phase separation temperature 

through the familiar extrapolation procedure. With increasing d-water fraction, the spinodal 

temperature is seen to decrease, disappear at low temperatures then reappear at high 

temperatures. 
597

10

100

1000

104

0.001 0.01 0.1 1

 4% PEO, Mw = 90,000 g/mol 

 UCST 

LCST 

Spinodal Temperatures (K)

d-Water Fraction

UCST

 LCST

Figure 3: Variation of the spinodal phase separation temperature with increasing d-water 

fraction (in d-water/d-ethanol mixtures). Estimated spinodal temperatures were obtained 

through extrapolation. 

Similar SANS measurements were made for PEO in other mixed solvent pairs in which dwater is one of the solvents. The same non-ideal mixing trend was observed in d-water/dmethanol and d-water/d-ethylene glycol (Hammouda, 2006). 
598

0.5

1

1.5

2

2.5

3

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

4% PEO/d-solvents,

 Mw= 42,900 g/mol, T = 50 oC

PEO/d-water/d-methanol

PEO/d-water/d-ethanol

PEO/d-water/d-ethylene glycol

Solvation Scattering Intensity (cm -1

)

d-Alcohol Fraction

PEO/d-water

PEO/d-alcohol

Figure 4: Variation of the solvation intensity with increasing d-alcohol fraction. 

3. PEO/D-WATER/D-METHANOL TERNARY MIXTURES

PEO/d-methanol solutions are also characterized by crystalline behavior for low 

temperatures. In order to avoid PEO crystallization, SANS data at the elevated temperature 

of 50 oC are discussed here. To analyze the SANS data from PEO in mixed (d-water/dmethanol) solvents, the ternary Random Phase Approximation (RPA) model is used. This 

mean-field model is highly approximate for polymer solutions. The ternary RPA equations 

are not reproduced here. The three components are chosen as: A = PEO, B = d-methanol and 

C = d-water. 

The degrees of polymerization and specific volumes are: 

nA = 975, nB = nC = 1 

vANav = 38.94 cm3/mol, vBNav = 40.54 cm3/mol, vCNav = 18.07 cm3/mol, 

Note that Avogadro’s number (Nav = 6.02*1023 /mol) was used to multiply the specific 

volumes. The scattering lengths are:

bA = 4.1326*10-13 cm, bB = 3.9133*10-12 cm, bC = 1.9145*10-12 cm.
599

The volume fractions for the three components are defined as A, B and C. The two limiting 

cases of binary mixtures 4 % PEO/d-water and 4 % PEO/d-methanol (at T = 50 oC) are 

considered first. For the case of 100 % d-water (B = 0), the fit to the SANS data gives for the 

solvation intensity C = 1.171 cm-1. This gives a Flory-Huggins interaction parameter of 

AC/V0 = PEO/d-water/v0 = 0.0106 mol/cm3. For the case of 100 % d-methanol (C = 0), the fit 

to the SANS data gives C = 0.860 cm-1 which yields AB/v0 = PEO/d-methanol/v0 = 0.0268 

mol/cm3. The third Flory-Huggins interaction parameter BC/V0 is obtained (for example) 

from the case of 4 % PEO in 40 % d-methanol/60 % d-water solvent (A = 0.04, B = 

0.96*0.4 = 0.384, C = 0.96*0.6 = 0.576). For this case C = 0.6648 cm-1 which yields BC/V0 

= d-methanol/d-water/v0 = 0.0099 mol/cm3. 

This exercise shows that even-though it is highly approximate, the mean-field RPA approach 

provides a model for fitting SANS data from polymers in mixed-solvents. 

4. DNA SOLVATION IN MIXED SOLVENTS

Salmon DNA undergoes a helix-to-coil transition in d-water at 94 oC and in d-ethylene glycol 

at 38 oC for 4 % DNA volume fraction and with the addition of 0.1 M NaCl. In order to 

investigate solvation in mixed solvents, SANS data were taken from DNA (same conditions) 

in d-water/d-ethylene glycol mixed solvents. Variation of the solvation intensity C with 

increasing d-ethylene glycol fraction at three temperatures (25 oC, 50 oC and 75 oC) shows 

clearly the helix phase and the coil phase (Hammouda-Worcester, 2007). 

Note that the helix phase is characterized by ideal solvent mixing around the DNA phosphate 

groups whereas the coil phase is characterized by non-ideal solvent mixing just as for the 

PEO solution case. 
600

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

4 % DNA in d-water/d-ethylene glycol mixtures, 

0.1 M NaCl

25oC

50oC

75oC Solvation Intensity C (cm-1

)

d-Ehylene Glycol Fraction

DNA/d-water

DNA/d-ethylene glycol

helix coil 

helix

coil 

helix

Figure 5: Variation of the SANS solvation intensity with increasing d-ethylene glycol 

fraction for DNA in d-water/d-ethylene glycol solvent mixtures shown for three 

temperatures. 

REFERENCES

B. Hammouda, “Solvation Characteristics of a Model Water-Soluble Polymer”, J. Polymer 

Science, Polymer Phys. Ed. 44, 3195-3199 (2006)

B. Hammouda and DL Worcester, “The Denaturation Transition of DNA in Mixed 

Solvents”, Biophysical Journal 91, 2237-2242 (2006)

QUESTIONS

1. Is a PEO/d-water solution characterized by a Lower Critical Solution Temperature or an 

Upper Critical Solution Temperature? How about a PEO/d-ethanol solution? 

2. What are the two main characteristic features of SANS data from water-soluble 

macromolecules?

3. How can one tell whether two solvents are mixed randomly or not around 

macromolecules?
601

4. Why is the DNA helix phase characterized by ideal solvent mixing?

5. Most polymers tend to dissolve better in solvent mixtures than in individual solvents. 

Name an exception to this. 

ANSWERS

1. A PEO/d-water solution is characterized by a Lower Critical Solution Temperature; i.e., it 

phase separates upon heating. On the other hand, a PEO/d-ethanol solution is characterized 

by an Upper Critical Solution Temperature; i.e., it phase separates upon cooling. 

2. SANS data from water-soluble macromolecules are characterized by a low-Q clustering 

feature and a high-Q solvation feature. 

3. Two solvents are mixed randomly if the solvation intensity varies linearly between the two 

single solvent limits. Non-ideal mixing is characterized by nonlinear variation. 

4. The DNA helix phase is characterized by random (ideal) solvent mixing because only the 

phosphate groups are in contact with the solvent. Different solvents interact similarly with 

the phosphate groups. The other groups (sugars and amine bases) are not in direct contact 

with the solvent. 

5. Poly(N-isopropyl acrylamide) also referred to as PNIPAM does not dissolve in water and 

in solvents like methanol, ethanol or THF, but dissolves fine in mixtures of such solvents 

with water. 
602

Chapter 55 - CLUSTERING IN MACROMOLECULAR MEDIA

1. INTRODUCTION

The clustering phenomenon is observed in many water-soluble macromolecular systems such 

as synthetic polymers and biological macromolecules. It has also been observed in mixtures 

of polar solvents as well. Clustering shows up as major aggregation in the sample on the 

micrometer length scale which produces a strong low-Q signal with SANS. It also shows up 

as a “slow” mode in Dynamic Light Scattering (DLS). A figure shows SANS data from a 

series of macromolecular systems. Note the low-Q clustering signal and the high-Q solvation 

signal. The solvation signal is either of a decreasing Lorenzian type shape or characterized by 

a correlation peak. 

0.1

1

10

0.01 0.1

5% Poly(acrylic acid)/d-water

5% Poly(ethylene oxide)/d-water

5% DNA/d-water 

5% Poly(lysine)/d-water 

Scattered Intensity (cm-1

)

Scattering Variable Q (A -1)

High-Q 

solvation 

feature

 Low-Q 

clustering 

feature

Figure 1: SANS data from two synthetic polymers and two biological macromolecules 

dissolved in deuterated water at ambient temperature (25 oC). 

In order to understand possible cause(s) of clustering, the simplest water-soluble polymer, 

poly(ethylene oxide) or PEO is investigated in d-water. 
603

2. CHAIN END CLUSTERING 

A set of three PEO polymer chains (Mw=51,700, Mn=48,500 g/mol), were used in which 

chain ends were either both –OH, both –OCH3 or –OH at one end and –OCH3 at the other. 

Each of these three different polymers was dissolved in three different solvents, d-water 

(which is hydrophilic), d-benzene (which is hydrophobic), and d-methanol (which is 

amphiphilic) respectively. SANS measurements were taken from 4 % PEO in each one of 

these solvents and with one of the three differently end-capped polymers (Hammouda et al, 

2004). Since PEO crystallizes in d-methanol at low temperature, SANS data are presented for 

50 oC sample temperature for which crystallization has melted. 

0.1

1

0.01 0.1

4% PEO in d-benzene

Mw= 50,000 g/mol, T = 50 oC

HO-...-OH

CH3O-...-OH

CH3O-...-OCH3 Scattered Intensity (cm-1

)

Scattering Variable Q (Å-1)

Figure 2: SANS intensity for the 4 % PEO/d-benzene case at 50 oC and for the three different 

chain end groups. Note that clusters disappear completely in the CH3O-PEO-OCH3 case. 

In order to analyze the SANS data, the following empirical model for the scattering intensity 

is used:

B

1 (Q )

C

Q

A

I(Q)

n m



 

  . (1)
604

The first term describes Porod scattering from clusters and the second term describes 

scattering from dissolved polymer chains. This second term characterizes the 

polymer/solvent interactions and therefore the thermodynamics of mixing and phase 

separation. The two multiplicative factors A and C, the incoherent background B and the two 

Porod exponents n and m are used as fitting parameters. The clustering strength is defined as 

A/Qn where Q = 0.004 Å-1is taken to be the lowest measured Q value for the used instrument 

configuration. The solvation strength is defined as parameter C. 

The clustering strength (A/Qn) is plotted for the three polymers in each of the three solvents. 

High clustering strength corresponds to networks (where both chain-ends stick to other 

chains), and low clustering strength corresponds to dissolved chains (no chain-end sticking). 

The intermediate case corresponds to branched structures (only one chain-end is tethered to 

other chains). As shown in the figure, the clustering strength is high when the end-group is 

solvent-phobic (such as for –OCH3 in water or –OH in benzene) and low when the end-group 

dissolves well (such as for –OCH3 in benzene or –OH in water). The clustering strength is 

small for both types of end-groups when using the amphiphilic methanol solvent. 

-2

-1

0

1

2

3

Varying PEO End-Groups

PEO/d-Benzene

PEO/d-Water

HO-(CH CH 2 O)

2 n

-H

HO-(CH CH 2 O)

2 n

-CH3

CH O-(CH 3 CH 2 O)

2 n

-CH3

PEO/d-Methanol

Clustering Strength

4 % PEO/d-solvents

Mw= 50,000 g/mol, T = 10oC and 50 oC

Figure 3: Variation of the clustering strength (obtained for the low-Q feature in the 

scattering) with PEO varying end-groups. 

Given these SANS results, the cause of the chain-end form of clustering becomes clear. 

Clusters form when chain ends cannot stay dissolved because they are solvent phobic. 
605

Figure 4 Schematic rendering (not to scale) of the PEO/d-water clusters in the case with 

different chain ends (-OCH3 and –OH). The –OH end group stays dissolved in water 

whereas the –OCH3 end group ends up sticking to other hydrophobic (CH2CH2) groups on 

the PEO chain. PEO chains become tethered at one end. 

The anionic form of polymerization uses hydrophobic initiators which end up as chain ends. 

Hydrophobic chain ends are prevalent even for water-soluble polymers. 

Another (more important) form of clustering is discussed next. 
606

3. CLUSTERING DUE TO MONOMER STICKING INTERACTIONS

In order to assess the dominant form of clustering in macromolecular systems, a series of 

SANS measurements were performed from PEO/d-water solutions (Hammouda, 2009). The 

PEO molecular weights were Mw = 100 000 g/mol and Mn = 96 000 g/mol corresponding to a 

polydispersity index of Mw/Mn = 1.04. A set of seven samples were measured. These 

correspond to PEO volume fractions of 0.5 %, 1 %, 2 %, 3 %, 4 %, 5 %, and 10 %. The 

measured temperatures were 10 oC, 30 oC, 50 oC, 70 oC, and 90 oC. A figure shows 

representative data from the 5 % PEO sample. The low-Q clustering feature and the high-Q 

solvation features are clearly observed. 

0.1

1

10

0.01 0.1

5% Poly(ethylene oxide) in d-Water 

90 oC 

70 oC 

50 oC 

30 oC 

10 oC 

Scattering Intensity (cm-1

)

Scattering Variable Q (Å -1)

High-Q 

solvation 

feature

 Low-Q 

clustering 

feature

Figure 5: SANS from 5 % poly(ethylene oxide) in d-water for various temperatures. The 

low-Q clustering feature and the high-Q solvation feature can be clearly observed. 

Another figure compares the low-Q clustering intensity A/Qnand the high-Q solvation 

intensity C for the 5 % PEO/d-water sample in the measured temperature range. Clustering is 

seen to decrease while solvation increases with temperature. These two trends are opposite 

pointing to different driving forces for these two phenomena. 
607

0

1

2

3

4

5

6

7

0 20 40 60 8 0 100

5% poly(ethylene oxide) in d-water 

clustering

solvation

Scattering Intensity (cm-1

)

Temperature ( oC)

Figure 6: Variation of the clustering intensity A/Qnand the solvation intensity C with 

increasing temperature. A low-Q value of 0.004 Å-1is used for the clustering intensity. 

Smooth curves have been included as a visual guide. 

Increase of the solvation intensity with temperature characterizes a lower critical solution 

temperature (LCST) phase behavior in which phase separation occurs upon heating. 

Composition fluctuations increase when the phase boundary is approached leading to an 

increase in the scattering intensity (fitting parameter C). A plot of 1/C vs 1/T (where T is the 

absolute temperature) is characterized by a linear trend. Extrapolation to 1/C = 0 (solvation 

intensity “blows up”) yields an estimate for the so-called spinodal temperature (98 oC for the 

5 % PEO/d-water sample). Note that some other polymers in solution phase separate upon 

cooling and are characterized by an upper critical solution (UCST) instead. 

In order to understand the low-Q clustering part, we consider the PEO monomer to be an 

alternating copolymer of an ethylene (–CH2CH2–) block and an oxygen (–O–) block and use 

the RPA model for regularly alternating block copolymers in solution. This is a ternary 

system containing the oxygen blocks (component 1), the ethylene blocks (component 2) and 

d-water (component 3). The ternary RPA equations are summarized here. 

In the thermodynamic (Q = 0) limit, the scattering cross section (previously referred to as 

solvation intensity C) is given by:
608

22 1 2 12

2

11 2

2 C 1 S S 2 S

d

d

       





. (7)

The relevant contrast factors are:

 

2

1 3

2

1    

(8)

 

2

2 3

2

2     .

The partial structure factors for the fully interacting mixture are given by:

   

     

0

2 2 2 1

0

2 1 1 1

0

1 2 2 2

0

1 1 1 2

0

2 2 2 2

0

2 1 1 2

0

1 2 2 1

0

1 1 1 1

0

2 2 2 1

0

2 1 1 1

0

1 2

0

2 2 2 2

0

2 1 1 2

0

1 1

1 1

1 v S v S 1 v S v S v S v S v S v S

S 1 v S v S S v S v S

S

      

   



(9)

   

     

0

2 2 2 1

0

2 1 1 1

0

1 2 2 2

0

1 1 1 2

0

2 2 2 2

0

2 1 1 2

0

1 2 2 1

0

1 1 1 1

0

1 1 1 2

0

1 2 2 2

0

2 1

0

1 1 1 1

0

1 2 2 1

0

2 2

2 2

1 v S v S 1 v S v S v S v S v S v S

S 1 v S v S S v S v S

S

      

   



   

     

0

2 2 2 1

0

2 1 1 1

0

1 2 2 2

0

1 1 1 2

0

2 2 2 2

0

2 1 1 2

0

1 2 2 1

0

1 1 1 1

0

1 2 2 1

0

1 1 1 1

0

1 2

0

1 2 2 2

0

1 1 1 2

0

1 1

1 2

1 v S v S 1 v S v S v S v S v S v S

S v S v S S 1 v S v S

S

      

    



Excluded volume factors are defined in terms of the three Flory-Huggins interaction 

parameters (12, 13, and 23) as:

0

13

0

33

11

v

2

S

1

v



  (10)

0

23

0

33

22

v

2

S

1

v



 

0

23

0

13

0

12

0

33

12

S v v v

1

v











  .

The reference volume v0 is expressed in each case as the square root of the product of the 

relevant volumes. The non-interacting scattering factors for this alternating copolymer 

solution are given by:

1 1 1

0

S11  n  v

(11)

2 2 2

0

S22  n  v

1 1 1 2 2 2

0

S12  n  v n  v

3 3

0

S33   v .
609

The volume fractions are related by 1+2+3 = 1. The polymer volume fraction is P = 

1+2. The individual volume fractions are expressed as 

n v (n v n v ) 1  P 1 1 1 1  2 2

and 2

= P-1. 

The following sample information is used:

n1 = 2273 (12)

v1 = 2.35*10-23 cm3

1 = 2.47*10-6 Å

-2

n2 = 2273

v2 = 4.12*10-23 cm3

2 = -4.04*10-7 Å

-2

.

v3 = 3.03*10-23 cm3

3 = 6.35*10-6 Å

-2

.

Nonlinear least squares fits are performed using the ternary RPA model. SANS data files 

containing two columns (P, C) for each temperature are used to back out the three FloryHuggins interaction parameters 12, 13, and 2 for that temperature. In order to improve the 

fits, composition-dependent Flory-Huggins interaction parameters are considered. Linear best 

fit results of the temperature dependence are given by:

12 P

T

3014 7.14

T

27 0.46 











   











   

(13)

13 P

T

1477 3.51

T

31 0.59 











   











  

23 P

T

1425 11.25

T

50 0.54 











  











   .

These results are summarized in the following figure.
610

-0.4

-0.2

0

0.2

0.4

0.6

0.0026 0.0028 0.003 0.0032 0.0034 0.0036

Poly(ethylene oxide) in D2O, 

P

 = 0.01 



 



 



 

y = 0.64905 - 64.083x R= 0.96931 

y = 0.55323 - 45.859x R= 0.97876 

y = -0.52835 + 56.78x R= 0.81476 

Flory-Huggins Interaction Parameters

Inverse Temperature (K -1)

a

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0.0026 0.0028 0.003 0.0032 0.0034 0.0036

Poly(ethylene oxide) in D2O, 

P

 = 0.05 



 



 



 

y = 1.0991 - 121.1x R= 0.94985 

y = 0.41281 - 104.93x R= 0.99698 

y = -0.81375 + 177.32x R= 0.88695 

Flory-Huggins Interaction Parameters

Inverse Temperature (K -1)

b 

-0.5

0

0.5

1

1.5

0.0026 0.0028 0.003 0.0032 0.0034 0.0036

Poly(ethylene oxide) in D2O, 

P

 = 0.1 













y = 1.6616 - 192.36x R= 0.94066 

y = 0.23728 - 178.77x R= 0.98891 

y = -1.1705 + 328x R= 0.88141 

Flory-Huggins Interaction Parameters

Inverse Temperature (K -1)

c 

-0.4

-0.3

-0.2

-0.1

0

0.1

0.0026 0.0028 0.003 0.0032 0.0034 0.0036

Poly(ethylene oxide) in D2O 



12

, 

P

 = 0.01 



12

, 

P

 = 0.05 



12

, 

P

 = 0.1 

y = -0.52835 + 56.78x R= 0.81476 

y = -0.81375 + 177.32x R= 0.88695 

y = -1.1705 + 328x R= 0.88141 

Flory-Huggins Interaction Parameters

Inverse Temperature (K -1)

d 

Figure 7: Variation of the Flory-Huggins interaction parameters between the three 

components: 12 (oxygen/ethylene), 13 (oxygen/d-water) and 23 (ethylene/d-water). Case 

(a) corresponds to a PEO volume fraction of P = 0.01, case (b) corresponds to P = 0.05 and 

(c) case (c) corresponds to P = 0.1. Case (d) summarizes 12 for increasing PEO fraction. 

