Table Of ContentMETHODS OF
EXPERIMENTAL PHYSICS
Robert Celotta and Judah Levine, Editors-in-Chief
Founding Editors
L. MARTON
C. MARTON
Volume 23
Neutron Scattering
PART C
Edited by
Kurt Skold
Institute for Neutron Research
Uppsala University
Studsvik, Nykoping, Sweden
David L. Price
Materials Science Division
Argonne National Laboratory
Argonne, Illinois
@
1987
ACADEMIC PRESS, INC.
Harcourt Brace Jovanovich, Publishers
Orlando San Diego New York Austin
Boston London Sydney Tokyo Toronto
COPYRIGHQT 1 987 RY ACADEMIPCR ESSI,N C.
ALL RIGHTS RESERVED.
NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR
TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC
OR MECHANICAL, INCLUDING PHOTOCOPY. RECORDING. OR
ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT
PERMISSION IN WRITING FROM THE PUBLISHER.
ACADEMIC PRESS, INC.
Orlando. Florida 32x87
United Kingdom Edition published by
ACADEMIC PRESS INC. (LONDON) LTD.
24-28 Oval Road, London NWI 7DX
Library of Congress Cataloging in Publication Data
(Revised for Part C)
Neutron scattering.
(Methods of experimental physics; v. 23)
Includes indexes.
1. Neutrons-Scattering. 2. Condensed matter.
I. Skold, Kurt. 11. Price, David L. (David Long),
Date. Ill. Series.
QC793.5.N4628N496 1987 539.7’21 3 86-1 128
ISBN 0-12-475968-8 (pt. C: alk. paper)
PKINTI<U IN THE UNITED STATES OF AMF.RICA
878~8990 9 8 7 6 5 4 3 2 1
PREFACE
The neutron scattering technique for measuring the structure and dy-
namics of condensed matter has developed over the 50 years ofthe neutron’s
history into a widely used tool in physics, chemistry, biology, and materials
science. Since the early diffraction studies in the 1940s and the first measure-
ments of inelastic scattering in the 1950s, developments in experimental
methods have greatly increased the sensitivity and range of applications of
the technique. Thus, while the early measurements probed distances on the
order of interatomic spacings (-3 A) and times on the order of typical
periods of lattice vibrations (- I ps), the current range of neutron scattering
experiments covers distances from 0.1 to 10,000 A, and times from 10 fs to
1 ps. This has been achieved by expanding the range of neutron energies
available to the experimenter from a few milli-electron-volts (at cold sources
in research reactors) to several electron-volts (at pulsed spallation sources),
and by using a variety of novel detection methods such as position-sensitive
detectors and back-scattering and spin-echo techniques. As a result, the areas
of investigation have expanded from the conventional crystal structures and
lattice dynamics (and their magnetic analogs) of 30 years ago to high-resolu-
tion studies of the atomic spacings in amorphous thin films, biological struc-
tures on a cellular scale, unraveling of long chains of polymers, and transi-
tions between energy levels in molecular solids.
Along with these developments, the community of neutron users has
expanded and diversified. Whereas 30 years ago neutron scattering was
practiced largely by solid-state physicists and crystallographers, the users of
present-day centralized neutron facilities include chemists, biologists, ce-
ramicists, and metallurgists, as well as physicists of diverse interests ranging
from fundamental quantum mechanics to fractals and phase transitions.
The neutron centers have developed from essentially in-house facilities at
the national nuclear research laboratories into centralized facilities orga-
nized for use by the general scientific community at an international level.
The pioneer of this mode of operation was the Institut Laue-Langevin in
Grenoble, France, operated since 1972 by Britain, France, and Germany as a
user-oriented facility for scientists from these and other countries. Similar
modes ofoperation are now being established at other major reactor facilities
like those at Brookhaven and Oak Ridge in the United States, and the pulsed
spallation sources that have recently come into operation at Argonne in the
United States, the KEK Laboratory in Japan, and the Rutherford Labora-
ix
X PREFACE
tory in Britain have been set up from the beginning with this mode of
operation. The current population of users of these and other neutron facili-
ties has been recently estimated* to be 500 in the United States, 1250 in
Western Europe, and about 200 in Japan.
The aim of the present book is to describe the current state of the art of
application of neutron scattering techniques in those scientific areas that are
most active. The presentation is aimed primarily at professionals in different
scientific disciplines, from graduate students to research scientists and uni-
versity faculty members, who may be insufficiently aware of the range of
opportunities provided by the neutron technique in their area ofspecialty. It
does not present a systematic development of the theory, which may be
found in excellent textbooks such as those of Lovesey or Squires, or a de-
tailed hands-on manual of experimental methods, which in our opinion is
best obtained directly from experiencedpractitionersat the neutron centers. It
is rather our hope that this book will enable researchers in a particular area to
identify aspects of their work in which the neutron scattering technique
might contribute, conceive the important experiments to be done, assess
what is required to carry them out, write a successful proposal for this
purpose for one of the centralized user facilities, and carry out the experi-
ments under the care and guidance of the appropriate instrument scientist.
