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PHYSICS IN CANADA
LA PHYSIQUE AU CANADA
Vol. 62 No. 5
September / October 2006
septembre / octobre 2006
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Vol. 62 No. 5
PHYSICS IN CANADA
LA PHYSIQUE AU CANADA
TABLE
OF
CONTENTS / TABLE
September / October 2006
septembre / octobre 2006
DES MATIÈRES
Editorial : Neutron and X-Ray Scattering at Major Facilities, by J. Katsaras, Guest Editor
Éditorial : Diffusion des neutrons et des rayons X dans les grandes installations de recherche,
par J. Katsaras, rédacteur honoraire
225
226
PHYSICS AND EDUCATION / LA PHYSIQUE ET L’ÉDUCATION
Small-Angle Neutron Scattering and Biomolecules, by/par J. Katsaras et al.
233
Neutrons and Transition Metal Oxides: A Match Made in Heaven, by/par J.E. Greedan
241
Stop That Corrosion - If You Can, by/par Z. Tun et al.
249
FEATURE ARTICLES / ARTICLES DE FOND
Quantum Magnetism and Superconductivity, by/par W.J.L. Buyers and Z. Yamani
257
Polarized Neutron Reflectometry as a Unique Tool in Magnetization Reversal Studies of Thin Films
and Multilayers, by/par H. Fritzsche et al.
265
Diffraction Studies of Gas Hydrates with an Emphasis on CO2 Hydrate, by/par B.H. Torrie et al.
273
Phase Transitions in Organic-Inorganic Perovskites, by Ian Swainson
279
Revealing the Microstructure of Polymeric Materials using SANS, by B. Frisken
285
Use of Neutron Diffraction for Development of Metal Hydrides: Case of BCC Alloys, by J. Huot et al.
289
Neutrons and Muons as Complementary Probes of Exotic Magnetism and Superconductivity, by C. Wiebe
295
Status of the Canadian Macromolecular Crystallography Facility: Design and Commissioning of the 08ID-1
Beamline at the Canadian Light Source, by P. Grochulski et al.
301
Phonon Spectroscopy and X-Ray Scattering using Synchrotron Radiation, by J.S. Tse and D.D. Klug
305
Synchrotron Advances at the Frontiers of Food Physics: Studies of Edible Fats such as Chocolate
Under Shear, by G. Mazzanti et al.
313
DEPARTMENTS / RUBRIQUES
Erratum and Letter / Erratum et communication
229
FRONT COVER / COUVERTURE AVANT
2006 Congress / Congrès 2006
230
News (IUPAP Prize) / Informations (Priz de l’UIPPA)
232
From bottom left, clockwise, a selection of
figures from the articles by Fritzsche et al.
(fig. 6, pg. 268), Greedan (fig. 10, pg. 246),
Swainson (fig. 3, pg. 280), Huot et al.
(fig. 1, pg. 289), and Katsaras et al. (fig. 1,
pg. 234). Cover design by Alastair McIvor,
National Research Council, Chalk River.
Call for Suggestions/Nominations (Council/CNILC)
Appel de candidatures (Conseil/CNILC)
256/272
News (37th Olympiad) / Informations (37ième Olympiad)
278
Institutional, Corporate and Sustaining Members /
Membres institutionnels, corporatifs, et de soutien
284
Books Received / Livres reçus
Book Reviews / Critiques de livres
321
322
Employment Opportunities & Commercial Ads
/ Postes d’emplois et publicités commerciales
331
Art of Physics / Art de la physique
IBC
Advertising Rates and Specifications (effective January 2006) can
be found on the PiC website (www.cap.ca - PiC online). / Les tarifs
publicitaires et dimensions (en vigueur depuis janvier 2006) se trouvent sur le site internet de La Physique au Canada (www.cap.ca PiC Électronique).
En partant du coin en bas à gauche, dans
le sens des aiguilles d’une montre, une
sélection de figures tirées des articles de
Fritzsche et al., (fig. 6, p. 268), de Greedan
(fig. 10, p. 246), de Swainson (fig. 3,
p. 280), de Huot et al. (fig. 1, pg. 289) et de
Katsaras et al. (fig. 1, p. 234). Montage par
Alastair McIvor, Conseil national de
recherches, Chalk River.
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EDITORIAL
PHYSICS IN CANADA
LA PHYSIQUE AU CANADA
The Journal of the Canadian Association
of Physicists
La revue de l'Association canadienne des physiciens et physiciennes
ISSN 0031-9147
EDITORIAL BOARD / COMITÉ DE RÉDACTION
Editor / Rédacteur en chef
-- E D I T O R I A L /
ÉDITORIAL
--
NEUTRON AND X-RAY SCATTERING AT
MAJOR FACILITIES
DIFFUSION DES NEUTRONS ET DES RAYONS X DANS
LES GRANDES INSTALLATIONS DE RECHERCHE
Béla Joós, P.Phys.
Physics Department, University of Ottawa
150 Louis Pasteur Avenue
Ottawa, Ontario K1N 6N5
(613) 562-5800x6755; Fax:(613) 562-5190
e-mail: [email protected]
Associate Editor / Rédactrice associée
Managing / Administration
Francine M. Ford
c/o CAP/ACP
Book Review Editor / Rédacteur à la critique de livres
Andrej Tenne-Sens
c/o CAP / ACP
Suite.Bur. 112, Imm. McDonald Bldg., Univ. of / d' Ottawa,
150 Louis Pasteur, Ottawa, Ontario K1N 6N5
(613) 562-5614; Fax: (613) 562-5615
Email: [email protected]
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Michael Steinitz, PPhys
Department of Physics
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Antigonish, Nova Scotia B2G 2W5
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Email: [email protected]
Board Members / Membres du comité :
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(613) 225-6156
Email: [email protected]
René Roy, phys
In 1895 Wilhelm Conrad Röntgen, a professor of physics and the director of the
Physical Institute of the University of Würzburg (Germany) discovered a new form
of radiation, which he called X-rays. Years later, James Chadwick, a professor of
physics at Cambridge University discovered the neutron, a neutral particle in the
nucleus of an atom. For their discoveries, Röntgen and Chadwick were awarded,
respectively, in 1901 and 1935, the Nobel prize in physics.
The theoretical beginnings of synchrotron radiation go back to Thomson's discovery of the electron in 1897. Subsequently, Larmor derived an expression from classical electrodynamics describing the power radiated by an accelerated charged particle, which Liénard extended for a relativistic particle undergoing centripetal
acceleration in a circular trajectory. The radiated power of a relativistic particle is
proportional to (E/mc2)4/R2, where E is particle energy, m is the rest mass, and R
is the radius of the trajectory. By 1945, Julian Schwinger – yes, Schwinger of relativistic quantum electrodynamics fame - worked out the classical theory of radiation from accelerated relativistic electrons. On 24 April 1947, Elder, Gurewitsch,
Langmuir and Pollock observed the bluish-white light of synchrotron radiation
using General Electric’s 70 MeV (mega electron volt) electron synchrotron at
Schenectady, New York.
David J. Lockwood, PPhys
In 1942, Enrico Fermi supervised the design and assembly of an "atomic pile" graphite blocks, uranium, and cadmium control rods - in the squash courts beneath
the University's football stadium. The world’s first controlled nuclear reaction took
place on December 2, 1942. Interestingly, 18 months earlier in Canada, George
Laurence, then on staff at the National Research Council (NRC), constructed his
own atomic pile in the basement of the NRC’s laboratories at 100 Sussex Drive. But
for a small amount of boron impurity in the graphite, Canada’s atomic pile might
have been the first in the world to go critical. In June of 1944, the National Research
Council established the “NRC Atomic Energy Project” at Chalk River.
ANNUAL SUBSCRIPTION / ABONNEMENT ANNUEL:
The first controlled nuclear reaction in Canada took place on September 5, 1945,
when the Zero Energy Experimental Pile (ZEEP) reactor went into operation. The
ZEEP reactor was designed to produce only a few watts of heat, and was used to
provide data for the design of the 25 MW (mega watts) NRX (National Research
eXperimental) reactor, for a time the most powerful reactor in the world. Ten years
later, the completion of the National Research Universal (NRU) reactor in 1957 was
a landmark achievement in Canadian science and technology. Operating at 120
MW of power it was the most powerful reactor of its time, and even today, produces some of the most intense thermal neutron beams from a core flux of 3x1018
neutrons/m2/s2. In 1994, Bertram N. Brockhouse was awarded the Nobel Prize in
physics for his development of neutron spectroscopy and the triple-axis spectrometer while using the intense neutron beams provided by the NRX and NRU reactors.
Département de physique, de génie physique et d’optique
Université Laval
Cité Universitaire, Québec G1K 7P4
(418) 656-2655; Fax: (418) 656-2040
Email: [email protected]
Institute for Microstructural Sciences
National Research Council (M-36)
Montreal Rd., Ottawa, Ontario K1A 0R6
(613) 993-9614; Fax: (613) 993-6486
Email: [email protected]
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PHYSICS
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CANADA
Neutron and X-ray scattering have become indispensable tools in a wide range of
condensed-matter research fields (e.g. crystallography, structure of surfaces and
interfaces, disordered systems, etc.). A highlight of note is the important role X-rays
have played in macromolecular structure determination starting with the structure
The contents of this journal, including the views expressed above, do not necessarily represent the views or policies of the Canadian Association of Physicists.
Le contenu de cette revue, ainsi que les opinions exprimées ci-dessus, ne
représentent pas nécessairement les opinions et les politiques de l'Association
canadienne des physiciens et des physiciennes.
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PRÉFACE
of DNA and continuing to the recent Nobel prize awarded to
Roger Kornberg for his studies of the molecular basis of eukaryotic transcription using the Stanford Synchrotron Radiation
Laboratory.
The 13 articles in this issue cover a range of scientific topics.
What they all have in common is neutron and X-ray scattering
carried out at major facilities for condensed matter research. The
Canadian facilities used were the Canadian Light Source (CLS),
the Tri-University Meson Facility (TRIUMF) Centre for
Molecular and Materials Science and the Canadian Neutron
Beam Centre (CNBC).
In Canada, the Canadian Light Source (CLS), located at the
University of Saskatchewan, is a 2.9 GeV (giga electron volt)
state-of-the-art, third generation synchrotron light source estimated to meet ~ 90% of the current and future demands of the
Canadian research community. The CLS consists of a 250 MeV
electron Linac, a booster to ramp the beam to 2.9 GeV, and the
main ring, which is designed to operate at 2.9 GeV and currents
up to 500 mA. Presently there are 7 beamlines being commissioned, with another 7 under construction.
TRIUMF, located at the University of British Columbia, is one of
the three subatomic research facilities in the world that specialize in producing intense beams of protons. The cyclotron, the
biggest in the world, is capable of accelerating 20 000 billions
particles/second (~ 300 Microamps). The proton beams produced by the cyclotron, strike different kinds of targets, producing neutron, pion, and muon beams, which can be used for different types of experiments. More recently, the ISAC (Isotope
Separation and ACceleration) facility allows for the production
of some of the most intense beams of exotic ions (isotopes) in the
world, making TRIUMF an internationally-recognized centre for
the study of nuclear astrophysics.
NRC operates today’s CNBC at Chalk River. Presently, the facility is comprised of 5 thermal instruments, which are located at
the NRU reactor. The University of Western Ontario recently led
12 other universities in procuring funds from the Canada
Foundation for Innovation to add a new, world-class neutron
reflectometer to the CNBC facility. The reflectometer is expected
to begin operation in 2007, to support a wide range of nanotechnology applications never before possible in Canada.
The CNBC primarily operates as an accessible international user
facility, supported in part by a Natural Sciences and Engineering
Research Council of Canada (NSERC) Major Resources Support
(MRS) grant, which is presently administered through McGill
University.
Neutrons, which possess a magnetic moment (spin ½ particles),
are ideally suited to the study of magnetic structures and magnetic fluctuations. The article by Fritzsche et al. describes how
polarized neutrons are used to study magnetic thin films and
multilayers, while Buyers and Yamani describe how neutrons
have played a pivotal role in the discovery of new phases of matter in quantum gapped systems, a highly-correlated heavyfermion system, and superconductors. Adding to the neutron/magnetism flavour, is Wiebe’s article on neutrons and
muons as complementary probes for exotic magnetism and
superconductivity.
Powder diffraction, because of its ease of use and quick turnover
of samples, is one of the most widely used techniques to study
structural properties of materials. Neutron powder diffractometers, although less commonly available than their x-ray counterparts, are “workhorse” instruments found in practically all major
neutron laboratories. As a result, it is not surprising to see 4 articles in this issue based on neutron powder diffraction for their
science (i.e., Greedan, Swainson, Torrie et al., and Huot et al.).
Before small-angle neutron scattering (SANS), studies of polymer structure were, for the most part, limited to light and smallangle x-ray scattering techniques. The unique role of SANS in the
case of hydrogenous materials lies in the difference in the coherent scattering length between hydrogen (bH = -0.37 × 10-12 cm)
and deuterium (bD = 0.67 × 10-12 cm). This difference results in a
marked difference in scattering power (contrast) between
hydrogenous and deuterated monomer units. Canada has SANS
expertise, even though we lack a proper SANS instrument. Here
the articles by Katsaras et al. and Frisken demonstrate the versatility of SANS, and the power of contrast variation.
Neutron reflectometry is a technique used to probe the structure
of surfaces, thin-films and buried interfaces, as well as processes
occurring at surfaces, e.g. corrosion. In their article, Tun et al.
report on neutron reflectometry studies of passive oxide layers
on titanium and zirconium, materials of interest to the nuclear
industry.
March 2006 saw an important milestone for the CLS, and the
Canadian crystallography community, with the first results coming off the Canadian Macromolecular Crystallography Facility
(CMCF) beamline. The images were of a protein crystal, PEP carboxykinase. In this issue, Grochulski, Blomqvist and Delbaere of
the CMCF team describe, in detail, the 08ID-1 beamline which
was used to collect the data. The report by Grochulski et al. is followed by the articles of Tse and Klug, and Mazzanti et al.
describing diverse topics such as X-ray phonon spectroscopy and
synchrotron X-ray studies of chocolate under shear.
John Katsaras
National Research Council, Chalk River
Guest Editor, Physics in Canada
DIFFUSION
DES NEUTRONS ET DES RAYONS X
DANS LES GRANDES INSTALLATIONS DE
RECHERCHE
En 1895, Wilhelm Conrad Röntgen, professeur de physique et
directeur de l'Institut de physique de l'Université de Würzburg
(Allemagne), découvre une nouvelle forme de radiation qu'il
nomme rayons X. Plusieurs années plus tard, James Chadwick,
professeur de physique à l'Université Cambridge fait la découverte du neutron, une particule électriquement neutre qui est un
constituant du noyau de l'atome. Röntgen et Chadwick ont reçu
le Prix Nobel de physique pour leurs découvertes, respectivement en 1901 et 1935.
Les premières théories sur le rayonnement synchrotron remontent à la découverte de l'électron par Thomson en 1897. Par la
suite, Larmor calcule une expression au moyen de l'électrodynamique classique décrivant le rayonnement produit par une
particule chargée accélérée, que Liénard applique à une particule
relativiste soumise à une accélération centripète dans une trajectoire circulaire. La puissance de rayonnement d'une particule relativiste est proportionnelle à (E/mc2)4/R2, où " E " représente
LA PHYSIQUE AU CANADA
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l'énergie des particules, " m ", la masse au repos et " R ", le rayon
de la trajectoire. Vers 1945, Julian Schwinger - oui, " LE
Schwinger " qui a connu la gloire grâce à son travail sur l'électrodynamique quantique relativiste - met au point la théorie classique du rayonnement produit par des électrons relativistes
accélérés. Le 24 avril 1947, Elder, Gurewitsch, Langmuir et
Pollock observent une lumière blanche bleuâtre émise par le rayonnement synchrotronique en utilisant un synchrotron à électrons de 70 MeV (mégaélectrovolts) à Schenectady, New York.
En 1942, Enrico Fermi dirige la conception et la construction
d'une " pile atomique " - en superposant des briques de graphite,
de l'uranium, et des barres de commande de cadmium - sur les
courts de squash situés sous les gradins du stade de football de
l'Université de Chicago. Le 2 décembre 1942, il réussit la première réaction nucléaire contrôlée. Fait intéressant, 18 mois plus
tôt, le Canadien George Laurence, alors membre du personnel
du Conseil national de recherches du Canada (CNRC), avait
construit sa propre pile atomique dans le sous-sol des laboratoires du CNRC au 100, promenade Sussex. N'eut été d'une
petite quantité d'impureté de bore, la pile atomique construite
au Canada aurait été la première pile à fonctionner au monde.
En juin 1944, le Conseil national de recherches mettait sur pied
le " NRC Atomic Energy Project " (Projet de l'énergie atomique
du CNRC) à Chalk River.
Au Canada, la première réaction nucléaire en chaîne contrôlée a
lieu le 5 septembre 1945, lorsque le réacteur de recherche ZEEP
(pile expérimentale d'énergie zéro) est mis en service. Le ZEEP a
été conçu pour produire uniquement quelques watts de chaleur
et a été utilisé pour fournir des données pour la conception du
réacteur NRX (réacteur national de recherche expérimental) de
25 MW (mégawatts), qui a été, pendant un certain temps, le réacteur le plus puissant au monde. Dix ans plus tard, la construction du réacteur national de recherche universel (NRU) fut une
réalisation historique en 1957 dans le domaine des sciences et de
la technologie au Canada. Ce réacteur qui fonctionne à des
niveaux de puissance de 200 MW était le réacteur le plus puissant de l'époque, et même aujourd'hui il produit certains des
faisceaux de neutrons thermiques les plus intenses dans un flux
de 3x1018 de neutrons/m2. En 1994, Bertram N. Brockhouse remporte le Prix Nobel de physique pour le développement de la
spectroscopie neutronique et l'invention du spectromètre neutronique à trois axes en utilisant les faisceaux de neutrons intenses produits par les réacteurs NRX et NRU.
La diffusion des neutrons et des rayons X est devenue un outil
essentiel à une vaste gamme de champs de recherche sur la
matière condensée (p. ex., la cristallographie, la structure des
surfaces et interfaces, les systèmes désordonnés, etc.). Un fait
marquant digne d'être souligné est le rôle important qu'ont joué
les rayons X dans la détermination de la structure macromoléculaire, en commençant par la structure de l'ADN, jusqu'aux
études de Roger Kornberg sur le fondement moléculaire de la
transcription eukaryote entreprises au Laboratoire de rayonnement synchrotron de Stanford (Stanford Synchrotron
Radiation Laboratory) qui lui ont valu le Prix Nobel en 2006.
Les 13 articles publiés dans ce numéro traitent de divers sujets
scientifiques qui ont tous en commun la diffusion des neutrons
et des rayons X, effectuée principalement dans des grandes
installations de recherche dans le cadre de la recherche sur la
matière condensée - notamment le Centre canadien de rayonnement synchrotron (CCRS), le Centre for Molecular and
Materials Science de la Tri-University Meson Facility (TRIUMF)
et le Centre canadien de faisceaux de neutron (CCFN).
228
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IN
CANADA
Au Canada, le Centre canadien de rayonnement synchrotron
(CCRS), situé à l'Université de la Saskatchewan, est une source
de rayonnement synchrotron de troisième génération à 2,9 GeV
(gigaélectrovolts) à la fine pointe de la technologie pouvant satisfaire selon les estimations près de 90 p. 100 des demandes
actuelles et futures de la communauté des chercheurs canadiens.
Le CCRS comprend un accélérateur linéaire d'électrons de 250
MeV, un injecteur permettant d'accélérer l'énergie du faisceau
jusqu'à 2,9 GeV et l'anneau principal conçu pour fonctionner à
2,9 GeV, et alimenté par des courants pouvant atteindre 500 mA.
Il y a actuellement 7 faisceaux dont la mise en service est en
cours et 7 autres qui sont en construction.
Le grand accélérateur TRIUMF, situé à l'Université de la
Colombie-Britannique, est l'une des trois installations de
recherche subatomique au monde qui se spécialisent dans la
production de faisceaux intenses de protons. Le cyclotron - le
plus gros au monde -peut accélérer 20 000 milliards de particules
par seconde (~ 300 Microamps). Le faisceau de protons produit
par le cyclotron heurte différents types de cibles produisant des
faisceaux de neutrons, de pions et de muons, pouvant être utilisés pour effectuer différents types d'expériences. Plus récemment, ISAC (pour " Isotope Separation and ACceleration ", ou
séparation et accélération d'isotopes) permet la production de
faisceaux d'ions exotiques (isotopes) parmi les plus intenses au
monde, faisant de TRIUMF un des centres de recherche mondiaux pour l'astrophysique nucléaire.
Le Conseil national de recherches du Canada exploite le Centre
canadien de faisceaux de neutrons (CCFN) à Chalk River.
L'installation comprend 5 instruments thermiques, qui se trouvent dans le réacteur NRU. L'Université de Western Ontario à la
tête d'une demande impliquant 12 autres universités a obtenu
récemment des fonds de la Fondation canadienne pour l'innovation en vue d'installer au CCFN un nouveau réflectomètre à neutrons de calibre mondial. Le réflectomètre devrait entrer en service en 2007, afin de soutenir une vaste gamme d'applications
des nanotechnologies qu'il était impossible de réaliser auparavant au Canada.
Le CCFN est exploité principalement comme une installation
accessible aux utilisateurs de divers pays et soutenue en partie
par la subvention Accès aux grandes installations (AGI) octroyée
par le Conseil de recherches en sciences naturelles et en génie
(CRSNG), qui est administrée par l'intermédiaire de l'Université
McGill.
Les neutrons, qui ont un moment magnétique (particules à spin
½), sont idéaux pour étudier les structures et les fluctuations
magnétiques. L'article de Fritzsche et coll. décrit de quelle
manière les neutrons polarisés sont utilisés pour étudier les films
minces magnétiques et les multicouches, alors que Buyers et
Yamani expliquent le rôle crucial qu'ont joué les neutrons dans
la découverte des nouvelles phases de la matière dans les systèmes à fossé quantique, un système de fermion lourd à forte
corrélation et des supraconducteurs. L'article de Wiebe sur
les neutrons et les muons comme étude supplémentaire sur le
magnétisme exotique et la supraconductivité contribue à la connaissance des neutrons et du magnétisme.
En raison de la facilité de son utilisation et du renouvellement
rapide des échantillons, la diffusion des poudres est l'une des
techniques les plus couramment utilisées pour étudier les propriétés structurales des matériaux. Les diffractomètres à neutrons, bien qu'ils soient moins accessibles que les diffractomètres
à rayon X, sont des instruments d'une très grande utilité que l'on
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PRÉFACE
trouve dans pratiquement tous les grands laboratoires de neutrons. Il n'est par conséquent pas étonnant de voir que 4 articles
publiés dans ce numéro soient basés sur une telle diffusion de
neutrons dans le cadre d'études scientifiques (Greedan,
Swainson, Torrie et coll., et Huot et coll.).
faces enfouies ainsi que les processus se produisant à la surface
comme la corrosion. Dans leur article, Tun et coll. présentent
leurs études de réflectométrie des neutrons des couches d'oxyde
passive appliquées au titane et au zirconium, des matériaux
intéressants pour l'industrie nucléaire.
Avant que n'advienne la diffusion de neutrons à petit angle
(DNPA), les études sur la structure des polymères étaient pour
la plupart limitées aux techniques de diffusion de la lumière et
de diffusion radiologique à petit angle. Le seul rôle de la DNPA
en ce qui a trait aux matériaux hydrogéniques réside dans la différence entre la longueur de diffusion cohérente de l'hydrogène
(bH = -0,37 × 10-12 cm) et du deutérium (bD = 0,67 × 10-12 cm). Cet
écart entraîne une différence marquée dans la diffusion par la
poudre (contraste) entre les unités d'hydrogène et les unités
monomères deutérées. Le Canada bénéficie de l'expertise de la
DNPA bien qu'il ne possède pas un instrument approprié pour
la DNPA. Dans leurs articles, Katsaras et coll. et Frisken mettent
en évidence la versatilité de la DNPA et la puissance de la variation des contrastes.
En mars 2006, le CCRS et la communauté canadienne des cristallographes ont franchi une étape importante avec les premiers
résultats obtenus par le faisceau lumineux du Canadian
Macromolecular Crystallography Facility - CMCF (installation
canadienne de cristallographie macromoléculaire). Les images
étaient celles d'un cristal de protéine, la PEP carboxykinase. Dans
ce numéro, Grochulski, Blomqvist et Delbaere de l'équipe du
CMCF présentent en détail le faisceau lumineux 08ID-1 utilisé
pour recueillir les données. Le rapport de Grochulski et coll. est
suivi d'articles rédigés par Tse et Klug, et Mazzanti et coll. portant
sur divers sujets, tels que la spectroscopie des phonons par
rayon X et des études par rayons X du chocolat soumis à une
pression de cisaillement émis par synchrotron.