This figure shows that two of the Flory Huggins interaction parameters, 13 (oxygen/d-water) 

and 23 (ethylene/d-water) characterize an LCST phase behavior (phase separation upon 

heating) while the third one 12 (oxygen/ethylene) characterizes a UCST phase behavior 

(phase separation upon cooling). The oxygen and ethylene groups, however, cannot phase 

separate since they form the PEO monomer (are covalently bound). This produces a 

“frustrated” system where the ethylene group is forced to remain next to the backbone 

oxygen while it “prefers” to be close to another ethylene group. Ethylene groups remain 

dissolved but use any opportunity to stick to other ethylene groups on adjacent chains each 

time they get close enough. This produces physical crosslinks that form large clusters. Chain 
611

entanglements in semidilute and concentrated solutions produce favorable sites for close 

proximity of ethylene groups. The clustering process is kinetically driven. 

Figure 8: Schematic representation of dissolved PEO chains showing two clustering sites. 

REFERENCES

B. Hammouda, D. Ho, and S. Kline, “Insight into Clustering in Poly(ethylene oxide) 

Solutions”. Macromolecules 37, 6932-6937 (2004)

B. Hammouda, “The Mystery of Clustering in Macromolecular Media”, Polymer 50, 5293-

5297 (2009)
612

QUESTIONS

1. Define the various terms of the empirical SANS data analysis model given by 

B

1 (Q )

C

Q

A

I(Q)

n m



 

 

2. What could produce a strong low-Q SANS signal?

3. What characterization methods could detect clustering in macromolecular media?

4. What are polar interactions? Are these the same type of interactions that produce 

clustering?

ANSWERS

1. In the empirical SANS data analysis model given by: 

B

1 (Q )

C

Q

A

I(Q)

n m



 

 

, the first 

term A/Qn represents the low-Q clustering feature and C represents the intermediate-Q 

solvation feature. A and C are scale factors, B is an incoherent background level, n and m are 

Porod exponents and  is a correlation length (length beyond which correlations die out). 

2. A strong low-Q SANS signal could be due to crystallization, phase separation, 

inhomogeneities in the sample, aggregation or clustering. 

3. Various characterization methods could detect clustering. These include microscopy 

(optical or electron microscopy), scattering methods (SANS, SAXS, DLS) and rheology 

(clustering produces shear thickening). 

4. Polar interactions are between delocalized electrons in molecules. Polarity exists even in 

neutron molecules. Polarity is often represented by + or - on molecules. Same sign charges 

repel while opposite sign charges attract. Polar interactions are different from thermodynamic 

(hydrophobic, hydrophilic) interactions. 
613

Chapter 56 – SANS FROM POLYMERIC MATERIALS

Materials Science is one of the major SANS research areas. A host of Materials Science 

projects are performed using the SANS technique. Here, a couple of projects in which this 

author was involved are described. Focus is on macromolecular orientation in polymeric 

materials associated with various types of deformation and processing. Macromolecular 

orientation affects the mechanical properties of polymeric materials. 

1. MATERIALS AND METHOD

The projects described here use partially deuterated polystyrene to monitor macromolecular 

chain deformations associated with specific sample treatments. Deuterated atactic 

polystyrene (dPS) of Mw = 338,000 g/mol and Mn = 239,000 g/mol and hydrogenated 

(normal) polystyrene (hPS) of comparable molecular weight have been synthesized and 

characterized by GPC. A 5 % dPS weight fraction was mixed to hPS in solution. The 

homogeneous mixture was then dried. Various partially deuterated PS plates of uniform 

thickness were produced using the melt-pressing method. 

The SANS technique was used to monitor macromolecular chain orientation associated with 

various sample treatments such as hot-stretching, injection molding and shear band 

formation. The SANS instrument used (University of Missouri Research Reactor) uses a 

neutron wavelength of 4.75 Å. This wavelength is obtained using a pyrolytic graphite 

monochromator with a 90odiffraction angle. SANS data analysis consisted in a standard 

sector averaging method as well as in the elliptical averaging method which results in radii of 

gyration parallel and perpendicular to the orientation direction. The birefringence method 

was also used in some cases in order to compare chain orientations. The birefringence 

method averages over inter-chain contributions and can be measured over small spot sizes 

whereas the SANS technique is more sensitive to single-chain orientation. SANS sample 

thickness varied between 1 and 2 mm and the spot size diameter was 1 cm. Single-chain 

scattering was represented by the familiar Debye function for Gaussian polymer coils. 

2. MACROMOLECULAR ORIENTATION WITH HOT STRETCHING

The 5 % dPS/hPS homogeneous polymer mixture was compression molded into bars that 

were hot stretched in an Instron machine for stretching ratios from 1 to 4 at a temperature of 

110 oC; i.e., slightly above the softening glass-rubber transition temperature Tg for PS. Upon

reaching the desired External Draw Ratio (EDR) the samples were quenched in water at 23 

oC. 

SANS measurements were made from the hot stretched PS samples after and before hot 

stretching (Hammouda et al, 1986). Radii of gyration Rgy and Rgx along and perpendicular to 

the stretch direction after hot stretching and Rg0 before hot stretching were obtained. Isointensity contour maps displayed elliptical asymmetry. The elliptical eccentricity  is defined 
614

as the ratio  = Rgx/Rgy. The Molecular Draw Ration (MDR) is defined as MDR = Rgy/Rg0. 

Assuming that the molecular volume is conserved during deformation and that the x and z 

directions are equivalent, the MDR would also be given by 

gx Rg0 MDR  R / . 

20

40

60

80

100

0 0.02 0.04 0.06 0.08

Hot Stretched Polystyrene 

SANS Data and Fit 

Scattered Intensity 

Q* (Å-1) 

Figure 1: Typical SANS data from a hot stretched sample along the stretching direction and 

fit to the Debye function.

A table summarizes the hot stretching conditions that produced the five samples and the radii 

of gyration results obtained from SANS measurements at ambient temperature. The 

birefringence measurements have also been included. 

Table 1: Hot stretched samples and chain orientation results obtained from SANS and 

birefringence measurements

Sample Load

(lbs)

Stretch 

Rate 

(inch/min)

Temp

(

oC)

EDR Eccent. 



Rgy

(Å)

Rgx

(Å) MDR

Birefring.

*103

CDS1 12.8 10 110 2.0 .45 249 112 1.70 5.74

CDS2 15.1 10 110 1.5 .75 170 127 1.21 1.78

CDS3 11 10 110 2.75 .33 293 97 2.09 8.80

CDS4 10 5 110 4 .30 330 99 2.23 10.02

CDS5 -- 10 110 1.75 .54 214 116 1.51 4.21
615

A plot of the MDR vs EDR shows that chain deformations become nonlinear (i.e., nonaffine) for large EDR values. Note that the trends obtained from SANS and birefringence 

measurements are similar. 

1.2

1.4

1.6

1.8

2

2.2

2.4

0

2

4

6

8

10

12

1 1.5 2 2.5 3 3.5 4 4.5

Hot Stretched Polystyrene 

SANS 

Birefringence

Molecular Draw Ratio from SANS 

Birefringence (*1000) 

External Draw Ratio 

Figure 2: Plot of the MDR for varying EDR. Results from SANS and birefringence 

measurements are included. The line with a slope of 1 is also shown. 

The SANS technique has been used more recently to investigate residual orientation in 

injection molded polymer samples (Healy et al, 2006). 

3. HOT STRETCHING REVISITED

Similar SANS measurements were performed on another series of partially deuterated hot 

stretched polystyrene bars (Schroeder, 1991). The deuteration level was increased to 20 % 

for increased sensitivity. The MDR were obtained and plotted with increasing EDR. The 

SANS results were compared to rubber elasticity models. The simplest model assumes an 

affine deformation whereby 

Rgy Rg0*EDR

and 

Rgx Rg0/ EDR

where Rgy, Rgx and 

Rg0 are the radii of gyration parallel, perpendicular and before stretching respectively. The 

prediction for the direction perpendicular to stretching falls on top of the data while the 
616

prediction parallel to the stretching direction is far from the data points. Polynomial fit to the 

data parallel to the stretch direction yielded a good fit to a second order polynomial. 

0

1

2

3

4

5

0.5 1 1.5 2 2.5 3 3.5 4 4.5

Hot Stretched Polystyrene 

Parallel Data and Fit

Affine Model Prediction

Perpendicular Data and Fit

 Molecular Draw Ratio 

 External Draw Ratio 

Parallel Data Fit 

Y = M0 + M1*x + ... M8*x8 + M9*x9

M0 0.059124

M1 1.0452

M2 -0.10817

R 0.99998

Figure 3: Variation of the MDR with EDR along with predictions of a rubber elasticity 

model. 

More precise elliptical fitting of the iso-intensity contour maps showed that the eccentricity 

depends on the scattering variable Q, and therefore on the length scale in direct space. 
617

4. MACROMOLECULAR ORIENTATION WITH INJECTION MOLDING

Injection molding is the most widely used method for polymer processing. It is characterized 

by high repetition rates, low cost and high precision. During the injection, packing and 

cooling stages of the molding process, flow and thermal stresses develop resulting in 

preferential orientation of the polymer chains. Macromolecular orientation associated with 

injection molding was investigated using the SANS technique on the same 5 % dPS/hPS 

polystyrene mixture (Hammouda et al, 1986). 5 cm diameter disks and 2 cm wide by 15 cm 

long bars were injection molded using specific conditions. 

Table 2: Injection molding conditions

Cold Conditions Hot Conditions

Melt Temperature 167 oC 229 oC

Pressure 3700 psi 2500 psi

Injection Time 8 s 8 s

Cooling Time 30 s 30 s

Band Temperature 176 oC 246 oC

Mold Temperature 24 oC 65 oC

Injection molded bars and disks were 3 mm thick. This thickness was chosen in order to be 

able to mill down a few spots in order to observe chain orientation close to the skin surface or 

deep inside the bulk. The milled down spots had an optimal thickness of 1.5 mm appropriate 

for SANS. 
618

Figure 4: Injection molded disk and bar.

SANS measurements were performed and eccentricity factors () and orientation angles of 

the iso-intensity contour maps were obtained. Representative results are included here. 

Table 3: Eccentricity and orientation angle for one of the injection molded samples (cold 

molded disk). 

Eccentricity Orientation Angle

(degrees)

Spot A .63 5

Spot B .77 9

Spot C .87 24

Spot D .79 38

Spot Z .92 40

Analysis of the SANS results yielded the following conclusions: there is more 

macromolecular chain orientation close the injection gate than away from it, close to the 

sample surface than deep inside, in open parts of the sample than in corners. Moreover, the 

“cold” condition injection molding showed more orientation than the “hot” condition. 

Injection 

Gates

A

B

C

Z D

A

B

C

Z

A,B,C,D

Z

Thicknesses

1.5 mm

1.5 mm

y

x
619

5. MACROMOLECULAR ORIENTATION IN SHEAR BAND DEFORMATIONS 

Polymeric materials have two main modes of deformation: the elastic and the plastic modes. 

The plastic mode is of interest since it involves irreversible mechanisms of deformation. 

Shear bands form in notched and compressed samples. Birefringence has been the main 

technique to investigate macromolecular chain orientation within shear bands. It was found 

that shear band packet propagates at 38ofrom the compression axis and that there is 

formation of a diffuse shear zone which propagates ahead of the shear band at 45o. 

In order to investigate chain orientation within shear bands using the SANS technique, a 2 

mm thick plaque of 5 % dPS/hPS mixed polystyrene was molded. Identical rectangular plates 

(2 cm*4 cm) were cut out, sharply notched (notch was less than 1 mm deep) and compressed 

in a special device (Bubeck et al, 1986). Cases corresponding to fast and slow compression 

were considered. After compression schemes taking of order of minutes at various 

temperatures, the device was locked at the target strain, cooled to ambient temperature after 

which SANS data were taken. A new sample was used for each case. The compression was 

along the y axis making the x-y plane parallel to the scattering plane. 
620

Figure 5: Representation of the device used to create shear bands. 

Four holes for heater 

probes

Cadmium 

internal lining

Thin aluminum windows

Side spacers

Notched sample

Screws to block 

the device at the 

desired strain

Ruler to 

measure strains

Holes for 

thermocouples

Disk to center the device 

on the sample tray
621

Figure 6: Representation of the sample plane (in direct space) and of the scattering plane 

(reciprocal space). Note that the neutron area detector is 64 cm*64 cm whereas the sample is 

2 cm*4 cm. 

Results of the SANS data analysis are summarized in a table. The radii of gyration parallel 

(y’) and perpendicular (x’) to the chain orientation axis were obtained using fits to the Debye 

functional form. An overall radius of gyration was also obtained as 

R (R 2R )/ 3

2

gx'

2

g



gy' 

. The eccentricity was obtained as the ratio 

gx' Rgy'

  R / . 

Table 4: Conditions used to create shear bands and SANS macromolecular orientation 

results. 

Temperature 

(

oC)

Strain 

(%)

Strain 

Rate (s-1)

Eccentricity 



Orientation 

Angle 

(degrees)

Rgy’

(Å)

Rgx’

(Å)

Rg

(Å)

80 7.5 5*10-3.90 80 138 124 129

60 7.5 5*10-3.90 82 168 151 157

40 7.5 5*10-3.90 84 165 148 154

90 7.5 5*10-4.82 79 148 131 137

70 7.5 5*10-4.84 78 158 133 142

50 7.5 5*10-4.82 78 170 139 150

y

x

compression

axis

polystyrene 

sample

shear 

band 

axis

contour map on 

area detector

2 cm

4 cm

notch

chains 

orientation 

angle 

beam 

spot

y’

x’
622

30 7.5 5*10-4.82 84 181 148 160

90 12.5 4.6*10-4.74 80 152 112 127

80 12.5 4.6*10-4.72 78 157 113 129

70 12.5 4.6*10-4.74 78 160 118 133

50 12.5 4.6*10-4.73 77 160 117 133

40 12.5 4.6*10-4.72 78 161 116 133

20 12.5 4.6*10-4.78 76 154 120 132

Some of the conclusions from the SANS results are included here. Macromolecules 

participating in the shear band are oriented at about 81ofrom the compression axis regardless 

of the compression conditions (temperature, strain and strain rate). This is in agreement with 

the birefringence measurements showing a 38o+45o = 83oorientation angle. Only glide 

modes of deformation of elongated coils are observed in the shear bands. Diffusion modes 

are not prominent in our measurement conditions. Non-affine chain deformation was 

observed. These are plastic modes of deformation. Saturation was observed for the high 

strain rate case. 

REFERENCES

B. Hammouda, R.A. Bubeck and D.F.R. Mildner, “Macromolecular Orientation in Hot 

Stretched and Injection Molded Polystyrene”, Polymer 27, 393-397 (1986)

R.A. Bubeck, B. Hammouda and H. Kaiser, “Macromolecular Orientation Associated with 

Shear bands in Compressed Polystyrene”, Polymer Communications 27, 354-356 (1986)

J.R. Schroeder, B. Hammouda, R.A. Bubeck and J.W. Chang, “SANS from Hot Stretched 

Polystyrene Revisited”, J. Polym. Sci., Polym. Phys. Ed. 29, 1607-1612 (1991)

J Healey, G.H. Edward and R.B. Knott, “Residual Orientation in Injection-Molded Samples”, 

Physica B 385-386, 620-622 (2006)

QUESTIONS

1. How to hot stretch a polymer bar? What instrument is used for that?

2. How is injection molding performed? What are the determining factors?

3. What are shear bands? How to create them?

4. Why is the Debye function adequate for the description of polymer chains in the melt?

5. Name two methods that are sensitive to macromolecular chain orientation. 
623

ANSWERS

1. A polymer bar is hot stretched by heating slightly above the softening glass-rubber 

transition temperature (Tg), stretching to the desired external draw ratio and quenching in 

cold water. The Instron machine is used for hot stretching. 

2. Injection molding is performed by melting the polymer, injecting the melt into a mold, 

then cooling down. Temperature, pressure and injection rate are the determining factors. 

3. Shear bands are a form of plastic deformation in polymeric materials. Shear bands are 

obtained by notching and compressing a sample which is confined in order to avoid buckling. 

4. Polymer chains follow Gaussian chain statistics in the melt state; i.e., they follow a 

random walk with no excluded volume. 

5. Two methods that are sensitive to macromolecular chain conformation are optical 

birefringence and SANS. SANS is more sensitive but requires partially deuterated samples. 
624

Chapter 57 - NEUTRON SCATTERING WITH SPIN POLARIZATION

The neutron spin affects nuclear and magnetic scattering. Polarized neutrons are useful to 

investigate magnetic systems as well as in the process of separation of incoherent and 

coherent scattering. In what follows, the spin dependent elastic neutron scattering cross 

section is derived and discussed. The approach used by Moon et al (1969) has been followed. 

1. THE SPIN DEPENDENT ELASTIC NEUTRON SCATTERING CROSS SECTION

Consider an elastic scattering experiment with incident neutron spin s and nuclear spin I in 

the sample. Within the first Born approximation of quantum mechanics, the elastic neutron 

scattering cross section (per atom) is given in terms of the average of the interaction potential 

V(Q) between the initial state |i> and final state |f>. 

   



















i,f

2

i 2

2

2

P | f | V(Q) | i |

N

1

2

m

d

d (Q)



. (1)

Here m is the neutron mass. Averaging over the initial state and summing up over the final 

state have been performed. Pi is the probability of finding the neutron-nucleus system in the 

initial state. V(Q) is the Fourier transform of the Fermi pseudo-potential V(r):

   















 



i

i i i

2

(b 2Bs.I ) (r r )

N

1

m

2

V(r)

    

. (2)

Here bi is the scattering length of the ith nucleus, 

s



is the neutron spin operator. 

i

I



and 

i

r



are 

the spin operator and the position of the ith nucleus. B is related to the spin-incoherent 

scattering length bincoh by the defining relation:

I(I 1)

b

B

incoh



 . (3)

The initial and final states are separated as |i>=|sI> and |f>=|s’I’> where |s> and |s’> are the 

initial and final neutron spin states and |I> and |I’> are the initial and final nuclear spin states. 

The spin-dependent cross section is given by:

  













 

I,I'

2

I 2

ss'

V(Q) |sI |

2

m

P | I's'|

dΩ

dσ (Q)



(4)

     

I,I'

2

i

I i i i

exp(iQ.r )(b 2Bs.I ) |sI |

N

1

P | I's'|

   

.

Neglecting isotopic incoherence makes bi and 

i

I



independent of the i index. Therefore:
625

       

I,I'

2

i

I i

ss'

exp(iQ.r ) |I |

N

1

P | I' s'| (b 2Bs.I) |s

dΩ

dσ (Q)    

. (5)

In order to perform the neutron spin averages, we introduce the Pauli spin matrices 

2s

 

 

defined as:

















 

1 0

0 1

x 













 

 

i 0

0 i

y 















 

0 -1

1 0

z

. (6)

The neutron eigenstates |s> and |s’> correspond to the spin up 

|  

and spin down 

|  

states which form a complete basis set:

















 

0

1

|

and

















 

1

0

| . (7)

Noting that 

x x y y z z 2s.I  .I   I   I   I



 



and assuming that the neutron spin directions are 

along the z axis, the various averages can be worked out as:

b BIz   | b  B.I |    





(8)

b BIz   | b  B.I |    





| b B .I | B(I iI )       

x  y





| b B .I | B(I iI )       

x  y





.