With this object in view, each chapter relating to a particular field of science
has been written by a leading practitioner or practitioners of the application
of the neutron methods in that field.
Volume 23, Part A, of this work starts out with a brief survey of the
theoretical concepts ofthe technique and establishes the notation that will be
used throughout the book. Chapters 2 and 3 review the fundamental hard-
ware of neutron scattering, namely, sources and experimental methods, and
Chapter 4 discusses fundamental physics applications in neutron optics. The
remaining chapters of Part A treat various basic applications of neutron
scattering to studies of the atomic structure and dynamics of materials. The
Appendix contains a compilation of neutron scattering lengths and cross
sections that are important in nearly all neutron scattering experiments.
Volume 23, Part B, contains surveys of the application of neutron scatter-
ing techniques to nonideal solids, such as solids with defects, two-dimen-
sional solids and glasses, and to various classes of fluids. Finally, Volume 23,
Part C, treats neutron scattering investigations of magnetic materials, solids
undergoing phase transitions, and macromolecular and biological struc-
tures. In recognition of the expanding use of neutron scattering in technol-
* Current Status of Neutron-Scattering Research and Facilities in the United States (National
Academy Press, Washington, D.C., 1984).
PREFACE xi
ogy, the last chapter in Part C is devoted to a survey of industrial applica-
tions.
We wish to thank the authors for taking time out of their busy schedules
for contributing these chapters, Dr. R. Celotta for inviting us to undertake
this work, and the staff of Academic Press for their encouragement and
forbearance.
KURTS K~LD
DAVIDL . PRICE
LIST OF SYMBOLS
Bound scattering length
Coherent scattering length
+
Scattering length for I f state
Scattering length for I - f state
Incoherent scattering length
Spin-dependent scattering length
Velocity of light = 2.9979 X loLoc m sec-l
Mass density
Equilibrium position of atom in unit cell
Magnetic interaction operator
Differential cross section
Double differential cross section
Incident, scattered energy
Energy lost by neutron (Eo- El)
Charge on the electron = 4.8033 X lo-” esu
Polarization vector of normal mode j [ei (q) for non-Bravais crystal]
Structure factor for unit cell
Form factor
+
Space-time correlation function [Gd(r,t ) G&, t)]
“Distinct” space-time correlation function
“Self” space-time correlation function
Pair distribution function [p(r)/p,,]
Planck’s constant/2n = 1.0546 X erg+ s ec
Intermediate scattering function [I,(Q, t) &(Q,t )]
Intermediate “distinct” scattering function
Intermediate “self” scattering function
Angular momentum operator for nucleus
Incident, scattered wave vector
Boltzmann’s constant = 1.3807 X erg K-l
Position of unit cell
Mass of atom
Mass of neutron = 1.0087 u
Mass of electron = 9.1095 X g
Number of unit cells in crystal
Avagadro’s number = 6.0220 X loz3
Radial distribution function (4n r2p(r)]
Scattering vector (k,, - k,)
Reduced wave vector (Q - 7)
Number of atoms in a unit cell
Classical electron radius (e2/m,c2)= 2.8179 X cm
Spin operator for ion or atom
Static structure factor I(Q, 0)
Coherent scattering function
lncoherent scattering function
Spin operator for electron
...