La réflectrométrie des neutrons est une technique utilisée pour
étudier la structure des surfaces des films minces et des inter-
John Katsaras
Conseil national de recherches du Canada, Chalk River
Rédacteur en chef invité
ERRATUM
IN MEMORIAM - PHILIP R. WALLACE
P.R. Wallace on the band structure of graphite was published in
1949. The year should have read 1947.
2006 JULY/AUGUST PIC
This obituary, which was printed in the 2006 July/August issue,
contains a factual error.
Using a quote from an article by MMR Williams published in Prog.
Nucl. Energy in 2000, it was stated that the now classic paper by
This Physical Review article is now gaining great prominence
because of graphene, the name given to a single sheet of graphite.
The material’s unique band structure and ease of fabrication are
creating great enthusiasm for its potential in generating exciting
new quantum physics and new electronic applications.
LETTER / COMMUNICATION
BOOK REVIEW - “AGAINST
THE
TIDE”
2006 JULY/AUGUST PIC
In a recent volume of your Journal (Vol. 62, No. 4, 2006) there
appeared what seemed at first to be a long-delayed review of my
autobiography, Against the Tide (IOP Publishing, 2000), written by
Professor T.W. Johnston, a plasma physicist, researching into controlled fusion in the Université du Québec. In his first paragraph his
‘verbatim’ quotation from the Publisher's review is rather spoilt by
some undisclosed editing, for example by removing references to
Rhodes and to the Chairman of Oxford University's Mathematical
Institute. However by his third paragraph it is clear that the Professor
is really engaged on another mission of much greater importance.
Now I have admired and frequently referenced the comprehensive
work The Particle Kinetics of Plasmas (1964), written by the Professor
and two past colleagues, Drs. Shkarofshy and Bachynsky, although
disappointed by some of the unphysical mathematical constructs in
Chapters 9 and 10. These speculations help me to understand why,
instead of genuinely reviewing my autobiography and giving the
Reader even the briefest explanation of my tokamak theory, the
Professor diverts attention to my incidental and entirely orthodox
statements about plasma pressure. But after this detour he enthusiastically joins the Fusion Lobby with an ex cathedra declaration that my
theory of transport is completely wrong. It seems that the wide range
of observational support that he could have hardly avoided seeing,
counts for nothing − not even a single mention in dispatches!
So what is wrong with Chapter 10, entitled “Collisionless Plasmas in
Strong Magnetic Fields"? By Newton's second law of motion, particles
that do not collide conserve momentum and can not exert any sort of
force. But scalar pressures parallel and perpendicular to the magnetic field are found in this “overreached" chapter, spoiling an otherwise
scholarly text. There is no problem with magnetic pressure and even
if the perpendicular pressure pz is somehow due to magnetic pressure in disguise, what about pk2? Particles moving parallel to the magnetic field hit nothing and are not affected by the magnetic field B, so
where is the pressure force? Dr. Ware's biassed review of my
Principles of Magnetoplasma Dynamics (OUP, 1987), to which
Professor Johnston alludes with approval, was also preoccupied with
denying Newton's second law (see Article 5 on my website).
Incidentally there is a spectacular blunder in Chapter 9; the free space
permeability μ0 and a magnetization current jm are not good bed-fellows; recall that jm = L H M where M / B(μ0-1 - μ-1).
Professor Johnston confidently asserts “The equations used for basic
tokamak theory (including neo-classical theory) are derived from fundamental kinetic equations by very careful approximation and there is
no justification for inserting such ad hoc (more properly ad hominem)
forces derived from one man's intuition". Logical positivism in attack
mode! The positivist Pierre Duhem was probably too polite to use similar language about Maxwell's displacement current, imaginatively
derived from elasticity theory, but Maxwell's great success with the
speed of light must have annoyed him. See pp. 80-86 of The Aim and
Structure of Physical Theory, Atheneum, New York, 1962.
(cont’d on pg. 232)
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2007 CONGRESS
2007 CAP CONGRESS
Joint with CLS Users’ Advisory Group
Meeting, 2007 June 15-17
June 17-20
University of Saskatchewan
Saskatoon, Saskatechewan
CALL FOR ABSTRACTS
The 2007 congress of the Canadian Association of Physicists (CAP) will be held from June 1720, 2007 at the University of Saskatchewan in Saskatoon, Saskatchewan, in conjunction with
the CLS Users’ Advisory Group meeting from June 15-17. The CAP Divisions are working hard
to establish a very exciting program with talks and posters in the following topics of physics :
Medical and Biological Physics
Nuclear Physics
Optics and Photonics
Particle Physics
Physics Education
Plasma Physics
Theoretical Physics
Women in Physics
Atmospheric and Space Physics
Atomic and Molecular Physics and Photon
Interactions
Condensed Matter and Materials Physics
History of Physics
Industrial and Applied Physics
Instrumentation and Measurement Physics
Abstract submission forms can be found through the CAP's website
at http://www.cap.ca
Deadline: March 1, 2007
The HERZBERG MEMORIAL PUBLIC LECTURE, to be held on Sunday evening, June 17th, will be given
by the 2001 Nobel Prize winner, Dr. Carl E. Wieman, of the University of Colorado (as of January
1, Dr. Wieman will be located at the University of British Columbia).
Bookmark the CAP's congress site and keep visiting for details of the technical sessions, invited speakers, and other congress arrangements as they unfold.
WE
LOOK FORWARD TO SEEING YOU IN
SASKATOON
IN
JUNE !!
FUTURE CAP CONFERENCES
2008 Annual Congress
June 8 - 11, 2008 (to be confirmed)
Laval University, Québec, QC
WWW.CAP.CA
230
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IN
CANADA
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CONGRÈS 2007
CONGRÈS DE L’ACP 2007
17-20 juin
Université de la Saskatchewan
Saskatoon, Saskatechewan
Conjointement avec la réunion du
<< CLS Users’ Advisory Group >>,
15-17 juin 2007
APPEL DE RÉSUMÉS
Le congrès de 2007 de l’Association canadienne des physiciens et physiciennes (ACP) se tiendra du 17 au 20 juin 2007 à l’Université de la Saskatchewan, à Saskatoon, Saskatchewan. Ce
congrès aura lieu conjointement avec la réunion du << CLS Users’ Advisory Group >> du 15 au
17 juin. Les divisions de l’ACP travaillent fort à établir un programme très excitant, comprenant
des exposés et des affiches sur les sujets suivants en physique :
Physique atmosphérique et de l’espace
Physique atomique et moléculaire et interactions
aves les photons
Physique de la matière condensée et matériaux
Histoire de la physique
Physique industrielle et appliquée
Physique des instruments et des mesures
Physique médicale et biologique
Physique nucléaire
Optique et photonique
Physique des particules
Enseignement de la physique
Physique des plasmas
Physique théorique
Les femmes en physique
On peut trouver les formulaires de soumission de résumé sur
le site de l’ACP à l’adresse http://www.cap.ca
Date limite : 1er mars 2007
Le CONFÉRENCE PUBLIQUE COMMÉMORATIVE HERZBERG, qui aura lieu le dimanche soir, 17 juin, sera
donnée par le récipiendaire du Prix Nobel en 2001, Dr. Carl E. Wieman, de l’Université du
Colorado (dès le 1er janvier, il sera à l’Université de la Colombie-Britannique).
Ajoutez le site du congrès de l'ACP à vos signets et venez y consulter les détails des sessions techniques, la liste des conférenciers invités et les renseignements pratiques à mesure qu'ils se complètent.
AU
PLAISIR DE VOUS ACCUEILLIR À
SASKATOON
EN JUIN!!
PROCHAINS
CONGRÈS DE L’ACP
Congrès annuel 2008
8 - 11 juin 2008 (à confirmer)
Université Laval, Québec, QC
WWW.CAP.CA
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NEWS
NEWS / INFORMATIONS
INTERNATIONAL UNION OF PURE AND APPLIED PHYSICS
- YOUNG SCIENTIST PRIZE IN COMPUTATIONAL PHYSICS
We would like to draw attention to the Young Scientist Prize in
Computational Physics which has recently been established by the
C20 Commission on Computational Physics of IUPAP. The award
is aimed at recognising outstanding achievements of scientists at
an early stage of their careers in the broad field of computational
physics.
Name of the Award:
International Union of Pure and Applied Physics Young Scientist
Prize in Computational Physics
Frequency/Venue:
C Triennially, up to three International Union of Pure and Applied
Physics [IUPAP] Young Scientist Prizes in Computational
Physics will be awarded.
C They will be announced and presented at the annual Conference
on Computational Physics (CCP).
C It is intended that one award be made each year. However, in
any given year, the selection committee may, at its discretion,
may decide not to make an award. If so, multiple awards may be
made in the following year.
C [It is proposed that the first award be made at CCP2007 in
September 2007, in Brussels]
Criteria for selection:
C The recipients of the awards in a given year should on January 1
of that year have a maximum of 8 years of research experience
(excluding career interruptions) following their PhD.
C The recipient should be the principal performer of original work
of outstanding scientific quality in Computational Physics.
C A previous recipient will not be eligible for another award.
Nomination procedure:
C The awards will be advertised electronically by the C20 Commission on its web page [see www.iupap.org] and elsewhere.
C The deadline for nominations is 1st March.
C Self-nominations will not be considered.
C Nominations shall be made to the Chairman of the C20
Commission by electronic mail [[email protected]] and
should include the following:
B A letter of not more than 1,000 words evaluating the nominee's achievements and identifying the specific work to be
recognised.
B A Curriculum Vitae including all publications.
B A brief biographical sketch.
The selection committee:
The selection committee consists of the Members and Associate
Members of the C20 Commission The selection committee may
consult with appropriate external assessors.
Type of Awards:
C The Awards will be US$1000 each, plus a medal and certificate
to be provided by IUPAP.
C The award money will normally be given as a contribution
towards the expenses for attending the CCP.
C The winner will be invited to present a paper at the CCP.
LETTER / COMMUNICATION (CONT’D FROM PG. 229)
These so-carefully derived equations yield energy losses ~ 10-4 times
those observed − the so-called neo-classical ‘correction' reduces this
factor to a mere 10-2 or so. But are the fundamental equations really
to be trusted? Boltzmann's famous kinetic equation is both fundamental and wrong − at least for the purpose of finding energy losses from
tokamaks. His collision integral is valid only to first order in the
Knudsen number expansion; at higher orders the hypothesis of
molecular chaos on which he based his theory fails. See Article 3 on
my website.
Turbulence is the scapegoat for fusion theorists' 40-year failure to discover why − so far as conserving mass and energy is concerned −
tokamaks resemble canoes made from barbed wire. There are several varieties of turbulence and all need the help of ‘adjustable' constants; but what is wrong with the turbulence escape? Just the incidental fact that it is not supported by observations. Sufficient microturbulence to increase the thermal diffusivity in the electron gas by the
required several orders of magnitude would, by Spitzer's extension of
Lorentz's theory of conductivity, have a similar effect on electrical
resistivity, yet observations of the voltage drop around a tokamak
torus give values only a little larger than predicted by classical theory.
A pity, for by choking the toroidal current to negligible values, a good
dose of turbulence would have put paid at an early stage to tokamak
research and saved the vast sums of public money being lavished on
this 50 year-old international sport, speculated to deliver energy to the
grid as soon as 2045. The only recognition of this simple fact about
turbulence that I have found is in Kikuchi et al. (Nuclear Fusion, 30(2),
341, 1990); the authors come very close to heresy. Referring to observations in tokamak JT60 they remark: “However, it is surprising to
observe such classical behaviour in the ‘diffusion-driven' current when
other transport coefficients are anomalous."
Your reviewer claims that my theory involves “major surgery of the
fundamental equations." It certainly does not; all the basic MHD equa-
232
PHYSICS
IN
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tions are unchanged. He failed to notice that I have merely extended
the usual constitutive equation (Fourier's law) for the heat flux vector
q − an approximation valid to first order in the Knudsen number Kn −
to second-order in Kn to obtain for either of the ion or electron fluids,
q = − κ∇T −
o
5 kB p
τ B × ∇ v i ∇T,
2
2Q B
(1)
where LE v is the deviator of the fluid velocity gradient, kB is
Boltzmann's constant, p is the pressure, Q is the particle charge, B is
the magnetic induction, T is the temperature and τ is related to the
bounce time of the particles trapped in the tokamak field. In tokamak
conditions the last term in (1) is dominant by orders of magnitude and
allowed me to explain many apparently distinct tokamak observations:
6H
thermal diffusivity, energy confinement time, L and H modes, the L6
transition, internal transport barriers, sawtooth oscillations, major disruptions, etc., with numerical values close to those observed without
any empirical fiddling. No other theory manages to explain convincingly any of these phenomena. There is a corresponding successful
theory for particle diffusion, based on second-order (in Kn) viscosity.
See Theory of Tokamak Transport, Wiley-VCH, 2006, the Preface
of which is on my website.
For a brief account of the fusion saga including the physical principles
involved in my theory, the Reader is referred to Article 1 on my website, which I wrote for the 2006 Balliol College Annual Record, the
Editor of which insisted that it should be intelligible even to classicists
and other arts graduates. To see how the lieutenants of Fusion
Research protect their status and incomes, Article 9 is instructive.
There is much more to say of course, but I shall close by thanking the
Journal Editor for accepting this rebuttal and Professor Johnston for
the glimpse of my obituary he allowed me in his final paragraph.
L. C. Woods
Emeritus Fellow of Balliol College, Oxford, UK
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LA PHYSIQUE ET L’ÉDUCATION ( SMALL-ANGLE NEUTRON ... )
SMALL-ANGLE NEUTRON SCATTERING
AND BIOMOLECULES
by J. Katsaras, T.A. Harroun, J. Pencer, T. Abraham, N. Kučerka and M.-P. Nieh
S
oft materials, both polymeric and biologically relevant, are
“heavy” atoms. In the case of polymeric materials, neurich in hydrogen. By coincidence, neutrons have the unique
trons are used to precisely locate hydrogen atoms [9,10].
capability of scattering differently from hydrogen (coherent
scattering length of hydrogen, bH = -0.37 × 10-12 cm) com(iii) 1H has a negative scattering length giving it “contrast”
pared to its isotope deuterium (bD = 0.67 × 10-12 cm). As a
when surrounded by other, positive scattering length
result of this marked difference in scattering power (contrast)
atoms. For biological samples intrinsically rich in hydrobetween native hydrogenated materigen, judicious substitution of 2H for 1H
als and their counterparts synthesized
a powerful method for selectAs a result of a marked dif- provides
from deuterated monomer units, neuively tuning the contrast of a given
tron scattering techniques have proven ference in scattering power macromolecule. By doing so, one can
to be powerful tools for the study of
accentuate, or nullify, the scattering
soft condensed matter systems. Here, (contrast) between native from particular parts of a macromolecuwe will discuss the small-angle neu- hydrogenated materials and lar complex. This powerful technique is
tron scattering (SANS) technique,
commonly referred-to as “contrast variwhich is presently playing a pivotal their counterparts synthe- ation” [11-13].
role in extracting unique structural sized
from
deuterated
information from intrinsically disor(iv) Neutron energies are similar to the
monomer units, neutron
dered systems.
energies of atomic and electronic
processes, i.e. meV to eV range.
scattering techniques have
This allows for the study of the
NEUTRONS
proven to be powerful tools
various dynamic properties (i.e.,
Neutrons are electrically neutral, subtranslations, rotations, vibrations
atomic, elementary particles, found in for the study of soft conand lattice modes) exhibited by
all atomic nuclei, except hydrogen densed matter systems.
molecules and eV transitions with(1H). They are approximately 1,840
in the electronic structure of matetimes more massive than an electron
rials [14-16].
and have a nuclear spin of 1/2. Neutrons are only stable
when bound by an atomic nucleus, while unstable free neu(v) Because they possess a magnetic moment (spin 1/2 partrons have a mean lifetime of approximately 900 s, decaying
ticles), neutrons are ideally suited to the study of magnetinto a proton, an electron, and an antineutrino [1,2].
ic structures (short- and long-range) and short wavelength magnetic fluctuations. It is important to note that
Because neutrons interact with atomic nuclei the scattering
the cross-sections for magnetic scattering are of the same
“power” (cross-section) of an atom is not strongly related to
magnitude to those for nuclear scattering [17-18].
its atomic number. Neighbouring elements in the periodic
table can therefore, have substantially different scattering
SMALL ANGLE NEUTRON SCATTERING (SANS)
cross sections [3]. More importantly, the interaction of a neutron with the nucleus of an atom allows neutrons to interact
Small angle neutron scattering (SANS) probes structure
differentially with an element’s isotopes. The classic example
in materials of length scales ranging from tens of angstroms
is the isotopic substitution of 1H for deuterium (2H) in poly(10-9 m) to hundreds of nanometers (10-7 m) [19]. The length
[4,5]
meric materials
. As a result of their intrinsic properties,
scale, d, is determined by the neutron wavelength, λ, and the
neutrons are used as follows:
scattering angle, θ, through the relationship
(i) Since they interact weakly with atomic nuclei, neutrons
are highly penetrating. This feature allows neutrons to
probe samples in complex sample environments, without
the need to engineer neutron “windows” or ports into the
sample enclosure. This enables the measurement of bulk
processes under realistic conditions [6-8].
(ii) Because the scattering ability of an atom is not strongly
related to its atomic number, neutrons are used extensively to locate “light”, low atomic number atoms among
λ = 2d sin θ/2,
John Katsaras <[email protected]>a,b,c,
Thad A. Harrounc, Jeremy Pencera, Thomas Abrahama,
Norbert Kučerkaa, and Mu-Ping Nieha; aCanadian Neutron
Beam Centre, National Research Council Canada, Chalk
River Laboratories, Chalk River, ON, K0J 1J0;
bBiophysics Interdepartmental Group, Guelph-Waterloo
Institute for Physics, Guelph, ON, N1G 2W1; cDepartment
of Physics, Brock University, St. Catharines, ON, L2S 3A1
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commonly referred-to as Bragg’s Law. Through the use of
cold (i.e. long wavelength) neutrons and the appropriate
beam collimation, length scales approaching tens of micrometers are possible [20,21].
In general, SANS can provide information regarding a particle’s size and shape, distribution of scattering inhomogeneities, conformational changes and molecular associations in solution. More importantly, because of the properties of neutrons individual components within a macromolecule can be systematically manipulated either through isotopic labelling or the judicious use of solvents. Below we will
discuss SANS instrumentation and provide a few examples
of SANS data obtained from lipid/water and
surfactant/water systems.
SANS INSTRUMENTATION
Figure 1 shows a schematic of a typical SANS instrument
located at a neutron source capable of producing long wavelength (λ ~ 5 – 20 Å) or commonly referred-to, “cold neutrons” [22]. Velocities (i.e., wavelengths) of the incoming neu-
trons are chosen by a mechanical velocity selector, basically, a
high-speed rotor. The helically twisted rotor blades are coated with 10B, a neutron absorbing material, and reasonably
monochromatized neutrons (bandpass, Δλ/λ of ~ 10%) are
obtained by varying the rotor speed (revolutions/minute,
rpm). Those neutrons whose velocities are not synchronized
to the rotor speed are absorbed by the 10B coated blades.
Monochromatic neutrons are then transported over meters
and are collimated through a series of nickel-coated guides,
which take advantage of the wave-like properties of neutrons.
The propagation characteristics of neutrons involves the
refractive index (n) of the medium. Since the critical angle (θc)
depends on the refractive indices of the media that the neutrons traverse, when nmedium < nair neutrons are transported
along the length of the guide by a mechanism known as
total external reflection (θincident < θc).
For neutrons
n = 1 – (λ2p/2π) and the scattering length density, ρ is equal
to Σbi/V, where bi is the coherent scattering length and V is
the sample volume. Neutron guides are made of optically flat
glass whose interior is generally coated with nickel or its isotope 58Ni (larger θc and increased Δλ/λ). Since, for neutrons,
the index of refraction of 58Ni is slightly less than one, then
all neutrons with an angle < θc (i.e., < 0.5° for λ = 5 Å neutrons) are transported.
Recently developed supermirrors made up, for example,
of Ni/Ti multilayers can increase the effective θc by up to
a factor of 3, compared to pure Ni [23]. They do so not only
by utilizing the total external reflection component, but also
the superimposed constructive interference (Bragg reflection)
from the successive layers of Ni, effectively extending
the plateau of total external reflection. The desired energy
neutrons impinge on the sample, which when scattered, are
usually detected by a 3He-filled two-dimensional (2D) detector.
SANS INFORMATION AT A GLANCE: FRACTAL
DIMENSIONALITY
Fig. 1
234
Schematic of a typical SANS instrument utilizing
long wavelength or commonly referred-to cold neutrons. Velocities (i.e., wavelengths) of incoming neutrons are chosen by a mechanical velocity selector (a).
For a given cylinder length, L, and a spiral pitch, p, if
the cylinder spins on its axis at an angular velocity ω
ω/2π
π) are transmitonly neutrons of velocity, V (= pω
ted. Reasonably monoenergetic cold neutrons are
transported over meters and collimated through a
series of nickel-coated or supermirror (e.g., Ni/Ti
multilayers) guides (b). The evacuated guides, which
transport cold neutrons via total external reflection,
are made of optically flat glass and their interiors are
coated with either nickel, or its isotope 58Ni (larger
λ), or multilayers of
critical angle, θc, increased Δλ/λ
Ni/Ti, which offer an even greater θc. Neutrons then
interact with the sample (c), which scatters neutrons
usually detected by a 3He-filled 2D detector (neutron
+ 3He 6 3H + 1H + 0.76 MeV) (d). The flight path in
which the 2D detector is housed is evacuated, resulting in a reduced background.
PHYSICS
IN
CANADA
For objects with a radius of gyration, RG , and Q << 1/RG
where Q = 4π/λ sin θ/2, plotting ln[I(Q)] vs Q2 results in a
straight line of slope –RG2/3, commonly referred to as a
Guinier plot. However, when Q >> 1/RG I(Q) decays as Q-α,
where α is the fractal dimension of the scattering object. In
this case, fractal refers to a complex structure made up of geometrical objects (self-similarity). The magnitude of α permits
for the geometry (i.e. morphology) of the scattering object to
be determined. In the case where the Q-range of the scattering data is sufficiently large (over one decade in Q) [24], one
can estimate α by simply determining the slope of the line
from a log-log plot of I(Q) vs Q. Table I shows the fractal
dimensions corresponding to various morphologies adopted
by biomolecules and polymeric systems.
SANS can also be used to characterize the stability of biological membranes interacting with additive molecules. Of special interest are pharmacologically important molecules that,
in appropriate concentrations help to either stabilize the lipid
bilayer or cause it to undergo structural change (e.g., lamellar
to hexagonal transition). For example, non-ionic surfactant
molecules such as, N-dodecyl-N,N-dimethylamine (DDAO)
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destabilize
dioleoyl
phosphatidylcholine
(DOPC) bilayers forming mixed micelles whose
shape changes, as a function of increasing
DDAO concentration, result in rod-like particles
(e.g., tubular or cylindrical micelles) and hard
sphere objects (e.g., globular micelles) [25].
TABLE 1
lipid, such as dihexanoyl
phosphatidylcholine
FRACTAL EXPONENTS FOR VARIOUS
(DHPC) [44,45]. In this sysMORPHOLOGIES
tem, a typical saturated
acyl chain lipid, such as
dimyristoyl phosphatidylcholine (DMPC, di-14:0
hydrocarbon
chains),
MORPHOLOGIES OF “BICELLE” MIXforms a disk-shaped bilayTURE LIPIDS DETERMINED BY SANS
er whose edges are stabiAmphipathic phospholipids are one of the main
lized by a curved monolaycomponents of biological membranes. They are
er of detergent [46]. Since
composed of hydrophobic fatty acid chains and
their discovery, bilayered
hydrophilic headgroups (Fig. 2), and along with
micelles have been used in
cholesterol are the primary constituents of cell
a number of studies
membranes. In purified forms, lipid/water sysattempting to elucidate the
tems form a variety of interesting structures
structure of proteins under
(e.g., lamellar, cubic and hexagonal phases,
physiologically relevant
micelles, etc.) (Fig. 3) which for a number of reac o n d i t i o n s [39,45,47,48].
sons have been the focus of both experimenHowever, as we will show, the bilayered micelle morpholotal [26-32] and theoretical interest [33-38]. Many of these strucgy is just one of many structures that these lipid mixtures are
tures exhibit features on length scales ranging from nanomecapable of adopting.
ters to microns.
In the recent past there has been a great deal of scientific
activity in a system forming bilayered micelles, or commonly
referred-to ‘‘bicelles’’ [39-40]. As we shall show below, neutron scattering has proven extremely useful in characterizing
these systems. Although bicelles were commonly formed in
aqueous solutions of ionic surfactants and alcohols [41-43], for
biologists a more pertinent system is where the detergent
molecules have been substituted by a short chain phospho-
Fig. 2
Chemical composition and space-filling
model of 1-stearoyl-2-oleoyl-sn-glycero-3phosphocholine (18:0-18:1 PC). This lipid is
composed of a hydrophilic phosphorylcholine headgroup, a glycerol backbone, and
two hydrophobic hydrocarbon chains.