One specific spin-dependent cross section is as follows:

     



I,I' i

2

I z i

exp(iQ.r ) | I |

N

1

P | I'| (b BI )

dΩ

dσ (Q) 



 (9)

        exp(iQ.r ) | I 

N

1

exp(iQ.r ) | I' I'| (b BI )

N

1

P I | (b BI )

j

I i I' j

I z i z



 

.

Using the sum rule: 

  

I'

| I' I'| 1

, this expression can be simplified to the form:

         



I i,j

2 i j

2

I z

exp iQ.(r r )

N

1

P I | (b BI ) | I

dΩ

dσ (Q)  



. (10)

Assume non polarized nuclei with 

P 1 (2I 1)

I

 

, define 

ij i j

r r r

  

  , and use the following 

property:
626

  I(I 1) | I 

3

1

I | I

3

1

I | I

2 2

z



. (11)

On the left hand side 

2

I



is a spin operator while on the right-hand side I(I+1) is its 

eigenvalue for eigenstate |I>. Note that in 

z

2

z

2 2 2

(b  BIz)  b  B I  2bBI

only the first two 

terms contribute because 

P I | I z| I 0

I

 I   .

Therefore:

   



i,j

2

2 ij incoh

2

b

3

1

exp(iQ.r )

N

1

b

dΩ

dσ 



. (12)

The other partial spin dependent cross sections can be obtained similarly as:

   



i,j

2

2 ij incoh

2

b

3

1

exp(iQ.r )

N

1

b

dΩ

dσ 



(13)

2

bincoh 3

2

dΩ

dσ





2

bincoh 3

2

dΩ

dσ





.

Use was made of the definition of B through 

2

incoh

2 B I(I 1)  b

. One can rewrite these 

results in terms of non-spin-flip (NSF), spin flip (SF), incoherent and coherent cross sections 

as follows: 

NSF coh dΩ incoh

dσ

3

1

dΩ

dσ

dΩ

dσ













 











 











(14)

SF dΩ incoh

dσ

3

2

dΩ

dσ













 











.

Note that the coherent cross section depends on Q while the incoherent one does not. 

2. SEPARATE-OUT THE COHERENT AND INCOHERENT CROSS SECTIONS

The previous results can be written in the following alternative form:

coh NSF dΩ SF

dσ

2

1

dΩ

dσ

dΩ

dσ













 











 











(15)
627

incoh dΩ SF

dσ

2

3

dΩ

dσ













 











.

These equations are used to separate out coherent from incoherent scattering using spin 

polarization and polarization analysis. 

3. SUM OF THE SPIN-DEPENDENT CROSS SECTIONS

Defining P+ and P- as the probabilities of finding the incident neutron in the up or down state, 

the sum of all spin-dependent cross sections is given by:

















  















 

 



 



dΩ

dσ

dΩ

dσ

P

dΩ

dσ

dΩ

dσ

P

dΩ

dσ

. (16)

For unpolarized neutrons (

P P 1/ 2

) or for polarized neutrons (either P+=1, P-=0 or 

P+=0, P-=1), we have:

   

i,j

2

2 ij incoh

2

exp(iQ.r ) b

N

1

b

dΩ

dσ 



. (17)

Note that the double summation of the phase factors still needs to be averaged over atomic 

positions and orientations (this average is noted 

 ... 

). The incoherent scattering length is 

referred to as bincoh instead of bi (usual notation) in order to avoid confusion with the “i” 

summation index. The cross sections  can be obtained from the differential cross sections 

by integration over all solid angles as 

 











 

dΩ

dσ

dΩ

. For instance, the scattering cross 

section integrates to the sum of the coherent and incoherent cross sections: 

4 (b b )

2

incoh

2 s

   . 

The treatment described here concerns only spin-incoherence. Contributions from isotopic 

and composition (disorder) incoherence are easily performed. 

4. NEUTRON SPIN POLARIZATION FACTOR

The final neutron spin polarization Pf is defined as:

















  















 

 



 



dΩ

dσ

dΩ

dσ

P

dΩ

dσ

dΩ

dσ

P P

dΩ

dσ

f

. (18)

Assuming incident neutrons polarized in the up direction (P+=1, P-=0) or down direction 

(P+=0, P-=1), one obtains:
628

2

incoh

ij 2 ij

2

2

incoh

ij 2 ij

2

f

exp(iQ.r ) b

N

1

b

b

3

1

exp(iQ.r )

N

1

b

P

  

  











. (19)

For non-polarized neutrons (P+ = P- = 1/2) one obtains the trivial result of Pf = 0. 

With the possibility of using a spin polarizer and spin analyzer, one could separate out the 

coherent from the spin-incoherent cross sections.

5. NUCLEAR SPIN POLARIZATION

Polarization of the nuclear spins is very difficult to achieve. It has been performed in rareearth metals at very low temperatures (in the mK range). These are magnetic systems 

whereby strong electron spin moments transfer polarization to the nuclear spins. The low 

temperatures are required to damp out phonon modes thereby reducing strong lattice 

coupling which makes nuclear polarization difficult. Spin polarization has also been achieved 

through optical pumping of electronics states in He-3. Nuclear spin polarization is then 

achieved through collisions. 

6. SUPERMIRROR NEUTRON SPIN POLARIZER

SANS polarizers are used in transmission geometry. These consist of supermirror coatings 

that transmit one spin state and reflect the other spin state. The large difference in the critical 

angles of the supermirror coating for the down spin state (c~2Ni) and the up spin state 

(c~3Ni) permits the polarization of a wide cross section neutron beam over a wide neutron 

wavelength range. Ni is the critical angle for neutron reflection for nickel. 

A specific supermirror used on the NG3 30 m SANS instrument at NIST consists of Fe/Si 

coating on 1 mm thick silicon plates aligned to form a V (1.92 oangle between the two 

plates) inside a copper-coated neutron guide. This polarizing cavity is 1.2 m long and 

polarizes a 4*5 cm2neutron beam in a wavelength range from 5 Å to 15 Å. Immediately 

following the polarizing cavity is a flat coil  spin flipper for reversing the direction of 

polarization. Permanent magnets maintain a 500 gauss vertical field to magnetize the 

supermirror coating and a 50 gauss field from the supermirror to the sample is used to 

maintain neutron polarization. The polarizer is characterized by a measured polarization of 

89 % for neutrons of wavelength  = 8 Å and  = 0.15. The flipper polarization under 

spin reversal was measured to be 96 % for the same wavelength condition. 
629

Figure 1: Schematic top view representation of the supermirror polarizing cavity. This figure 

is not to scale. 

7. HE-3 GAS NEUTRON SPIN ANALYZER

Spin polarization analysis is also performed in transmission geometry in SANS instruments. 

The wide angular divergence of the scattered beam (as much as 20 degrees) prohibits the use 

of a supermirror polarizing cavity. A large number of these supermirrors would have to be 

used, each slightly misaligned in the vertical and horizontal directions in order to cover the 

wide angular divergence. This scheme is too complex to be of practical use. Instead, 

transmission geometry polarizing “bottles” can be used for polarization analysis. These 

contain a polarized gas that transmits only one spin state from a neutron beam with broad 

angular divergence. 

A neutron spin analyzer uses polarized helium-3 (He-3) gas. He-3 has a large spindependence of the neutron absorption (capture) cross section. The absorption cross section is 

negligible for neutrons with spins parallel to the He-3 nuclear spins. Other orientations get 

absorbed. Two methods of polarizing He-3 are used. (1) The spin-exchange optical pumping 

method consists of polarizing electrons in rubidium (Rb) atoms (using circularly polarized 

laser light) that then transfer their polarization to He-3 nuclei. Rubidium is added to He-3 in 

very small amounts (ratio of Rb to He-3 is around 10-5). (2) The metastability-exchange 

optical pumping method consists of using an electric discharge and a laser light to excite and 

polarize electrons in He-3 atoms that then transfer their electronic polarization to the He-3 

nuclei. Pressures used in polarizing/analyzing bottles are a few bars of He-3 gas pressure. An 

axial holding magnetic field of 50 gauss is used to maintain polarization. If not constantly 

maintained, there is a slow exponential decay of the spin polarization inside the polarizing 

bottle over a period of a few hours to a couple of days. Reasonable neutron polarization has 

been achieved with the first generation of He-3 analyzer used on the NG3 SANS instrument 

at the NIST CNR (Gentile et al, 2000). 

4 cm

1.2 m

Fe/Si supermirror coating 

on Si substrate 

Cu-coated guide

transmitted down 

neutron spins 1.92 o

reflected up 

neutron spins

m a g n e t s

m a g n e t s
630

REFERENCES

R.M. Moon, T. Riste, and W.C. Koehler, “Polarization Analysis of Thermal Neutron 

Scattering”, Physical Review 181, 920-931 (1969)

T.R. Gentile, G.L. Jones, A.K. Thompson, J. Barker, C.J. Glinka, B. Hammouda and J.W. 

Lynn, J. Appl. Cryst. 33, 771-774 (2000)

QUESTIONS

1. What is the value of the neutron spin? Does that make the neutron a Fermion or a Boson? 

2. What is the first Born approximation? What type of neutron scattering does this 

approximation apply to?

3. What type of scattering are polarized neutrons used for?

4. What is the advantage of using polarized neutrons and polarization analysis? 

5. What particles’ spins are polarized in magnetic materials? Is it easy to polarize the nuclear 

spins?

6. Is the incoherent scattering cross section related solely to the non spin flip (NSF) or spin 

flip (SF) cross section?

ANSWERS

1. The value of the neutron spin is 1⁄2. This half integer value makes the neutron a Fermion. 

Full integer spins characterize bosons (for example photons). 

2. The first Born approximation is a simple method for solving the Schrodinger equation. It 

consists in expanding the Green’s function representation of the solution and keeping only 

the first term. This approximation applies to all types of neutron scattering where multiple 

scattering is not significant. The only notable exception is neutron reflectometry whereby the 

refraction limit (not diffraction limit) is used instead. 

3. Polarized neutrons are used mainly to investigate spin-dependent magnetic scattering. 

4. The main advantage of using polarized neutrons and polarization analysis is the sorting out 

of the various (++, --, +-, and -+) spin-dependent cross sections. These partial cross sections 

allow for instance the separation of the coherent and incoherent cross sections. 

5. The electrons’ spins are polarized in magnetic materials. It is so difficult to polarize the 

nuclear spins that it is an unpractical option. It involves very low temperatures and huge 

magnetic fields. 

6. The incoherent scattering cross section is related solely to the spin flip (SF) cross section 

as 

incoh dΩ SF

dσ

2

3

dΩ

dσ













 











. 
631

Chapter 58 – OTHER SANS TOPICS 

This chapter summarizes some other projects that do not fit into the general categories of 

polymers, complex fluids, and biology. 

EFFECT OF FILLER NANOPARTICLES ON POLYMER CHAIN 

CONFORMATIONS

Nanoparticle fillers are often added to polymeric materials to modify their physical 

properties. The effect of nanoparticle addition is, however, not entirely understood at the 

molecular level. This motivated a research effort geared toward a better understanding of the 

molecular structure of polymer/nanoparticle composites. A basic question to be answered is 

as to how polymer chain conformations are modified in the presence of nanoparticles? And 

what are the controlling parameters? For example, how does adding nanoparticles to 

polymers alter their mechanical response? Answering this question may ultimately lead to a 

molecular theory of elasticity for these polymer nanocomposites. 

A summary of the relevant results related to the modification of polymer chain 

conformations obtained so far is included in chronological order. Monte Carlo simulations 

predicted some polymer chain swelling in the presence of nanoparticles (Yuan et al, 1996). 

Such nanoparticles would act as a plasticizer or solvating agent. This finding motivated a 

series of small-angle neutron scattering (SANS) measurements from isotopic mixtures of 

polymers in the presence of nanoparticles. An investigation (Nakatani et al, 2001) focused 

on blend mixtures of dPDMS (deuterated poly dimethyl siloxane) and hPDMS in the 

presence of soft polysilicate nanoparticles. The soft nanoparticles were surface treated in 

order to enhance compatibility with the polymer and avoid phase separation. Only small 

nanoparticles (around 1 nm in radius) were used. 

They used a contrast variation series whereby the overall polymer scattering length density 

matches the nanoparticle scattering length density. This method consists in measuring 

samples with fixed total polymer fraction but varying the relative fraction of deuterated 

polymer. This approach is often used when the deuterated and unlabeled chains have the 

same degree of polymerization (matched molecular weights). With unmatched molecular 

weights, one can obtain the single-chain scattering factors for both the dPDMS and hPDMS 

polymers simultaneously but more samples need to be measured. 

This method allows the measurement of single chain dimensions (radius of gyration) 

precisely without resorting to extrapolation to infinite dilution. Keeping the nanoparticles 

contrast matched at all times, measurements corresponding to two different polymer 

molecular weights (called 100 PDMS and 1000 PDMS) were performed. The 100 PDMS 

series had matched h/d molecular weights and the 1000 PDMS had unmatched ones. 

Five different nanoparticle filler fractions were investigated. For each filler content, a series 

of dPDMS to hPDMS ratios were measured to back out the two single-chain scattering 

factors 

S (Q)

H

S

and 

S (Q)

D

S

. The polymer radius of gyration 

Rg

was then obtained in each 

case using either the Zimm approach or fitting to a Gaussian chain model. Nakatani et al 
632

found polymer chain expansion (increase in 

Rg

) upon nanoparticle loading for chain sizes 

larger than the nanoparticle size (

Rg  Rp

). Up to 60 % chain expansion was observed for 

40 % nanoparticle loading. Nanoparticles are acting as a swelling agent (solvent) in 

agreement with the Monte Carlo simulations. They also found phase separation for 

Rg  Rp

. 

0.8

1

1.2

1.4

1.6

1.8

0 0.1 0.2 0.3 0.4 0.5 0.6

hPDMS

dPDMS

Normalized Radius of Gyration Rg/Rg0

Nanoparticle Mass Fraction 

Figure 1: Radius of gyration for the dPDMS and the hPDMS chains (normalized relative to 

the unfilled case) in the 1000 PDMS blend (unmatched molecular weight) as function of 

polysilicate nanoparticle fraction. Deviation from 1 indicates chain swelling. 

Another effort at investigating this issue was also performed (Sen et al, 2007). They used 

polystyrene polymer loaded with silica particles. Three sets of isotopic mixtures of dPS and 

hPS were measured by SANS at the contrast match condition with the nanoparticles (of 14 

nm radius). Since the molecular weights of the deuterated and unlabeled polymers were 

matched, scattering was proportional to the single chain scattering factor except at low 

scattering variable Q where spurious scattering (due to voids and sample inhomogeneities) 

was observed. Krakty plots (Q2*I(Q) vs Q2) allowed the estimation of the radius of gyration 

at high-Q. It is noted that this method relies on careful subtraction of the incoherent 

background level. Sen et al found no observable effect of nanoparticle loading on the 

polymer chains radius of gyration for up to 27 % loading in the range 

Rg  Rp

. This is in 

agreement with Polymer Rotational Isomeric State Model (PRISM) calculations. They also 

found phase separation for 

Rg  Rp

. 
633

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0 5 10 15 20 25 30

90 kg/mol

250 kg/mol

620 kg/mol

Normalized Radius of Gyration Rg/Rg0

Nanoparticle Volume Fraction 

Figure 2: Radius of gyration for the PS chains (normalized relative to the unfilled case) for 

the three molecular isotopic blends measured. No effect is observed. 

Yet another investigation (Tuteja et al, 2008) used dPS/hPS blend mixtures loaded with 

crosslinked polystyrene nanoparticles. The isotopic polymer mixture was contrast matched to 

the nanoparticles. Three unmatched polymer molecular weight pairs and four different 

nanoparticles were measured. Reasonable effort was made to obtain homogeneously mixed 

samples. Guinier and Kratky plots as well as fits to the Debye function (Gaussian chain) were 

performed. They observed chain expansion by as much as 20 % for 10% nanopartricle 

loading for 

Rg  Rp

. Samples were not miscible for 

Rg  Rp

. 
634

0.95

1

1.05

1.1

1.15

1.2

1.25

0 0.02 0.04 0.06 0.08 0.1

25 kg/mol

52 kg/mol

135 kg/mol

Normalized Radius of Gyration Rg/Rg0

Nanoparticle Volume Fraction 

Figure 3: Radius of gyration for the PS chains (normalized relative to the unfilled case) for 

the three molecular isotopic blends measured. Chain swelling was observed. 

A more recent investigation (Crawford et al, 2013) revisited this topic. They used six sets of 

isotopic mixtures of matched molecular weight polystyrene and surface treated silica 

nanoparticles (13 nm diameter). All samples were prepared in the contrast match condition 

which corresponds to 0.68 volume fraction hPS and 0.32 volume fraction of dPS. Electron 

micrographs showed good dispersions. The SANS scattered intensity was due to single chain 

scattering. Kratky plots were used to obtain the radius of gyration. The isotopic polymer 

mixture and the three to four particle loadings showed no effect on the radius of gyration for 

all polymer molecular weights and for 

Rg  Rp

in agreement with Sen et al. Measurements 

were made on different SANS instrument to confirm these findings. Phase separation occurs 

for 

Rg  Rp

. 
635

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

493 kg/mol / 480 kg/mol

194 kg/mol / 231 kg/mol

637 kg/mol / 692 kg/mol

Normalized Radius of Gyration Rg/Rg0

Nanoparticle Volume Fraction 

Figure 4: Radius of gyration for the PS chains (normalized relative to the unfilled case) for 

three of the six measured isotopic blends. To avoid crowding, only three molecular weights 

are presented. These are named using the hPS/dPS molecular weights. No effect was 

observed within statistics. 

The SANS technique has proven valuable for the investigation of single chain conformations 

in polymer nanoparticle composites. Contrast matching the isotopic polymer blend mixtures 

to the nanoparticles has permitted the reliable measurement of the polymer chains radius of 

gyration. A couple of polymers and a couple of different nanoparticles have been 

investigated. These include surface treated hard silica and soft crosslinked polymer 

nanoparticles. Results obtained so far have been reasonable. Nakatani et al used soft and 

small nanoparticles and found substantial polymer chain swelling. Tuteja et al used larger 

nanoparticles consisting of crosslinked polystyrene in a contrast matched isotopic 

polystyrene blend and found smaller polymer swelling effect. Sen et al and Crawford et al 

used isotopic polystyrene blend mixture and surface treated silica particles and found no 

effect on polymer chain dimension. All investigations found no dispersion for 

Rg  Rp

but 

found different levels of polymer swelling depending on the polymer used and the softness of 

the nanoparticles. It seems that there is no simple universal behavior (either swelling or no 

swelling) and not a unique determining factor (such as the ratio 

g Rp R /

or the nanoparticle 

fraction). Nanoparticle surface treatment, polymer/nanoparticle interactions, etc. may be 

contributing factors as well. There is a need for more experiments and better focused 

simulations. Likely, there will be more to come on this topic. 
636

2. INFLUENCE OF ORGANIC LIQUIDS ON CELLULOSE NANOSTRUCTURE

Cellulose is used to extract ethanol from biomass through the fermentation process. This is 

produced by enzymatic digestion. The crystalline nature of cellulose components is an 

impediment to such digestion; enzymes act on amorphous regions, but cannot easily access 

crystalline ones. The goal of this research is to break the crystalline regions using solvent 

pretreatments (Watson et al, 2012). 