Xlll
xiv LIST OF SYMBOLS
Atomic mass unit = 1.6606 X g
Vibrational amplitude (uf, for non-Bravais crystal)
Volume of sample
Volume of unit cell
Exponent of Debye- Waller factor (in cros’s section)
Density of phonon states
Gyromagnetic ratio of neutron = 1.9 132
Debye temperature
Brag angle
Magnetic moment of ion or atom
Nuclear magneton (eh/2mpc=) 5.0508 X erg G-’
Bohr magneton (eh/2mec) = 9.2741 X erg G-I
Average number density
Pair density function [G(r, 0) - &r)]
Bound total cross section (scattering plus absorption)
Bound coherent scattering cross section
Bound incoherent scattering cross section
+
Bound scattering cross section (17, a,)
Spin operator for neutron
Reciprocal lattice vector (2n[(h/a)(,k /b),( Ilc)])
Neutron flux (n cw2s ec-I)
Scattering angle (=28 for Brag reflection)
Susceptibility
Generalized susceptibility
Solid angle
Frequency of normal modej
18. PHASE TRANSITIONS
R. A. Cowley
Department of Physics
University of Edinburgh
Edinburgh EH9 3JZ Scotland
18.1. Introduction
18.1.1. History
Phase transitions occur when the structure of a system changes. In magnetic
materials this change may be from a high-temperature disordered phase in which the
atomic magnetic moments are randomly oriented, to a low-temperature phase in
which the magnetic moments are ordered in a well defined structure. In crystals there
may be a structural phase transition from a high-temperature, high-symmetry, crystal
structure to a distorted structure of lower symmetry. An essential part of the study of
phase transitions is the determination of these different structures, and especially in
the case of magnetic materials neutrons have played an essential role, as described
in Chapter 19. Indeed one of the first applications of neutron diffraction techniques
was the determination of several antiferromagnetic structures.' In this chapter the
intent is not to review the contributions that neutron scattering techniques have made
to the study of phase transitions through the study of structures but will concentrate
on the contributions that have been made by studying the fluctuations in the structure
that occur particularly close to continuous phase transitions. Indeed, it is the under-
standing of these fluctuations that is at the heart of the modem theory of phase
transitions, and without the results of neutron scattering experiments it would have
been difficult for the subject to have developed so rapidly.
The first neutron scattering experiment in which the scattering by the critical
fluctuations was observed was that of Latham and Cassek2 Using an accelerator-
based source of neutrons, they measured the total scattering cross section of various
materials and found that the total scattering cross section of iron increased at
temperatures close to the Curie temperature for the onset of long-range ferromagnetic
order. They correctly identified this extra scattering as arising from scattering by the
magnetic fluctuations associated with the onset of the magnetic long-range order, by
analogy with the phenomenon of critical opalescence at the liquid-gas critical points.
The experiment was soon repeated using a more intense reactor source by Squires:
who obtained more accurate results.
METHODS OF EXPERIMENTAL PHYSICS Copyright 0 19x7 by Academic Press. Inc.
Vol. 23, Part C All rights of reproduction in any form reserved.
2 R. A. COWLBY
A major step forward was the work of Van Hove: who developed the theory of
the scattering from the magnetic fluctuations close to a phase transition by using the
classical or Omstein-Zemike thee$ to describe the fluctuations. We shall review
this theory in Section 18.2. At least partly as a result of this development, there were
then a number of experiments6 performed to measure the scattering by the critical
fluctuations as a function of wave vector and frequency. At this time, however, the
experiments were sufficiently difficult that the results are of interest more for the
development of the neutron scattering techniques than for the information they
provided about critical fluctuations. In my view, the modem study of critical fluctua-
tions began with the work in 1967 of Als-Nielsen and Dietrich’ on p-brass. They
studied the order-disorder transition and clearly showed that neutron scattering
techques could be used to perform quantitative measurements of the temperature
dependence of the order parameter and critical fluctuations. They measured three of
the critical exponents (Section 18.2), and all of them had values clearly different from
their classical or mean field theory values and in agreement with the results predicted
by series expansions.
This experiment led to other studies, particularly on magnetic systems, and the
results are discussed for pure materials in Section 18.3 and for disordered materials
in Section 18.4 The utility of the neutron scattering technique was greatly extended
by the growth of single crystals of magnetic materials in which the magnetic ions
were coupled not only in three-dimensionala rrangementsb ut also in two-dimensional
sheets and one-dimensional lines. These have enabled experiments to be done to test
the theories of phase transitions in two dimensions and of statistical mechanics in
one dimension. Since the properties of phase transitions are very dependent on the
dimensionality, these systems have been very important in extending the usefulness
of the neutron scattering technique. In Sections 18.3 and 18.4 we review some of the
results obtained in two- and three-dimensional systems, while one-dimensional sys-
tems are discussed in Chapter 20.
Although the most detailed tests of the theory of phase transitions have been
performed using magnetic systems, neutron scattering techniques have also made a
significant contribution to structural phase transitions. The development of the un-
derstanding of structural phase transitions began in a way different from that of
magnetic transitions. In part this is because the high-temperature phase is often an
ordered structure that becomes distorted below the structual phase transition, whereas
in the magnetic case the transition is from a disordered hign-temperature phase to
an ordered low-temperature phase. In the structural case, Cochran’ developed the
“soft mode” concept, in which the phase transition arises as an instability of the
crystal against a particular normal mode of vibration of the high-temperature phase.
One of the first experimental tests of this was provided by neutron scattering
measurements’ of the ferrcelectric soft mode in SrTi03. Since then many similar
observations have been made in other materials, but the observations have shown
that a simple soft mode cannot describe all of the behavior, as discussed in Sec-
tion 18.5