Fig. 3
COMPLETE UNBINDING OF LAMELLAE:
FORMATION OF UNILAMELLAR VESICLES
Figure 4 shows SANS profiles of varying wt% DMPC/DHPC
lipid mixtures doped with the negatively charged lipid,
dimyristoyl phosphatidylglycerol (DMPG) [49]. A 25 wt%
sample was diluted in single steps, at 45oC, to final wt% concentrations of 18.0, 12.5, 9.0, 5.0, 2.5, 1.25, 0.5, and 0.1. The
profiles for lipid concentrations, clp $ 2.5 wt% exhibit quasiBragg maxima, characteristic of equidistant lamellae (i.e.,
multibilayers). As a function of increasing amounts of water,
the lamellar repeat spacing (d-spacing) varied linearly with
changes in clp-1, from 104 to 1348 Å with the lamellar reflec-
Example morphologies adopted by lipids: (a) Prolate micelle;
(b) Inverse prolate micelle; (c) Hexagonal; (d) Inverted hexagonal (e) Micelle; (f) Inverted micelle; (g) Unilamellar vesicle; (h)
Bilayered micelle; (e) Bilayer; (j) Cubic.
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range, representative of isolated bilayers. The data
were fit using a model of noninteracting polydisperse ULV.
Fig. 4
SANS profiles of (DMPC/DHPC/DMPG) samples prepared and
diluted at 45 °C. For all samples, the molar ratios of ([DMPC]+
[DMPG])/[DHPC] and [DMPG]/[DMPC] were fixed at 3.2 and
0.01, respectively. Bragg maxima are evident for 2.5 wt % # clp #
25 wt % samples, the result of multibilayers with a precise lamellar periodicity, d-spacing. For samples < 2.5 wt %, the multilamellar stacks unbind forming variable size ULV. Note that the SANS
profiles decay monotonically and lack the oscillations which are
characteristic of monodisperse ULV.
So what happens if we take some of these morphologies, cool them down to 10oC and reheat back
to 45oC? Figure 5 includes SANS data of 1.25 and
2.5 wt% samples. At 10oC the data do not show
any sharp peaks and are well described by the
bicelle morphology. The data can be best fit to a
bilayered disk morphology using a combination of
the core-shell-discoidal (CSD) model and the
Hayter-Penfold structure factor, SHP(Q), resulting
in a disk core radius, R, of 590 and 220 Å for the
1.25 and 2.5 wt% samples, respectively. Not surprisingly, both samples have the same bilayer
thickness (42 Å). On reheating to 45oC, the lamellar morphology is recovered in the case of the
2.5 wt% sample. However, of greater interest is
that on reheating the 1.25 wt% sample to 45oC, the
scattering pattern shows an oscillatory behavior as
a function of Q, the fingerprint of monodisperse
ULV, instead of the monotonic decay seen initially
at 45oC. The data were fit to a ULV model with a
SHP(Q) structure factor and a Schulz size distributionyielding an average core radius<Ri>of ~300 Å,
a bilayer thickness of 33 Å, and a polydispersity of
0.14. Whereas the ULV were initially large and
highly polydisperse (Fig. 5), after temperature
cycling they became smaller and more monodisperse. The formation of polydisperse ULV from
lamellae is not surprising, since the unbinding of
the bilayers does not select any particular length
scale. However, the situation is very different
when ULV are formed from bicelles, whereby the
bilayered micelle morphology dictates the size of
ULV formed. Figure 6 pictorially summarizes the
various morphologies observed by Nieh et al. [49].
SANS AND CONTRAST VARIATION
Fig. 5
SANS profiles of 2.5 and 1.25 wt% samples prepared at 45°C (top
curve) on cooling to 10°C (middle), and on reheating to 45°C (bottom). The arrow represents the sequence of temperatures. The
lamellar phase is recovered in the 2.5 wt % sample, whereas for
the 1.25 wt % sample, initially polydisperse ULV become
monodisperse on reheating (profile exhibits an oscillatory behaviour as a function of Q). The solid lines are fits to the data.
tions moving systematically to lower values of Q. However,
at clp # 1.25 wt% the lamellar reflections disappear, the
result of a complete unbinding transition whereby, the
extended lamellar stacks have disintegrated forming variable
radii unilamellar vesicles (ULV). The scattered intensity for
clp # 1.25 wt% follows a Q-2 dependence over an extended Q
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For polymeric materials rich in hydrogen, the use
of contrast variation and SANS makes for a powerful combination. By judiciously exchanging the
molecule’s hydrogen atoms for deuteriums, or by
changing the solvent’s scattering length density
(ρ), one can enhance the “visibility” of a molecule’s
moieties. For example, the optimum contrast conditions for studying the overall bilayer structure
are a fully hydrogenated lipid in 100% D2O solvent. On the other hand a solvent composed of
50:50 D2O:H2O provides the best contrast for lipids
with perdeuterated chains while the same lipid in
a pure D2O provides information mainly about the
lipid’s headgroup. The data obtained from these
experiments can then be analyzed using either
model dependent or model independent methods.
A model independent method based on the Guinier approximation (i.e., low Q region) provides a reasonably straightforward procedure for extracting the bilayer’s structural parameters [13]. By analyzing the SANS data obtained at several different contrast conditions, the average bilayer scattering
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LA PHYSIQUE ET L’ÉDUCATION ( SMALL-ANGLE NEUTRON ... )
Fig. 6
Schematic summary of the morphological transformations observed by Nieh et al. [49]. On
diluting below a critical lipid concentration clpu at T > TM (chain melting transition of
DMPC), extended bilayer sheets unbind into a polydisperse ULV dispersion. On cooling
below TM and clpu $ clp $ 1.25 wt %, polydisperse ULV transform into an isotropic bicellar
solution, which on reheating to T > TM, gives rise to monodisperse ULV. For clp # 0.5 wt %
polydisperse ULV are trapped and cannot, at low T, transform into bicelles. Monodisperse
ULV can also be obtained by diluting the bicellar phase below clpu at T < TM, followed by
heating above TM. In the case of very dilute mixtures, i.e., clp # 0.1 wt % and T < TM, bilayered micelles do not reform. Instead, oblate ellipsoids are created. The dashed lines indicate
plausible transformations not probed by the experiments carried out by Nieh et al. [49].
length density is evaluated from the quadratic
dependence of the
intensity at the origin
versus the solvent scattering length density.
The radius of gyration
(RG) is evaluated from
the slope of the KratkyPorod plot and then
plotted against the
inverse of the difference
between the solvent and
bilayer average scattering length densities. In
such a graph, one can
obtain RG at infinitely
large contrast corresponding to a point at
the graph’s origin.
Compared to a single
SANS
measurement,
this value - obtained
from multiple contrast
variation experiments is a more precise measure of the bilayer’s
apparent thickness and
can be used to study the
relative changes in a
bilayer using a modelfree approach.
Contrast variation experiments analyzed using a
model-based approach enables one to increase
the number of independent model parameters
leading to more realistic models with better
resolved structural features. Scattering curves
obtained at different contrast conditions (Fig. 7)
are used to capture the different features of the
bilayer. A single molecular model of the bilayer
is then used to simultaneously fit the different
contrast scattering curves. This model is made
up of the probability distributions corresponding to the different functional groups (e.g.,
choline headgroup, hydrocarbon chains, etc.) of
a bilayer (inset to Fig. 7).
MORPHOLOGY OF GEMINI SURFACTANT AGGREGATES
Fig. 7
SANS curves obtained at different contrast variation conditions.
ULV composed of fully hydrogenated DPPC and DPPC with
perdeuterated hydrocarbon chains (d62-DPPC) were prepared in
three different D2O:H2O (100%, 70% and 50%) mixtures. A molecular model of the bilayer is shown in the inset to the figure. The
bilayer profile is represented by probability distribution functions
corresponding to solvent molecules, the PC headgroup, and the CH2
and CH3 making up the lipid’s hydrocarbon chains.
The aggregation behaviour of Gemini surfactants is another problem that has been examined
with SANS. Gemini surfactants are composed of
two or more pairs of hydrophilic and hydrophobic groups connected to each other with a spacer
(Fig. 8). In order to modify the surface tension of
a solution, only small amounts of Gemini surfactants are required as their critical micellar concentration (cmc) in aqueous solutions is much
lower than the cmc of conventional surfactants
having the same hydrophilic and hydrophobic
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PHYSICS AND EDUCATION ( SMALL-ANGLE NEUTRON ... )
Fig. 8
α’The molecular structure of α,α
[2,4,7,9-tetramethyl-5-decyne-4,7-diyl]
ω-hydroxyl-polyoxyethylene].
bis-[ω
groups. One example of what SANS can
achieve in studying the structure of such surfactant systems is the molecule α,α’-[2,4,7,9tetramethyl-5-decyne-4,7-diyl]-bis-[ω-hydroxyl-polyoxyethylene] (Fig. 8), which contains
10 ethylene oxide (EO) segments. Conclusions
from previous studies were that the system
underwent two possible transitions namely a
monomer 6 micelle I and a micelle I 6 micelle
II at 0.9 and 2 wt%, respectively [50-53].
However, recent SANS data, outlined below, Fig. 9
have contradicted these findings [54].
SANS scattering curves of Gemini surfactants at concentrations varying from 0.5 to 5 wt%.
Figure 9 shows SANS patterns for various surfactant concentrations [54]. At low concentration (0.5 wt%) and a Q-regime
of < 0.02 Å–1, I(Q) decays as Q-4 decay indicating the presence
of large particles (> 50 nm) in solution - denoted later on as
“clusters”. Between 0.03 and 0.1 Å-1 I(Q) plateaus and
decays as Q-2 for Q > 0.1 Å-1, characteristic of particles with a
much smaller length scale, possibly monomer surfactant molecules. As the surfactant concentration increases to 1 wt%, the
slope of the scattered intensity decreases at small Q values,
indicative of scattering contributions from larger sized “clusters”. Moreover, the intensity plateau starts to decay earlier
than that seen in the 0.5 wt% sample, implying that the smaller aggregates are getting larger at higher concentrations, presumably due to micellation. The SANS data of the 1 wt%
sample also shows a slight upturning at very low Q
(< 0.005 Å–1), implying either the coexistence of micelles with
small amounts of clusters, or that the size of the clusters, at
this concentration, are so large that they are beyond the
SANS detecting limit. Above 2 wt% this low Q behaviour
disappears completely, indicating that either the clusters
have become too large to detect or that they no longer exist.
Analysis of the SANS data can reveal the size and aggregation number of the surfactant. For Q· RG # 1 (corresponding
to a Q range of 0.01 < Q < 0.04 Å-1) I(Q) can be related to the
radius of gyration, RG , aggregation number, ns and the second virial coefficient, A2 (an index for interparticle interaction), as follows
238
PHYSICS
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Fig. 10 Zimm plot constructed for SANS data of surfactants
with concentrations between 1.1 to 5 wt%. The Q
values range from 0.008 to 0.1 Å-1.
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LA PHYSIQUE ET L’ÉDUCATION ( SMALL-ANGLE NEUTRON ... )
⎛ 1
+ 2A 2 φ +
) ⎜⎝ vsn s
φ
1
=
I( Q) Δρ 2 ( 1 − Q 2 R 2G / 3 +
∼
( 1 + Q 2 R G2 / 3 +
Δρ
)⎛
1
+ 2A 2 φ +
⎜
⎝ vs n s
2
⎞
⎟
⎠
⎞
⎟
⎠
6.
7.
8.
where νs is the volume of one aggregate and Δρ is the difference in scattering length density between the solvent
(i.e., D2O) and the surfactant. This equation is the basis of the
Zimm plot. By plotting the extrapolated φ=0 (justifies
neglecting intermolecular interferences i.e., structure factor)
values from φ/I(Q) versus Q2 plots, a straight line is obtained
9.
R G3
. On the other hand, by plotting
3 ⋅ v s n s Δρ 2
10.
the extrapolated Q=0 (justifies the treatment of intramolecular interference i.e. form factor) values, a line is obtained with
11.
slope
12.
with slope
2A 2
.
Δρ 2
A Zimm plot (Fig. 10) can therefore be
(cφ+Q2),
constructed with φ/I versus
where c is an arbitrary
constant allowing for the various φ lines to be separated. The
obtained RG , ns and A2 values for micelles are (14.7 ± 3.5) Å,
(17.2 ± 0.2), and (-2 ± 6) x 10-5 mol/cm3, respectively. From
the analysis of the SANS data the two morphological transitions identified were clusters/monomers 6 clusters/micelles
and clusters/micelles 6 micelles.
CONCLUDING REMARKS
With regards to polymeric materials, SANS is arguably the
single most important neutron scattering technique. It is
used routinely to probe the size, shape and conformation of
macromolecular complexes whose size ranges from ten to
one thousand Ångstroms. In the case of biologically relevant
materials, SANS is employed to study molecules and molecular assemblies under physiologically meaningful conditions, and also allows for the study of disordered materials
that are difficult or impossible to crystallize. It should be
pointed out that the use of SANS has flourished in the last
decade from the production of cold neutrons (i.e., 5 – 20 Å)
whose long wavelengths have greatly facilitated the interrogation of materials with large unit cells (e.g. proteins).
13.
14.
15.
16.
17.
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NEUTRONS AND TRANSITION METAL OXIDES:
A MATCH MADE IN HEAVEN
by J.E. Greedan
T
ransition metal oxides (TMO) have been perhaps the most
ments of similar Z due to the somewhat random variation of
intensively studied class of inorganic solids for the past two
b with Z. Several useful situations exist, for example the trio
decades. It is common to attribute this sustained interest in
of elements important in organic and bio materials
part to the high Tc cuprate phenomenon which emerged in
C (b = 6.65 fm), N (b = 9.36 fm) and O (b = 5.80 fm) and the
the late 1980’s and the colossal magneto resistance manthird row p-block elements Al (b = 3.45 fm), Si (b = 4.15 fm),
ganates of the 1990’s. In fact new oxide
P (b = 5.13 fm), S (b = 2.85 fm) and
materials seem to emerge at regular
(b = 9.58 fm). For the 3d transition
metal oxides Cl
intervals, for example the recent dis- Transition
elements a particularly useful pattern
covery of superconductivity in (TMO) have been perhaps exists as seen in Fig. 1, where the b’s for
NaxCoO2 • yH2O [1].
In addition
adjacent elements vary widely, permitTMO’s such as LiCoO2 are the basis of the most intensively stud- ting facile discrimination which would
the science and technology of lithium ied class of inorganic solids be very difficult with x-rays. Also, note
batteries as cathode or positive electhat, unlike for x-rays, neutrons can
trode materials, another very active for the past two decades. have negative scattering amplitudes.
research area, currently. Neutron scat- The pairing of transition The physical interpretation of this is
tering has been an essential tool for
that the neutron wave changes phase by
probing the complex physics underly- metal oxides and neutron π upon scattering. While neutron scating the behaviour of the materials diffraction extends back to tering lengths can not be calculated
which exhibit these phenomena but in
from first principles, all of the relevant
fact the pairing of transition metal the beginnings of neutron values have been measured and tabuoxides and neutron diffraction extends
lated [4]. Examination of such tables
beam science in the early shows
back to the beginnings of neutron
another useful property of neubeam science in the early 1950’s. 1950’s.
trons in that different isotopes of the
Significantly, the first experiments to
same element can scatter neutrons very
confirm Néel’s prediction of antiferrodifferently. The most famous example
magnetism were performed by Shull, Strauser and Wollan
is provided by the stable isotopes of hydrogen, 1H and 2H, for
using neutron powder diffraction on MnO, FeO, CoO and
which the scattering amplitudes are – 3.74 fm and 6.67 fm,
NiO in 1951[2,3]
respectively.
Why are neutrons so indispensable in the study of transition
metal oxides? The basic reasons stem from the composition
of the neutron and the nature of the interaction between neutrons and matter. The neutron is comprised of two down
quarks and one up quark, so, it is electrically neutral and has
a spin of one-half. Thus, neutrons are not scattered by electron density fields, as are x-rays, but are scattered only by
other nucleons, i.e. the nuclei of atoms and, as will be discussed shortly, by magnetic fields. Neutrons are, thus, highly penetrating and the scattering amplitude of the neutron nucleus interaction is to a first approximation independent of
atomic number but strongly dependent on the composition of
the nucleus of a given element. These amplitudes are called
“scattering lengths”, b. An important implication is that
“light” or low Z elements scatter neutrons as efficiently as
“heavy” or high Z elements. For example the elementally
averaged scattering lengths for O (b = 5.80 fm) and Ba
(b = 5.06 fm) or W (b = 4.77 fm) are comparable. [Note:
1 fm = 10-15m]. This is far from the case with x-ray scattering
where the amplitude depends directly on Z.
Even
7Li (b = – 2.22 fm), which is nearly invisible to x-rays, especially in a powder experiment, can be seen easily with neutrons. As well, it is often possible to discriminate between ele-
Fig. 1
Variation of the neutron scattering
length versus atomic number (Z) for
the 3d transition elements.
Due to the
above reasons
neutron diffraction is an
important tool
in the determination
and
refinement of
crystal structures, especially those containing both
light elements
(such as lithium,
boron,
oxygen, etc.)
and heavy elements
and,
J.E. Greedan <[email protected]>, Brockhouse
Institute for Materials Research & McMaster University,
Dept of Chemistry, London, ON L8S 4M1
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PHYSICS AND EDUCATION ( NEUTRONS AND TRANSITION METAL OXIDES ... )
where appropriate, elements with similar Z. For diffraction
studies the key quantity is the structure factor defined below
for both x-rays and neutrons:
F(hkl) = Σ fi exp 2πi(hxi + kyi + lzi)
[ x-rays]
F(hkl) = Σ bi exp 2πi(hxi + kyi + lzi) [neutrons]
where h, k and l are the Miller indices of a given reflection, fi
is the x-ray scattering amplitude, xi , yi and zi are the fractional coordinates of atoms within the unit cell and the summation, Σ, includes all atoms within the cell. At this stage it
should be noted that fi is strongly attenuated as the scattering angle increases but that bi does not change with angle
due to the fact that the nucleus is a point scatterer. In general the neutron diffraction pattern of a transition metal oxide
will contain more information than an x-ray diffraction pattern covering the same range of momentum transfer with
comparable resolution and overall counting statistics.
But, the singular reason that neutrons and transition metal
oxides constitute such a perfect pairing is that neutrons are
sensitive to the magnetic fields which result from the presence of unpaired electrons at the transition metal site. The
variety of magnetic phenomena which follow from the presence of these “local” magnetic moments and their interactions can be probed with unique precision and accuracy
using neutron scattering. The key quantity is the so-called
magnetic scattering length, p, defined as:
p = e 2γ/2mc2
where e and m are the electronic charge and mass, c is the
speed of light and γ is the nuclear magneton. In appropriate
units p = 2.696 fm, i.e., the same order of magnitude as the
nuclear scattering lengths of most elements and is interpreted as the scattering amplitude associated with a magnetic
moment of one Bohr Magneton at zero scattering angle. One
Bohr Magneton is the magnetic moment of one unpaired
electron spin. Thus, the total magnetic scattering amplitude
is obtained by multiplying by the magnetic moment for the
transition metal ion in question which for the spin only case
(usually a good approximation for 3d electrons) is pgSf and
where orbital magnetism cannot be ignored (usually the case
for 4f electrons), pgJJf. In these expressions g = 2.00, S is the
total spin quantum number, gJ is the Lande factor for f electron configurations and J is the total angular momentum
quantum number. The common factor, f, is called a “form
factor” and arises due to the fact that the magnetic fields are
electronic in origin and thus, there is a strong attenuation
with increasing scattering angle. This is similar to the situation for the atomic scattering factor in x-ray diffraction but in
most cases the attenuation is even more severe. Thus, given
that the amplitudes for the magnetic and nuclear scattering
are on the same scale for neutrons, the magnetic component
is easily detected. This is unlike the case for x-ray magnetic
scattering where the magnetic component is only of order
10-4 or less of the electron density scattering.
scattering angle and contributes to the background at low
scattering angles due to the strong influence of the form factor. When the local magnetic moments are ordered to form an
infinite magnetic lattice, i.e., long range magnetic order, the
magnetic scattering concentrates in Bragg peaks which reflect
the dimensions and symmetry of the “magnetic” unit cell.
One can define a magnetic structure factor as below:
F(hkl) = Σ pigSifi exp 2πi(hxi + kyi + lzi)
[magnetic neutron diffraction]
where the sum is now over all of the magnetic atoms in the
magnetic unit cell. The total neutron diffraction pattern
(assuming an unpolarized neutron beam) for a magnetically
ordered material such as a transition metal oxide is thus a
superposition of the chemical and magnetic contributions.
This is expressed in the equation below:
F2(hkl)Total = F2(hkl)Chem + q2F2(hkl)Magn
where the factor q2 = sin2 α and α is the angle between the
scattering vector (normal to the (hkl) plane) and the magnetic moment vector. This factor arises because what is actually
measured in a magnetic neutron diffraction experiment is the
component of S or J normal to the scattering vector. Often the
magnetic cell has either larger dimensions or different symmetry than the “chemical” cell, so the magnetic Bragg peaks
are easily detected. In principle both the chemical and the
magnetic structure, i.e. the magnitude and spatial distribution
of the magnetic moments, can be determined simultaneously
from a neutron diffraction pattern.
EXAMPLES FROM EXPERIMENTS AT THE
CANADIAN NEUTRON BEAM CENTRE AND/OR BY
CANADIAN SCIENTISTS
The following examples involve studies using the powder
neutron diffraction technique either carried out at the
[b]
[a]
Fig. 2
The details of the magnetic contribution depend on whether
the local magnetic moments are random or ordered in the
solid. The former case is called paramagnetism and the
resulting magnetic scattering is incoherent with respect to
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(a) The ideal cubic perovskite unit cell showing
the A-site ions at the corners (spheres) and the
octahedral coordination of the B-site ions (polyhedra). The oxygen atoms are small spheres at the
corners of the octahedra; (b) The unit cell of
La1/3NbO3 showing the ordering of La3+ vacancies and the doubling of the ideal perovskite cell
in one direction.
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Canadian Neutron Beam Centre or by Canadian scientists at
international sites concentrating mainly, but not exclusively,
on transition metal oxides. No attempt is made to provide a
comprehensive survey.
The structure of defect perovskites of the type Ln1/3MO3.
Perovskites represent a vast family of oxide materials of general composition ABO3. The ideal, cubic structure is shown
in Fig. 2a wherein the small cation B is octahedrally coordinated by O2-. These octahedra share corners to form a three
dimensional network and the large cations, A, reside in the
large interstices in this network. It has been known for some
time that some perovskites can tolerate an unusually large
concentration of vacancies on the A-site, for example materials of the type Ln1/3MO3, where Ln is a trivalent lanthanide
ion and M is a pentavalent transition metal ion such as Nb5+
or Ta5+ [5]. The earliest studies, using x-ray diffraction, indicated that the Ln3+ ions were not distributed randomly over
the A-sites in the pseudo-cubic cell but ordered to produce a
super cell requiring the doubling of one pseudo cubic axis as
shown in Fig. 2b. Fig. 3a shows an x-ray powder diffraction
pattern from a fairly recent study of La1/3TaO3 which is consistent with the structure shown in Fig. 2b [6]. In this case the
TaO3 octahedra are well-aligned with no
evidence of tilting away from the crystallographic axes, i.e., the Ta – O – Ta angles
are all exactly 90o or 180o. Of course the
x-ray data are sensitive to the very heavy
elements, La and Ta and there is relatively little information about the oxygen
atoms.
A very different picture arises when neutron diffraction data are considered as
done by Dr. C.A. Bridges and the author
at McMaster University [7]. In Figs. 3b
and 3c are displayed the neutron powder
diffraction data obtained for the very similar perovskite, Ce1/3NbO3, at a so-called
“spallation source”, the Intense Pulsed
Neutron Source at the Argonne National
Laboratory. Fig. 3b shows the result of a
refinement of the neutron data using the
model found from an analysis of the x-ray
powder data. Note the presence of several strong reflections which are not
accounted for by the model. Upon further analysis, it was discovered that these
new reflections result from subtle tiltings
of the NbO3 octahedra which leave the Fig. 3
Nb5+ and Ce3+ positions essentially
unchanged but which result in significant
shifts in the O2- positions relative to those
in Fig. 2b.
Fig. 3c shows a refinement of the neutron
powder data taking these octahedral tiltings into account and the fit is seen to be
much improved. In Fig. 4, the structures
derived from the x-ray data (4a) and the
neutron data (4b) are compared which
illustrate the subtle shifts in the oxygen positions which are
easily detected by neutrons but are essentially invisible to xrays.