Phosphoric acid was first used to first solubilize Avicel type of cellulose, followed by 

washing with favorable organic solvents like methanol, ethanol, propanol, acetone or 

ethylene glycol. This step was compared to washing in water. FTIR measurements showed 

that washing in the organic solvents favored interchain hydrogen bonding crosslinks in the 

cellulose structure. This form of washing produced a gel-like precipitate. Washing in water, 

on the other hand, favored a network dominated by intrachain hydrogen bonding and 

produced an opaque white precipitate. 

The SANS technique was used to characterize the structure of cellulose washed in deuterated 

water of in some of these (deuterated) solvents. A simple power law fitting of the data 

yielded a Porod exponent at low-Q that pointed to a mass fractal. This exponent was used to 

characterize the tightness of the cellulose network. It was found that water was the worse 

solvent (exponent of 2.4) while ethylene glycol was the best (exponent of 2.09). 
637

Figure 5: Small angle neutron scattering from cellulose washed with selected deuterated 

liquids. The scattering by each sample was divided by the contrast factor, (Δρ2), to normalize 

for contrast variations enabling comparison. 

Ethanol biofuels have been produced industrially though fermentation processes that convert 

agricultural sugars (from corn or sugarcane) into alcohol. Biofuels can potentially be 

produced from biomass which is abundant and cheap. Partial breaking up of crystalline 

biomass cellulose was investigated here. This remains a challenge before industrial 

production of ethanol from biomass becomes viable. 

3. LARGE SCALE STRUCTURES IN THF/WATER MIXTURES

Mixtures of tetrahydrofuran (THF)/deuterated water show a strange form of clustering by 

many characterization methods such as dynamic light scattering (DLS) and SANS (Cheng et 

al, 2011). This shows up as a slow mode in DLS and a low-Q signal in SANS. It should be 

emphasized that this is just the mixture of two solvents with no macromolecules added. 

Many possible causes for this form of clustering have been discussed. One possible origin is 

discussed here. 

Commercial THF solvent is not stable over time. Butylhydroxytoluene (BHT) antioxidant 

is added to make THF stable and remain so over months to years. Despite the fact that BHT 

is added in small amount (10-5 weight fraction) to THF, it’s causing the undesired clustering 

effect. Since BHT is hydrophobic, the THF/d-water/BHT system forms micelles with the 

hydrophobic BHT forming the core part (even in trace amount), and the THF/d-water 

mixture forming he shell region of the micelles. 
638

Figure 6: SANS from 4.5 % mole fraction THF/d-water solution with trace amount of BHT 

at different temperatures.

In order to make sure that trace amounts of BHT are causing the observed clustering, pure 

THF (obtained through distillation) was used instead. This distilled THF is not stable over 

long time periods but is stable enough over a couple of days (long enough to make 

measurements). Using this pure THF removed the clustering. 

4. MESOSCALE INHOMOGENEITIES IN MIXTURES OF SMALL AMPHIPILLIC 

MOLECULES AND WATER

Small amphiphilic molecules (called hydrotropes) are too small to form micelles in aqueous 

solutions. However, their amphiphilic character forms inhomogeneous clusters (Subramanian 

et al, 2013). For instance, tertiary butyl alcohol (TBA) is one such hydrotrope. DLS, SANS 

and molecular dynamics (MD) simulation were used to investigate these clusters. 
639

Figure 7: SANS data from TBA–heavy water solutions at T = 25 °C for three mixing 

fractions. Triangles: 3.5 mol % TBA. Squares: 5 mol % TBA. Circles: 7.4 mol % TBA. 

MD simulation shows that the inhomogeneous clusters are short ranged (~nm size) and short 

lived (10 to 50 picosec) and are interpreted as micelle-like (or droplet-like) structural 

fluctuations. These contain a hydrophobe-rich core (pure TBA in this case) surrounded by a 

hydrogen-bonded shell (TBA/d-water molecules) which separates the oily core from the 

aqueous bulk phase. 

5. PROBING NANOSCALE THERMAL TRANSPORT IN SURFACTANT 

SOLUTIONS

Thermal transport across micellar structures formed of surfactants in solution was 

investigated by SANS and thermal conductivity measurements. Solutions of AOT surfactant 

in water and d-octane with varying surfactant mass fractions were measured. It was found 

that a minimum in thermal conductivity correlates well with the critical micelle concentration 

(CMC) which is the onset of micelle formation (Cao et al, 2015). At low surfactant fractions, 

the interfacial region between the surfactant hydrophilic headgroups and the octane 

molecules hinders the thermal transport thus reducing the thermal conductivity (below the 

CMC). Once the micelles have formed, the hydrogen bonding between the adsorbed water 

molecules and the surfactant headgroups provide rapid thermal transport thereby increasing 

the thermal conductivity (above the CMC). 
640

Figure 8: SANS data from AOT/d-octane/water with increasing AOT fraction. 

Figure 9: Micellar radius was obtained from fits to SANS data. A jump in the micelles radius 

is observed at the CMC. 
641

Figure 10: A minimum in the thermal conductivity is observed at the CMC. 

REFERENCES

F. Cao, J. Xu, Y. He, B. Hammouda, R. Qiao and B. Yang, “Experimental Probing 

Nanoscale Thermal Transport in Surfactant Solutions”, Scientific report, Nature 5:16040 

(2015)., Advances in Mechanical Engineering, 7, 1-8 (2015).

M. Crawford, R.J. Smalley, G. Cohen, B. Hogan, B. Wood, S.K. Kumar, Y.B. Melnichenko, 

L. He, W. Guise and B. Hammouda, Phys. Rev. Lett. 110, 196001-1 to -5 (2013)

Z. Li, H. Cheng, J. Li, J. Hao, L. Zhang, B. Hammouda and C.C. Han, “Large-Scale 

Structures in Tetrahydrofuran-Water Mixture with a Trace Amount of Antioxidant 

Butylhydroxytoluene (BHT)”, J. Phys. Chem. B 23, 7887-7895 (2011)

A.I. Nakatani, W. Chen, R.G. Schmidt, G.V. Gordon and C.C. Han, Polymer 42, 3713 (2001)

S. Sen, Y.P. Xie, S.K. Kumar, H.C. Yang, D.L. Ho, L. Hall, J.B. Hooper and K.S. Schweizer, 

Phys. Rev. Lett. 98, 128302 (2007)

D. Subramanian, C.T. Boughter, J.B. Klauda, B. Hammouda and M.A. Anisimov “Mesoscale 

Inhomogeneities in Aqueous Solutions of Small Amphiphilic Molecules”, Faraday 

Discussions 167, 217-238 (2013)

A. Tuteja, P.M. Duxbury, and M.E. Mackay, Phys. Rev. Lett. 100, 077801 (2008)
642

B.J. Watson, B. Hammouda, R.M. Briber and S.W. Hutcheson, Influence of Organic Liquids 

on the Nanostructure of Precipitated Cellulose, J. of Applied Polymer Science, 127, 2620-

2627 (2012)

Q.W. Yuan, A. Kloczkowski, J.E. Mark, and M.A. Sharif, J. Polym. Sci. Polym. Phys. Ed. 

34, 1647 (1996)
643

Part L – EVEN LOWER SANS SCALES

Chapter 59. SANS Resolution with Slit Geometry

59.1 Variance of the Q Resolution

59.2 Minimum Q with Slit Geometry

59.3 Application to a Specific Case

59.4 Slit Smearing Correction

References

Questions

Answers

Chapter 60. The VSANS Technique

60.1 Multiple Circular Converging Collimation

60.2 Specific Case of Multiple Circular Converging Collimation

60.3 Multiple Slit Converging Collimation

60.4 Performance of the Various VSANS Configurations

60.5 Overkill Apertures 

60.6 Specific Case for Overkill Apertures

60.7 Scattering Angle for Multiple Converging Collimation

60.8 Discussion

References

Questions

Answers

Chapter 61. The USANS Instrument

61.1 The USANS Instrument Components

61.2 The USANS Instrument Characteristics

61.3 The Fraunhoffer Diffraction

61.4 Typical USANS Data

References

Questions

Answers
644

Chapter 59 - SANS RESOLUTION WITH SLIT GEOMETRY

Slit geometry is sometime used in order to obtain high resolution in one direction. This 

enhances the flux-on-sample but introduces drastic smearing effects in the other direction. 

The two instruments that use slit geometry are the VSANS instrument (“V” is for “very) and 

the Bonse-Hart USANS instrument (“U” is for “ultra”). The resolution function for slit 

geometry is discussed here. 

1. VARIANCE OF THE Q RESOLUTION

Recall the following result that was derived for circular aperture geometry (MildnerCarpenter, 1984):

  2

2

2

y

2

x

2

2

geo

2

Q

L

4 σ  σ





 

(1)

with:

3

2

2

2

2

1

1 2

1

2

2

1

2 2

x x x

L

L L

x

L

L

     













 

   















 

3

2

2

2

2

1

1 2

1

2

2

1

2 2

y y y

L

L L

y

L

L

     













 

   















 

slit source 

aperture slit sample

aperture

y1

L1 L2

2D area 

detector

x

y

x1

y2

y3
645

Figure 1: SANS slit geometry with rectangular source and sample apertures and 2D area 

detector cells. This figure is not to scale. Cartesian coordinates are used to characterize the 

three vertical (source, sample and detector) planes. 

In the case of slit geometry, the various averages are calculated as follows. The horizontal slit 

openings for the source and sample apertures are defined as x1 and x2 and the vertical slit 

openings are defined as y1 and y2. x3 and y3 represent the detector cell dimensions. The 

various averages can be readily calculated. 

12

Δx

2

Δx

3

1

dx

dxx

x

2

1

2

1

Δ x /2

Δ x /2

Δ x /2

Δ x /2

2

1

2

1

1

1

1

 











   









(2)

12

Δx

x

2

2

2

2

  

; 

12

Δx

x

2

3

3

2

  

12

Δy

y

2

1

1

2

  

; 

12

Δy

y

2

2

2

2

  

; 

12

Δy

y

2

3

3

2

   .

The collimation contribution for slit geometry is similar to that for circular apertures with 

slightly different terms. Note that the gravity term appears only in the vertical y direction. 

2

Qy

2

Qx

2

Q    

(3)

2

2

x

2

3

2

2

2

1

1 2

2

1

2

1

2

2

2

2

Qx 6

1

Q

12

x

12

x

L

L L

12

x

L

L

L

2

































 



















 









































 

2

2 4

2

2

2

2

2

2

y

2

3

2

2

2

1

1 2

2

1

2

1

2

2

2

2

Qy 3

2

A

L

4

6

1

Q

12

y

12

y

L

L L

12

y

L

L

L

2























 































 



















 









































 

 

2

2

2 1 2

2h

gm A  L L  L .

Here L1 and L2 are the source-to-sample and sample-to-detector distances. 

Note that only the <x2> terms are different from the pinhole geometry case. 
646

2. MINIMUM Q WITH SLIT GEOMETRY

The minimum Q achieved with slit geometry has horizontal and vertical components. The 

horizontal component is the lowest because collimation is often tightened in that direction. 

Slits are aligned in the vertical direction to avoid gravity effects. The Qmin values are similar 

to the case of pinhole geometry and are summarized here. 

2

X min

min L

2 X

Q







and 

2

Y min

min L

2 Y

Q







(4)

2

x

2

x

L

L L

2

x

L

L

X

2 3

1

1 1 2

1

2

min





 























  

0 2 Ymin Ymin 2A

2

y

2

y

L

L L

2

y

L

L

Y

2 3

1

1 1 2

1

0 2

min





 





 .

Gravity affects the vertical direction which is of no value because it is highly smeared due to 

the open collimation in that direction. 

3. APPLICATION TO A SPECIFIC CASE

Consider the following instrument configuration with slit geometry:

x1 = 0.25 cm (5)

y1 = 2.5 cm

x2 = 0.05 cm

y2 = 0.5 cm

x3 = 0.05 cm

y3 = 0.5 cm

L1 = 15 m

L2 = 15 m

 = 12 Å

λ

Δλ

= 15 %.

Therefore:

A = 0.0138 cm/Å2(6)

x

2 = 0.00625 cm2

y

2 = 0.625 cm2

.

So that:
647 2

x

2 10 Qx  7.61*10  0.0037 Q



(in units of Å-2) (7)

2

y

2 8 Qy  8.34*10  0.0037 Q



(in units of Å-2). 

Moreover, 

X -1 Qmin  0.00014 Å

(8)

Y -1 Qmin  0.0016 Å . 

In this case, the horizontal resolution is very good but the vertical one is poor. 

10-5

0.0001

0.001

0.01

0.0001 0.001 0.01 0.1

slit geometry 

vertical direction 

horizontal direction 

Standard Deviation 

Q (Å-1

)

Q (Å-1)

vertical resolution 

Qy

horizontal resolution 

Qx

Qmin

X

Qmin

Y

Figure 2: Standard deviation of the instrumental resolution for slit geometry. Resolution is 

tight in the horizontal direction and “opened up” in the vertical direction to enhance flux on 

sample. The values of Qmin are also indicated. 
648

4. SLIT SMEARING CORRECTION

When correcting data with slit smearing, the horizontal and vertical directions are decoupled 

as follows:

R(Q,Q’) = R(Q-Q’) = R(Qx-Q’x) R(Qy-Q’y) (9)















 





 

2

Qx

2

x x

2

Qx

x x

2σ

(Q Q' )

exp

2 σ

1

R(Q Q ')















 





 

2

Qy

2

y y

2

Qy

y y

2σ

(Q Q' )

exp

2 σ

1

R(Q Q ') .

The resolution integral becomes:

u n smeared

x y

x x y y

-

y

-

x

smeared dΩ

dΣ(Q' ,Q' )

dQ' dQ' R(Q Q ')R(Q Q ')

dΩ

dΣ(Q)























     











 (10)

Slits are usually very small in the horizontal direction so that R(Qx-Q’x) = (Qx-Q’x) where 

is the Dirac Delta function. In the vertical direction the resolution is sometime replaced by a 

uniform “box” function (Barker et al, 2005):

R(Qy-Q’y) = 0 for |Qy-Q’y| <

2

 Qy

or |Qy-Q’y| >

2

Qy

(11)

R(Qy-Q’y) =

ΔV

1

for 

2

 ΔQy



|Qy-Q’y| 



2

ΔQy

.

Within this “infinitely thin slit” approximation, the resolution integral becomes simpler.

u n smeared

x y

Δ Q

0

y

smeared y dΩ

dΣ(Q ,Q' )

dQ'

ΔQ

1

dΩ

dΣ(Q) y















  











(12)

u n smeared

2

y

2

x

Δ Q

0

y

y dΩ

dΣ( Q Q' )

dQ'

ΔQ

1

y

















 

 

.

We have made use of the following property of the Dirac Delta function:

 

u n smeared

x y

u n smeared

x y

x x

-

x

dΩ

dΣ(Q ,Q' )

dΩ

dΣ(Q' ,Q' )

dQ' Q Q'

 

















 











   . (13)
649

The desmearing procedure becomes a simple 1D integration. 

REFERENCES

D.F.R. Mildner, and J.M. Carpenter, ”Optimization of the Experimental Resolution for 

SAS”, J. Appl. Cryst. 17, 249-256 (1984) 

J.G. Barker, C.J. Glinka, J.J. Moyer, M.H. Kim, A.R. Drews, and M. Agamalian,” Design

and Performance of a Thermal-Neutron Double-Crystal Diffractometer for USANS at 

NIST”, J. Appl. Cryst. 38: 1004-1011 (2005).

QUESTIONS

1. What is the main difference in the variance of the resolution function between the cases 

with circular apertures and with slit geometry?

2. What are the main advantage and disadvantage of slit geometry?

ANSWERS

1. The main difference in the variance of the resolution function Q between the cases with 

circular apertures and with slit geometry is in the averaging process involved in the 

calculation of the geometry contribution; for a circular aperture of radius R1, the average is 

4

R

x

2

1

1

2

  

, whereas for a slit of width x1, it is 

2

1

1

2

2

x

3

1

x 









 

   . 

2. The advantage of slit geometry is increased flux-on-sample in the relaxed collimation 

direction. The disadvantage of the slit geometry is the drastic smearing effect. 
650

Chapter 60 - THE VSANS TECHNIQUE

Very small-angle neutron scattering (VSANS) pushes the SANS small-Q limit down by an 

order of magnitude. It consists in using tighter collimation and a higher resolution detector 

combined with the usual long flight paths and the use of a velocity selector. SANS 

collimation can be tightened in one of two ways: either through multiple circular converging 

apertures or through multiple slit converging collimation. This last option enhances flux-onsample but introduces slit smearing. VSANS falls between regular SANS and Ultra SANS 

(USANS). The three major figures of merit (minimum Q, flux on sample, and Q resolution) 

are discussed for possible VSANS configurations for the VSANS instrument at NIST. 

1. MULTIPLE CIRCULAR CONVERGING COLLIMATION

Resolution

Multiple circular (also called pinhole) converging collimation is appealing because it allows 

an improvement in the resolution without too much loss in the flux on sample. Overkill 

apertures are required in order to eliminate neutron cross collimation. 

Figure 1: Multiple circular converging collimation. Intermediate apertures are placed to 

avoid cross collimation and keep neutrons in the same aperture channel. This figure is not to 

scale. Vertical scale is of order centimeters while horizontal scale is of order meters. 

multiple

circular

source 

apertures

multiple

circular

sample

apertures

L1

r1

x

y

z
651

The main change to the variance of the Q resolution Q

2

in going from a single large circular 

aperture to multiple small circular converging apertures is to change the radii of the source 

and sample apertures from R1 and R2 (large radii) to r1 and r2 (small hole radii) respectively. 

Everything else remains the same and will not be repeated here (Mildner-Carpenter, 1984; 

Mildner et al, 2005). 

Similarly, the minimum scattering variable, Qmin for the single aperture and multiple 

apertures collimations are the same provided that the small circular aperture radii are used. 

Resolution with Focusing Lenses

The addition of focusing lenses to the multiple converging collimation geometry allows the 

opening up of the sample apertures without penalty in resolution. Modification of the Q 

resolution equations to incorporate lenses involves replacing the sample aperture term 

4

r

L

L L

2

2

1

1 2















 

by the following term 

4

r

L

L L

λ

Δλ

3

2

2

2

2

1

1 2

2















 













 where 

















is the 

wavelength spread and L1 and L2 are the source-to-sample and sample-to-detector distances. 

This corresponds to the condition where the neutron detector is located at the source image. 

Similarly, modification of the Qmin equations to incorporate lenses involves replacing the 

sample aperture term 

2

1

1 2

r

L

L L















 

by the equivalent term 

2

1

1 2

r

L

L L

2















 

















. 

Flux-on-Sample

The neutron flux on sample can be approximated by the following estimate (based on 

measurements made at the NG3 30 m SANS instrument at the NIST CNR). 

2

1

1

4 2

15

L

a

λ

Δλ

λ

30.25

exp

λ

1.507 *10 ( ) 























    n/cm2

.s. (1)

 is the neutron wavelength,  is the wavelength spread, a1 is the area of the source (small) 

aperture, and L1 is the source-to-sample distance with a1 = r1

2

. The neutron current (or rate) 

on sample is given by:

2

1

1 2

4 2

15

2

L

a a

λ

Δλ

λ

30.25

exp

λ

1.507 *10 ( ) ( )a 





















        . (2)

a2 is the area of the sample aperture (a2 = r2

2

). For n small apertures, () and () are 

multiplied by n. 
652

Neutron Trajectories Constraint

Neutrons follow parabolic trajectories due to gravity. Neutrons climb up after crossing the 

source aperture, reach an apex half-way between the source and sample apertures, and fall 

back down to make it through the sample aperture. One constraint to consider for multiple 

circular converging collimation is the fact that all the neutron paths must make it through the 

overkill apertures, especially the middle overkill aperture. After the sample aperture, 

neutrons keep on falling down till they reach the detector plane. 

Figure 2: Trajectories of successfully transmitted neutrons corresponding to different 

wavelengths. Slower neutrons climb higher between the source and sample apertures. 

Neutrons that are successfully transmitted through the source and sample apertures follow 

trajectories that depend on neutron wavelength. The maximum vertical beam spread occurs at 

mid-point between these two apertures. Neutrons follow parabolic trajectories that are 

described by the following parametric equation:

source 

aperture

sample

aperture

L1 L2

2D area 

detector

y

z

0
653

geo grav y(z)  [y(z)] [y(z)]

(3)

[y(z)] B (z zL )

1

2 2

grav

    .

with:

9 1 -2

2

2

3.073*10 cm .Å

2h

gm B

 

  . (4)

Note that the related neutron fall constant at the detector location (i.e., for z = L1+L2) is 

defined as:

2

2

2 1 2

2h

gm A  L (L  L ) . (5)

The geometry part of the variance of the Q resolution corresponds to neutron trajectories 

without gravity effect. The effect of gravity is non-negligible for long wavelengths. The top 

neutron trajectory corresponds to + whereas the bottom trajectory corresponds to -. 

Constraining all neutrons within the wavelength spread to pass through the middle overkill 

aperture is performed by constraining the vertical neutron spread. The vertical beam spread 

corresponds to:

[ y(z)] B[( ) ( ) ](z zL )

1

2 2 2  grav

          (6)

B4 (z zL )

1

2 2

 















   .

And at the midpoint between the sample and source apertures (apex point where z = L1/2), it 

is equal to:

2

1

2

[ y(L1 2)]grav B L















   . (7)

The constraint that all neutrons within the wavelength spread make it through the middle 

overkill aperture (located at z = L1/2) can be stated as:

m 1 geo 1 grav 2r  [y(L 2)] [y(L 2)] . (8)

rm is the radius of the middle overkill aperture. This constraint translates to the condition:
654















 







2

B L

2

r r

r

2

1

2

1 2

m

. (9)

This is a constraint on the size of the middle overkill aperture rm in terms of the radii of the 

source and sample apertures r1 and r2, the source-to-sample distance L1, the neutron 

wavelength  and wavelength spread . An alternative criterion (not considered here) 

could be to constrain the variance of the beam spread instead. 

2. SPECIFIC CASE OF MULTIPLE CIRCULAR CONVERGING COLLIMATION

Without Lenses

Consider the following possible VSANS instrumental conditions:

Source circular aperture (hole) radius: r1 = 0.3 cm (10)

Sample circular aperture (hole) radius: r2 = 0.15 cm

Number of apertures (holes): n = 18

Detector cell horizontal size: x3 = 0.1 cm

Detector cell vertical size: y3 = 0.1 cm

Source-to-sample distance: L1 = 20 m

Sample-to-detector distance: L2 = 20 m

Neutron wavelength:  = 8.5 Å

Wavelength spread: 

λ

Δλ

= 0.13.

This corresponds to a source aperture of 15 cm * 6 cm and a sample aperture (and a sample 

size) of 7.5 cm * 3 cm. 

Therefore: A = 0.0246 cm/Å2so that x

2 = y2 = 0.0458 cm2

and:

2

x

2 9 Qx  6.26*10  0.0028Q



(in units of Å-2) (11)

2

y

2 8 Qy 1.11*10  0.0028Q



(in units of Å-2).

For this multiple circular converging collimation configuration, 

X -1 Qmin  0.00024 Å

(12)

Y -1 Qmin  0.00041Å . 

Neutrons fall by 1.78 cm in the detector plane (z = L1+L2). At the apex position (z = L1/2), 

neutron height corresponds to y(L1/2) = 0.22 cm. This is the amount by which the middle 

overkill aperture has to be raised. Other overkill apertures are raised proportionally. 
655

It is noted that a VSANS instrument with multiple circular converging collimation operates 

at preset discrete wavelengths because each wavelength requires different height adjustments 

for the overkill apertures. Based on the constraint criterion used, the middle overkill aperture 

must have a radius of rm  0.283 cm for  = 8.5 Å. 

The neutron flux and count rate on sample are estimated for a neutron wavelength and 

wavelength spread of  = 8.5 Å and / = 0.13. Using the source and sample areas of a1 = 

0.283 cm2and a2 = 0.0707 cm2, and n = 18, one obtains:

(8.5 Å) = 31,400 n/cm2.s (13)

(8.5 Å) = 2,218 n/s. 

These numbers are for a possible VSANS configuration characterized by a Qmin which is an 

order of magnitude lower than the similar SANS configuration. 

With Lenses

When using focusing lenses, the sample aperture can be opened up. Consider now r2 = 0.5 

cm. This gives slightly lower resolution:

2

x

2 9 Qx  3.57*10  0.0028Q



(in units of Å-2) (14)

2

y

2 9 Qy  8.43*10  0.0028Q



(in units of Å-2).

X -1 Qmin  0.00022 Å

Y -1 Qmin  0.00040 Å . 

The neutron current is higher since the source aperture is larger.

(8.5 Å) = 31,400 n/cm2.s (15)

(8.5 Å) = 24,649 n/s.

Use MgF2 lenses (one stack of lenses per small aperture) and recall the following basic 

equations for MgF2 lenses:

5 2

1 2 c

1 2

2

6.25*10 Å

ρb

π

L L

L L

R

N

  



















. (16)

Consider lenses of radius of curvature R that are thin at the center (0.5 mm thickness) in 

order to keep neutron transmission high. A multiple aperture configuration with L1 = L2 = 20 

m yields a focal length of:
656

1000 cm

L L

L L

f

1 2

1 2



















 . (17)

Therefore:

2

2

625Å

R

N





. (18)

For R = 0.5 cm and  = 8.5 Å, it takes about 4.33 lenses to achieve the desired focal length. 

In order to enhance flux-on-sample, multiple-slit converging collimation is discussed next.

3. MULTIPLE SLIT CONVERGING COLLIMATION

Multiple slit converging collimation has the advantage of increasing flux on sample by 

opening up the vertical resolution while tightening the horizontal resolution. The effect of 

gravity becomes irrelevant. 

Figure 3: Multiple converging slit collimation. Intermediate apertures are placed so as to 

avoid cross collimation and keep neutrons in the same aperture channel. 

The Q resolution for multiple slit converging collimation is similar to that for multiple 

circular converging collimation but with different “geometry” contributions x

2

and y

2

. 

Also, the averages <x2>1 and <x2>2 over the beam defining apertures for slits are different 

from those for circular apertures. Recall that:

source slits

sample slits

L1

y

1

x

1
657 2

1

1

2

2

x

3

1

x 









 

  

for slit of width x1 (19)

4

R

x

2

1

1

2

  

for circular aperture of radius R1

Similarly, the Qmin value for multiple converging slit collimation in the horizontal (x-) 

direction is the same as that for multiple circular converging collimation. The same 

expression for the flux-on-sample applies for multiple circular collimation and multiple slit 

collimation. 

4. PERFORMANCE OF THE VARIOUS VSANS CONFIGURATIONS

Comparison of the performance of various VSANS configurations for circular or slit 

apertures with or without lenses is presented in a table. These predictions assume the 

following: 

Source-to-sample distance: L1 = 20 m

Sample-to-detector distance: L2 = 20 m

Neutron wavelength:  = 8.5 Å

Wavelength spread: 

λ

Δλ

= 0.13.

Source aperture of 15 cm * 6 cm 

Sample aperture (and sample size) of 7.5 cm * 3 cm. 

Table 1: Prediction of the performance of the VSANS instrument for various configurations

Circular Apertures Slit Apertures

Small 

Aperture 

Sizes (cm)

Performance

Small 

Slit 

Sizes (cm)

Performance

Without 

Lenses

r1 = 0.3

r2 = 0.15

n = 18

Qmin

X = 0.00024 Å-1

Qmin

Y = 0.00041 Å-1

(8.5 Å)=31,400 n/cm2.s

(8.5 Å) = 2,218 n/s

x1 = 0.6

y1 = 15

x2 = 0.3

y2 = 7.5

n = 3

Qmin

X = 0.00024 Å-1

Qmin

Y = 0.0057 Å-1

(8.5 Å)=1.67*105

n/cm2.s

(8.5 Å) = 3.75*105 n/s

With 

Lenses

r1 = 0.3

r2 = 0.5

n = 18

Qmin

X = 0.00022 Å-1

Qmin

Y = 0.00040 Å-1

(8.5 Å)=31,400 n/cm2.s

(8.5 Å) = 24,649 n/s

x1 = 0.6

y1 = 15

x2 = 1.0

y2 = 7.5

n = 3

Qmin

X = 0.00022 Å-1

Qmin

Y = 0.0037 Å-1

(8.5 Å)=1.67*105

n/cm2.s

(8.5 Å) = 1.25*106 n/s
658

The configuration for circular apertures with lenses yields a slightly lower Qmin

X

. This Qmin

X

is maintained when moving to slit collimation in the horizontal direction. Using lenses 

increases the neutron current. Using slits enhances the flux-on-sample. This, however, 

requires dealing with slit smearing corrections. Note that the neutron flux on sample and 

neutron current were estimated based on predictions for the NG3 guide.

The two main figures of merit (variance of the Q resolution and Qmin) are compared in a 

figure. 

10-9

10-8

10-7

10-6

10-5

0.0001 0.001 0.01 0.1

 VSANS Resolution 

x-circular-without lenses

x-circular-with lenses

x-slits- with lenses

y-slits-with lenses

Variance of the QX Resolution 

Q

2 (Å-2

)

 Q (Å-1)

Qmin

X

Qmin

Y

Qmin

X

Figure 4: Variation of the variance of the Qx resolution and Qmin for various VSANS 

configurations. The x- and y- axes are along the horizontal and vertical directions (the zdirection is along the neutron beam). 

5. OVERKILL APERTURES 

Spacing Scheme
659

Uniformly spacing the overkill apertures is not the most effective way of distributing them. 

Another scheme is discussed here. A similar scheme has been used to place disks on 

multidisk velocity selectors. 

Figure 5: Top view of the distribution scheme proposed to place the overkill apertures along 

the neutron path. The neutron source aperture is located on the left and the sample aperture is 

located on the right. 

The proposed scheme is illustrated in a figure. It consists in placing a series on m overkill 

apertures starting from the sample aperture on the right side. A line AB is drawn to the 

middle of an absorbing region. It cuts line CD at a point E. This gives the location of aperture 

m-1 a distance Dm-1 from the source aperture (left side). 

Dk

Dm=L1

A

B

E

C

D

source slit

sample slit

I J

M N

neutrons
660

Figure 6: Open slits and blocked regions on aperture k. 

Defining the sizes of the open (slit) and of the blocked parts of aperture k as Ok and Bk

respectively, one can express the blocked-to-open ratio as:

k

k

m

m

O

B

O

B

w   . (20)

The distance Dm-1 can be calculated from distance Dm based on a scaling argument but 

working backward.

2

w

1

1

O

O

2

B

O

O

NB

M E

JB

IE

AJ

AI

D

D

m

m-1

m

m

m-1

m

m-1







    (21)

K

O

O

D D

m

m-1

m1 m

Where:

2

w

1

1

K



 . (22)

Dm is the full source-to-sample distance (usually referred to as L1). Similarly, one obtains for 

the location of subsequent slits:

Ok

Bk

aperture k
661

2

m

m 2

m 2 m K

O

O

D D 





















(23)

The general formula is:

k

m

m k

m k m K

O

O

D D 





















. (24)

The position of overkill apertures follows a geometric progression. This scheme ensures that 

no unwanted open channels are left open and guarantees more than single coverage of the 

blocked channels. In practice this scheme is used for a reasonable number of overkill slits. 

The transmission factor for the apertures is given by:

















1 w

1

T

(25)

Slit Sizes

The following scaling relation applies:

m m

1 m m 1 m m 1

m

m 1

O D

O (D D ) O D

O

O       . (26)

O1 is the slit (open) size of the first (source) aperture and Om is the slit (open) size of the last 

(sample) aperture. Replacing Dm-1=Dm 













 

m

m 1

O

O

K, one obtains the following slit sizes 

relationship:

m

1 m

1

m 1

O

O O

1 K

O

O





  . (27)

Furthermore:

m

k 1 m

1

m k

O

O O

1 K

O

O





  . (28)
662

This scheme would work for multiple slit converging apertures as well. For multiple circular 

converging apertures, vertical apertures sizes would have to be corrected for the gravity 

effect using the constraint discussed previously. 

6. SPECIFIC CASE FOR OVERKILL APERTURES

Consider the following specific case for converging circular apertures without lenses. 

Slit opening on the source aperture O1 = 0.6 cm.

Blocked area between slits on the source aperture B1 = 0.6 cm.

Slit opening on the sample aperture Om = 0.3 cm.

Blocked area between slits on the sample aperture Bm = 0.3 cm.

Source-to-sample distance Dm = 20 m. 

Consider m = 9; i.e., a total of 9 apertures. 

In this case:

Blocked-to-open ratio w = 1.

Apertures transmission T = 1/2 = 50 %.

Factor K = 2/3 = 0.667

Factor 

1

O

O O

m

1 m





.

The apertures are located at the following distances from the source aperture:

D9 = 20 m (sample aperture)

D8 = 16.00 m

D7= 12.32 m

D6 = 9.15 m

D5 = 6.61 m

D4 = 4.66 m

D3 = 3.24 m 

D2 = 2.22 m 

D1 = 0 m (source aperture). 

This is a possible placement sequence for the collimation apertures. This sequence can be 

used either in the forward or backward directions starting from the neutron source aperture. 

Slit sizes in each aperture are as follows:

O9 = 0.30 cm (sample aperture slit).

O8 = 0.36 cm

O7 = 0.42 cm

O6 = 0.46 cm

O5 = 0.50 cm
663

O4 = 0.53 cm

O3 = 0.55 cm 

O2 = 0.57 cm 

O1 = 0.6 cm (source aperture slit). 

This is a possible sequence of slit sizes on the apertures. 

7. SCATTERING ANGLE FOR MULTIPLE CONVERGING COLLIMATION

An issue worth discussing is whether the scattering angle would be the same for the various 

neutron scattering rays involved in the multiple converging collimation. As shown 

graphically, all of the marked scattering angles are equal within the first order (small-angle) 

approximation. Higher order corrections may be needed when scattering angles are not small. 

Figure 7: The marked angles are all equal to the scattering angle  within the first order 

(small-angle) approximation. 

8. DISCUSSION

The VSANS instrument combines the VSANS and SANS measurement ranges chosen in 

turn. Use of single-aperture collimation and coarse detector resolution covers the standard 

SANS range. Insertion of multiple apertures and of the high resolution neutron detector 

covers the VSANS range. The use of focusing lenses tightens the neutron beam spot on the 

detector while opening up the sample aperture. The use of multiple slits helps increase the 

sample

aperture

 area 

detector

source 

aperture
664

neutron current substantially. This feature, however, works best with taller (rectangular) 

samples. 

The VSANS instrument will use a couple of discrete wavelengths (for example 6 Å and 8.5 

Å). These wavelengths must be higher than the Bragg cutoff (around 5 Å) for MgF2 used for 

focusing lenses. 

The VSANS instrument will use a “regular” area detector with resolution around 5 mm. A 

row of linear position sensitive detectors will replace the area detector by covering the larger 

angle areas. Linear “tube” detectors have the advantage of high count rate and robustness. 

Banks of such tube detectors will be placed on both sides of the main detector as well as at 

the top and bottom. This would cover a wide area. 

The VSANS option requires a high resolution neutron detector as well. The technology for 

building high resolution (1 mm or 2 mm) detectors is improving. The Millimeter-resolution 

Large Area Neutron Detectors (MILAND) project is an international focus group for 

improving such technology. Imaging plates and scintillation detectors can achieve that 

resolution. Imaging plates are sensitive to gammas and scintillation detectors can produce 

undesired gamma background. 

The lens system for multiple slits could be designed by incorporating holes in a slab cassette. 

Figure 8: Schematic representation of the vertical lenses cassette that would comprise a 

number of slits. The cassette consists essentially of a slab with equally spaced vertical holes 

that are lined up. Two half holes make up one row of lenses, two half holes and four full 

holes in-between make up five rows of lenses. 

1.0 cm

7.5 cm

5 cm

lens cassette
665

The VSANS instrument would benefit from the use of two velocity selectors, one with 

typical and the other with high resolution respectively.

Figure 9: Two velocity selectors can be used, one characterized by  ~ 0.15 and the other 

one with  ~ 0.05. 

Typical low-Q and high-Q configurations cover the following SANS Q range: 0.003 Å -1 < Q 

< 0.3 Å -1. The USANS Q range is typically 3*10-5 Å

-1 < Q < 0.005 Å -1

. A figure shows 

SANS and USANS data taken from 4 % poly(ethylene oxide) in d-ethanol. USANS requires 

high scattering cross section samples. The SANS data were acquired over a period of 30 

minutes and the USANS data were acquired over a period of 5 hours. At the very low-Q, 

USANS statistics are very good, but for Q > 0.0004 A-1, the USANS statistics become poor. 

This is where VSANS will improve data quality (counting statistics) in that region. The 

VSANS instrument will cover the Q range 3*10-4 Å

-1 < Q < 0.009 Å -1

. The NIST CNR 

VSANS instrument is at an advanced stage of construction (Barker, 2007). A VSANS 

instrument was built and another is under operation at Saclay (Brulet et al, 2008). 

vertical 

windows

Two Velocity Selectors
666

0.01

1

100

104

106

108

1010

10-5 0.0001 0.001 0.01 0.1 1

4% PEO/d-Ethanol, Mw=42,900 g/mole, T=25oC

USANS Data

SANS Data

Scattered Intensity (cm-1

)

Q (Å-1)

USANS range

SANS range

VSANS range

Figure 10: Compound plot of SANS data and USANS data from a 4 % poly(ethylene oxide) 

sample in d-ethanol. The Q range of the proposed VSANS technique is shown.

REFERENCES

D.F.R. Mildner, and J.M. Carpenter, ”Optimization of the Experimental Resolution for 

SAS”, J. Appl. Cryst. 17, 249-256 (1984).

D.F.R. Mildner, B. Hammouda, and S.R. Kline, “A Refractive Focusing Lens System for 

SANS”, J. Appl. Cryst. 38, 979-987 (2005).

J.G. Barker, “VSANS Conceptual Design Report”, NIST Center for Neutron Research 

Internal Report (2007)

A. Brulet, V. Thevenot, D. Lairez, S. Lecommandoux, W. Agut, S.P. Armes, J. Du and S. 

Desert, “Toward a New Lower Limit for the Minimum Scattering Vector on the VSANS 

Spectrometer at Laboratoire Leon Brillouin”, J. Appl. Cryst. 41, 161-166 (2008). 
667

QUESTIONS

1. What are the main components that make VSANS possible?

2. What is the main difference in the variance of the resolution function for circular and slit 

apertures?

3. What term changes in the variance of the resolution function when using focusing lenses?

4. What is the modification of the sample term in Qmin when using focusing lenses?

5. Why is the VSANS option with multiple slits characterized by so much higher neutron 

current? What is the main drawback of using slit collimation?