Cation Order/Disorder in the Battery Cathode Material
Li(Ni1/3Co1/3Mn1/3)O2
As already mentioned, transition metal oxides are the cathode materials of choice in modern lithium battery technology. The first lithium battery to be a commercial success, marketed by SONY, uses LiCoO2 as the cathode or positive electrode. Lithium battery cathodes must be capable of storing
and releasing Li+ ions reversibly at room temperature as
illustrated by the electrochemical half reaction below:
LiCoO2 ] Li+ + “CoO2” + eIn practice not all of the Li+ can be extracted from LiCoO2 due
to the inherent instability of Co4+ under ambient oxygen partial pressure. The reversibility of this reaction is due to the
crystal structure which is of the layered NaCl type, Fig. 5a, in
which it is seen that the Li+ and Co3+ ions do not randomly
occupy the Na+ sites in the NaCl structure but instead order
in separate layers which are normal to the body diagonal of
the NaCl cell. Li+ is removed reversibly from the Li layers
and re-inserted during charging and discharging of the bat-
(a) Rietveld fit of x-ray powder diffraction pattern for La1/3NbO3 to the
model shown in Fig. 2b. The fit is clearly excellent [Ref 6, reproduced with
permission of Elsevier]; (b) Fit to the neutron powder data for Ce1/3NbO3
using the model derived from x-ray powder diffraction, illustrated in
Fig. 2b. The crosses are the data, the solid line is the fitted model, the vertical tic marks locate Bragg reflections and the bottom curve is the difference
between the data and the model. The inset shows major peaks not accounted for in the model and the fit is seen to be poor [Ref. 7, reproduced with
permission of the International Union of Crystallography]; (c) Fit to a model
derived from an analysis of the neutron diffraction data in which the NbO3
octahedra tilt along the unit cell axes and the light atoms (oxygen) move
from the ideal positions. The fit is seen to be much improved [Ref 7, reproduced with permission of the International Union of Crystallography].
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Fig. 4
(a) Projection of the crystal structure of
Ce0.33NbO3 from x-ray powder diffraction
data compared with the result from neutron
power diffraction. The Ce3+ and O2- ions are
shown as spheres while the NbO3 corner
sharing octahedra are shown in polyhedra
representation. Note the subtle tilting of the
octahedra which shift the oxygen atoms from
their idealized positions (Fig. 2b). These small
shifts are easily detected in the neutron experiment but missed in the x-ray data due to the
low scattering power of oxygen.
Fig. 5
tery. There are nonetheless many problems with LiCoO2,
due mainly to the toxicity and cost of Co but also to the instability of Co4+ as mentioned earlier, so there is considerable
impetus to replace this material. One promising candidate is
Li(Co1/3Mn1/3Ni1/3)O2. The structural questions concerning
this oxide revolve around issues such as whether the three
elements are randomly distributed within the transition
metal layer or ordered, whether the Li+ ions can mix with the
transition metal ions as is the case in LiNiO2 and whether
changes in structure occur when large amounts of Li are
removed. In fact two possible cation ordering schemes had
been proposed [8], which are illustrated in Fig. 5. This is of
course an ideal problem for neutron diffraction, given the
ease of discrimination among the three transition elements
(Fig. 1) and the Li+ positions can be determined with great accuracy as well. These
questions were recently addressed by
two Canadian groups, at NRC
(P.S. Whitfield and I.J. Davidson) and the
University of Waterloo (L.F. Nazar) [9,10]
using both synchrotron x-ray diffraction
and neutron diffraction methods.
By comparing calculated neutron diffraction patterns for the two cation ordering
schemes and a random model with the
observed data, the NRC group were able
to show, Fig. 6, that neither cation ordering scheme could be detected in the neutron data, indicating that long range
order does not exist. A more detailed
analysis of the results showed a very
minor, 2%, mixing between Li+ and Ni2+. Fig. 6
The Waterloo group studied the effect on
structure upon electrochemical removal
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The ordered NaCl structure of LiCoO2 is shown in (a). The
large open spheres are oxide ion, the smaller black and
hatched spheres are Co2+ and Li+. Two possible cation ordering schemes are proposed for Li(Ni1/3Co1/3Mn1/3)O2. In (a)
ordered occurs within the transition metal ion layers and in
(b) the transition metal ions order into separate layers [8]
[Ref. 9, reproduced with permission of Elsevier].
of Li, i.e. the structures of Lix(Ni1/3Co1/3Mn1/3)O2. An important result from their research that a structure change occurs
for very low Li contents (x = 0.04) as well as a significant contraction of the c-axis in strongly Li depleted phases. Both
observations explain the problems of reversibility and electrochemical cycling in this composition range.
Nonlinear Optical (NLO) Materials – Heavy Metal
Borates
This is an example from the laboratory of Professor J. Barbier
of McMaster University. In this group the crystal and structural chemistry of metal borates is investigated as part of a
search for new NLO compounds. For such materials a noncentrosymmetric structure is necessary and this often can be
achieved by incorporation of so-called “lone pair” cations
(a) Calculated neutron powder patterns for the cation ordered models for
Li(Ni1/3Co1/3Mn1/3)O2 and a random model compared with (b) the actual
data for λ = 2.37 Å neutrons. Clearly, there is no evidence for long range
cation ordering [Ref. 9, reproduced with permission of Elsevier].
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such as Pb2+ or Bi3+. A recent case concerns BaBiBO4
for which only powder samples were available [11].
The accurate location of the B atoms or even O in the
presence of very high Z elements such as Ba and Bi
presents a significant challenge. The heavy atoms
could be located from analysis of x-ray powder data
and approximate positions for B and O were also
suggested. However, a detailed structure refinement
could only be carried out using neutron data. (11B
enriched starting materials are generally available to
avoid the strong neutron absorber, 10B). The detailed
analysis was necessary to determine if the structure
was indeed non-centrosymmetric. Fig. 7 shows a socalled Rietveld refinement of the high angle neutron
data for BaBiBO4 and the resulting crystal structure. Fig. 8
Refinement in a non-centrosymmetric model was
shown to be slightly but significantly superior to that
in a centric model. This was confirmed by measuring
the optical properties in which a high second harmonic generation efficiency was observed.
Two possible models for the structure of ammonium cyanate,
NH4CNO, involving (a) N – H – O hydrogen bonds and (b) N –
H – N hydrogen bonds. The carbon atoms are not shown
[Ref. 14, reproduced with permission of the American Chemical
Society].
Hydrogen Bonding in Ammonium Cyanate, NH4CNO
While clearly not a transition metal oxide, this example is
nonetheless compelling in its illustration of the singular
power of neutrons to detect light atoms (H or D) and to distinguish between atoms of similar Z (N and O). This work is
a collaboration among several laboratories including that of
Prof. R.R. Tykwinski of the University of Alberta and
J.D. Dunitz of the ETH in Zürich [12]. The solid state transformation of ammonium cyanate to urea, H2N – C O – NH2,
observed by Wöhler, a pioneer of organic chemistry, was first
studied more than 170 years ago [13]. The mechanism is still
Fig. 9
Fig. 7
Rietveld refinement of the neutron powder diffraction pattern of BaBiBO4 on a non- centrosymmetric
model consistent with NLO properties and a model
of the refined structure. The Ba2+ and O2- ions are
shown as large and small spheres, the BO3 groups as
planar triangles and the unusual distorted five-fold
pyramidal site for Bi3+ is apparent [Ref. 11, reproduced with permission of Elsevier].
Rietveld refinement of neutron powder data for
ND4NCO. The refined structure is consistent with
the model in Fig. 8b, i.e., with N – H – N hydrogen
bonding [Ref. 14, reproduced with permission of the
American Chemical Society].
under investigation. It is of course important to understand
the structure of ammonium cyanate, the starting compound.
Two possible models for this structure, focussing on the environment about the NH4+ (ammonium) ion, are shown in
Fig. 8. There is a clear choice between (a) N – H – O and (b) N
– H – N hydrogen bonding linkages. Earlier studies of this
problem using laboratory x-ray powder data, supported by
calculations, had supported formulation (a) [14]. However,
subsequent analysis of high quality synchrotron x-ray data
showed that there was in fact no significant difference
between the two models and that x-ray data, even of the highest quality, could not solve this problem. To quote the authors
“Chastened by our experiences with the use of x-ray powder
diffraction to tackle this problem, we turned to neutron powder diffraction of a deuterated sample ND4NCO.” Fig. 9
shows the Rietveld refinement of the data on model (b).
Attempts to refine on model (a) were significantly inferior.
Thus, the solid state structure of ammonium cyanate has been
established, unequivocally, by neutron diffraction.
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MAGNETIC NEUTRON
DIFFRACTION
unit cell must be twice the value
of the chemical or nuclear unit
cell along the c-axis direction.
We conclude this survey of
The temperature dependence for
activity in Canadian neuthe strongest magnetic Bragg
tron powder diffraction
peak is shown in Fig. 12 to be
with some examples in
consistent with a Tc = 155K, just
which magnetic scattering is
at the peak in the χT vs T plot. A
highlighted. Most of these
Rietveld refinement of the full
involve projects from the
data set using models for both
author’s laboratory somethe crystal and magnetic structimes in collaboration with
tures yields the result shown in
others.
the inset in which it is seen that
within the layers the Fe and Re
Magnetic Structure of the
spins are coupled ferrimagnetiQuasi-Two Dimensional
cally while overall the layers are
Ferri-magnets, the Pillared
coupled antiferromagnetically.
Perovskites La5Re3MO16,
Magnetic moments in the
M = Mn, Fe, Co,Ni
ordered state can be refined for
each atom and are 1.53(13) μB for
Oxides of this unusual comRe5+ and 3.10(15) μB for Fe2+.
position were discovered
Note that the spins for both ions
about 10 years ago, indepoint directly along the c-axis
pendently,
by
two Fig. 10 The crystal structure of La5Re3MO16. Note the corner direction. This is determined by
sharing octahedral layers (light grey and dark grey)
groups [15,16].
While the
the absence of reflections of the
of composition Re5+M2+O77- which are pillared by
chemical formula appears
type (0 0 ½) which indicates that
edge-sharing dimeric Re2O99- units(black). The La3+
complex, the crystal strucions are shown as white spheres.
the moment direction must be
ture is easy to understand,
along c. Now, the bulk properties
Fig. 10. It consists of infinite
can be understood. Upon coolsheets of corner sharing octahedra, as found in the perovskite
ing
ferrimagnetic
correlations
develop within the layers as
structure of composition Re5+M2+O77-, which are “pillared”
manifested
in
the
sharp
increase
in χ just above 155K, below
by edge-sharing octahedral dimeric units, Re2O98-. The
which the layers begin to couple antiferromagnetically which
dimeric pillaring units are diamagnetic so the magnetic ions
gives rise to the sharp drop with decreasing temperature.
are in the perovskite-like layers which are > 10.3 Å apart. A
typical example is La5Re3FeO16. The bulk magnetic properThe work described above is exemplary of that which has
ties are very unusual, showing a sharp in the susceptibility
been carried out over a period of several years in the author’s
near 155K followed by a
sharp decrease. To understand the microscopic magnetic structure, neutrons
are needed.
Fig. 11 shows the low angle
part of the diffraction pattern for two temperatures,
160K, 13K and the difference plot for 13K – 160K.
Difference plots are very
useful devices for locating
magnetic reflections which
are present only below the
“critical” or ordering temperature, Tc. In the inset to
Fig. 11, four such magnetic
reflections are indexed.
That these peaks are seen
only at low scattering
angles is a reflection of the
form factor, f, discussed
previously. Note that the l Fig. 11
index is half integral, indicating that the magnetic
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Low angle part of the neutron powder diffraction pattern for La5Re3FeO16 at 160K, 13K
and the difference, 13K – 160K, showing the magnetic contribution and indexation of the
magnetic peaks. Note that the c-axis for the magnetic cell is doubled relative to the chemical cell.
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Fig. 13 (a) The “pyrochlore” lattice consisting of a three
dimensional array of corner-sharing tetrahedra;
(b) Geometric frustration of three spins on a triangle
and four spins on a tetrahedron.
Li+ into the cubic spinel LiMn2O4 as below:
Fig. 12 Temperature dependence of the strongest magnetic
reflection ( -1 +/-1 ½) showing Tc = 155K and the
refined magnetic structure of La5Re3FeO16 which
consists of ferrimagnetic Fe 8 Re 9 layers coupled
antiferromagnetically.
laboratory by undergraduates, exchange students, graduate
students and postdoctoral fellows including Thomas Langet,
Aurilien Gourrier, Andrew Green,
Chris Wiebe, Heather Cuthbert and
Lisheng Chi.
Unexpected Two Dimensional
Spin Correlations in the
Three Dimensional Frustrated
Spinel Oxide, Li2Mn2O4
This study was done in collaboration with Dr. A.S. Wills of
University College London. Spinel
oxides have attracted much attention lately in the context of the
study of geometrically frustrated
magnetic materials. The formula of
a spinel oxide is AB2O4 where A
ions occupy tetrahedral sites and B
ions octahedral sites in a cubic close
packed array of O2- ions. The B sites
form what is called a “pyrochlore”
lattice consisting of a three dimensional array of corner sharing tetrahedra as shown in Fig. 13. If the Fig. 14
nearest neighbour spin coupling
constraint is for anti parallel spins,
this lattice is “frustrated” in that
only 2 of the 4 spin pairings can be
satisfied simultaneously. As part of
a long established and ongoing
study of frustrated systems, the
spinel Li2Mn2O4 was prepared by
room temperature introduction of
Li – n butyl (hexane) + LiMn2O4(s) 6
Li2Mn2O4(s) + n- octane(hexane)
Thus, a metastable form of “LiMnO2” can be prepared with
Mn3+ ions on a slightly distorted (Mn3+, 3d4, is a so-called
Jahn-Teller ion and the structure is distorted from cubic to
tetragonal) pyrochlore lattice. The stable form of LiMnO2 has
a different structure. Again the bulk susceptibility of this
material is unusual,with a broad maximum near 200 K and a sharp increase
below ~ 50K with no discernable paramagnetic regime [17]. Again, neutron diffraction is needed to probe the magnetic
correlations and the results appear in
Fig. 14. Remarkably, the magnetic reflections have the asymmetric shape, called
the Warren line shape, associated with
correlations in two dimensions, yet the
Mn3+ sublattice is three dimensional!
The two observed peaks can be indexed
as (20) and (13) on a particular type of
two dimensional magnetic structure
called the /3 x /3 Kagomé structure.
(The Kagomé lattice consists of a planar
array of corner sharing triangles and the
pyrochlore lattice is built up of alternating Kagomé and triangular planar lattices.) See the original paper for more
details [17].
Low angle part of the neutron diffraction pattern of Li2Mn2O4 showing the
development of the asymmetric
“Warren” peaks with decreasing temperature. The two peaks can be
indexed as (20) and (13) suggesting
short range order on the Kagomé
planes in the pyrochlore structure
[Ref. 17, reproduced with permission
of the American Chemical Society].
LA PHYSIQUE AU CANADA
A correlation length for the short range
two dimensional ordering can be
derived from a detailed fitting of the
Warren peak shape and the temperature
dependence is shown in Fig. 15. Note the
sharp, first order like increase below 45
K and that the length scale remains finite
at ~ 90 Å. There is still no detailed model
to explain this remarkable result.
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blet (100)/(001) arises from an induced
moment on the Nd3+ sites and the higher
one (110)/(011) is due to the Ti3+ spins.
This enhanced resolution makes possible
for the first time the unambiguous determination of the magnetic structure of
NdTiO3, shown on the right hand side of
the figure.
CONCLUSION
An attempt has been made to show the
scope of problems which can be addressed
using neutron diffraction, mainly the powder method, in the study of transition
metal oxide materials using examples carried out by Canadian scientists at the
CNBC or elsewhere. Neutron diffraction is
not merely complementary to x-ray diffraction but is an essential tool in the study
of these systems. Perhaps B.N. Brockhouse
said it best, “If neutrons did not exist they
would need to be invented.”
Fig. 16 The two strongest magnetic reflections for NdTiO3 showing clear
Fig. 15 The temperature developresolution into doublets. This perment of the two dimensional
mits the unambiguous determinaspin-spin correlation length
tion of the magnetic structure
in Li2Mn2O4. Note the first
shown at the right.
order like increase near 45K
[Ref. 17, reproduced with
REFERENCES
permission of the American
Chemical Society].
1. K. Takada, H. Sakurai, E. Takayama-Muomachi,
The Magnetic Structure of the Perovskite NdTiO3 using
High Resolution Neutron Powder Diffraction
2.
3.
The rare earth titanium perovskites, LnTiO3, where Ln3+ is a
lanthanide, have been studied for more than 25 years due to
a remarkable set of electronic properties. The parent compounds are Mott-Hubbard insulators, rather than metals, as
expected from band theory due to a strong electron-electron
correlation energy. Unique among rare earth transition metal
perovskites, the sign of the spin-spin correlations between
the transition metal ions changes from antiferromagnetic to
ferromagnetic as the size of the rare earth ion decreases
across the lanthanide series. The materials for Ln = La – Sm
are antferromagnetic while those for Ln = Gd to Lu are ferromagnets. Considering the antiferromagnetic series, there are
two important issues – one is the anomalously low ordered
moment at the Ti3+ site of ~ 0.5 μB where ~ 1 μB is expected
for a S = ½ ion and the second is the exact magnetic structure
in the antiferromagnetic state. Surprsingly, the detailed magnetic structure is not known with certainty for the LnTiO3
antiferromagnets due to two factors – lack of true single crystals due to micro-twinning and the general use of low resolution data in older studies in which key magnetic reflections
are not measured separately. To remedy this situation neutron diffraction data were obtained on nearly stoichiometric
NdTiO3 using high resolution conditions (ë = 2.37 Å) at the
CNBC [18].
4.
Data were obtained at 3.8K (Tc = 88K for the sample studied).
The magnetic reflections are very weak relative to the structure reflections ( 1% < of the most intense structure peak)
due of course to the very small magnetic moments involved.
In Fig. 16 the magnetic reflections are isolated and the resolution into two sets of doublets is clear. The lower angle dou-
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5.
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September / October 2006
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STOP THAT CORROSION - IF YOU CAN
by Zin Tun, Jamie Noël and Dave Shoesmith
I
technique offers some unique capabilities. This brings us to
n Spring 1836, Michael Faraday received a letter from
the topic of this article. At Chalk River we have an ongoing
Prof. Schoenbein, Professor of Chemistry at the University of
research program to study passive oxide layers with neutron
Bâle [1]. Schoenbein lamented how slow German periodicals
reflectometry (NR). Instead of iron, our experiments have
were in getting scientific papers published, and apologized
been on titanium (Ti) and zirconium (Zr), two metals that are
for taking the liberty of writing directly to Faraday.
of interest to the nuclear industry.
Schoenbein further presented observations he had made
recently by dipping iron wires into a
NEUTRON REFLECTOMETRY
strong nitric acid where, under certain
conditions, iron seemed to be fully The scientific study of pasprotected from the acid. Faraday sive oxide on metals started NR is a relatively new neutron scattering technique that allows determination
repeated Schoenbein’s experiments for
of near-surface layer structure of a flat
in
1836
continues
to
this
verification, carried out investigations
of his own, and proposed tentatively day. It is fair to question if sample to a resolution of 1 or 2 nanometers in the direction normal to the samthat the passivation of iron was caused
by the growth of a thin oxide layer on there is anything more to ple. As the measurements are limited to
the wire. This body of work was pub- discover in such an old area small scattering angles, individual
atoms are not observed by NR. Instead,
lished
in
the
Philosophical
data analysis yields a function known
[2]
of
research.
After
all,
we
Magazine later in the year.
as the scattering-length density (SLD).
The scientific study of passive oxide on now know Faraday’s pro- The SLD is the product of coherent neumetals thus started continues to this posal is basically correct [3]. tron scattering-length (usually denoted
by b) and number density of constituent
day. It is fair to question if there is
isotopes of a particular layer. If a layer
anything more to discover in such an
contains more than one isotope, the
old area of research. After all, we now
layer SLD is the all-inclusive sum performed over all isotopes
know Faraday’s proposal is basically correct [3]. The continpresent. We herein report specular reflectivity measurements
ued interest in the phenomenon is due to several factors, each
where the scattering vector q is kept perpendicular to the surproviding a compelling reason. First, the problem is interestface of the sample, taken to be the z-direction. Consequently,
ing in its own right, involving the not well-understood solidspecular reflectivity provides no in-plane sensitivity. The
state ion transport. Also there remains some fundamental
SLD profiles we report are functions of z only, and they are
issues not fully resolved. For instance, even the composition
the xy-sum of SLD contributed by all isotopes that lie at the
and structure of the oxide are still matters of controversy [3].
particular depth z.
Second, we all realize that metal passivity, if it were fully
understood and controlled, would have major impact, not
Neutron scattering cross-sections are small compared to those
only for industry but for the whole society. This promise
drives researchers to continue looking for a unifying model or
that give rise to X-ray or electron diffraction. Consequently,
explanation, such as a recipe for growing an passive oxide
diffraction signals from a nanometer scale layer would be too
that would effectively stop corrosion of any metal under any
weak to measure with neutrons. NR overcomes this difficulenvironmental condition. We are fully aware that a general
ty by making use of total external reflection of neutrons when
recipe may not exist given that intrinsic properties of metals
they strike the sample surface at a small enough grazing
(alloys included) and their oxides vary enormously.
angle. Just as for visible light, below the critical angle, θc, the
Nevertheless, even if we do not achieve such a lofty goal, the
reflectivity of the interface is unity, i.e. the entire incident
research will still be worthwhile since we will no doubt come
beam is reflected. Once the grazing angle exceeds θc, the
across some interesting phenomena as we work in this rich
reflectivity drops quickly but this drop can be measured with
field of research. Finally, the very fact that a thin layer of
great precision for at least 5 or 6 orders of magnitude. The
atoms, often not more than 10 crystallographic unit cells if it
details of this drop are what conveys the information about
were an ordered structure, is capable of stopping this
the layer structure. For instance, for a sharp interface
omnipresent agent of destruction - corrosion - is intriguing. It
between two different media, the drop follows the well
has captured many researchers’ interest in the past and is likely to do so in the foreseeable future.
CHANCE FAVOURS THE PREPARED MIND, BUT …
The chance of observing an interesting phenomenon increases if one uses a new experimental technique, especially if the
Z. Tun <[email protected]>, National Research
Council, Chalk River, ON K0J 1J0; J. Noël and
D. Shoesmith, Dept.. of Chemistry, Western Ontario,
London, ON N6A 5B7
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Fig. 2
Fig. 1
Fresnel equation (dashed curve) plotted as a function
of the scattering vector q. This is the expected variation of reflectivity if the interface causing the reflection is a sharp boundary between two media. For
this plot we assume the media to be air and Si. In
the case where there is a thin layer of a third material
on the surface (assumed to be Au for this example)
an interference pattern will be observed (solid
curve).
known Fresnel equation. In the case of a more complicated
example, if the sample consists of several layers, each with a
well defined thickness and SLD, an interference pattern
appears superimposed on the Fresnel Law (Figure 1). Broad
interfaces, in general, can be thought of as a series of “microlayers” whose SLD follows a profile defined by a sigmoidal
function of a finite width. For data analysis, one proposes a
layer profile model whose calculated reflectivity is at least
approximately similar to the observed reflectivity. Much in
the same way as for crystallographic structure refinement,
the residual discrepancies between Robs and Rcal are then
minimized by least-squares refinement of the model parameters, i.e. the thickness, SLD, and interface width of the layers
are refined.
One advantage of NR that is very much relevant to our
research is its ability to detect the presence of hydrogen or Hcontaining species. 1H (or natural H which is 99.99% 1H) is
one of the few isotopes whose neutron scattering length, b, is
negative, meaning the Fermi pseudo potential between a proton and a neutron is attractive instead of repulsive.
Therefore, during an experiment where an in-situ chemical
reaction is in progress, a decrease in the SLD of a layer, for
example, may signal H ingress into a layer. One can verify
this tentative result by repeating the experiment in a deuterated environment, for 2H (or D), unlike H, has a very large
positive b. If the hypothesis about the H-ingress is correct,
the SLD of the layer should increase in the deuterated experiment.
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Two possible geometries of performing neutron
reflectometry on a thin metal film (e.g. Ti film of several tens of nanometers) deposited on a Si substrate.
The circular substrate of 100 mm diameter and 6 mm
thickness is shown edge-on, and the arrows represent
incoming and reflected neutron beams.
Another advantage of NR is that the scattering geometry
makes it particularly easy to combine NR with many other
experimental techniques. The θ/2θ scan performed for NR is
almost a straight-through geometry (often 2θ < 10o), providing un-obscured frontal view of the sample surface and direct
access for other probes for simultaneous investigation. The
geometry is depicted in Figure 2a. On the other hand, if the
sample needs to be enclosed in a special environment (controlled humidity, high vacuum, etc.) one can easily design an
enclosure with two small windows. Yet another possibility is
to use a substrate that is highly transparent to neutrons to
support the thin film sample. Most popular substrates are Si
or sapphire. In that case, the beam can be arranged to strike
the thin-film sample from the substrate side as shown in
Figure 2b. Note that the space away from the neutron beams
is then entirely clear and open for whatever experiments one
wishes to perform simultaneously. Figure 3 is an example of
such an experiment where a specially designed cell allows
simultaneous performance of NR and electrochemistry on a
working electrode. These possibilities make NR very versatile and have produced many ingenious experiments worldwide.