6. What type of detectors could achieve 1 mm to 2 mm spatial resolution? 

ANSWERS

1. The main components that make VSANS possible are: tight collimation through the use of 

multiple apertures and high resolution detector. 

2. The main difference in the variance of the resolution function is in the averaging of the 

geometry contribution, <x2>1 = R1

2

/4 for a circular aperture of radius R1 whereas <x2>1 = 

(x1/2)2/3 for a slit of width x1. 

3. The only term that changes in the variance of the resolution function is the sample aperture 

term which involves < x2+y2>2. When using focusing lenses, this term is reduced; i.e., it is 

multiplied by the term (2/3)(/)

2 which is small. 

4. The sample term in Qmin is multiplied by 2(/) when using focusing lenses. 

5. The VSANS option with multiple slits is characterized by a much higher neutron current 

because collimation is opened up in the vertical direction. The main drawback of this is slit 

smearing of the data. 

6. Imaging plates and scintillators could achieve sub-millimeter spatial resolution. These, 

however, are characterized by gamma background issues. 
668

Chapter 61 - THE USANS INSTRUMENT

In order to achieve much lower Q ranges, an entirely different instrument concept is used. 

Instead of choosing long wavelengths and long flight paths (as done in conventional SANS 

instruments), thermal wavelength neutrons and perfect single crystals are used in the USANS 

instrument. The USANS instrument uses the Bonse-Hart method for achieving very narrow 

collimation through multiple bounce monochromator (and analyzer) reflections. 

1. THE USANS INSTRUMENT COMPONENTS

Figure 1: Schematic representation of the USANS instrument at the NIST CNR. 

USANS INSTRUMENT SCHEMATIC

sapphire filter

graphite filter

thermal neutrons

pre-monochromator

beam monitor monochromator

sample

isolation

table

transmission

detector

analyzer

detectors
669

Since the USANS instrument at the NIST CNR uses thermal neutrons, it is located inside the 

confinement building (not in the guide hall). It uses the silicon (220) reflection peaked at a 

neutron wavelength of  = 2.4 Å. Neutron filters (sapphire and pyrolytic graphite) are used to 

filter out higher components of the energy spectrum. These eliminate neutrons with fractional 

wavelengths (/2, /3, etc). A pre-monochromator is used to reduce the radiation level on the 

monochromator and other components. Channel cut perfect silicon crystals are used for 

monochromator and analyzer. The three crystals (for the triple bounce) are cut from the same 

large silicon ingot. A set of five end-window counters are used as neutron detectors to stepscan the angular range (Barker et al, 2005). 

Figure 2: Triple bounce monochromator and analyzer single crystals. The three pieces are 

carved in the same single crystal silicon ingot. 

2. THE USANS INSTRUMENT CHARACTERISTICS

The USANS instrument uses triple reflections from the single crystal silicon monochromator 

and analyzer in order to achieve very narrow  and very high collimation (by reducing 

the beam angular divergence). It uses the slit geometry whereby the Q resolution is very tight 

in the horizontal direction and opened up in the vertical direction. The standard deviation of 

the Q resolution function is of order:

hor 5 1

Q 2.25*10 Å

   

(1)

ver 1

Q 0.022 Å

   .

The coarse vertical instrumental resolution allows the use of large sample sizes (5 cm 

diameter) and therefore large neutron beam currents. 

Perspective View Top View
670

The silicon (220) reflection is characterized by the  = 2.4 Å (operating) USANS 

wavelength. Silicon has very narrow mosaic spread (a few arcsec FWHM). This low 

wavelength minimizes multiple scattering so that standard (1 mm or 2 mm) sample thickness 

can be used. Low wavelength also minimizes gravity effects on the neutron trajectories. 

The reflectivity profile from one silicon piece drops down like R(Q) ~ 1/Q2. For three 

reflections in a row (triple bounce), R(Q) ~ 1/Q6. The wings of the beam profile are 

substantially depressed making the Bonse-Hart adequate for ultra small-angle measurements. 

The USANS instrument covers an ultra low-Q range: 4*10-5 Å < Q < 0.01 Å. This 

corresponds to the size range 7.8 m > d-spacing > 628 Å that overlaps with optical 

microscopy. Microscopy measurements are performed in real space whereas scattering 

measurements are performed in reciprocal space. Optical methods do not work on opaque 

samples whereas neutron scattering can measure opaque samples. 

The main drawback of using slit geometry is the introduction of slit smearing of the data. The 

slit smearing integration of the scattering cross section (scattering intensity) is as follows:



 





 

















d

d ( Q Q )

dQ

Q

1

d

d (Q)

2

y

2 Q

0

y

smeared ver

ver

. (2)

Here the vertical integration window Qver is proportional to the Q resolution in the vertical 

direction Q

ver. Integrating up to 2 or 3 standard deviations is reliable. 

Figure 3: Rectangular binning for the slit geometry and the spherical symmetry of the 

scattered neutrons. Scattered intensity is summed up over the rectangular bin. 

Qy

Q

Qver

Qhor

Q

Qy
671

Two other figures of merit are the neutron current and the signal-to-noise ratio to insure low 

(acceptable) background level. The neutron current is  = 25,000 n/sec for a vertical sample 

slit size of y2 = 5 cm. The signal-to-noise ratio is fairly low 5*105despite the fact that the 

USANS instrument at the NCNR is located inside the confinement building. 

Assuming a Guinier-type function as a simple model for the scattering cross section at lowQ:















 











3

Q R

exp

d

d (0)

d

d (Q)

2

g

2

. (3)

Here Rg is a characteristic radius of gyration. The smearing integral becomes:















  









 

















3

(Q Q )R

exp

d

d (0)

dQ

Q

1

d

d (Q)

2

g

2

y

2 Q

0

y

smeared ver

ver

. (4)

The following variable change is made 

t  QyRg 3

and the following integral is used:

 

2

dt exp exp t

0

2 

  



. (5)

Assuming that the resolution window is wide-enough, one obtains the following low-Q limit. 

smeared 2 QverRg

3

d

d (0)

d

d (0)









 















. (6)

Another simple function to model the scattering cross section is the Lorentzian function:

2 2

1 Q

1

d

d (0)

d

d (Q)

  









. (7)

Here also, making the variable change 

t  Q'

y 

and using the normalization integral:

1 t 2

1

dt

0

2











. (8)

The following result is obtained for the low-Q limit:
672

 







 















smeared d 2 Qver

d (0)

d

d (0)

. (9)

3. THE FRAUNHOFFER DIFFRACTION

Fraunhoffer diffraction appears when aperture sizes become small. A circular aperture of 

radius R yields the following fringe pattern:

2

1

QR

2J (QR)

I(Q) I(0)













 . (10)

Here J1 is the cylindrical Bessel function or order 1 and I(0) constitutes the diffracted beam 

intensity through the aperture. Fraunhoffer fringes start at the edge of the direct beam on the 

detector bank. A slit aperture of width L yields the following fringe pattern:

2

QL/ 2

sin(QL/ 2)

I(Q) I(0)













 . (11)

These fringes constitute unwanted background when their Q range becomes comparable to 

the scattering features from the sample. This is the case of the USANS instrument whereby 

sub-millimeter slits are used and multi-micron size scales are probed. 

Recall that 

2

1

QR

2J (QR)













and 

2

QL/ 2

sin(QL/ 2)













are the form factors for a scattering disk of 

radius R and of a slab of width L provided that they are oriented perpendicular to the neutron 

beam. It is to be expected that the scattering from a particle of a specific shape be the same as 

the scattering from a sample of uniform density (think solvent) but with a beam defining 

mask of the same shape as the particle. This holds provided that the mask dimension is 

comparable to the size of the particle (nanometer to fraction of a micrometer scale). 

4. TYPICAL USANS DATA

SANS and USANS data were taken from a 4 % poly(ethylene oxide) in d-ethanol. PEO 

crystallizes into lamellae that form a sponge-like structure. No Guinier region was observed 

in the SANS data, but its onset is visible in the USANS data. The scattering cross section is 

very large at low-Q, which yields good statistics. At intermediate-Q, the cross section is 

lower and the statistics are poor. It took 30 minutes to acquire the SANS data and 5 hours to 

acquire the USANS data. 
673

0.01

1

100

104

106

108

1010

10-5 0.0001 0.001 0.01 0.1 1

4% PEO/d-Ethanol, Mw=42,900 g/mole, T=25oC

USANS Data

SANS Data

Scattered Intensity (cm-1

)

Q (Å-1)

Figure 4: SANS and USANS data taken from a sample of 4 % poly(ethylene oxide) in dethanol. 

More SANS and USANS data were taken from the PEO/d-ethanol at higher polymer weight 

fractions. The Guinier region is better defined and the high-Q oscillations from the lamellar 

structure occur at lower Q values (Ho et al, 2006). The high-Q Porod exponent is m = 3.5 

which points to a mass fractal (sponge-like structure). 
674

10-4

10-2

100

102

104

106

108

1010

10-5 0.0001 0.001 0.01 0.1

20 wt%

10 wt%

4 wt%

Scattered Intensity (cm-1

)

Q (Å-1)

PEO/d-Ethanol, Mw = 90 Kg/mol, T = 23 oC 

-3.5

SANS

USANS

Figure 5: SANS and USANS data taken from crystalline PEO/d-ethanol. The incoherent 

background level has been subtracted. 

A confocal optical micrograph was taken from 4 % hPEO/h-ethanol at ambient temperature. 

The sponge-like structure can be readily observed. 
675

Figure 6: Confocal optical micrograph for a 4 % hPEO/h-ethanol sample. This picture 

represents data taken 28 m underneath the sample surface. The scale bar represents a 20 m 

length scale. 

SANS and USANS data from a crosslinked gel are described next (Kim et al, 2006). Here 

also, the USANS data showed scattering information not observed by SANS. This system 

consisted of 1 % cetyltrimethylammonium 4-vinylbenzoate (CTVB) surfactant in d-water. 

Divinyl benzene crosslinker (0.8 mol crosslinker per 1 mol surfactant) was added to form a 

polymerized micelles soft gel. A slice of the gel was then equilibrated with excess oil (octane 

or toluene). The gel turns more opaque and white as it solubilizes oil. The USANS range 

shows that there are large "droplets" of oil present that cannot be seen by SANS. Clearly, the 

gel can solubilize more toluene than octane.
676

0.0001

0.01

1

100

104

106

108

1010

10-5 0.0001 0.001 0.01 0.1 1

Crosslinked CTVB Micelles 

gel with no excess oil 

gel with excess octane 

gel with excess toluene 

Scattered Intensity (cm-1

)

Q (Å-1)

Figure 7: SANS and USANS data from CTVB wormlike micelles that were crosslinked to 

form a soft gel then swollen in oil. 

REFERENCES

J.G. Barker, C.J. Glinka, J.J. Moyer, M.H. Kim, A.R. Drews, and M. Agamalian,” Design 

and performance of a thermal-neutron double-crystal diffractometer for USANS at NIST”, J. 

Appl. Cryst. 38: 1004-1011 (2005).

D.L Ho, B. Hammouda, S.R. Kline and W-R Chen, “Unusual Phase Behavior in Mixtures of 

Poly(ethylene oxide) and Ethyl Alcohol”, J. Polym. Sci., Polym. Phys. Ed. 55, 557-564 

(2006)

T-H Kim, S-M Choi and S.R. Kline, “Polymerized Rodlike Nanoparticles with Controlled 

Surface Charge Density”, Langmuir 22, 2844-2850, (2006)

QUESTIONS

1. Does the USANS instrument use cold or thermal neutrons? 
677

2. What is the Q range for the USANS instrument? What is the corresponding d-spacing size 

range?

3. What is the main advantage of the USANS instrument? 

4. What is the main difference between conventional SANS instruments and the USANS 

instrument? 

ANSWERS

1. The USANS instrument uses thermal neutrons in order to avoid multiple scattering 

characterizing cold neutron wavelengths. 

2. The USANS instrument covers the following Q range: 4*10-5 Å < Q < 0.01 Å. This 

corresponds to 7.8 m > d-spacing > 628 Å. 

3. The USANS instrument can observe structures in the optical (micrometer) size range in 

opaque samples. Optical methods cannot probe opaque samples. 

4. Conventional SANS instruments use velocity selectors for monochromation and long 

flight paths for collimation whereas the USANS instrument uses single crystal silicon for 

monochromation and triple bounce reflections (monochromator and analyzer) for improved 

resolution. 
678

Part M – FINAL ISSUES

Chapter 62. Gallery of SANS Data Images

62.1 Sheared Multi-Layer Vesicles

62.2 The Butterfly Pattern

62.3 Packed Spheres

62.4 Multi-Phase Aluminum Texture

62.5 Kangaroo Tail Tendon 

62.6 Twinned Crystal

62.7 Correlations in a Multiphase Composite

62.8 Sheared Spherical Micelles

62.9 Peptide Orientation in Membranes

References

Questions

Answers

Chapter 63. Brief History and Future Prospect

63.1 Brief History Timeline

63.2 SANS User Statistics

63.3 SANS Publications

63.4 Future Prospect

63.5 The NCNR Expansion

63.6 Final Words

63.7 Disclaimer
679

Chapter 62 - GALLERY OF SANS DATA IMAGES

This author has collected, over the years, a gallery of SANS data images from oriented 

samples. Some of these images are included here purely for their esthetic (artistic) value. A 

reference has been included in each case. This reference does not necessarily include the 

same SANS images, but is representative. 

1. SHEARED MULTI-LAYER VESICLES

Multilayer vesicles (MLV) have an onion-skin type of structure. AOT surfactant in brine 

(i.e., salty) water solution forms MLVs. When sheared, MLVs yield characteristic SANS 

images dominated by orientation of the lamellae (Bergenholtz-Wagner, 1996). Couette shear 

is effective at orienting the lamellar structures. Oscillatory shear produces more orientation 

than simple shear. The tangential view (whereby the neutron beam is parallel to the shear 

direction) measures the shear gradient and neutral directions. 

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.10 -0.05 0.00 0.05 0.10

-0.10

-0.05

0.00

0.05

0.10

3.0

2.8

2.6

2.4

2.2

2.0

1.8

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.10 -0.05 0.00 0.05 0.10

-0.10

-0.05

0.00

0.05

0.10 2.5

2.0

1.5

1.0

0.5

0.0

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.10 -0.05 0.00 0.05 0.10

-0.10

-0.05

0.00

0.05

0.10

4.0

3.5

3.0

2.5

2.0

1.5

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.10 -0.05 0.00 0.05 0.10

-0.10

-0.05

0.00

0.05

0.10

3.0

2.5

2.0

1.5

1.0

Figure 1: AOT in brine/D2O multilayer vesicles sheared in a Couette shear cell. Top left: 

radial view under simple shear and low shear rate (0.025 rps), Top right: tangential view 

under simple shear and high shear rate (7.0 rps), Bottom left: radial view under oscillatory 
680

shear and high shear rate (15 rps), Bottom right: tangential view under oscillatory shear and 

high shear rate (15 rps). The two SANS data sets shown in the bottom were taken at high 

temperature (50 oC). 

2. THE BUTTERFLY PATTERN

SANS measurements are made in reciprocal space. When samples are oriented along the 

horizontal direction, they yield SANS patterns oriented along the vertical direction. This is 

due to the fact that reciprocal Q space and direct space form a conjugate pair. The exception 

to this is the case of the so-called “butterfly” pattern whereby orientation in direct space and 

in reciprocal space are along the same (here horizontal) direction. Cross linked polymer 

networks are characterized by a butterfly scattering pattern. Here, a SANS image from a 

crosslinked and stretched poly(dimethyl siloxane) gel is included (Mendes et al, 1996).

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.02 -0.01 0.00 0.01 0.02

0.02

0.01

0.00

-0.01

-0.02

2.7

2.6

2.5

2.4

2.3

2.2

2.1

Figure 2: Butterfly SANS pattern from stretched poly(dimethyl siloxane) gel consisting of a 

mixture of crosslinked and (deuterated) linear polymer chains. 

3. PACKED SPHERES

Highly packed silica particles in D2O solution can form a “single crystal” texture 

characterized by bright diffraction spots under gentle shear (Butera et al, 1996). The SANS

image shows 6-fold symmetry pointing to a cubic structure (body centered cubic). Four 

orders of diffraction spots are visible before the instrumental smearing becomes 

overwhelming (at high-Q). 
681 120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.008 -0.004 0.000 0.004

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Figure 3: Single crystal diffraction pattern obtained from highly packed silica particles under 

gentle shear and in D2O. 

4. MULTI-PHASE ALUMINUM TEXTURE

SANS diffraction pattern obtained from a multi-grain aluminum sample is shown. Three 

major grains can be resolved. Note the dark blue spot on the middle-left part of the image. 

This is a damaged spot on the neutron detector produced by neutron over-exposure. 
682 120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.04 -0.02 0.00 0.02 0.04

-0.04

-0.02

0.00

0.02

0.04 4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

Figure 4: SANS diffraction pattern from multi-grain aluminum. 

5. KANGAROO TAIL TENDON 

Collagen from kangaroo tail tendon is a highly oriented fiber with crystalline structure along 

the fiber. Five order reflections can be resolved. Note that the second order reflection is 

extinct. 

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.08 -0.04 0.00 0.04

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08 3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0
683

Figure 5: Scattering pattern from collagen from a kangaroo tail tendon showing the strong 

first and third reflection peaks as well as weak higher order reflection peaks; the second 

reflection peak is not allowed. The ordered structure is along the fibers and has a d-spacing 

of 667 Å.

6. TWINNED CRYSTAL

SANS from a twinned single-crystal of NdBa2Cu3O7 (high Tc superconductor) is shown at 

100 K. The oxygen content can be changed from O7 to O6. The O6 system is 

antiferromagnetically ordered, tetragonal, and insulating, while the O7 system is 

orthorhombic, and is superconducting (Tc around 90 K). The twinned crystal grew along two 

orthogonal directions. Crystal boundaries occur when two crystals inter-grow with a highly 

symmetrical interface, often with one crystal being the mirror image of the other; atoms are 

shared by the two crystals at regular intervals. The twinning was produced by the tetragonal 

to orthorhombic distortion. Scattering is mostly from nuclear (not magnetic) scattering. 

SANS data for a related system YBa2Cu3O7 have been published (Keimer et al, 1993). 

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.02 -0.01 0.00 0.01 0.02

0.02

0.01

0.00

-0.01

-0.02

5.5

5.0

4.5

4.0

3.5

3.0

2.5

Figure 6: SANS data from the high Tc superconducting NdBa2Cu3O7 cuprate at 100 K. The 

oriented structures characterizing the two crystals forming the twinned crystal are orthogonal 

(i.e, have orthorhombic symmetry) yielding the cross-like SANS patterns. 

7. CORRELATIONS IN A MULTIPHASE COMPOSITE
684

SANS data image taken from a multiphase aluminum oxide composite (Adolphs et al, 2002) 

is included. Distortion of the structure is observed. Further details are not available. 

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.01 0.00 0.01

0.01

0.00

-0.01

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

Figure 7: SANS from a multiphase aluminum oxide composite. 

8. SHEARED SPHERICAL MICELLES

Anisotropic SANS data from P85 Pluronics micelles sheared in a Couette shear cell are 

shown. The characteristic hexagonal peak pattern (six fold symmetry) points to a cubic 

structure formed by the spherical micelles for 25 % mass fraction P85 in D2O solutions. P85 

is a triblock copolymer of poly(propylene) which is hydrophobic as the middle block and 

poly(ethylene oxide) which is hydrophilic as the outside blocks (PEO-PPO-PEO). P85 

micelles are well formed at ambient temperature. Shearing helps the packing of the spherical 

micelles into a face centered cubic structure (Slawecki et al, 1998). 