PASSIVE OXIDE ON Ti
Using a cell based on the same design as the one depicted in
Figure 3, Wiesler & Majkrzak [4,5] carried out NR on a Ti thin
film in contact with (0.1 mole/dm3) H2SO4 solution. In this
pioneering work, the authors grew anodic oxide on the
already existing passive oxide in two ways – either by applying anodic potential abruptly or by slowly ramping (1 mV/s)
the potential to its final value. They found that slightly
denser packing results if the potential is slowly ramped, and
the denser oxide thus formed dissolves slower in the acid.
They also made observations on the loading of H into the
oxide and the underlying metal by reversing the bias to
cathodic polarity.
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anodization, and used up some 38 D of the underlying metal
in the process. These changes correspond to anodization
ratio
α = (tE – tOC)/(E – EOC) = 25.4
D/V
and
the
Pilling Bedworth ratio (volume of oxide produced per volume of metal consumed) RPB = 1.72 ± 0.04. These values
compare well with literature values: the most commonly
accepted value for α for Ti is 25 ± 5 D/V [9], while RPB,
depending on the crystallographic structure of the oxide, is
expected to be 1.77, 1.82, and 1.96, respectively, for rutile,
brookite and anatase structures. We note that the observed
RPB was very low, barely acceptable compared to the formation of rutile, the highest density form of TiO2. This suggests
the resultant oxide was free of major voids and cavities.
Fig. 3
Schematic of a cell to perform neutron reflectometry
and electrochemical reactions simultaneously. The
working electrode (WE) is a metal thin film deposited on low-resistance Si, while a Pt foil serves as the
counter electrode (CE). A saturated calomel electrode
(SCE), connected to the main body of electrolyte via a
capillary and a porous ceramic junction acts as the
reference electrode (RE) of the cell.
In the 1990s, research groups at AECL’s Whiteshell
Laboratories had a strong interest in the corrosion properties
of Ti because the metal was being considered as a container
material for underground spent nuclear fuel disposal
(Canadian Nuclear Fuel Waste Management Program).
Wiesler & Majkrzak’s work inspired us and we set out to do
similar NR experiments combined with insitu electrochemistry. However, there is one
major difference: instead of H2SO4 we chose
to work with aqueous NaCl solution
(0.27 mole/dm3) as the containers would
come into contact with near-neutral pH,
saline underground water. In addition to a
better understanding of the corrosion mechanism, we needed a good estimate of the
threshold polarization that would initiate H
adsorption into Ti. Our results have been
reported in several papers [6-8]. One of the
highlights of the work is being able to shed
light on the mechanism of ion transport
across an existing oxide layer during
anodization.
Figure 4 shows the SLD curves of the Ti film
exposed to the electrolyte solution both
before (i.e. at the open circuit potential, EOC)
and after being anodized to E = 2 V. Ti is
another one of the few isotopes with negative
scattering length, and hence the SLD of the
metal layer is below zero. The scan at 2 V
was performed several hours after setting the Fig. 4
potential to ensure we measured the long
time equilibrium behaviour of the oxide
layer. As expected, the oxide layer, originally at tOC = 47 D, thickened to tE = 112 D by
Another important result depicted by Figure 4, as deduced
from least-squares refinement with various models, is that
the outer layer of the anodized oxide (i.e. the part that was in
contact with the electrolyte) has a lower SLD than the original oxide. Three possibilities present themselves to explain
the low SLD region: a) this region of oxide is porous, b) it is
oxygen deficient, i.e. TiOx with x < 2, or c) it contains some
form of hydrogen. Possibility a) can be ruled out on the basis
that the observed RPB is very low, indicating the resultant
oxide is highly compact. Since the Ti oxides are insoluble in
water, an argument such as “RPB is actually high but it
appears low as part of the oxide had dissolved away before
NR was started” is not credible. Possibility b) can be rejected
on the grounds that oxygen deficiency, if any, should occur
deep within the oxide layer, not on the surface. This leaves
possibility c) which is most likely since incorporation of H (as
OH species) during anodization is indeed a known phenomenon (e.g. the observation of absorbed OH by angle resolved
XPS as reported by Tun et al. [8]).
Scattering-length density (SLD) profile of a Ti thin-film deposited on a
Si substrate. The SLD of the metal layer is negative since the scattering
length of Ti is negative. The oxide layer, 47 Å prior to anodization
(open-circuit EOC = – 0 56 V ), thickened to 112 Å after acquiring equilibrium at 2 V applied potential.
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mobility), and react with Ti. Meanwhile, Ti migrates
out (with 35% mobility) and reacts with O (as OH or
water) at the oxide/electrolyte interface. Thus we end
up with an inner layer of H-free oxide and an outer
layer of H-containing oxide. If we accept this picture of
oxide growth, the fact that the H-containing region ends
just at the right depth so that the H-free region happens
to have the same thickness as the original air-grown
oxide is an accident. While there is nothing preventing
such a fortuitous outcome, we thought it was desirable
to look for alternative models of solid-state oxide
growth where the amount of H-free oxide is necessarily
conserved.
The alternative model we favour is the point-defect
model (PDM) [11]. The basic idea is depicted in Figure 6.
According to this model, field-driven migration of one
of the species, say O, is initiated when an O atom near
Fig. 5 Visualization of field-assisted ion transport across an existthe metal/oxide layer hops into a defect site in the
ing oxide layer. Most of potential drop between the metal of metal. This creates a vacancy in the oxide, which is
anode and cathode takes place within the oxide which is an
filled by an O atom at a shallower depth of the oxide,
insulator or a large-gap semiconductor. The resulting elecleaving its original site vacant. This sequence of vacantric field within the oxide layer drives cations and anions in
cy creation and subsequent filling by a nearby ion propopposite directions.
agates outward through the existing oxide until, finally,
the vacancy created at the oxide/liquid interface is filled
Having thus explained the low SLD region, we are confrontby an O atom (or an OH ion) derived from the solution. The
ed with yet another question: Is the high SLD region the origmetal ions can also migrate through the existing oxide by a
inal oxide? From the peak value of SLD and the thickness,
similar vacancy transfer process running in the opposite
the high SLD region seems to be very similar to the original
direction.
oxide; it is tempting to think that they are one and the same.
One could argue that the high SLD region is the original
The O in the original air-grown oxide is H-free and, since this
oxide that has “sunk” closer to the Si/Ti interface because
number of O atoms is conserved according to PDM, the thicksome amount of metal has been removed and transported
ness of H-free oxide would be conserved. The newly incorpoacross the original oxide to form new oxide (the low SLD
rated O by anodization is contaminated to some extent with
region) at the oxide/electrolyte interface. Note also that this
H, causing the extra thickness of the oxide produced by
consideration is valid regardless of which of the three possianodization to have a lower SLD. The fact that we do not
bilities as explained above is responsible for the low SLD
need to invoke a fortuitous outcome makes the PDM more
region. However, such an interpretation is not consistent
attractive. However, we recognize that a more direct experiwith known transport numbers for anodization of Ti.
mental evidence is needed to decidedly settle this issue.
Exclusive outward growth of the oxide would require very
high Ti mobility. Using α-particle emitters implanted prior
to anodization, Khalil and Leach [10] determined from the
energy-loss spectrum that the emitter layer was buried within the thickened anodized oxide, at a depth of 35% of the
increase in the oxide layer thickness. This corresponds to
35% metal and 65% oxygen mobility. Clearly, an alternative
explanation is needed.
It is important to first recognize that our desire to talk about
the original oxide comes from the field-assisted ion transport
(FAIT) model. The basic concept of this mechanism is depicted in Figure 5. Differently charged ions migrate through an
existing oxide and react with respective counter ions at the
metal/oxide or oxide/liquid interface. According to this
simple-minded picture of FAIT, the original air grown oxide
is merely a passive screen.
Within the FAIT, we are then led to the following picture at
the molecular level: What is attracted towards the anode is
OH. Presumably H is stripped by the very large potential
gradient within the existing oxide, H remains within the shallower depths while O continues to pass through (with 65%
252
PHYSICS
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Fig. 6
September / October 2006
Visualization of point-defect model of ion transport.
When an ion within the existing oxide hops to a
neighbouring lattice vacancy (defect), it creates a
vacancy at its original site.
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recorded over a frequency range spanning from a few mHz
to ~100 kHz. Currents as low as a few nano-amperes are
recorded. The data are then analyzed in terms of an equivalent circuit where a capacitor, for example, would represent
the charge storage capacity of the double-layer on the electrode surface and a resistor parallel to it would represent
charge leakage whose origin could be an electrochemical
reaction or an electronic or ionic current. Only a preview of
our NR and EIS results is provided here as we intend to publish a full account of these experiments elsewhere.
Fig. 7
Scattering-length density (SLD) profile of a Zr thinfilm deposited on a Si substrate before anodization
(labeled EOC), after attaining equilibrium at 1 V
(E = +1V) and subsequently at 3 V (E = +3V).
The SLD curves of our Zr thin film sample obtained at open
circuit (EOC = –0.17 V) and two anodizing potentials are
shown in Figure 7. Actually, NR (and EIS) data were collected at anodic potentials starting from 0 to 3 V in steps of 0.5 V.
To ensure long time equilibrium behaviour was recorded, the
NR scans, each typically taking 6 h, were repeated until no
further change was discernable in the measured reflectivity.
Depending on whether the NR scan was initiated immediately after the potential was applied or delayed by several
hours, it took two or three repeats to achieve stability. EIS
scans were always started immediately after setting the
Repeating NR with a solution made
of D2O instead of H2O may provide
further evidence, but a fresh look
with an entirely different technique
may provide a more convincing outcome.
PASSIVE OXIDE ON Zr
Another metal that is of interest to
the nuclear industry is Zr. Our
studies of Zr are more extensive
than those on Ti since we managed
to carry out electrochemical impedance spectroscopy (EIS) concurrently with NR. Combining NR and EIS,
though simple in theory, is very
challenging because of electrical
interference. Following the timehonoured approach of trial and
error, we learned not only that the
cell must be electrically isolated
from the common ground of the
neutron instrumentation, but also
that no metal part of the cell must
come close (within about 2 inches)
to any grounded object (e.g. the top
metal plate of the θ rotary drive)
even when these parts themselves
are electrically isolated from the EIS
circuit.
EIS is a familiar technique for electrochemists. By superimposing a
small AC ripple of variable frequency on the DC bias applied for anodic or cathodic polarization of the
working electrode, the AC response Fig. 8
of the electrode (i.e. current) is
Impedance of the thin-film Zr electrode before it was anodized (labeled EOC),
and after being anodized to a potential of 1 V (1V) and 3 V (3V).
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Fig. 9
Equivalent circuit proposed for the analysis of
observed frequency dependence of Zr electrode
impedance.
potential. Each typically taking ~ 2 h to complete, we collected 12 to 18 completed EIS scans at each potential.
Just as Figure 4 does for Ti, Figure 7 shows thickening of
oxide at the expense of the Zr metal. The numerical values of
α and RPB, 34 D/V and 1.57 respectively, are in good agreement with the literature values [12,13]. However, a two-oxide
model such as that used for the anodized Ti oxide does not
give a stable or significantly improved least-squares refinement at any of the potentials where we have made measurements. We concluded that there is no basis for invoking two
types of oxide, and adopted a single oxide layer throughout
for reporting the NR results.
The impedance of the cell as measured by EIS at the same
three potentials is shown in Figure 8. While many equivalent
circuits with different levels of sophistication could be proposed, the circuit shown in Figure 9 is one of the simplest for
an electrochemical cell. It consists of a capacitor, Cox, to represent charge storage capacity across the oxide layer and a
parallel resistor, Rox, included to model possible charge leakage. The series resistor Rs represents the solution (electrolyte) resistance. For a good insulating oxide layer and an
ionic solution such as the one we used Rox >> Rs. At the lowest frequency (almost DC), Cox is not conducting since its
magnitude of impedance 1/ωC64, and we essentially have
the two resistors Rox and Rs in series. We identify |Z| . Rox,
and expect the phase, Θ, to be zero. As the frequency increases, Cox begins to conduct making the |Z| fall and Θ 6 –90°.
Once Cox is fully conducting (frequency ~ 0.1 Hz) the circuit
is essentially Rs and Cox in series, whose real part is Rs and
the imaginary part (–ι/ωCox). As the frequency increases further, the real part remains constant but the imaginary part
decreases. Eventually, as the imaginary part becomes comparable to, and then smaller than Rs in magnitude, Θ rotates
back from –90° to zero and |Z| . Rs.
Figure 10 shows the impedance Z of the equivalent circuit
whose components Rs, Rox and Cox have been adjusted to
obtain the best fit to the EOC data. The calculated Z shows all
the expected trends discussed above. The calculated Z vs.
frequency is in good agreement with the EOC curve of
Figure 8 for all frequencies below ~ 5 kHz. At higher frequencies, however, the observed Z deviates from the model
both in magnitude and in phase. This feature is likely to be
an artefact, as it is not reproduced by EIS (ex-situ) measurements with simplified cell geometry. For instance, using a
cell where the distance between the Zr working
electrode and the counter electrode is reduced
(~1 cm), and no capillary for the reference electrode in place, Z is observed to behave exactly as
depicted in Figure 10 up to 1 MHz in frequency.
Fig. 10 The impedance of the proposed equivalent circuit whose parameters have been least-squares fitted to the EOC result depicted in
Figure 8.
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We are now ready to examine how NR and EIS
could complement each other and lead to results
that could not be obtained by using either of them
alone. To this end, we need to first find correlations between the changes recorded by NR and
EIS at various stages of the experiment. The EOC
curve of Figure 8 shows that, prior to anodization,
the low-frequency |Z| was very high, indicating
the original passive oxide layer was continuous
and provided good protection against corrosion.
The high |Z| persisted as we increased polarization to anodic potentials of 0 V, 0.5 V and 1 V
(only 1 V curve is shown in Figure 8 for clarity).
At the next higher potential, 1.5 V, the low-frequency |Z| suddenly dropped and never recovered. As the 1.5 V curve is similar to the 3 V curve,
only the latter is shown in Figure 8. The oxide
layer thickness determined by NR, on the other
hand, increased monotonically with the potential
as shown in Figure 11. Note that there is no discernable discontinuity or anomaly of the oxide
thickness at or around 1.5 V. It turned out that the
expected discontinuity occurred for the SLD of the
oxide layer. While the oxide SLD decreased with
every potential increase (as seen in figure 7), the
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Unexpectedly, the above monotonic trend revealed an underlying difference between the behaviour probed by NR and
EIS. The solid-state ion transport across existing oxide no
doubt continues many hours after the anodization potential
was set. The behaviour of the cell current monitored by EIS,
on the other hand, is distinctly different: it shoots up at the
moment of the application of the potential, but decays to a
small value within minutes. Over longer periods stretching
beyond the first hour, the current is either very stable or drifts
slightly. The drifts, typically very slow (over tens of minutes),
may be in either the positive or the negative direction. This
behaviour of the cell current, thus, stands in stark contrast
with the unidirectional thickening of the oxide layer.
Fig. 11 Chronological variation of oxide layer thickness
observed by neutron reflectometry at EOC (–0.17 V),
and at set anodic potential increased in steps from
0 to 3 V at intervals of 0.5 V. Time sequence of
repeated scans at a given potential is denoted by
Δ), squares (~), and circles (o).
triangles (Δ
largest drop occurred for the potential increase from 1 to
1.5 V. Moreover, the second largest drop of oxide SLD
occurred for the next potential increase, from 1.5 to 2 V.
Higher subsequent potentials brought about further decreases of the oxide SLD but they were all very small. The explanation that is consistent with both the EIS and NR observations is that a significant number of cracks developed within
the oxide layer at 1.5 V, and they were immediately filled
with H2O. This inrush of water led to a step in the oxide SLD,
as well as a large drop in Rox as even a few saline water pathways can effectively short-circuit the oxide capacitor.
A more detailed inspection of Figure 11 reveals a trend that
very convincingly demonstrates the reliability of the NR
data. The figure shows the results obtained by repeated NR
scans using different symbols; triangles (Δ) are yielded by the
scans carried out immediately after potential increase,
squares (~) by the subsequent scan, and circles (o) by yet
another scan if carried out. Lapse time between these scans
is typically 6 h. Circles are the best estimate we have for the
equilibrium oxide layer thickness, i.e. the thickness we would
get at infinitely long time. A circle represents the very first
point of the plot at EOC since the oxide layer does not thicken
by just filling the cell. Thus chronologically encoded, the plot
shows a remarkable trend – the change is monotonic with the
symbols appearing always in the order Δ, ~, o. This shows
that the variation of dox detected by NR are real, even when
the thickening is only a matter of a few Angstroms (between
~ and o).
In retrospect, the fact that NR and ESI expose different
processes is not surprising. Anodization involves mass
and/or charge transport between layers of the working electrode, arising from motion of neutral atoms, electrons or ions.
Movement of neutral atoms is detected only by NR, electrons
only by EIS, and ions by both. This is perhaps the most compelling reason for performing NR and EIS concurrently in the
study of passive oxide layers.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
M. Faraday, Experimental Researches in Electricity, Vol. 2,
pp. 234–239, Dover Publications, Inc., New York, (1965);
republication of the original publication by Richard & John
Edward Taylor in 1844.
M. Faraday, Lond. and Edinb. Phil. Mag. 4, 53 (1836).
H.H. Uhlig, “Passivity of Metals”, in Proceedings of the
4th International Symposium on Passivity, pp. 24–25,
R.P. Frankenthal and J. Kruger editors, Electrochem. Soc.
Inc., Princeton, New Jersey (1978).
D.G. Wiesler and C.F. Majkrzak, Physica B198, 181–186
(1994).
D.G. Wiesler and C.F. Majkrzak, Mat. Res. Soc. Symp. Proc.
376, 247–257 (1995).
J.J. Noël, B.M. Ikeda, N.H. Miller, S.R. Ryan, D.W.
Shoesmith, S. Sunder and Z. Tun in Surface Oxide Films,
Proceedings Vol. 96-18, pp. 246–257, J.A. Bardwell editor,
Electrochem. Soc. Inc., Pennington, New Jersey (1996).
Z. Tun, J.J. Noël and D.W. Shoesmith, Physica B Condensed
Matter 241-243, 1107–1109 (1998).
Z. Tun, J.J. Noël and D.W. Shoesmith, J. Electrochem. Soc.
146, 988–994 (1999).
See Tun et al. (1999) [Ref. 8] and references therein.
N. Khalil and J.S.L. Leach, Electrochim. Acta 31, 1279–1285
(1986).
D.D. Macdonald, J. Electrochem. Soc., 139, 3434–3449 (1992).
P. Meisterhahn, H.W. Hoppe and J.W. Schultze, J lectroanal.
Chem. 217, 159–185 (1987).
Values of specific gravity for Zr, ZrO2 (zirconia), and ZrO2
(baddeleyite) are 6.49, 5.6, and 5.8 respectively, giving
RPB = 1.56 for zirconia and 1.50 for baddeleyite.
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Nominations are invited to fill one position (term ending Dec. 31, 2006) on the Canadian National IUPAP Liaison Committee
(CNILC) for a term of three years commencing January 1, 2007 (ends Dec.31/09). Although there are no restrictions on who is
nominated, efforts will be made to ensure that there is a broad representation on the Committee covering the areas of geographic
location, physics sub-discipline, and language requirements. The final decision remains with the CNILC Secretariat.
The current members of the Committee are:
G.W.F. Drake (Chair)
L. Marleau (term ends Dec.31/06)
E. Hessels (term ends Dec.31/08)
P. Hawrylak (Secretary)
J.W. McDonald (term ends Dec.31/07)
C. Gale (term ends Dec.31/08)
Ex-officio IUPAP Commission members (Sept.2005 to Sept.2008) are:
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(Plasma Physics)
P. Hawrylak
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(Physics Education)
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(Computational Physics)
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ARTICLE DE FOND ( QUANTUM MAGNETISM ... )
QUANTUM MAGNETISM AND SUPERCONDUCTIVITY
by William J.L. Buyers and Zahra Yamani
T
magnetic moment that precesses around the sum of the static
he spin of the neutron allows neutron scattering to reveal
and dynamically varying field of its neighbours. This is the
the magnetic structure and dynamics of materials over
site-based picture of localized spins.
nanometre length scales and picosecond timescales. Neutron
scattering is particularly in demand in order to understand
In metals the situation is entirely different. The conduction
high-temperature superconductors, which lie close to magarises from band-based electrons in which it is the electron
netically ordered phases, and highly correlated metals with
momentum that is well defined at the
giant effective fermion masses, which
Fermi surface rather than the electron
lie close to magnetic order or pass
through a mysterious phase of hidden In this article, examples are position. Although embedded in a liqorder before becoming superconduct- given of quantum phenome- uid of high-velocity conduction electrons, local spins may still behave indeing. Neutron scattering also is the
probe of choice for revealing new na where neutron scattering pendently provided their energy scale,
phases of matter and new particles, as has played a defining role given by the exchange coupling, J, is
much less than the eV bandwidth of the
seen in the surprising behaviour of
challenges current fast conduction electrons. The conducquantum spin chains and ladders that
electron spin responds adiabaticalwhere mass gaps and excited triplons understanding
of con- tion
ly to the motion of the slow local spins.
replace conventional spin waves.
This is the picture for the rare earth
Examples are given of quantum phe- densed matter.
metals, except for a few mixed-valent
nomena where neutron scattering has
examples.
The decoupling works
played a defining role that challenges
because the small-radius 4f magnetic shell lies inside the 5s
current understanding of condensed matter.
shell and so is shielded from the destructive influence of its
neighbours. Even in this weakly coupled system the spin
INTRODUCTION
excitations of the f electrons are not eigenstates - the indirect
Magnetism is at the heart of fundamental processes. The way
coupling through the conduction electron sea shortens their
in which black holes suck in matter from neighbouring stars
lifetime. They acquire a relaxation rate, seen as a spectral line
is a fundamentally magnetic process and not just caused by
width, proportional to the imaginary part of the conduction
gravity[1]. The magnetic moment of the neutron allows scienelectron (Lindhard) spin susceptibility χ”(q,ω) because coutists to study magnetism in materials at the nanoscale and
pling, I, of the local f-spin to the conduction electron spin
below. The pattern of neutron scattering and its velocity discauses an indirect (RKKY) exchange between f-moments of
tribution reveals the structure and dynamics of the atomic
the form J(q,ω)=I2χ(q,ω). The same indirect coupling, now
magnetic moments or spins. Spins in condensed matter
through the charge susceptibility, gives phonons in metals a
belong to the overall electron system and arise from unpaired
spectral broadening, and this is only removed for energies
electrons in one of the outer orbital shells of the atom. The
below the pairing gap in its conduction electron charge
ground state of the spin depends on whether it is surrounded
response when the metal becomes superconducting below
by and exchange coupled to other well-defined spins as in
Tc [2].
insulators, or whether the spins are embedded in a liquid of
conduction electrons that may screen their moment and
Nonetheless, when the coupling to conduction electrons is
damp out their excitations.
strong by exchange or by hybridization, the spins behave as if
they are free at high temperature but are progressively
In insulators an integral charge state is determined by chemiscreened on cooling by coherent reorganization of the concal valency and the environment allows the several unpaired
duction electrons. The effect is described as Kondo screening
electrons to form localized states of definite orbital and spin
when the spin of the atomic core and of the conduction elecangular momentum linked by spin-orbit coupling. Hund’s
tron can reorganize without substantial change of charge state
rule is king. At each site we have an independent atom the
and is described as mixed valency when hybridization
symmetry of whose orbit is lowered by the electrostatic field
changes the occupancy and effective charge.
of its neighbours. The Pauli exclusion principle leads to an
effective coupling, J, between neighbouring spins that may be
ferromagnetic (parallel) or antiferromagnetic (antiparallel).
The latter is more prevalent in nature because the magnetic
William J.L. Buyers <[email protected]> and
atoms in insulators establish superexchange bonds through
Zahra Yamani, Canadian Neutron Beam Centre, National
Research Council, Chalk River, Ontario, Canada K0J 1J0
shared non-magnetic neighbours such as O2- in MnO or F- as
in KMnF3. The atom behaves magnetically as if it has a fixed
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FEATURE ARTICLE ( QUANTUM MAGNETISM ... )
Examples of exotic or unconventional phenomena discovered with neutron scattering
include magnetic solitons [4,5], and the quantum gap (Haldane gap) between the ground
state and a triplet of massive spin particles that
appears for integer but not half-integer spin
chains. One-dimensional chains of spins are
created by chemically separating chains of
magnetic ions by ligands of non-magnetic
ions. Clever solid-state chemists are responsible for creating the wide variety of 1D, 2D and
3D magnetic systems where singlet ground
state, spin liquid and quantum phenomena
may be investigated.