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.05 0.00 0.05

-0.05

0.00

0.05

3.5

3.0

2.5

2.0

1.5

1.0

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.05 0.00 0.05

-0.05

0.00

0.05

3.5

3.0

2.5

2.0

1.5

1.0

0.5
685

Figure 8: SANS data from 25 % P85 Pluronic (PEO-PPO-PEO triblock copolymer) in D2O 

under Couette shear (5 Hz frequency) at 40 oC. The micelles form a cubic “single crystal” 

structure. Left: radial view. Right: Tangential view. 

9. PEPTIDE ORIENTATION IN MEMBRANES

Peptides that are embedded in membranes produce highly oriented structures and yield a 

good harvest of interesting SANS images. Two antibiotic peptides (alamethicin or magainin) 

were investigated extensively. These were oriented between quartz plates and embedded into 

phospholipid bilayers forming the membrane. Deuterated water fills the inter-layer space for 

enhanced neutron contrast. Peptides form inter-layer “pores” that can be clearly observed. 

Temperature and relative humidity were controlled in order to monitor hydration effects on 

the structures. Fully hydrated samples show no inter-layer correlation. Dehydrated samples 

show strong such correlation that shows up as rich anisotropy in the SANS pattern 

characteristic of “single crystal” structure. In order to sample both the in-plane and the outof-plane structure, the oriented membranes were tilted with respect to the neutron beam 

(Yang et al, 1998; Yang et al, 1999). 

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.2 -0.1 0.0 0.1

-0.2

-0.1

0.0

0.1

3.5

3.0

2.5

2.0

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.2 -0.1 0.0 0.1

-0.2

-0.1

0.0

0.1

4.5

4.0

3.5

3.0

2.5

2.0

1.5

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.2 -0.1 0.0 0.1

-0.2

-0.1

0.0

0.1

4.0

3.5

3.0

2.5

2.0

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.2 -0.1 0.0 0.1

-0.2

-0.1

0.0

0.1

4.0

3.5

3.0

2.5

2.0

1.5

Figure 9: SANS data from peptides embedded into membranes and oriented between quartz 

plates. The sample was oriented at 60 oto the neutron beam in order to observe structures 
686

both parallel and perpendicular to the membrane surface. Top left: 60 oalignment angle. Top 

right: 80 oalignment angle. Bottom left: 80 oalignment angle and different hydration level. 

Bottom right: -80 oalignment angle. All patterns were obtained at 28 oC sample temperature 

but with different hydration levels. 

These images have a number of bright spots and more smeared diffuse features. The 

interlayer spots can be easily distinguished (specular scattering) since these are the brightest. 

The other spots and diffraction features are from the peptide structure. The major elements of 

that structure can be resolved based on the various clues available. It looks like there are two 

characteristic d-spacings throughout; one from the inter-layer spacing and one from the interpeptide nearest neighbor (first coordination shell) d-spacing. This field of research has barely 

started (Yang et al, 1998; Yang et al, 1999). 

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.2 -0.1 0.0 0.1

-0.2

-0.1

0.0

0.1

4.0

3.8

3.6

3.4

3.2

3.0

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.2 -0.1 0.0 0.1

-0.2

-0.1

0.0

0.1

4.0

3.5

3.0

2.5

2.0

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.2 -0.1 0.0 0.1

-0.2

-0.1

0.0

0.1

3.5

3.0

2.5

2.0

1.5

120

100

80

60

40

20

0

0 20 40 60 80 100 120

-0.2 -0.1 0.0 0.1

-0.2

-0.1

0.0

0.1

3.5

3.0

2.5

2.0

Figure 10: More patterns obtained from peptides embedded into membranes and oriented 

between quartz slides. The main features are understood. These structures, however, have not 

all been resolved yet. 

Once the observed structures have been resolved in detail, one could think of using partially 

deuterated blocks within the peptides to nail down these structures in more detail. Nowadays, 

using deuterated amino acid sequences to synthesize specific peptides is possible. 
687

REFERENCES

T. Slawecki, C. Glinka and B. Hammouda, “Shear-Induced Micellar Crystal Structures in an 

Aqueous Triblock Copolymer Solution”, Phys Rev E 58, R4084-4086 (1998)

J Bergenholtz and N.J Wagner, “Formation of AOT/Brine Multilamellar Vesicles”, 

Langmuir 12, 3122-3126 (1996)

L. Yang, T.A. Harroun, W.T. Heller, T.M. Weiss and H.W. Huang, “Neutron Off-Plane 

Scattering of Aligned Membranes; Method of Measurement”, Biophysical Journal 75, 641-

645 (1998). 

L. Yang, T.M. Weiss, T.A. Harroun, W.T. Heller, and H.W. Huang, “Supramolecular 

Structures of Peptide Assemblies in Membranes by Neutron Off-Specular Scattering: Method 

of Analysis”, Biophysical Journal 77, 2648-2656 (1999). 

B. Keimer, F. Dogan, I.A. Aksay, R.W. Erwin, J.W. Lynn, and S. Sarikaya, “ Inclined-Field 

Structure, Morphology, and Pinning of the Vortex Lattice in Microtwinned YBa2Cu3O7”, 

Science 262, 83-86 (1993).

E. Mendes, R. Oeser, C. Hayes, F. Boue, J. Bastide, “SANS Study of Swollen Elongated 

Gels”, Macromolecules 29, 5574-5584 (1996)

R.J. Butera, M.S. Wolfe, J. Bender and N.J. Wagner, “Formation of a Highly Ordered 

Colloidal Microstructure upon Flow Cessation for High Shear Rates”, Phys. Rev. Lett. 77, 

2117-2120 (1996)

J. Adolphs, M.J. Setzer and P. Heine, “Changes in Pore Structure and Mercury Contact 

Angle of Hardened Cement Paste Depending on Relative Humidity”, Materials and 

Structures 35, 477-486 (2002).

QUESTIONS

1. When is the “butterfly” pattern obtained?

2. A diffraction pattern containing six-fold symmetry points to what possible structure? 

3. What is the difference between “single crystal” diffraction and “powder” diffraction?

4. Fiber diffraction is characterized by what type of pattern?

ANSWERS

1. The butterfly pattern is obtained when the direction of orientation of the iso-intensity 

contour plots is the same as the orientation of the anisotropy in the sample. 

2. The cubic and the hexagonal structures are characterized by six-fold diffraction patterns. 
688

3. Single crystal diffraction is characterized by spots while powder diffraction is 

characterized by diffraction rings. 

4. Fiber diffraction is characterized by spots aligned along one direction. 
689

Chapter 63 - BRIEF HISTORY AND FUTURE PROSPECT

Over the past century, we have come from the discovery of the neutron to the present stage of 

neutron sources, neutron scattering and small-angle neutron scattering (SANS) programs. 

Great progress has been made in the use of neutrons as a probe for nanoscale structures in a 

wide variety of research areas. A brief history timeline of major events leading to the present 

state is presented. Then a few kind comments about future prospects are offered. 

1. BRIEF HISTORY TIMELINE

-- 1932: Discovery of the neutron by Chadwick. 

-- 1942: First controlled nuclear reaction by Fermi’s team at the University of Chicago 

stadium (called Chicago Pile 1 or CP1). Other reactors were constructed at US National Labs 

(CP2 to CP5) over the following ten years. 

-- 1945: Sadly, first detonation and use of a nuclear weapon. 

-- 1953: Launch of the “Atoms for Peace” program by President Eisenhower. 

-- 1955: Construction of the first university-based nuclear research reactor at Penn State 

University.

-- 1950s: First neutron scattering experiments at Oak Ridge National Lab. 

-- 1960s: First power-producing nuclear reactors. 

-- 1972: First SANS instrument built at the ILL (Grenoble, France) using a cold neutron 

source. 

-- 1975: First spallation source demonstrated at the Argonne National Lab. 

-- 1980-1982: First SANS instruments built at the Oak Ridge National Lab, the National 

Institute of Standards and Technology and the University of Missouri. These instruments 

used thermal neutrons. 

-- 1985: First operating spallation source in the US at the Intense Pulsed Neutron Source at 

Argonne National Lab. 

-- 1985: First horizontal cold source (inside a beam tube) in the US at the Brookhaven 

National Lab.

-- 1998: First cold source at a spallation source in the US at the IPNS. 

-- 1990: First optimized cold neutron source (in the reflector region) in the US at NIST. 

-- 1994: Nobel Prize to Brockhouse and Shull for their pioneering work on neutron 

scattering. 

-- 2000’s: Major upgrades in most neutron scattering facilities. 

2. SANS USER STATISTICS

Following are some user statistics based on the number of SANS beamtime proposals 

submitted to the NCNR for each research category and for three proposal rounds. 
690

Proposal Rounds 10 15 20

Year 2000 2003 2006

------ ------ ------

Polymers 18 15 12

Complex Fluids, Chemistry 4 10 13

Biology 5 8 12

Materials Science 19 14 16

Condensed Matter Physics, Magnetism, Physics 14 8 4

Total number of beamtime proposals: 60 55 57

Figure 1: SANS user statistics at the NIST Center for Neutron Research for the years 1999, 

2003 and 2006. 

These are the total numbers of proposals that were allocated beamtime for three separate 

proposal rounds. Each proposal round covers a period of 7 months. This covers the full-time 

use of one 30 m SANS instrument and amounts to about 250 days per year. The numbers of 

submitted proposals were 2 to 3 times higher. Each accepted proposal gets usually between 2 

and 3 days of beamtime. 

1999 2003 2006
691

Figure 2: More recent (2016) breakdown of user proposals at NIST. 

3. SANS PUBLICATIONS

The NIST CNR has two 30 m SANS instruments in operation: one for the outside users 

program (through the proposal system) and one for the internal users program (through the 

beam time request system). During the past six years (2001 to 2007), a total of 522 papers 

were published resulting from the use of the NCNR SANS instruments. This is an average of 

70 to 80 SANS publications per year. Moreover, over the same period of six years, about 70 

PhD theses have been successfully defended by students that used the SANS technique. 

These are from various colleges mostly in the US. The SANS program is highly productive. 

A table summarizes the breakdown of SANS publications by research areas. 

Year 2001 2003 2006

------ ------ ------

Polymers 38 28 16

Complex Fluids, Chemistry 14 28 25

Biology 3 6 9

Materials Science 12 12 27

Condensed Matter Physics, Magnetism, Physics 2 2 4

Total number of SANS publications: 69 76 81

S33 - Approved Proposals

polymer science

materials science

complex fluids

biomolecular science

earth science

hard condensed matter

magnetic materials

materials chemistry
692

Figure 2: SANS publications resulting from the use of the NIST Center for Neutron Research 

for the years 2001, 2003 and 2006.

As of 2016, the same number of SANS papers are being produced by NIST users resulting 

from the use of the two 30 m SANS instruments. 

4. FUTURE PROSPECT

Future prospect for neutron scattering and its main SANS engine looks bright. In the US, the 

Spallation Neutron Source (Oak Ridge National Lab) went into operation in 2006, a 

horizontal cold source has been installed at HFIR along with a guide hall and two SANS 

instruments. The NIST Center for Neutron Research has undergone a major expansion 

including a second guide hall. Major upgrades are planned at most neutron scattering 

facilities in the world. In Europe, the ILL’s second guide hall has been fully equipped and the 

construction of the ISIS second (low-frequency) target is complete. In Asia, most neutron 

sources have undergone (or are undergoing) upgrades to acquire cold sources and guide halls. 

SANS research has traditionally been strong in the areas of polymer science and complex 

fluids. These two research areas have constituted the lion-share of the user community. Use 

of SANS in biology has been increasing steadily moving from 5 % of the beamtime 

proposals to 25 % in just 15 years at the NIST Center for Neutron Research. Biology 

research may become the primary focus of SANS research. Recent advances in the synthesis 

of peptide sequences and in the availability of deuterated amino acids are making a 

difference. Biologist are discovering the benefits of SANS. New areas of research using 

SANS include geology and investigations of porosity in rock structures, hydrogen fuel cell 

membranes, organic photovoltaic materials, etc. 

2001 2003 2006
693

5. THE NCNR EXPANSION

The NIST Center for Neutron Research has undergone a major upgrade that includes the 

construction of a new guide hall. Both guide halls look at the same optimized liquid 

hydrogen cold source. An assortment of new instruments will be constructed. These include a 

VSANS instrument that will cover the traditional SANS scattering range as well. Some 

instruments like the NG3 SANS instrument and the Neutron Spin Echo instrument have been 

moved from the old guide hall to the new guide hall. Moreover, the design and optimization 

of a liquid deuterium cold source is under way. 

Figure 3: Schematics of the old and new guide halls at the NIST Center for Neutron 

Research. The old guide hall is almost 20 years old and the new guide hall is under 

construction. Ultimately, there will be two 30 m SANS instruments, a 10 m SANS 

instrument, a 40 m VSANS instrument and a thermal neutrons Bonse-Hart USANS 

instrument. 

30 m SANS

Present Guide Hall

New Guide Hall

30 m SANS

40 m VSANS

10 m SANS

USANS

Confinement

Building
694

Figure 4: Photo of the old guide hall at NIST. The NG7 SANS instrument (red color) can be 

seen on the left side. 

Figure 5: Photo of the new guide hall at NIST. The purple color instrument of the NG3 

SANS instrument that got renamed NGB30 when it got moved to the new guide hall. 

This expansion will keep nanoscale research alive and well, and growing for the next two 

decades. 
695

6. FINAL WORDS

The SANS technique has gained maturity in many research areas. To quote my dear mentor 

Prof. Walter Stockmayer (one of the founders of polymer science) who once said: "Four 

great developments have made polymer science what it is today; these are the (1) advent of 

lasers and optical methods that followed, (2) Small-Angle Neutron Scattering, (3) the 

development of NMR, and (4) advances in computer speed for simulation purposes”. The 

SANS technique has managed to grow steadily over the past twenty-five years from a 

"follow the trends" technique to a sophisticated characterization method used for studies in 

morphology, thermodynamics, and rheology. Advances in the use of judicious sample 

environments (shear cells, magnets, pressure cells, temperature quench apparatuses, etc) have 

instilled new momentum. 

Other characterization methods are often brought to bear and complement the SANS 

technique. Some of these include Transmission Electron Microscopy (TEM), Wide-Angle XRay Scattering (WAXS), Ultra-Violet (UV) Absorption Spectroscopy, Dynamic light 

Scattering (DLS), Differential Scanning Calorimetry (DSC), densitometry, etc. 

The SANS technique has been a driving force justifying upgrades of neutron sources and 

enhancements of neutron scattering facilities. Its use has been ever growing. It has managed 

to generate new momentum through novel technological advances as well as better modeling 

capabilities. The need for higher fluxes and better resolution has kept us all in business for a 

good many years. SANS will undoubtedly outlive many careers. 

7. DISCLAIMER

Naming products or brand-names does not imply endorsement by the National Institute of 

Standards and Technology nor does it imply that the items are the best for this purpose. The 

content of the papers included in the literature review chapters is not guaranteed to be correct 

or of special significance.
696

Part N – APPENDICES

Appendix 1. - Useful Mathematical Expressions

1. Progressions

2. Useful Identity

3. Integration Trick

4. The Convolution Integral

5. Integration by Parts

6. Bessel Functions

7. The Gamma Function

8. Some Other Integral Functions

9. Laplace Transform of Trigonometric Functions

10. Fourier Transform Integrals

11. Dirac Delta Function

12. The Gaussian Distribution

13. Property of the Laplacian Operator

14. Basic Matrix Manipulations

15. Matrix Diagonalization

References

Questions

Answers

Appendix 2. – Elements of Quantum Mechanics

1. The Schrodinger Equation

2. The Fermi Golden Rule

3. The Bra-Ket Notation

4. The Heisenberg Time Evolution Operator

5. The Pauli Spin Matrices

References

Questions

Answers
697

Appendix 1 - USEFUL MATHEMATICAL EXPRESSIONS

This appendix summarizes some mathematical expressions that have been used in this book. 

1. PROGRESSIONS

Arithmetic progression:

2a (N 1)r

2

N

(a kr)

N 1

k 0

    





. (1)

First three terms:

2

N(N 1)

k

N

k 1



 



(2)

6

N(N 1)(2N 1)

k

N

k 1

2  

 



(3)

2 N

k 1

3

2

N(N 1)

k











 

 



. (4)

Geometric progression:

q 1

a(q 1)

aq

N N

k 1

k 1





 





; 

q  1

(5)

Arithmetico-geometric progression:

2

N N 1 N 1

k 1

k

(1 q)

rq(1 q )

1 q

[a (N 1)r]q

1 q

aq (a kr)q









 





  

 



; 

q  1

(6)

2. USEFUL IDENTITY

Consider the following identity:

   

   

n

k 1

n

i,j 1

F(| i j|) n 2 (n k)F(k) . (7)
698

In order to show this relation, consider an (i,j) plane for the summation variables where the 

limits are 1 and n for both axes. 

Figure 1: The (i,j) summation space. 

The summation space consists of a line i = j dividing the space into two equal square 

triangles. In order to scan one of these triangles entirely, one can choose another summation 

variable k = |i-j|. The i = j line gives the first term n and the double sum is for the two 

triangles. 

3. INTEGRATION TRICK

Consider the following integral trick that consists in switching the integration and 

differentiation steps: 

exp[ Ax]

A

2 dx(1 x)exp[ Ax] 2 dx 1

1

0

1

0

 















     

(8)

 































   















  

1

0

1

0 A

exp[ Ax]

A

dx exp[ Ax] 1

A

2 1











  











 















 

















  2 A

exp[ A] 1

A

exp[ A]

A

exp[ A] 1

2

A

exp[ A] 1

A

2 1











   



2 A

exp[ A] 1 A

2 .

This property holds as long as the variable A does not depend on x. Note that this is a way to 

derive to so-called Debye function that represents the form factor for Gaussian coils with A = 

Q

2Rg2

. 

i

j

1 n

|i-j|

i=j line

n

1
699

4. THE CONVOLUTION INTEGRAL

The convolution integral is used to smear a model function using the instrumental resolution. 

The following property has been used. 





 



 







 









d

d (Q' )

dQ' P (| Q Q' |)

d

d (| Q Q' |) dQ' P (Q' )

d

d (Q ) x

x 1D x x

x x

x 1D x

x

. (9)

This can be shown through a variable change. 

5. INTEGRATION BY PARTS

Integration by parts helps in the calculation of many integrals:

 

 

 

b

a

b

a

b

a

UdV UV VdU . (10)

Integration by parts has been used to perform the following integral with 

z N U R



and 

dV dRexp[ (z 1)R R ]    av

:

  







dRR exp[ (z 1)R R ]

av

z N

0

 

 

 

R dRR exp[ (z 1)R R ]

z 1

z N

R R exp[ (z 1)R R ]

av

z N

0

z N av av 0 av  





  







 

.

(11)

The first term is identically equal to zero. 

6. BESSEL FUNCTIONS

There are two kinds of Bessel functions. These are the cylindrical Bessel functions J0(z), 

J1(z), etc and the spherical Bessel functions j0(z), j1(z), etc. The two are related through the 

following relationship:

J (z)

2z

j (z) n n1/ 2



 . (12)

The cylindrical Bessel functions are of integer order and the spherical ones are of fractional 

order. 