Solitons in an Ising-like antiferromagnetic
chain, where the spins point up or down, must
be excited in pairs by the neutron flipping a
spin (Sz → -Sz), thereby creating two domain
walls or solitons costing energy 2J because
there are two wrong bonds with ferro- instead
of antiferro-orientation. This is because neutron scattering only connects states linked by
the spin operator – it is a spin-one probe. The
Ising spin exchange JSzi Szi+1 in the presence of
weak transverse coupling allows each soliton
to hop two sites at a time away from the initially localized soliton pair (Fig. 2). These solitons
can be visualized as a place where we have
Fig. 1 Energy levels of Co++ in the antiferromagnet KCoF3. The spin waves
twisted the rest of the chain through 180º to
are transitions from the ground to all excited states and form a lowest make a π soliton. Because the initial excitation
band up to the illustrated single-ion spin-flip frequency of 7 THz and carried spin one, we find that each soliton is a
down to a gap frequency set by the exchange mixing of higher spinspin one-half particle. A single soliton may be
orbit states [3].
thermally excited with an activation energy J,
half the spin wave energy. This simple examIn pure metallic systems spins normally condense into a state
ple from the Ising chain has given rise to the concept of spinons, the basic particle in the S=½ isotropic (or Heisenberg)
whose symmetry is lowered as a result of formation of magchain, later used for high-temperature superconductivity.
netic order, a spin-density wave, a charge density wave, or a
Because they are created in pairs, conservation of momentum
superconducting paired state, while some systems remain
ensures that there is a continuum of spinon excitations
paramagnetic to the lowest temperatures.
instead of sharp spin waves, a continuum that extends down
to a lower limit set by the Bethe ansatz.
A spin or orbital excitation appears as a collective excitation
of the ordered state whose energy-momentum dispersion
relation is a direct measure of the magnetic forces between
any two atoms. Their energies give information on the local
crystal field, the spin-orbit coupling and the interatomic
exchange as shown in Fig. 1 for the insulator KCoF3.
Although it has become customary in orbitally ordered materials such as manganites to treat the orbitons separately from
the spin waves, spin wave and orbital states are not distinct
as they are coupled by spin-orbit interaction. They together
form the collective magnetic dipole excitations of the electronic system and should be included in an extension of the
standard model [3].
EXOTIC PARTICLES
More than just measuring the strength of interactions, as may
be done in well-understood systems where the magnetism
appears in the form of well-defined spin waves, the neutron
is uniquely suited to discovering new phenomena that are
not contained within accepted textbook lore.
258
PHYSICS
IN
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Fig. 2
September / October 2006
Solitons hopping along a chain of S=½ spins [5].
Because the excitation is topological (half the spin
chain is turned over at each thickly marked wall) as
opposed to the sinusoidal spin-wave pattern, the
soliton response looked at with a Fourier probe such
as neutron scattering appears as a continuum.
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ARTICLE DE FOND ( QUANTUM MAGNETISM ... )
When the spins interact with isotropic rather than the above
Ising exchange, classical thinkers, and most scientists, expected that the spin spectrum would be gapless. Instead Haldane
conjectured [6] that the chains of integral-spins would exhibit
a mass gap but those of half-integral spins would not. The
Haldane gap was not expected, at least not by those that
wave their fingers to illustrate rotational invariance and at
the same time consider a long wavelength spin wave to be a
precession of a classical spin! Before Haldane, Kadanoff’s
postulate of Universality had wide acceptance because both
experimental results on different materials and theoretical
calculations with S=½ and S= 4 (i.e., classical result) gave
phase transitions with the same properties. Renormalization
group theory offered a mechanism for universal properties to
arise since under repeated renormalization transformations
some parameters are attenuated and become irrelevant while
others remain relevant. Haldane’s conjecture was controversial mainly because it contradicted Universality. While it is
sometimes valid to take a site-based view where the spin precesses in the field of its neighbours, this largely works only
when there is a static field (a magnet with long-range order),
in high dimensions, and in lattices without frustration (competing exchange fields). In one dimension there can be no
long-range order and the spins can attempt to form bond
order where pairs of spins form a singlet ground state. These
singlet pairs may then interact and it is not obvious a priori
whether this will give a lower overall ground state than the
site-based approach. Anderson showed that the 2D triangular lattice preferred a ground state of resonating valence
bonds over the Néel state, but the situation for one dimension
(1D) was unknown until the work of Haldane [6].
The Haldane conjecture remained controversial, as well as
being counter-intuitive, until the spin gap was discovered
directly in neutron scattering experiments at Chalk River
involving a University of Toronto student [7]. The isotropically coupled S=1 Ni2+ chains in CsNiCl3 were the test bed.
The conventional (linear spin wave theory) view was that the
lowest spin excitations were gapless Goldstone bosons but as
shown in Fig. 3 the integral Ni2+ spins exhibit a large gap,
about 40% of J. This result was soon confirmed in Europe on
an organic material [8] and in CsNiCl3 polarized neutron scattering showed that the gap states were triplets [9].
In recent years the unusual temperature dependence of the
gapped triplet states, which has given rise to the new name
for a particle, the triplon, have been fully explored by
Kenzelmann et al. [10]. The spin triplons increase their energy
on heating, whereas spin-wave energies decline in ordered
systems (Fig. 4). Within the non-linear sigma model, this is
because, to conserve the total moment, the triplon energies
must rise to counteract their increase in population through
thermal excitation.
A useful picture for a singlet with a gap to excited states is the
valence bond solid described in the review by Affleck [11], in
which each of the two electrons of an S=1 atom form a singlet
pair with one electron of a neighbouring atom, one to the left
and one to the right. This global singlet state is the exact solution of a closely related Hamiltonian. For S=½ the sole electron can form only one singlet bond, all to the left or all to the
right, but then there are two degenerate states, no singlet and
no gap.
The discoveries of the quantum gap presaged the large current body of research on singlet-to-triplet excitations or
Fig. 3
The Haldane spin gap discovered in the integer-spin chain system CsNiCl3 in its 1D
phase [7]. The Ni2+ (S=1) chains lie along the
hexagonal z direction [0 0 1]. If the excitations were conventional spin waves all freη,η
η, 1)
quencies along the 1D zone centre Q =(η
would lie at zero, since there is no longrange order, but the quantum disordered
ground state leads to a mass gap of 0.32 THz
to a triplet of spin excitations with only
short-range spin correlations. The weak
coupling perpendicular to the chains along
η,η
η, 0) leaves a residual in-plane dispersion.
(η
Fig. 4
Haldane gap triplet energies rise with temperature in accordance with the self-consistent non-linear sigma model [10].
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FEATURE ARTICLE ( QUANTUM MAGNETISM ... )
triplons, as found in even-leg spin ladders and in systems
formed from integer-spin triangle motifs. The solitons in
magnetic chains led to the now pervasive modern concept of
spinons.
HIDDEN ORDER
An enigmatic problem in the field of strongly correlated
heavy fermion systems is the nature of the hidden order that
sets in below the large specific heat jump at T0 = 17 K in
URu2Si2. In addition, a superconducting phase occurs in
URu2Si2 below 1.2 K. The heavy-fermion epithet stems from
the fact that the Sommerfeld specific heat coefficient, γ=C/T,
usually taken as a measure of the electronic density of states
at the Fermi surface, is large above T0, 160 mJ/mol-K2. This
is a hundred times larger than that of a simple metal like copper and suggests that the effective electron band mass is a
hundred times the free electron mass. Clearly the proximity
to a magnetic or exotic transition is causing the large mass,
through spin or hybridization effects. Since evidence of
heavy charge masses has been seen in de Haas-van-Alphen
experiments, the strong spin response must be producing a
slowing of the electron velocities, although the spins can
themselves contribute to the giant specific heat. The charge
and spin spectra must be renormalized downward to a few
meV in energy to add to specific heat.
Although a second order transition occurs at 17 K with substantial associated entropy, the nature of the order has
remained a mystery for over 20 years [12,13]. Landau theory
tells us that the small antiferromagnetic moment of 0.03 μB
that develops below 17 K cannot possibly explain the large
specific heat jump (entropy) associated with a second order
local spin transition. It would require ordering of a moment
Fig. 5
260
~100 times larger! Antiferromagnetism therefore cannot be
the hidden order parameter. The system seems rather to have
condensed into a new phase of matter for which the order
parameter and associated symmetries differ from conventional expectations. The properties are typical of ordering due to
broken symmetry but, since its origin has not yielded to practically every known experimental technique, we refer to it as
‘hidden’ order. Strong hybridization is expected between the
conduction and the 5f electrons and prevents application of
either a purely localized or itinerant electron model. Local
probes and pressure experiments suggest that the weak
moment may be a parasitical phenomenon that forms in a
very small volume fraction. The small moment may be simply a quixotic distraction from the real bulk order parameter
that causes the large loss of electronic density.
What is clearly a bulk property of the hidden order phase is
the unusual spectrum of magnetic excitations (Fig. 5).
Neutron scattering has shown [13] that they form well-defined
propagating collective modes over most but not all of the
Brillouin zone. Moreover they carry a large spin matrix element of 1.2 μB and are thus a property of the bulk or dominant phase. What is unusual is that the spin motion is entirely longitudinal along the tetragonal c direction. Contrast this
with a spin wave of a magnetically ordered system where the
motion is transverse to the moment. Also unusual is that
while the well-defined excitations suggest a localization of the
5f moment, along the tetragonal [0 0 1] direction the lifetime
shortens and damps out the response and so suggests decay
into an itinerant-electron continuum. Itinerant spins are also
suggested by long-range (RKKY) exchange that produces the
several extrema in the dispersion relation. Over the last two
decades many searches have been carried out for the hidden
order and the evidence is either absent or contrary to models
involving charge-density wave formation, quadrupolar
ordering, multispin correlators [14], crystal fields [15], or
orbital currents [16].
The frequency of gapped spin excitations versus wave vector in URu2Si2 at 4 K well within the hidden order
phase [13]. Long-range exchange through the electron liquid
causes several minima with the minimum gap at
Q = (1, 0, 0). For directions within the tetragonal basal
plane the excitations are long-lived, but those propagating
in the c direction along (1, 0, ζ) are damped out at large
wave vector.
PHYSICS
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September / October 2006
Fig. 6
The temperature dependence of the incommensurate fluctuations at (1.4, 0, 0) and E=0.25 THz
(~1 meV) energy transfer [17]. The fit gives an
activation temperature of 110±10 K, the coherence temperature for the charge transport not the
spin excitation energy.
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ARTICLE DE FOND ( QUANTUM MAGNETISM ... )
A significant new result came from a search for an exotic
form of magnetism predicted to arise below T0 from orbital
currents where the weak moment would result from electron
currents flowing around the atoms in a unit cell [16]. The
orbital moment theoretically predicted was not observed.
Wiebe et al [17] made a more important observation, however.
Above the 17 K transition the spectral weight moves to the
incommensurate wave vector (1.4, 0, 0) of the second minimum (Fig. 5) and the spectrum becomes gapless. The onset
of the collapse of the gap was measured by probing the fluctuations at 0.25 THz, well below the gap frequency of
1.2 THz. The important result is that the incommensurate
scattering is activated with a temperature T* = 110 K, the
coherence temperature (see Fig. 6). Thus in the precursor
phase to hidden order there are gapless incommensurate
spin fluctuations over a finite region of the Brillouin zone.
These can give rise to a term in the specific heat linear in T
that can be misconstrued as electronic specific heat. The specific heat will jump at T0 and decrease below as the spin gap
is formed. The large linear-in-T specific heat then may be
thought of as coming from the spin fluctuations rather than
from a Aheavy-electron@ charge band. Theorists often like to
work with the Aone band does all@ approach with a Hubbard
model that tries to reproduce both the charge and the spin
response. Whereas most focus on the fermions determining
the charge transport properties, the new results require more
attention to the bosons of the spin response. The hidden
order phase is robust and persists to a field of 35 T [18]. These
exotic results have led to exotic theories, most recently to the
suggestion that a Pomeranchuk instability of the electron liquid has grossly changed the Fermi surface [19].
SUPERCONDUCTORS
Neutron scattering is particularly well suited to explore the
intimate relation between magnetism and high-temperature
superconductivity. The superconductors consist of squarelattice CuO2 planes of S=½ copper spins, into which holes
have been created in the plane by the chemical removal of
electrons from oxygen ligands, a process known as doping.
Neutron scattering can show how the spin spectrum evolves
as the superconducting transition temperature increases and
then decreases as the electronic doping is increased beyond a
critical value, pc~5%, into the phase known as the superconducting dome. The S=½ holes sit equally on the four oxygen
neighbours and, from a distance, screen the copper moment
to form a spin singlet. The resonating valence bond (RVB)
ground state has been adduced to account for the precursor
state that connects a Mott insulator (LaCuO4 or YBa2Cu3O6)
to the hole-doped state where high-temperature superconductivity takes place [20]. The RVB state consists of sets of singlet pairs between copper spins at all distances with a symmetry similar to that of a superconducting pair. In conventional (phonon or S-wave) superconductors the pairing gap
occurs for all directions of Fermi momentum, kF. In contrast
the spins of the RVB pair lie on different atoms and the gap
has d-wave symmetry with nodes along the directions kx =
±ky.
Although there is a large amount of neutron beam research
on La2xSrxCuO4 and YBa2Cu3O6+x, most is for relatively highly doped materials. In recent years attention has shifted in
three continents to underdoped materials where superconductivity is weaker but magnetic fluctuations are
stronger [21,22,23]. With the advent of high quality crystals
from University of British Columbia it has been possible to
study highly-ordered ortho-II crystals that display greater
electronic order and thus a larger Tc for the same oxygen
doping. With these crystals the hour-glass spectrum of
incommensurate spin modulations at low energy, a resonance localized in Q and in ω, and a cone of damped highenergy spin waves has been well-established in recent work
at Chalk River and at ISIS in the UK [24,25].
Here we focus on systems that lie much closer to the critical
onset of superconductivity where the destruction of spin
order and spin wave propagation seems the most crucial
requirement for the onset of this anisotropic superconducting
charge pairing. A recent study [26] has shown that high-temperature superconductors close to the edge of the superconducting dome behave quite differently from both their more
highly doped counterparts and from their antiferromagnetic
parent compounds. Although no Bragg peak, and so no long
range order, is observed for lightly doped superconductors,
subcritical 3D antiferromagnetic correlations are formed.
This is evident from fact that the spin scattering is centred at
integer values of L for zones (½,½,L) (Fig. 7 ). Thus the
doped holes prevent the formation of the long range ordering
but there is a memory of the phase that would be formed by
further reduction of the hole content.
Compared to the higher doped materials with a high-energy
resonance (33 meV for YBCO6.5) at a commensurate position
and no elastic central mode [25], the energy spectrum of lightly doped superconductors consists (Fig. 8) of a central mode
coupled to a broad inelastic peak with a relaxation of
Fig. 7
In YBCO6.35 with Tc=18 K, the antiferromagnetic
correlations coupling the planes extend over only
15 Å along the c-axis [27].
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insulator. It suggests a frozen glass
state that inhibits
the transition to
magnetic
longrange order and
provides the random spin environment that allows
superconductivity.
Fig. 8
Two magnetic energy scales near the onset of
superconductivity in YBa2Cu3O6.35, a narrow
central mode with FWHH<0.08 meV, and faster
relaxational excitations peaked at ~2 meV. The
line is from a model where the soft relaxational
magnetic mode of the superconducting phase is
coupled to an elastic (central) mode and drives
up its intensity to divergence as a quantum
phase transition to the ordered magnetic phase
is approached [26]. The nearly-elastic mode arises from the slow tumbling of about a hundred
copper spins that are nearly ordered.
One of the most
remarkable
features of the cuprate
superconductors is
the characteristic
spin
excitation
energy known as
the Aresonance@. It
tracks Tc as the
doping is varied
(Fig. 11). The spin
spectrum exhibits
a peak whose enerFig. 9 The central peak grows on cooling
gy scales as Eres~
with no change at Tc = 18 K as if the
6kBTc. Inclusion of
spins ignore superconducting transithe results [26] for
tion.
p=0.06 (YBa2Cu3
O6.35) shows that
the inelastic spin energy, albeit reduced by an order of magnitude in energy from that of optimally doped YBCO and
heavily damped (Fig. 8), is a critical spectral feature of superconductivity. Fig. 11 shows it is the soft mode of the superconducting dome.
2.5 meV. Both are centred on the commensurate antiferromagnetic position but are broad in momentum. Correlation
lengths associated with both modes are short ranged (longer
in the basal plane than along the c-axis). The intensity of the
central mode increases on cooling from 80 K and saturates at
a low temperature of order of 10 K with no suppression at Tc
(Fig. 9).
The general behaviour as holes are added is that the strong
antiferromagnetism of the parent insulators is rapidly broken
up, carriers form to conduct electricity and heat, and the spin
excitations evolve into strongly damped paramagnons [25].
Long-range antiferromagnetism has been destroyed and a
superconducting phase is entered with only ~5% of hole doping. This is much less than the percolation limit of ~50%
localized vacancies for a dilute 2D lattice, and clearly shows
that holes produce a large spatial extent of weakened AF coupling. Possibly local ferromagnetic correlations ensue
(Fig. 10).
Perhaps the most surprising property, observed with polarized neutron scattering [26], is that the spin orientation is
isotropic, unlike the XY order of the insulator. We can infer
that the superconductor is in a spin ‘hedgehog’ phase. Such
preservation of spin rotational invariance is a very different
topology than the collinear spins of the antiferromagnetic
262
PHYSICS
IN
CANADA
Fig. 10 A doped hole on the oxygen neighbours puts the
CuO4 into a singlet state and may cause ferromagnetic bonding. Even a low doping of ~5% destroys the
antiferromagnetic order because every hole affects
many sites.
September / October 2006
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ARTICLE DE FOND ( QUANTUM MAGNETISM ... )
Fig. 11 The characteristic energy of the inelastic spin response tracks
the superconducting transition temperature Tc(p) as the doping p in the CuO2 planes is increased.
In cuprate superconductors the reason for the remarkable
tracking of the superconducting transition temperature with
the resonance energy (Fig. 11) has not been explained satisfactorily. While a number of theories based on a Fermi liquid
coupled to a spin susceptibility have attempted to explain
individual experiments at large, near-optimum doping
where the electrons form a Fermi liquid, these theories are
unlikely to work in the region near the lower edge of the
superconducting dome that we have studied. There the electronic hole density is small, there is considerable doubt as to
whether a sharp Fermi surface exists, and the resistivity is
insulator-like, falling with increasing temperature, thus mirroring the decrease in χ”(q,ω) with frequency. Moreover the
spin fluctuations are so strong (recall from Fig. 9 they ignore
Tc) that a description based more on states, RVB or otherwise,
that pre-exist at a less-than-critical doping would seem a better starting point.
In this regard we suggested that the resonance can be regarded [24] as an image of the two-particle pairing states, states
that are allowed in the particle-hole spectrum detected by
neutrons only by dint of the superconducting order. Because
the pairing gap is d-wave of the form cos(kx)-cos(ky), the
spectrum of spin states coupled incoherently to all electron
momenta should exhibit an anisotropic rise to a peak at the
maximum d-wave gap followed by a sudden fall. This asymmetric resonance spectrum is very close to what was
observed in an oxygen-ordered crystal of YBCO6.5 (Fig. 12
based on [24]) and may be a fingerprint of superconducting
pairing. Moreover, in the normal phase almost half the resonance weight has already formed on cooling to just above the
superconducting transition temperature. We believe that this
fingerprint shows that incoherent superconducting pairs are
present in the normal phase. By contrast conventional
phonon-mediated superconductors show an extremely nar-
Fig. 12 The spin resonance peaked at 33 meV in
YBCO6.5 in its superconducting phase (8 K)
and in its normal phase (85 K) above its superconducting transition temperature of 59 K. The
π,π
π) selects spin
two-dimensional wave vector (π
fluctuations that have opposite sign (are of antiferromagnetic symmetry) between neighbouring Cu atoms in the square lattice. For this AF
phasing there is no gap in an ordered antiferromagnet. In the superconductor with its d-wave
gap for pairing charge carriers, the spin
response is shifted upward. The presence of a
similar but weakened spectrum above Tc. indicates that local incoherent pairs have already
formed in the normal phase [24] within vortexantivortex fluctuations.
row temperature range for critical fluctuations. When we
reduce the doping to the edge of the superconducting phase
we have seen that the spin fluctuations are strong and in this
interpretation are dominated by incoherent pairs, so much so
that they show little change in their growth rate on cooling
through Tc. Needless to say this concept is highly controversial, for it would suggest that the lightly-doped but nonsuperconducting antiferromagnet would carry some of the
same local pairing symmetry as the superconductor.
CONCLUDING REMARKS
The power of neutron scattering is that it provides direct
access to the energy, momentum and spin of the fundamental
particles in condensed matter systems. Other spectroscopic
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FEATURE ARTICLE ( QUANTUM MAGNETISM ... )
techniques are generally less direct, such as the local probes
of muon spin resonance and NMR. The positive muon traps
and interacts strongly on the large eV scale with its immediate electronic environment drawing a screening electron
around it; the field it measures may in some systems be different on the meV scale of spin fluctuations than the unperturbed field of the system. Other probes give an average of
the charge but not spin spectra, or are averages over many
particles such as thermal and electrical conductivity and specific heat. Neutron scattering has allowed new phases of
matter to be discovered as we have seen for quantum gapped
systems, for a highly-correlated heavy-fermion system, and
for the quantum antiferromagnet doped to form a superconductor.
12.
13.
14.
15.
16.
17.
ACKNOWLEDGEMENTS
WJLB benefited as a member of the Canadian Institute for
Advanced Research and both authors recognize technical
and scientific support from CNBC, Chalk River, and NIST,
Gaithersberg, MD. We are grateful to C. Stock, and to many
colleagues, for their insight and help over several years.
18.
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ARTICLE DE FOND ( POLARIZED NEUTRON REFLECTOMETRY ... )
POLARIZED NEUTRON REFLECTOMETRY
AS A UNIQUE TOOL IN MAGNETIZATION REVERSAL
STUDIES OF THIN FILMS AND MULTILAYERS
by H. Fritzsche, Z. Yamani, R. Cowley and R.C.C. Ward
T
understand the magnetization reversal process and to distinypically, the magnetization reversal behavior of thin magguish between a domain wall movement and a rotation
netic films and multilayers is studied with magnetometers
process because generally magnetic domains give rise to elec(e.g. a vibrating sample magnetometer or a superconducting
trical noise and reduce the performance of a sensor [13].
quantum interference device) or with the magneto-optical
Exchange-bias systems consisting of a ferromagnet in direct
Kerr effect. In this article we show how Polarized Neutron
contact with an antiferromagnet repreReflectometry (PNR) can be used as a
sent an essential component of thin-film
tool to measure hysteresis loops and to
study the magnetization reversal In this article we show how systems used as sensors. PNR shed
behavior of magnetic multilayers in Polarized Neutron Reflecto- light on the asymmetric magnetization
reversal of these
exchange-biased
more detail. We will discuss the
For the
instrumental setup needed to perform metry (PNR) can be used as multilayers [11,14,15,16,17].
PNR experiments along with the theo- a tool to measure hystere- Co/CoO system PNR experiments
revealed a rotation process of small
retical background of this technique.
In contrast to conventional magne- sis loops and to study the domains for increasing fields and a
motion for decreasing
tometers capable of measuring only
magnetization
reversal domain-wall
fields during the first cycle of a hysterethe average magnetization of a multilayer, PNR is able to determine the behavior of magnetic multi- sis loop.
magnetization profile and can distinIn this article we provide an introducguish between the magnetizations of layers in more detail.
tion to the theoretical description of
different magnetic layers in a multilayPNR along with some model simulaer. This feature of PNR being elementtions of reflectivity curves followed by a description of the
specific, will be demonstrated with the magnetization reveressential components needed to set up a polarized neutron
sal study of a (6 nm ErFe2 / 6 nm DyFe2) multilayer, where
reflectometry experiment. How PNR can yield layer-sensiwe were able to follow the magnetization reversal of the ErFe2
tive information for a magnetization reversal of a multilayer
and DyFe2 magnetizations independently.
consisting of two different ferromagnetic layers is shown for
INTRODUCTION
the case of a (6 nm ErFe2 / 6 nm DyFe2) multilayer. These
types of multilayers are of technological interest because they
During the last two decades Polarized Neutron Reflectometry
have potential applications for sensors and magnetic read
(PNR) has become a very powerful and popular technique in
heads.
the study of magnetic properties of thin films and multilayers. PNR has drawn a lot of attention of the scientific commuTHEORETICAL DESCRIPTION OF PNR AND
nity due to the study of the oscillatory exchange coupling in
MODEL SIMULATIONS
Fe/Cr [1,2], Co/Cu [3] or Fe/Nb [4,5] multilayers. PNR is the
method of choice to prove the antiferromagnetic coupling of
The scattering geometry of a typical reflectometry experiment
the magnetizations of ferromagnetic layers (e.g. Fe) separated
is shown in Fig. 1. The neutron beam hits the surface at an
by non-ferromagnetic layers (e.g. Cr) because the antiparallel
angle θ and the reflected intensity is simply measured as a
alignment of the magnetic layers gives rise to an additional
function of θ, which is typically in the range between 0 and 2°.
peak in the reflectivity curve. Even more complicated strucThe interfaces of the samples are arranged perpendicular to
tures such as a non-collinear 50° coupling [6] or a helical magthe scattering vector
netic structure [7] can be determined by PNR. Another area
4π
where PNR has been applied very successfully is the determi(1)
q = kf − ki =
sin(θ)
nation of the absolute magnetic moment of ultrathin Fe and
λ
Co films [8,9,10] to study surface and size effects. Here, the big
with k i and k f being the incoming and outgoing neutron wave
advantage of PNR is that the substrate does not contribute to
vector and λ the neutron wavelength. The neutron spins are
the magnetic signal unlike in conventional magnetometry
measurements where the diamagnetic signal of the substrate
dominates over the tiny magnetic signal originating from the
H. Fritzschea <[email protected]>,
sample.
a
b
b a
Only recently PNR was used to study the magnetization
reversal of thin films [11,12]. Thin magnetic films are used as
magnetic field sensors and therefore it is very important to
Z. Yamani , R. Cowley , R.C.C. Ward ; National Research
Council Canada, CNBC, Chalk River Labs, Chalk River,
ON, K0J 1J0, Canada; bOxford Physics, Clarendon
Laboratory, Parks Road, Oxford OX1 3PU, UK
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FEATURE ARTICLE ( POLARIZED NEUTRON REFLECTOMETRY ... )
Fig.2
The scattering length density (SLD) for spin-down neutrons
(left) and spin-up neutrons (right) for the case of bulk Fe. The
SLD is composed of a nuclear and a magnetic part.
Vj± =
Fig. 1
Scattering geometry for a neutron reflectometry experiment with the scattering angle θ,
scattering vector q,
P the external magnetic
P
field H
ext, and the magnetic moment of the
spin-up neutrons ( μP+) and spin-down neutrons ( μP-). The spin-flip (SF) axis as well as
the non-spin flip (NSF) axis is indicated
along with the crystallographic orientation
used for the experiments on the (ErFe2 /
DyFe2) multilayer.
oriented either parallel to the external field (spin-up neutrons) or antiparallel (spin-down neutrons) to the external
field which is in the film plane (see Fig. 1). The interaction
with the film is reduced to a one-dimensional problem,
which can be described for grazing incidences with an effective potential that is a sum of a nuclear and a magnetic component (the Zeeman interaction). For the case that the sample’s magnetization is parallel to the external field this potential Vj in layer j is given by:
Vj =
2π 2
N j b nuc
− μ ⋅ Bj
j
m
PHYSICS
IN
CANADA
(
)
(3).
PNR is very sensitive to magnetic structures because the
nuclear and magnetic SLDs are of the same order of magnitude. The nuclear SLD depends on the elements and their
isotopes in the sample [18,19]. A diagram of the SLD and the
simulated reflectivity curves for the case of bulk Fe saturated
along the external field (Nbnuc = 801.9 μm-2 and
Nbmag = 498.5 μm-2) is shown in Fig. 2 and Fig. 3, respectively. The solid line in Fig. 3 denotes R+, the reflectivity of spinup neutrons, the dashed line is R-, the reflectivity of down
neutrons. The reflectivity of the sample can be calculated by
solving the Schrödinger equation using the above mentioned
potential. The simulations presented here were calculated
with software based on the Parratt formalism [20]. The critical edge qc up to which total reflectivity (R=1) is observed is
different for spin-up and spin-down neutrons and depends
on the SLD in the following way:
(2)
where m and μ6 denote the neutron mass and magnetic
6
moment and Nj , bjnuc, and B j denote the atomic density,
coherent nuclear scattering length, and magnetic induction in
layer j. The first term in Eq. (2) results from the interaction of
the neutron with the nucleus, while the second term results
from the interaction of the neutron with the magnetization of
the sample. The product Njbjnuc is known as the nuclear scattering length density (SLD). The magnetic part of the potential depends on the orientation of the magnetic induction
with respect to the magnetic moment of the neutrons. It is
important to note that we are not sensitive to a perpendicular
magnetization component in the PNR setup sketched in
Fig. 1. The magnetic contribution to the potential can also be
expressed in terms of a magnetic SLD Njbjmag, which makes
it convenient to discuss the sample’s properties in terms of
SLDs. According to Eq. 2 the potential V+ for spin up and Vfor spin down neutrons is different and they can be expressed
in terms of a magnetic SLD as follows:
266
2π 2
N j b nuc
± N j b mag
j
j
m
September / October 2006
Fig. 3
Simulated reflectivity curves of a bulk Fe sample for spin-up neutrons (R+, solid line) and
spin-down neutrons (R-, dashed line).
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ARTICLE DE FOND ( POLARIZED NEUTRON REFLECTOMETRY ... )
to an additional peak at q = 2π/dAF. The
existence of this additional peak shows
immediately that the ferromagnetic layers are oriented antiparallel with respect
to each other. As the total magnetization
of the multilayer is zero, the critical scattering vector for spin-up and spin-down
neutrons are identical.
So far we only discussed cases where the
magnetization is collinear with the external field. The big advantage of PNR is
that it is also sensitive to in-plane magnetization components perpendicular to an
external field. These perpendicular components (parallel to the [001] direction in
Fig. 1) give rise to a so-called spin-flip
process, i.e. an incoming spin-down neutron will be reflected as a spin-up neutron
Fig. 4 Simulated reflectivity curves and magnetic structure of a ferromagnetically and vice-versa. The spin-flip scattering is
proportional to |μ6xB6| and is maximum
aligned (4 nm Fe / 1 nm Cr)20 multilayer in an external magnetic field Hext.
-1
Clearly observable is the chemical peak at q = 0.127 Å due to the chemical if the whole magnetization is perpendicumodulation of the multilayer.
lar to the external field or the neutron
spin, respectively. As shown in Fig. 6, the
antiferromagnetic structure of a Fe/Cr multilayer can be
observed as a peak in the spin-flip reflectivities R-+ and R+- if
q c± = 16 π N( b nuc ± b mag )
(4).
the magnetization of the Fe layers is perpendicular to the
external field. At the same time no antiferromagnetic peak
The simulated reflectivity curves of a ferromagnetically
can be observed in the non-spin flip reflectivities R++ and R--.
aligned (4 nm Fe / 1 nm Cr)20 multilayer along with the magThe two superscripts of the reflectivities denote the spin state
netic structure of the multilayer is depicted in Fig. 4. In this
of the neutrons before and after reflection from the sample.
case, the chemical period dchem is identical to the magnetic
This sensitivity of PNR to magnetization components parallel
period and leads to a Bragg peak at q = 2π/dchem. Analogous
and perpendicular to the external field has been exploited, e.g.
+
to the case of bulk Fe, the critical scattering vector qc for
to study the spin-flop transition in an antiferromagnetically
spin-up neutrons is larger than the critical scattering vector
coupled Fe/Cr multilayer [21]. At low fields the magnetizaqc- for spin-down neutrons. For the case of an antiferromagtion is collinear with the external field. When increasing the
netic alignment of the Fe layers, as shown in Fig. 5, the magexternal field the whole magnetic structure rotates at a certain
netic period dAF is now twice the chemical period and leads
field value from a collinear alignment like in Fig. 5 to a perpendicular arrangement as shown in Fig. 6.
So, where conventional magnetometers would simply measure
an averaged zero magnetization,
PNR can determine the magnetic
structure of the magnetic multilayer in more detail and even distinguish between the two types of
antiferromagnetic
alignment
shown in Figs. 5 and 6. More
detailed information on the theoretical description of PNR and fitting algorithms can be found elsewhere [22,23,24].
INSTRUMENTAL SETUP
Fig. 5
Simulated reflectivity curves and magnetic structure of an antiferromagnetically
aligned (4 nm Fe / 1 nm Cr)20 multilayer with its magnetizations collinear with the
external field. Clearly observable is the chemical peak at q = 0.127 Å-1 due to the
chemical modulation of the multilayer and the AF peak at q = 0.065 Å-1 due to the
magnetic modulation of the multilayer.
LA PHYSIQUE AU CANADA
The C5 triple axis spectrometer at
the neutron reactor NRU in Chalk
River was used to perform the
neutron reflectometry experiments
described in the next section. The
(111) reflection of a Cu2MnAl
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FEATURE ARTICLE ( POLARIZED NEUTRON REFLECTOMETRY ... )
As the polarizer delivers only
spin-down neutrons, we need a
device called “spin flipper”
(SF1) located after the polarizer
to convert spin-down neutrons
into spin-up neutrons in order
to be able to measure the reflectivity curve of spin-up neutrons
as well. Another spin flipper is
needed in front of the analyzer
(SF2) to determine the spin state
of the reflected neutrons. At
reactor sources Mezei-type spin
flippers [26] are commonly used.
They consist of two solenoids
with rectangular cross section.
One solenoid flips the neutron’s
spin and the other one compensates the stray fields. The field
produced by the compensation
Fig. 6 Simulated reflectivity curves and magnetic structure of an antiferromagnetically
coil is collinear with the guide
aligned (4 nm Fe / 1 nm Cr)20 multilayer with its magnetizations perpendicular to the field, whereas the flip field is
-1
external field. Clearly observable is the chemical peak at q = 0.127 Å in the nonperpendicular to the guide field.
spinflip reflectivities R- - and R++ due to the chemical modulation of the multilayer
Therefore, the neutron’s spin
-1
-+
+and the AF peak at q = 0.065 Å in the spin-flip reflectivities R and R due to the
precesses in the flip field and
magnetic modulation of the multilayer.
one has to adjust the magnetic
field produced by this solenoid
Heusler crystal is used to polarize and analyze the neutrons’
by tuning the current through the flipper coil in order to
spin state at a wavelength of λ=0.237 nm in combination with
achieve exactly a π rotation when the neutrons pass through
a pyrolitic graphite (PG) filter which reduces the higher order
the magnetic field region. A magnetic guide field of at least
contributions of a monochromator (λ/2, λ/3, etc.) [25] by a
a few Gauss is always needed along the neutron path in order
factor of about 1000. The distances from the monochromator
to preserve the spin polarization.
to the first slit, first slit to second slit, and second slit to samThe polarization of the beam can be determined by measurple were 0.16 m, 1.44 m, and 0.18 m, respectively. A sketch of
ing the flipping ratio F, i.e. F- = I --/I -+ for spin-down neuthe setup is displayed in Fig. 7. The components needed are
trons or F+ = I++/I+- for spin-up neutrons, respectively. The
a monochromator (M) and analyzer crystal (A), slit systems
polarization of the neutron beam can be deduced via:
(S1, S2, S3, and S4), spin flippers (SF1 and SF2), a PG filter (F),
a sample, and a detector (D). The slits S1 and S2 define the
I −− ( 1 − 1 / F )
I −− − I − +
F−1
(8).
collimation of the beam, whereas the slits 3 and 4 reduce the
P
=
=
=
−+
−−
−−
background. The detector is a 5 bar 3He gas detector.
I +I
I ( 1 + 1 /F )
F+1
The maximum achievable polarization by using the (111)
Bragg reflection of a Cu2MnAl Heusler crystal can be easily
calculated from the different structure factor for up neutrons
and down neutrons. The intensity I+ and I- for up and down
neutrons, respectively, is simply, ignoring element-specific
Debeye-Waller factors:
I+ ~ I0 × (bnuc, Mn – bnuc, Al + bmag, Mn)2 ~ I0 × 2.09
(5)
I-
(6)
~ I0 × (bnuc, Mn – bnuc, Al -
bmag, Mn)2
~ I0 × 249.7
where bnuc, Mn = -3.73 fm denotes the nuclear scattering
length of Mn, bnuc, Al = 3.449 fm denotes the nuclear scattering length of Al, and bmag, Mn = 3.2H2.695 fm denotes the
magnetic scattering length of Mn with 3.2 μB per Mn atom at
room temperature. The neutron beam polarization P is
defined as:
P=
I− − I+
I− + I+
(7)
Fig.7
and in the case of a perfect Heusler crystal the maximum
achievable polarization is 98.3%.
268
PHYSICS
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September / October 2006
Neutron reflectometry set-up using a polarizing
monochromator (M) and analyzer crystal (A), slit systems (S1, S2, S3, and S4), spin flippers (SF1 and SF2),
a PG filter (F), and a detector (D).
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ARTICLE DE FOND ( POLARIZED NEUTRON REFLECTOMETRY ... )
The square root is used here because the neutrons are reflected twice, from the monochromator and the analyzer, using
for both the same type of crystal, in our case a Heusler crystal. In our setup we typically achieve flipping ratios higher
than 25 corresponding to a polarization of about 96%.
The experiments on the (6 nm ErFe2 / 6 nm DyFe2)40 multilayers were carried out in a magnetic field of up to 6 T using
a cryomagnet because the rare earth / iron alloys are wellknown for their huge magnetic anisotropies and therefore, a
large external magnetic field is needed to saturate the sample’s magnetization along certain crystallographic directions.
The additional measures needed to maintain the neutrons’
polarization in the presence of huge magnetic stray fields of
a cryomagnet is described elsewhere [27].
MAGNETIZATION REVERSAL OF A (6 nm ErFe2 /
6 nm DyFe2)40 MULTILAYER
During the last decade there has been an increasing interest
in thin films and multilayers composed of rare earth materials because they have many potential technical applications
as sensors and magnetic read heads. In the (ErFe2 / DyFe2)
multilayer system both the DyFe2 and the ErFe2 layers are
hard magnets but with different easy axes in the bulk samples. The term “easy axis” describes the direction in which
the magnetization prefers to lie without an external field.
Bulk DyFe2 has the crystallographic <100> directions as easy
axes, whereas bulk ErFe2 has its easy axes along the <111>
directions. This can lead to very complex magnetic structures when considering that additional anisotropies such as
interface, shape, and strain anisotropy, are present in these
thin magnetic films [28,29,30]. On top of that there exists an
exchange interaction between the Fe atoms of adjacent layers
that strongly favors a parallel alignment of the magnetizations.
tion. The sample’s orientation with respect to the magnetic
field and the neutron spins is depicted in Fig. 1. After having
reached the negative saturation the magnetic field was
ramped down to zero and reversed to positive values. The
shown reflectivity curves were recorded at a field of 0.4 T (top
panel), 1 T (middle), and 6 T (bottom). Only the non-spin flip
reflectivities for down neutrons (R--, open circles) and upneutrons (R++, closed circles) are shown, the measured spinflip reflectivities R-+ and R+- were zero within errors. In the
top panel of Fig. 8 it can be clearly seen that the critical scattering vector up to which total reflectivity occurs, is larger for
R-- than for R++. That proves that the magnetization in a positive field of 0.4 T is still reversed, i.e. antiparallel to the
applied field. In positive saturation, as shown in the bottom
panel, the critical scattering vector for R-- is much smaller
than that for R++. The coercive field is about 1 T as can be
seen in the middle panel because the reflectivity curves for
spin-up and spin-down neutrons are very similar, especially
the critical scattering vectors for both cases are very close to
The magnetic structure of both the ErFe2 and DyFe2 is called
ferrimagnetic because the magnetic moment of the Fe atoms
is antiparallel and unequal to the magnetic moment produced by the rare earth atoms Dy and Er. Below room temperature the magnetic moment produced by the rare earths is
larger, and therefore the net magnetization of the layers
points always along the direction of the rare earth magnetization.
The sample was grown using the molecular beam epitaxy
facility in the Clarendon Laboratory, Oxford. Sapphire substrates with a (11⎯20) orientation were cleaned and 100 nm
niobium was deposited as a chemical buffer layer, followed
by a 2 nm iron seed to improve the crystal growth [31]. The
multilayer was then grown by co-deposition of the elementary fluxes with a layer thickness of 6 nm for both layers,
repeating the (ErFe2 / DyFe2) sequence 40 times. Finally the
multilayer was covered with a 10 nm thick yttrium layer as a
protection against oxidation. Both layers, DyFe2 and ErFe2,
had a (110) surface orientation.
In Fig. 8 a series of reflectivity curves are shown during a
magnetization reversal cycle at a temperature of T = 100 K.
Prior to the measurements the sample has been saturated in
a field of –6 T with the field along the in-plane [1⎯10] direc-
Fig. 8
Measured neutron reflectivities R++ (solid circles)
and R- - (open circles) of a (6 nm ErFe2 / 6 nm
DyFe2)40 multilayer along with the fits (solid and
dashed line, respectively) at a temperature T=100 K,
in an external magnetic field of 0.4 T (top panel), 1 T
(middle panel), and 6 T (bottom panel). Prior to the
measurements, the magnetization of the sample was
reversed in a field of –6 T, the field was always
applied along the [1⎯⎯1 0] in-plane direction.
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FEATURE ARTICLE ( POLARIZED NEUTRON REFLECTOMETRY ... )
Fig. 9
Magnetization component along the external field of
the DyFe2 layers (solid circles), ErFe2 layers (open circles), and the average magnetization of the (6 nm
ErFe2 / 6 nm DyFe2)40 multilayer (triangles) as a function of the external field at a temperature of 100 K.
Prior to the measurements, the magnetization of the
sample was reversed in a field of –6 T, the field was
always applied along the [1⎯⎯1 0] in-plane direction.
each other. For a completely demagnetized sample the spinup reflectivity would equal the spin-down reflectivity. An
important observation was that no increased spin-flip signal
was observed at 1 T. That means that there is no homogeneous in-plane rotation of the magnetization. The magnetization must be reversed either by domain wall movement or
a magnetization rotation perpendicular to the sample’s surface because in PNR we are not sensitive to magnetization
components perpendicular to the sample’s surface.
The reflectivity curves shown in Fig. 8 were fitted using the
Parratt formalism [20]. The solid line represents the fit to the
spin-up reflectivity, whereas the dashed line represents the
fit to the spin-down reflectivity. From the fit data we can
infer the magnetization component of the DyFe2 and ErFe2
layer parallel to the external field independently. The values
are shown in Fig. 9 with the closed circles representing the
magnetization component of the DyFe2 layers and the open
circles representing the magnetization component of the
ErFe2 layers. The average value, as it would be measured by
classical magnetometry is displayed as triangles. It can be
clearly seen that the behavior of the two magnetizations is
different. The magnetization curve of the DyFe2 represents
an easy axis loop with a very fast switching from negative to
positive saturation. In contrast, the ErFe2 magnetization
curve represents a hard axis loop where the magnetization
rotates continuously towards the direction of the magnetic
field. This nicely shows the capability of PNR to be elementspecific.
This property of PNR to be element-specific is underlined in
Fig. 10 where simulated reflectivity curves are shown for
three different cases of a magnetic structure giving a zero sig-
270
PHYSICS
IN
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Fig. 10 Simulation of R++ and R- - for different magnetic
structures in a (6 nm ErFe2 / 6 nm DyFe2)40 multilayer: in the top panel for the case where the magnetization of the DyFe2 layers is antiparallel to the external
field, the magnetization of the ErFe2 layers parallel to
the external field, the middle panel with swapped
magnetization directions, i.e. the DyFe2 layers parallel, the ErFe2 layers antiparallel to the external field,
and the bottom panel for the case of zero magnetization in both layers.
nal in a classical magnetometer. In the top panel the magnetization of the DyFe2 (MDyFe2 = -0.13 T) is antiparallel to the
external field, whereas the magnetization of the ErFe2 layer
(MErFe2 = 0.13 T) is parallel to the external field. This corresponds to the magnetic structure at the coercive field as
deduced from our PNR data at a magnetic field of 1 T (Fig. 8,
middle panel). As can be seen in the middle panel of Fig. 10,
where the directions of both magnetizations are swapped,
the intensity ratio of R++ to R- - at the Bragg peak position is
swapped as well. So, it is easy in a PNR experiment to distinguish between these two cases and it shows that the PNR
data in Fig. 8 (middle panel) could not have been fitted with
a magnetic structure where the directions of both layer magnetizations are reversed. For comparison, the case for zero
magnetization, where R++ = R- -, is displayed in Fig. 10 (bottom panel).
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ARTICLE DE FOND ( POLARIZED NEUTRON REFLECTOMETRY ... )
The reason for this element-specific magnetic information is
the different nuclear SLD of DyFe2 (Nbnuc = 729 μm-2) and
ErFe2 (Nbnuc = 554 μm-2). So, this type of element-specificity
of PNR would not work for systems where the nuclear SLDs
are identical or very close to each other.
CONCLUSION
The magnetization reversal experiments described here
demonstrate the unique capabilities of Polarized Neutron
Reflectometry for determining the magnetic structure of multilayers with different magnetic layers. PNR can determine
complicated magnetic structures inaccessible to standard
magnetometry techniques. PNR is capable of measuring element-specific hysteresis loops, i.e. PNR can distinguish the
magnetization of different magnetic layers, where standard
magnetometry techniques would only measure the average
magnetization.
9.
10.
11.
12.
13.
14.
15.
16.
ACKNOWLEDGMENT
17.
We are indebted to Zin Tun, CNBC, for his continuous efforts
during the past decade to install and improve the neutron
reflectometry set-up on C5 and for his time to explain to us
how to use his set-up properly. We appreciate the careful
reading of the manuscript by W.J.L. Buyers, Z. Tun, and
M. Saoudi.
18.
19.
20.
21.
22.
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CALL FOR NOMINATIONS
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MÉDAILLE HERZBERG DE L’ACP
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PRIX ACP/CRM DE PHYSIQUE THÉORIQUE ET MATHÉMATIQUE
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I
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ARTICLE DE FOND ( DIFFRACTION STUDIES OF GAS HYDRATES ... )
DIFFRACTION STUDIES OF GAS HYDRATES WITH
AN EMPHASIS ON CO2 HYDRATE
by B.H. Torrie, O.S. Binbrek, I.P. Swainson, K.A. Udachin, C.I. Ratcliffe and J.A. Ripmeester
U
nder the right conditions of high pressure and low temrounding water so knowledge of the density is critical for
peratures, water will form cages around a variety of simple
modeling this dispersal method. Gas hydrates are non-stomolecules. The cages are interconnected to form a crystal
chiometric compounds since the cage occupancies vary
structure and space filling requirements dictate that there be
depending on how the hydrate is formed and therefore the
more than one type of cage. Cage type is also determined by
density also varies. The purpose of the research described in
the size of the molecule that is enclosed. There are three comthe rest of this article is to develop a fuller understanding of
mon structures labelled I, II and H
how the CO2 molecules occupy the
with the structure of H being detercages in the hydrate, and to determine
mined by an NRC group using neu- Gas hydrates are a potential how practical it is to determine the dentron diffraction at Chalk River in the
sity of the samples using single crystal
source of huge quantities of and powder diffraction techniques.
1980’s [1].
natural gas and can also be
Why are gas or clathrate hydrates of
The structure of CO2 hydrate has been
interest? The right conditions for the used to store natural gas investigated in the past by a Japanese
[3]
formation of the hydrates are found in and
hydrogen and to group and a U.S. group working [4]at
coastal waters and under permafrost.
the Argonne National Laboratory ,
Natural gas hydrates have been found sequester carbon dioxide.
both using powder neutron diffraction.
off the west coast of Vancouver Island
A much more detailed structure was
and in the Mackenzie Delta and many
determined using single crystal X-ray
similar places around the world. The components of the natdiffraction at NRC [5].
ural gas come from two sources, the decay of organic matter
formed in the oceans or discharged from the continents,
BASIC DIFFRACTION THEORY
known as biogenic gas, or from the thermal cracking of
Since physics departments often neglect to teach basic crystalhydrocarbons that have migrated from deeper sources,
lography, a mini-course is included here so that you can betknown as thermogenic gas. It has been estimated that the
ter understand the rest of this paper. A comparison between
total fuel in gas hydrate form exceeds that from coal, oil and
diffraction from a grating and crystal diffraction should be
other sources of natural gas combined. Unfortunately these
helpful. If a plane wave is incident on a reflection grating,
hydrates are widely dispersed in awkward locations so gas
then diffraction from the grating lines produces cylindrical
hydrates are tomorrow’s fuel, not today’s. Of more immediwaves that combine to give a diffraction pattern made up of
ate interest is that synthetic gas hydrates can be formed from
lines of various orders at the detector. The more grating lines
natural gas. In the case of methane, the resulting crystals are
that contribute to the pattern, the sharper the diffraction lines
metastable for several days at ambient pressure just below
[2]
will be. Similarly, when a plane wave is incident on a crystal,
0C. Research in Norway
indicates that natural gas
the wavelets produced by the scattering centres combine to
hydrates can be formed, sent to markets in ships and decomgive a diffracted plane wave. Conditions are such that a scatposed to release the gas at a cost which is comparable to the
tering plane can be defined that acts like a mirror with the
cost of shipping liquid natural gas which is the alternative in
angle of incidence equal to the angle of reflection for the plane
current use when pipelines are not practical.
waves. The ‘mirror’ is called a Bragg plane. There are many
parallel Bragg planes in a crystal and, if the condition for conIn a similar vein, gas hydrates can be used to store hydrogen.
structive interference is satisfied, the plane reflected waves
There has been much hype about the hydrogen economy of
the future but, until a safe and economical method of storing
hydrogen has been developed, a hydrogen based economy is
Bruce Torrie <[email protected]>a, Omar Binbrekb,
not practical. Experimental and theoretical work at NRC has
Ian Swainsonc, Konstantin Udachind, Christopher
advanced the knowledge of storing clusters of hydrogen in
d and John Ripmeesterd; a Department of
Ratcliffe
hydrate cavities.