One definition of the cylindrical Bessel function is included here:
700 

 

  





0

n

n d cos(n )exp[iz cos( )] i

J (z) . (13)

The zeroth order and the first order functions are simply:





 





0

0 d exp[iz cos( )] 1

J (z)

(14)





  





0

1 d cos( ) exp[iz cos( )]

i

1

J (z) .

These two functions are related by the following relation:

J (a)

a

1

dzzJ (az)

1

1

0

0  

(15)

which in its general form reads:

J (a)

a

1

dzz J (az)

1

1

0

1

 



 

with Re > -1. (16)

The spherical Bessel functions obey the following recursive relation:

j (z)

dz

d

z

1

j (z) z 0

n

n

n 











  . (17)

The first three functions are:

z

sin(z)

j (z)

0 

(18)

z

cos(z)

z

sin(z)

j (z)

1 2

 

cos(z)

z

3

sin(z)

z

1

z

3

j (z)

2 3 2

 











  .

The following limit is known:

1

z

3j (z)

Lim 1

 











for 

z  0 . (19)
701

These are all Bessel function relations used in this book. 

7. THE GAMMA FUNCTION

The gamma function is defined as:





   

0

z 1

(z) dt exp[ t]t

for Re(z)>0. (20)

A few properties of the Gamma function follow:

(z 1)  z(z)  z! z(z 1)!

(21)

  













2

1

1 1

z! refers to factorial z. The incomplete Gamma function is defined as:

(z,U) = 



u

0

dt

exp(-t) tz-1. (22)

Note that 

(z)  (z,). 

8. SOME OTHER INTEGRAL FUNCTIONS

Some other integral functions are described here. These have been used in this book.

The Sine integral function Si(x) is defined as:





x

0

u

sin(u)

Si(x) du . (23)

The Dawson integral function D(U) is defined as:

    

 

U

0

2 2 D(U) exp U dt exp t . (24)

D(U) is sometime called F(U) instead. 
702

Normalization of the Gaussian (also called Error) function:

2

dt exp( t )

0

2 

  



. (25)

Normalization of the Lorentzian function:

1 t 2

1

dt

0

2











(26)

9. LAPLACE TRANSFORM OF TRIGONOMETRIC FUNCTIONS

Consider the following integrals that were used to calculate the polydispersity averages for 

dilute solution of spheres. 

 

A(A 4B )

2B

dX exp AX sin (BX)

2 2

2

0

2



  



(27)

 

A(A 4B )

A 2B

dX exp AX cos (BX)

2 2

2 2

0

2





  



 

2 2

0 A 4B

B

dX exp AX sin(BX) cos(BX)



  



10. FOURIER TRANSFORM INTEGRALS

The Fourier transform integral is similar to the Laplace transform integral except that the 

integration limits are from 

 

to 

  . 

The following Fourier transform integral was used to work out the Teubner-Strey model:

 





















 

























0

3

d

2 r

sin rexp

2 r

d

d r exp(iQ.r)





(28)

 





















 























    



0 

1

1

2

d

2 r

sin rexp

2 r

d

drr 2 d exp(iQr ) .

The orientational average simply yields:
703

Qr

sin(Qr)

d exp(iQr ) 2

1

1

   



. (29)

The following trigonometric relation is used:











 

  









 

  









 

d

2 r

cos Qr

d

2 r

cos Qr

d

2 r

sin(Qr)sin

(30)























 

  









 

 

d

2 ir

exp iQr

d

2 ir Re exp iQr .

Re{...} is the real part of a complex function. This helps in the integration steps as follows:





























 

 



   













 

 



 

 

d

2 ir iQr r

dr exp

d

2 ir iQr r

Re dr exp

0 0

(31)

2 2 2 2

d

2

Q

1

1

d

2

Q

1

1











 

  

































 

  

















































 

  



























 

  











  



2 2

d

2

1 Q

d

2

1 Q

d

2

2Q

2 2 4 4

2

2

2

2

3

2 Q Q

d

2

2

d

2

1

8

 

















  









 

  



























 





 .

This is the Fourier transform (scattering factor) of the Teubner-Strey correlation function. 

Similarly, the following Fourier transform integral was used to calculate the form factor for a 

uniform density sphere of radius R:





























 











 



 

 3

0

2

3 R

r

16

1

R

r

4

3

1

Qr

sin(Qr)

dr4 r

4 R

3

P(QR)

(32)

3

2

3

1

QR)

cos(QR)

(QR)

sin(QR)

QR

3

QR

3j (QR)





























  











 .
704

11. DIRAC DELTA FUNCTION

Some properties of the Dirac Delta function follow:

dx'f(x')(x  x')  f(x)







(33)

dr'f(r') (r r') f(r)

    

   

.

Integral representation of the Delta function:

dt exp i t

2

1

( )  



   





(34)

 

  





  dr exp iQ.r

2

1

(Q)

3



 

.

This last equation shows that the Dirac delta function is the 3D Fourier transform of unity. 

12. THE GAUSSIAN DISTRIBUTION

The Gaussian distribution is a peaked function of the form:



















  





 

2

2

2

( )

exp

2

1

P( ) . (35)

Here 



is the average value and  is the standard deviation of the distribution. The following 

moments can be calculated through integrations:

   

(36)





















     2

2

2 2

1



























     4

4

2

2

4 2

1 6 3 .

In the calculation of the SANS resolution function, the wavelength distribution outputted by 

the velocity selector can be assumed to be triangular or Gaussian. 
705

13. PROPERTY OF THE LAPLACIAN OPERATOR

In order to solve the Schrodinger equation for a square well potential, the following property 

of the Laplacian operator was used:

(r)

r

1

r

1 2

  











  













(37)

14. BASIC MATRIX MANIPULATIONS

Consider a [3x3] square matrix 

S

with elements:























31 32 33

21 22 23

11 12 13

S S S

S S S

S S S

S . (38)

The transpose matrix is defined as:























13 23 33

12 22 32

11 21 31

T

S S S

S S S

S S S

S . (39)

Matrix 

S

can be inverted if 

Det[S]  0. Det[S]

stands for the determinant of matrix 

S

. It is 

also called 

S

or . 

3 1 3 2

2 1 2 2

1 3

3 1 3 3

2 1 2 3

1 2

3 2 3 3

2 2 2 3

1 1 S S

S S

S

S S

S S

S

S S

S S

Det[S]  S    S  

(40)

        S1 1 S2 2S3 3 S2 3S3 2 S1 2 S2 1S3 3 S2 3S3 1 S1 3 S2 1S3 2 S2 2S3 1

.

The inverse of matrix 

S

is given by:





















   

    

   







13 23 33

12 22 32

11 21 31

1 1

S

(41)

The various co-factors are defined as:
706

2 2 3 3 2 3 3 2

3 2 3 3

2 2 2 3

1 1 S S S S

S S

S S

    , etc... (42)

The following properties apply to matrix inversion:

 

1 1 1 A B A B  

  

(43)

 

1 1 1 A.B B .A  



   

1 1 1 1 A B.C C A.C B   

   .

15. MATRIX DIAGONALIZATION

Consider a symmetric matrix 

U

that can be inverted (i.e., for which 

DetU 0

). There is a 

unique transformation whereby 

U

can be written as 

1 U A A

 

in terms of a diagonal 

matrix 



. The eigenvalue matrix 

A

obeys the relationship 

UA  A

. This diagonalization

of matrix 

U

helps in the calculation of 

N U

which becomes 

N N 1 U A A

 

. The diagonal 

elements of matrix 



are the eigenvalues 1, 2, etc. Matrix 

A

is composed of two unit 

eigenvectors 





and 





forming an orthogonal basis 

A  [,]





. These obey the following 

relations 

   

 

U 1

and 

   

 

U 2

. 

For example, consider the following matrix:











 



1 s

1 s

U . (44)

The two eigenvalues are:

2

(1 s) (1 s) 4 s

2

1

    

 

(45)

2

(1 s) (1 s) 4 s

2

2

    

  .

This formalism was used in the model describing the helix-to-coil transition in DNA. 

The bare minimum knowledge of Algebra and Calculus needed in this book has been 

included in this appendix.
707

REFERENCES

M. Abromowitz and I.A. Stegun, “Handbook of Mathematical Functions”, Dover 

Publications (New York, 1972)

I.S. Gradsteyn and I.M. Ryzhik, “Table of Integrals, Series, and Products”, Academic Press 

(1980). 

QUESTIONS

1. Give an example of where the useful identity 

   

   

n

k 1

n

i,j 1

F(| i j|) n 2 (n k)F(k)

has been 

used. 

2. Where was a convolution integral needed?

3. Which one is the “cylindrical” Bessel function, jn or Jn? How are the jn and Jn related?

4. Calculate (3)? How about (n)?

5. Where was the Sine integral function needed?

6. Where was the Dawson integral needed?

7. What is the main difference between the Laplace transform and the Fourier transform 

integrals?

8. How to invert a matrix product; i.e., how to calculate 

 

1 A.B

?

ANSWERS

1. The useful identity 

   

   

n

k 1

n

i,j 1

F(| i j|) n 2 (n k)F(k)

was used, for instance, to calculate 

the form factor for a Gaussian coil (Debye function). 

2. A convolution integral was needed in order to perform the smearing of a scattering model. 

3. Jn is the cylindrical Bessel function, whereas jn is the spherical one. These are related as 

follows 

J (z)

2z

j (z) n n1/ 2



 .

4. (3) = (2+1) = 2(2) = 2(1+1) = 2*1(1) = 2. Similarly, (n) = (n-1)(n-2)...1 = (n-1)!. 

5. The Sine integral function was needed to calculate the form factor for an infinitely thin 

rod. 

6. The Dawson integral function was needed to calculate the form factor for a Gaussian ring. 

7. The limits of the Laplace transform integral are from 

0

to 

 

whereas the limit of the 

Fourier transform integral are from 

 

to 

  . 

8. A matrix product is inverted as follows: 

 

1 1 1 A.B B .A  

 . 
708

Appendix 2 - ELEMENTS OF QUANTUM MECHANICS

Quantum mechanics tools are well suited to describe neutron scattering theory. Many 

elements of quantum mechanics have been used in this book and will be summarized here. 

1. THE SCHRODINGER EQUATION

The Schrodinger equation is expressed as follows:

H  E  

The scattering system Hamiltonian H contains a kinetic energy contribution and an 

interaction potential contribution:

V

2m

H -

2

2

  



. (2)

The momentum operator is defined as:

 







p i . (3)

The eigenfunction and eigenvalue E are solutions to the integral Schrodinger equation. 

The neutron current density is given by:

    

   

* *

2m

i

J . (4)

Where 

*



is the complex conjugate of 

 .

The scattering amplitude is given by:



 













  dr'exp( iQ.r')V(r)

2

m

f( )

2

 

 



. (5)

The scattering cross section is given by:

s 2

| f( ) |

d

d ( )

 



 

. (6)

2. THE FERMI GOLDEN RULE
709

To within the first order perturbation theory, the elastic scattering cross section is given by:

2

2

s

dr'exp( iQ.r')V(r') | i

2

m

s|

d

d ( )



  













 



   





. (7)

The quasielastic/inelastic double differential scattering cross section is more general:

V(Q) | i (E )

2

m

s|

k

k

dEdΩ

d σ

s i

2

2

i

s

2

    E  E













 



. (8)

Here |s> and |i> are the final and initial states. 

3. THE BRA-KET NOTATION

The <bra|ket> is a useful and more compact notation. Consider the following definitions:

r | k exp(ik .r)

i i



 



 

(9)

r | (r)

 

   

 r | V  V(r)

r | G | r' G(r r')

   

  

r | r' (r r')

   

    .

Define the following closure relations:



| r' dr' r'|1

  

(10)



| k  dk  k |1

  

.

The scattering amplitude is expressed as:

  













 

2 s ki k | V(Q) |

2

m

f( )

 



(11)

The scattering cross section is therefore given in terms of the transition probability 

 s| V | i 

as:

V(Q) | i (E )

2

m

s|

k

k

dEd

d (Q)

s i

2

2

i

s s

2

    E  E













 







. (12)
710

This result applies to elastic as well as quasielastic/inelastic scattering. It can also describe 

magnetic scattering provided that the spin coupling term is included in the interaction 

potential. 

4. THE HEISENBERG TIME EVOLUTION OPERATOR

The Heisenberg equation for a time-dependent operator 

r(t)



is stated as follows:

H,r(t)

t

r(t)

i





 





. (13)

Here 

H,r(t)



is the commentator of the Hamiltonian H and the position operator 

r(t)



and is 

defined as:

H,r(t) H.r(t) r(t).H

  

  . (14)

The Heisenberg time evolution operator helps represent the time dependence of an operator 

as follows:























 









 iHt

r(0)exp

iHt

r(t) exp . (15)

Note that two operators A and B are said to “commute” when: 

A,B  A.BB.A  0

. Most 

Quantum Mechanics operators do not commute. 

5. THE PAULI SPIN MATRICES

The Pauli spin matrices for the neutron spin 

2

1

s







are defined as 

2s

 

 

with:

















 

1 0

0 1

x 













 

 

i 0

0 i

y 















 

0 -1

1 0

z

. (16)

Some of their properties follow:



i  i

, Det(i

)  1, 1

2

i 

for any i = x,y,z (17)

  y z x  ,  2i ,   z x y  ,  2i ,   x y z  ,  2i

i xyz , 

x y z    i .
711

The spin up |> and spin down |> vector states are defined as:

















 

0

1

|

and

















 

1

0

| . (18)

These form an orthogonal basis such that 

  |     |  1

and 

  |    0 . 

They also obey the following relations: 

z|   |  

(19)

z|    |  

x|   |  

y|    i|  

, etc.

Given a nuclear spin operator 

I (I ,I ,I ) 

x y z



, the following averages can be calculated:

| .I | (I iI )      

x  y





(20)

| .I | (I iI )      

x  y





.

Assuming an eigenstate 

|  

for the nuclear spin operator 

I



, the following relations hold:

I

z

|    m |  

I |    I(I 1) |  

2



Here I is an eigenvalue and m is the “magnetic” number. 

REFERENCES

L.I. Schiff, “Quantum Mechanics”, McGraw Hill (1955). 

QUESTIONS

1. What are the two main parts of the Hamiltonian?

2. Write down the Schrodinger equation. Define the terms. 

3. Describe the Fermi golden rule. What is it used for?

4. What type of neutron scattering obeys the Fermi golden rule? Which type does not?

5. What are the Pauli spin matrices used for?

6. Given a spin operator 

I



with eigenstate 

|  

, where was the following eigenvalue relation 

I |    I(I 1) |  

2



used? 
712

7. Where were the Pauli spin matrices used? 

ANSWERS

1. The two main parts of a Hamiltonian are the kinetic energy and the interaction (potential) 

energy. 

2. The Schrodinger is given by 

H  E

where H is the Hamiltonian, 



is the eigenstate 

(wavefunction) and E is the eigenvalue (energy). 

3. The Fermi golden rule is a first order perturbation theory used to solve the Schrodinger 

equation. It is used to calculate the scattering cross section. 

4. Single scattering methods (SANS, diffraction, elastic, quasielastic/inelastic scattering) all 

follow the Fermi golden rule. Methods that involve multiple scattering (or refraction) such as 

reflectometry do not follow the Fermi golden rule. 

5. The Pauli matrices 

2s

 

 

are used to represent the neutron spin 

2

1

s





 . 

6. The eigenvalue relation 

I |    I(I 1) |  

2



was used when discussing spin incoherence 

during the scattering of a neutron and a nucleus. 

7. The Pauli spin matrices were used to derive the neutron scattering cross section with spin 

polarization. 
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LIST OF SYMBOLS AND NOTATION

The various symbols used throughout are listed here in the order they occur. 

J()

Neutron flux (units of neutron.cm-2.s-1)

()

Neutron current (units of neutron.s-1)

b Scattering length (units of fm)

v Specific volume (units of cm3)

v

b

 

Scattering length density (units of Å-2)



2 Contrast factor

 Scattering angle

Q Scattering variable (units of Å) 

d Solid angle

f() Scattering amplitude

I Nuclear spin





d

d (Q)

Microscopic differential scattering cross section 





d

d (Q)

Macroscopic differential scattering cross section 

 Microscopic scattering cross section 

  dd(Q) d

(units of barn)

 Microscopic scattering cross section 

  dd(Q) d

(units of cm-1)

 Neutron wavelength

E Neutron kinetic energy

k Neutron wavelenumber k = 2/





Relative neutron wavelength spread

2

Q

Variance of the Q resolution

L1 Source-to-sample distance

L2 Sample-to-detector distance

R1 Source aperture radius

R2 Sample aperture radius

x3 Detector cell size

Qmin Minimum scattering variable
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f Focal length for neutron lenses

T() Sample transmission

g Gravity constant g = 9.81 m.s-2

F(Q) Scattering form factor amplitude

2

P(Q) | F(Q) |

Scattering form factor



P(Q)  dr exp(iQ.r)P(r)

 







P(r)  dQexp(iQ.r)P(Q)





 



Probability distribution function

g(

r



)-1 = VP(

r



) Pair correlation function

p(r)

Qr

sin(Qr)

dr

R

1

P(Q)

R

0





(r)

R

r

p(r) 3

2

 













(r) the radial pair correlation function

E(Q) Propagation scattering factor for polymers

F(Q) Form factor amplitude

P(Q) Form factor

SI(Q) Inter-particle structure factor 

S(Q) Scattering factor for polymers

  



 dr exp iQ.r g(r) 1

V

1

S(Q)

 

 

g(r) 1 VP(r)



 

R Spherical particle radius

D Spherical article diameter

L Rodlike particle length

 Correlation length

n(r) Fluctuating density

n

Average density = n/V

VP Particle volume

v Specific volume

V Sample volume

N Number of particles or macromolecules in the sample

N

Number density = N/V

n Number of monomers per macromolecule (degree of polymerization)

 Volume fraction

a Statistical segment length for polymers
715

{...} Average over composition or polydispersity average

{Rn} n

th moment of R

<...> Statistical average 

j0(x) Spherical Bessel function of order 0

j1(x) Spherical Bessel function of order 1

J1(x) Cylindrical Bessel function of order 1

Rg radius of gyration

f(R) Polydispersity distribution

 Flory-Huggins interaction parameter in polymer blends

C(Q) Ornstein-Zernike direct correlation function

H(Q) Ornstein-Zernike total correlation function

zm Macroion electric charge

: Dielectric constant



: Debye-Huckel inverse screening length

T Temperature

kB Boltzman constant kB = 1.38*10-23 J.k-1

Nav Avogadro’s number Nav = 6.02*1023 mol-1



Planck’s constant 



= 1.055*10-34 J.s

Unit Conversions:

1 Å = 10-10 m

1 fm = 10-13 cm

1 barn = 10-24 cm2

1 bar = 1 torr = 14.7 psi = 760 mm Hg = 105 Pa

Acronyms of Characterization Methods

UV-Vis Ultra Violet and Visible Absorption Spectroscopy

IR Infra-Red Spectrocopy

CD Circular Dichroism

NMR Nuclear Magnetic Resonance

Mass-Spec Mass Spectroscopy

Dens-Meas Density Measurements

Visc-Meas Viscosity Measurement

DSC Differential Scanning Calorimetry

SEC Size Exclusion Chromatography

Micros Optical Microscopy

AFM Atomic Force Microscopy

Cryo-TEM Cryogenic Transmission Electron Microscopy
716

SALS Small-Angle Light Scattering

DLS Dynamic Light Scattering

Opt-Biref Optical Birefringence

SAXS Small-Angle X-Ray Scattering

WAXS Wide-Angle X-Ray Scattering (Diffraction)

SANS Small-Angle Neutron Scattering

WANS Wide-Angle Neutron Scattering
717