Another gas hydrate of interest is formed from water and
CO2. It has been proposed that this greenhouse gas, produced by the burning of fossil fuels, could be pumped into the
oceans at an appropriate depth where it would form a
hydrate that would sink to the ocean floor. Obviously this
will only happen if the CO2 hydrate is denser than the sur-
Physics, University of Waterloo, Waterloo, Ontario,
N2L 3G1; b Department of Physics, University of
Petroleum and Minerals, Dhahran, Saudi Arabia;
c National Research Council of Canada, Steacie Institute
for Molecular Sciences, Chalk River Laboratories, Chalk
River, ON, K0J 1J0; d Steacie Institute for Molecular
Sciences, National Research Council of Canada, Ottawa,
ON, K1A 0R6
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FEATURE ARTICLE ( DIFFRACTION STUDIES OF GAS HYDRATES ... )
will combine to give a Bragg reflection. The more planes that
contribute to the reflection, the sharper the reflection will be.
Scattering centres and Bragg planes are somewhat artificial
constructs but because a crystal is made up of three-dimensional repeating units, there are associated scattering centres
made up of atoms or groups of atoms that repeat in space and
through these centres it is possible to construct planes of various orientations that again repeat in space.
Three other concepts need to be introduced to give a basic
understanding of crystal diffraction. Bragg peaks will be
broadened by a combination of two instrumental effects. If
the incident wave is not plane, meaning that the incident
beam diverges or converges, then the peaks will be broadened if the incident wave if not monochromatic. Most
sources of neutrons, as found at Chalk River and other
places, tend to be weak compared to modern laboratory Xray sources or synchrotron X-ray sources so a more relaxed
geometry is used to increase the intensity but this also broadens the diffraction peaks. There are also sample dependent
effects to consider. If the spacing between the Bragg planes
varies because of strains in the crystal, then the peaks will
again be broadened. Also, the atoms in the crystal are in constant thermal motion and this alters the interference condition as a function of time. Diffraction measurements give a
time average of this motion which can be analyzed in terms
of a thermal ellipsoid since the amplitude of the motion
varies with direction depending on the nature of the interatomic bonding. Thermal motion reduces the intensity of
Bragg peaks at high scattering angles.
Bragg planes are labeled with three Miller indices, h k l, that
give the orientation and spacing between the planes. For a
cubic crystal, such as CO2 hydrate where the basic building
block is a cube, the planes parallel to the cube faces are
labeled (100), (010) and (001). More generally, the planes are
labeled (hkl) where h, k and l are the reciprocals of the intersection points of the Bragg plane with the cell sides. For
example, the plane parallel to the z-axis and intersecting the
x and y-axes at the length of one cell side away from the ori-
Fig. 1
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Small (dodecahedral) Cage of Type I Gas Hydrate
PHYSICS
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gin is labeled (110) and the separation between all such adjacent planes is l/sqrt(12 + 12 + 02) = 1/sqrt(2) as a fraction of a
cubic cell side. For the plane (hkl) the separation is l/sqrt(h2
+k2 + l2). Two planes will have the same separation if (h2 +k2
+ l2) is the same for both. An example of this is (334) and
(035). In a single crystal diffraction pattern the diffraction
spots for these reflections are well separated since the interplanar angle is 31.5o. What happens if a powdered sample is
used instead of a single crystal? An ideal powder contains
crystallites with all possible orientations so the diffraction
spots are expanded into rings which overlap completely in
our example. Since the rings have finite widths, for the reasons discussed above, there will also be partial overlap of
rings when the sums of the squares of the Miller indices are
only approximately equal. The density of rings increases
with scattering angle so there is considerable overlap at high
angles.
Let us look a little more carefully at what happens when a single crystal sample is replaced with a powder sample. There
will be a loss of information about interplanar angles since an
array of diffraction spots, located by two angles, has been
replaced by an array of diffraction rings, or more commonly
with a slice through the rings since the detector only operates
in a plane, that are located by one angle. For simpler unit cells
there is enough information in the low angle non-overlapping
powder peaks to determine the unit cell dimensions and
angles so the geometrical loss is not important. Details of the
structure, though, are contained in the peak intensities and
some of this information is lost due to peak overlap when a
powder is used rather than a single crystal.
MODELS FOR CO2 HYDRATE
CO2 hydrate forms a type I gas hydrate with a cubic structure
and a unit cell (basic repeat unit) containing 46 water molecules in a framework of two dodecahedral and six
tetrakaidecahedral cages as shown in Figures 1 and 2. Each
cage can hold one CO2 but the larger cages tend to be fully
occupied and smaller cages only partially occupied with the
Fig. 2
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merge. This is particularly true
of the carbon sites at the centre of
the large cage. The result will be
indistinguishable from what
would be obtained with a single
site occupied by an atom with
large thermal motion.
The
smearing is obviously more of
problem at high temperatures
for both the carbon and oxygen
atoms.
INSTRUMENTATION
The single crystal measurements
were made on a Bruker Smart
CCD X-ray Diffractometer as
found at the Steacie Institute for
Molecular
Sciences,
NRC,
Fig. 3 Possible locations for the oxygen
Ottawa.
Similar
instruments
can
Fig. 4 Possible Locations for the oxygen
atoms of CO2 in the small cage.
be
found
at
a
number
of
atoms
of
CO2
in
the
large
cage.
Black atoms are for the Argonne
Canadian universities. With this
Markings of atoms is the same as in
model, others for the NRC model
Figure 3.
instrument a single crystal difexcept for the clear atom that
fraction pattern was recorded at
marks the centre.
173 oK and the structural (NRC)
model was refined with the SHELXTL software package [6].
occupancy depending on preparation conditions. In the simpler (Argonne) model the carbon atoms sit in the centre of
The second instrument that was used is the C2 powder neuboth cages and oxygens are disordered among a collection of
tron diffractometer, also operated by the Steacie Institute, but
equivalent positions located one C-O bond length away from
located at Chalk River on the NRU nuclear reactor. Details of
the centres. The possible positions of the oxygen atoms in
this instrument can be found on the web, but the following
space is illustrated by the dark spheres in figures 3 and 4. In
differences from the first instrument are of note. X-rays are
the small cage the oxygens lie on a spherical shell and in the
scattered by electrons and scattering intensity increases in a
large cage the oxygens lie in a donut shaped ring. For the
regular fashion throughout the periodic table as the number
more complex (NRC) model, the carbon atoms still are locatof electrons increases. There are exceptions, but that is the
ed in the centres of the small cages but the oxygen atoms
general trend. Neutrons are scattered by nuclei and the scatoccupy three sets of equivalent positions. In other words,
tering intensity varies in an irregular fashion throughout the
there are three times as many possible positions for the oxyperiodic table. Light atoms are much easier to locate with
gens, 36 in total, as in the simpler model as shown by the
neutrons than X-ray but the most important light atom,
dark spheres in figure 3 but the occupancy of any one site has
hydrogen, has a large incoherent scattering cross-section so it
been reduced by a factor of three. For the large cages, the caris routine to use deuterated samples to remove most of the
bon atoms reside in a cluster of 16 equivalent positions surincoherent background. Although not of interest for the
rounding the centre and the oxygens reside in a broadened
present study, neutrons have a spin which interacts with
donut with 32 possible sites. The oxygen sites are shown by
atomic spins so magnetic materials can be studied. An
the spheres marked with quadrants in figure 4. In the
advantage and a disadvantage of neutrons is that the interacArgonne model there are two occupancies, one for each cage
tions with matter are weak. The advantage is that samples
type. In the NRC model there are 3 occupancies for the small
can be surrounded by cryostats and pressure chambers withcage and 2 occupancies for the large cage.
out a large reduction in signal strength but the disadvantage
is that scattering from the sample is weak although fairly uniformly distributed over the sample, i.e. there is little attenuaOBJECT OF RECENT NEUTRON EXPERIMENTS
tion of the incident beam within the sample due to scattering
Obviously the NRC X-ray experiment was very successful in
or absorption which require corrections to be applied to Xleading to the development of a very sophisticated model for
ray data.
CO2 hydrate but there was a desire for a faster method of
measuring occupancies. The growth of single crystal samThe C2 instrument was used to record a powder profile at the
ples is time consuming which is an impediment to making
same temperature of 173oK as was used for the single crystal
measurements on a number of samples prepared under difmeasurements. The sample was prepared using well
ferent conditions. Powder samples, on the other hand, can be
crushed frozen deuterium oxide in a metal reaction vessel
prepared much more readily but there is a loss of information
placed in a bath at -15oC before slowly adding CO2 gas.
in going from a single crystal diffraction pattern to a powder
Reaction conditions were 40 bars of CO2 at a temperature of
pattern as discussed above. With lower resolution the collec5oC for 24 hours to give a composition similar to that of the
tions of atomic sites shown in figures 3 and 4 will tend to
single crystal sample of reference 6.
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b
a
Fig. 5
a) X-ray powder profile (top) and neutron powder profile (bottom) in the range 1 to 2 Å; b) X-ray powder profile (top)
and neutron powder profile (bottom) in the range 2 to 3 Å.
ANALYSIS
As a test of what is possible the single crystal X-ray results
were transformed into an X-ray powder pattern using
GSAS [7], i.e. the atomic positions, occupancies and temperature factors were used as inputs to generate an idealized
powder pattern that is background and noise free. To do this
conversion it is necessary to specify instrumental parameters
for the X-ray powder diffractometer so these parameters
were taken from the X-ray powder diffraction example in the
GSAS manual. The assumed diffractometer is a conventional Bragg-Brentano type using CuKá radiation. Neutron and
X-ray powder profiles are compared in figure 5. The single
crystal diffraction results contain 478 unique peaks whereas
the derived powder pattern contains only 140 unique peaks
for a reduction of more than a factor of three. Similarly the
neutron powder pattern contains only 178 unique peaks for a
reduction of something less than a factor of three. These
reductions reflect the overlap mentioned above but the exact
numbers depend on the fact that the wavelengths and scattering angle ranges of the three instruments were somewhat
different.
Certain aspects of the structure are well known and will not
change significantly from one crystal to the next. For example, the CO2 molecules interact only weakly with the cages
that they occupy so the molecule should retain an O-C-O
angle of 180o and a typical C-O bond length of 1.16 Å. The
geometry of the cages is also well known and can be transferred from one diffraction study to the next with only minor
adjustments. Temperature factors should be similar since the
temperature was the same in all cases. .
The first question to address is whether the sophisticated
model used with the single crystal data can be applied to
powder data. To answer this question it is useful to use the
idealized X-ray powder profile. If the model fails with this
manufactured profile then it is unlikely to work with any real
X-ray or neutron powder profile. First of all the profile was
276
PHYSICS
IN
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tested against the fixed atom model that was used to generate
it. You might think that the fit would be perfect but there is
quantization noise that depends on scaling. The larger the
scale factor, the smaller the quantization noise. With a scale
factor that gives approximately the same number of counts
per peak as in the single crystal work, the fitting process gives
a κ2 = 0.1760E-1. Next step is to reduce the number of variables since the number of unique peaks is reduced as pointed
out above. This was done by fixing the cage structure and
using a rigid body representation of the CO2 molecule.
Occupancies and temperature factors tend to be highly correlated so temperature factors were included to test for correlation effects. Similar atoms should have similar thermal
motions so it was assumed that the framework oxygens,
hydrogens, carbons and oxygens in CO2 for each of the two
cage types had common temperature factors. Isotropic thermal motion was assumed to give a further reduction in the
number of variables. The total number of variables is now
24 compared to 108 in the single crystal study. Note that the
108 includes instrumental parameters that are not varied in
the analysis of the idealized powder pattern.
The resulting fit could be regarded as unsatisfactory or satisfactory depending on your goal. The exactitude of the single
crystal results has been lost. The centres of the molecules in
the large cage moved within the carbon blob in the centre of
the cage as is expected and the orientations of the molecules
in both cages changed significantly but still basically define a
spherical shell or donut as in figures 3 and 4. In spite of these
differences, the occupancies are in good agreement with the
single crystal values. The occupancies are 41%, 59% summing
to 100% (large cage) and 38%, 19% and 14% summing to 71%
(small cage) for both.
For completeness, the simpler Argonne model was also used
to fit the manufactured X-ray powder data. In this case the
carbon atom in the large cage is fixed in the centre so the carbon blob only manifests itself as a large temperature factor.
Again the orientations of the molecules change significantly
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but the occupancies are only slightly changed at 101% and
71%.
The above results gave us confidence that a powder pattern
can be used to determine occupancies but not the details of
arrangement of the CO2 molecules in the cages. Therefore
the analyses were repeated using the two models with a real
neutron powder pattern. From figure 5 it can be seen that the
real pattern differs from the idealized X-ray pattern in having
a background, Poisson noise and decreased resolution associated with increased peak widths. With the sophisticated
model, positional and orientational parameters again tend to
wander away from the single crystal values but the occupancies are only slightly different. For the large cage the occupancies are 42% and 60% summing to 102% and for the small
cage the occupancies are 38%, 17% and 17% summing to 72%.
With the simpler model the occupancies are 102% and 77%.
An anomalous feature of this set of results is that the isotropic temperature factors for the oxygen atoms in the small cages
is very high. It can also be seen from the correlation table in
GSAS that the occupancies and temperature factors are highly correlated. If the temperature factors of the oxygens in the
two types of cages are arbitrarily constrained to have the
same value then the occupancies become 101% and 73%. A
less arbitrary approach would be to make the measurements
at a lower temperature so that thermal motion is less pronounced and there would be less interaction between occupancies and temperature factors. We have done this with the
sample used here and other samples with the expected
decrease in the occupation factors for the small cages.
SUMMARY
Gas hydrates are of interest because they are a future source
of large quantities of natural gas. They also offer a possible
means of storing natural gas for shipment and hydrogen to
fuel the hydrogen economy. This article concentrated on one
aspect of using hydrates, which is for sequestering CO2 at the
bottom of the oceans. The molecules occupy cages in the
hydrate structure and occupancies depend on preparation
conditions. Cage occupancies were determined in diffraction
experiments and from this information the densities of CO2
hydrates can be calculated.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
J.A. Ripmeester, J.S. Tse, C.I. Ratcliffe, and B.M. Powell,
Nature 325, 135 (1987).
J.S. Gudmundsson, M. Mork, and O.F. Graff, 4th
International Conference on Gas Hydrates, page 997, Tokyo,
May 19-32 (2002).
T. Ikeda, O. Yamamuro, T. Matsuo, K. Mori, S. Torii,
T. Kamiyama, F. Izumi, S. Ikeda, and S. Mae, J. Phys. Chem.
Solids 60, 1527 (1999).
R.W. Henning, A.J. Schultz, V. Thieu, and Y.J. Halpern, J.
Phys. Chem. A 104, 5066 (2000).
K.A. Udachin, C.I. Ratcliffe, and J.A. Ripmeester, J. Phys.
Chem. B 105, 4200 (2001).
G.M. Sheldrick, Acta Crystallogr. A46, 467 (1990).
A.C. Larson and R.B. Von Dreele, General Structure Analysis
System, LAUR 86-. 748, The Regents of the University of
California.
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INTERNATIONAL PHYSICS OLYMPIAD
37TH INTERNATIONAL PHYSICS OLYMPIAD, SINGAPORE 2006
BY ANDRZEJ KOTLICKI AND GUILLAUME CHABOT-COUTURE
The 37th International Physics Olympiad (IPhO) was held in
Singapore from 7th to 17th of July, 2006. A total of 93 countries
participated in the competition making it the largest Olympiad
in our history. The number of participating countries increases almost every year with very rapidly increasing participation
from Asia but only 2 countries from Africa. Similarly to the
competitions held in Korea and Indonesia, the Olympiad was
quite clearly an event of primary importance to the
Singaporean government and its educational authorities. The
president of Singapore himself was scheduled to speak at the
opening ceremony but, he was force to cancel at the last
minute due to sickness (reported in the press). In his place several top government officials participated in the opening ceremony and stressed in their opening addresses the paramount
importance of science, technology and education for the
Singapore’s development.
Four Nobel Price Laureates and a Templeton Prize Laureate
gave lectures to the participants and socialized with them.
The social program was very entertaining and interesting with
visits to research centers, historical sites, a night safari and a
continuous “flow” of excellent Singaporean food. The students
were delighted by this exotic experience, most of them on their
first ever international trip.
The academic part of the competition was organized by faculty members from the two major Singaporean Universities: the
National University of Singapore (theoretical problems) and
the Nanyang Technological University (experimental problem).
The problems were quite interesting and well prepared. They
did not concentrate on extensive math but required creative
thinking and the ability to describe physical reality using
appropriate formulas and approximations. Marking by the
academic committee was very thorough and fair, and in most
cases, agreed closely with the marking of the leaders. The
marking moderations (the process of establishing the final
mark acceptable by both leaders and the local marking team)
were performed in a good collegial atmosphere with very few
real controversies.
Canada was represented by the following students:
Mr. Boris Braverman, from Sir Winston Churchill High
School, Calgary, Alberta
Mr. Lin Fei from Don Mills Collegiate Institute, Toronto,
Ontario
Mr. Patrick Kaifosh from University of Toronto Schools,
Toronto , Ontario
Ms Lu Liu from Waterloo Collegiate Institute, Waterloo,
Ontario
Mr Devin Trudeau from Dover Bay Secondary School,
Nanaimo , B.C.
The team leaders were: Dr Andrzej Kotlicki from the
Department of Physics and Astronomy of the University of
British Columbia and Guillaume Chabot-Couture, a former
member of the Canadian team at the IPhO in 2000, and at present, a PhD student at Stanford University.
278
PHYSICS
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The Canadian team had their best performance ever, winning
two gold medals (Boris and Lin), the forth and fifth gold
medals in the history of Canadian participation at the IPhO,
one bronze medal (Patrick) and a honorary mention (Devin).
Boris was 10th overall among over 400 participants. A total of
37 gold medals, 48 silver medals, 83 bronze medals and 81
honorary mentions were awarded.
The following 85 countries participated in the 37th
International Olympiad:
Albania, Argentina, Armenia, Australia, Austria, Azerbaijan,
Belarus, Belgium, Bolivia, Bosnia & Herzegovina, Brazil, Brunei
Darussalam, Bulgaria, Cambodia, Canada, China, Chinese Taipei,
Colombia, Croatia, Cuba, Cyprus, Czech Republic, Denmark,
Ecuador, Estonia, Finland, France, Georgia, Germany, Ghana, Great
Britain, Greece, Hong Kong , Hungary, Iceland, India, Indonesia,
Iran, Ireland, Israel, Italy, Japan, Jordan, Kazakhstan, Korea South,
Kuwait, Kyrgyzstan, Laos, Latvia, Liechtenstein, Lithuania, Macau,
Macedonia, Malaysia, Mexico, Moldova, Mongolia, The
Netherlands, Nigeria, Norway, Pakistan, Peru, Philippines, Poland,
Portugal, Romania, Russia, Saudi Arabia, Serbia, Singapore,
Slovakia, Slovenia, Spain, Sri Lanka, Suriname, Sweden,
Switzerland, Tajikistan, Thailand, Turkey, Turkmenistan, Ukraine,
USA, Uzbekistan and Vietnam
The following 8 countries send observers to the 37th
International Olympiad and plan to participate in the future:
Bangladesh, Cameroon, Montenegro, Myanmar, Nepal, New
Zealand, Puerto Rico, Zimbabwe.
The best score (47.2 points) was achieved by Mailoa Jonathan
Pradana from Indonesia, the absolute winner of the 37th IPhO.
The following limits (out of 50) for awarding medals and honourable mentions were established according to the Statutes:
Gold Medal 37 points, Silver Medal 29 points, Bronze Medal
21 points, Honourable Mention 14 points.
Acting on behalf of the organizers of the next International
Physics Olympiad, Prof. Sepehry Rad announced that the 38th
International Physics Olympiad will be organized in Isfahan,
Iran on July 13th – 21st , 2007. He showed a movie about the
preparation and site of the coming Olympiad and cordially
invited all the participating countries to attend the competition.
After the final national selection held at the end of May and
hosted by the University of Toronto had chosen the current
five best young Canadian physicists, they were invited to an
intensive training on the Canadian west coast at Department
of Physics and Astronomy of the University of British
Columbia before leaving for Singapore. This two day intensive training aimed at better preparing them for the international competition consisted of lectures and exams. The former
also included problem solving sessions on advanced topics in
physics while the latter were tailored to be 5 hours long to
match the duration of the exams at the IPhO and to make sure
that the students could adequately manage their time throughout the competition. After all the tricks and Olympiad wisdom
had been passed on, the team was off to their tropical destination on the other side of the world.
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PHASE TRANSITIONS IN
ORGANIC-INORGANIC PEROVSKITES
by Ian Swainson
T
he name perovskite refers to a mineral found over a large
Amines can be created with arbitrarily long carbon chains
region of (P-T) in the earth’s mantle, with predominant comand bulkier groups such as phenyl rings. Amines larger than
position MgSiO3. More generally, perovskite refers to the
MA, FA and TMA do not fit in the interstices of the ABX3
structure type consisting of octahedra that are fully cornerframework. They tend to crystallize as layer perovskites, typbonded and form a 3d-framework [1]. The general formula
ically in the A2BX4 stacking type (Fig. 1). Here, the interlayer
for this structure type is ABX3, in which the BX6 octahedra are
spacing is free to expand according to the cation size.
usually anionic so that counterions, A, are found in the interPerovskites with organic cations and an inorganic layer or
stices between the octahedra, which
framework have recently become
form variable geometry cages (Fig. 1). The name perovskite refers known as organic-inorganic perovskites
The term perovskite is often expanded
(OIPs), or hybrid perovskites [2]. Sn2+
further to include structures more cor- to a mineral found over halides have taken a special interest
rectly termed layered perovskite-relatof changes in the conductivity
a large region of (P-T) in because
ed structures, formed by concatenation
of the inorganic component at structurof corner-bonded octahedra in a single the earth’s mantle, with al transitions; semiconductor–metal
plane, leaving two unbonded, apical X
and semiconductor-insulator transiatoms protruding above and below the predominant composition tions are common, and some Sn2+-comlayers (Fig. 1). These structures will be MgSiO .
pounds can have a large electrical
3
loosely termed layer perovskites here.
mobility [2,3,4].
There are a large variety of additional
structural forms that are related in various ways to the fundaOne group of very famous transitions in perovskites is the set
mental perovskite structure [2], but we will not discuss them
of tilt transitions, which can be viewed as changes in the confurther here.
formation of the framework, driven by rotations of octahedra.
The highest symmetry form has space group Pm3m where
The most familiar perovskites have simple A cations, but it is
the macroscopic symmetry demands undistorted, untilted
possible to substitute organic molecular cations, most comoctahedra. In the case of the OIPs the amines have to be orimonly amines. Short chained protonated amines such as
entationally disordered in this symmetry as they sit on point
methylammonium (MA) of formula CH3NH3+, tetramethysymmetries that are higher than their molecular symmetry.
lammonium (TMA) of formula (CH3)4N+ and the planar forTilting of the octahedra about one of the pseudo-cubic axes
mamidinium ion (FA) of formula NH2-CH=NH2+ can be
causes antiferro- tilting in neighbouring octahedra connected
placed into the ABX3 structure. Neutron and synchrotron
in the plane normal to that axis (Fig. 2). In the next plane
radiation have proved to be essential to study the phase
below, to which the octahedra are corner-connected, combehaviour in these compounds.
mensurate tilting patterns may be in-phase or out-of-phase.
Fig 1
Untilted perovskites and layered perovskite-related
structure types. Left: isolated layers of corner-linked
octahedra stacked in a primitive tetragonal lattice,
typical of ABX4 structures, Middle: isolated layers
stacked in a body-centered tetragonal lattice characteristic of A2BX4. Right: octahedral frameworks in a
primitive cubic lattice, characteristic of ABX3 true
perovskite structures.
In the classic direct-space studies of tilt transitions of
Glazer [5,6] these tilt patterns were denoted with symbols such
as a0b+c -, where a superscripted + refers to in-phase rotations,
a superscripted - to out-of-phase rotations, and a 0 refers to no
rotation. a, b and c refer to differing magnitudes of rotation
about the three pseudo-cubic axes. More recently a group
theoretical study has shown that only two irreducible representations, labelled M3+ and R4+, are responsible for these two
commensurate styles of tilting, and both are associated with
the zone boundary of the Brillouin Zone (BZ) (Fig 2) of the
parent Pm3m structure [7,8]. This work demonstrated that
only 15 pure tilt structures exist, which had previously been a
matter of some debate. A third approach, using an idealized
form of lattice dynamics, called the Rigid Unit Mode (RUM)
Ian Swainson <[email protected]>, National
Research Council of Canada, Steacie Institute for
Molecular Sciences, Chalk River Laboratories, Chalk
River, ON, K0J 1J0
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initio code FOX [11] has proved very useful in solving the
structures. Use of the group analysis codes Isotropy [12] and
Isodisplace [13] greatly simplifies the relationship between
these OIP superstructures and the underlying tilt structures.
Fig 2
Top Left: One layer in a perovskite ABX3 framework.
Top Right: The act of rotating one octahedron clockwise about an axis perpendicular to the page is to
impose antiferro rotations about all neighbouring
octahedra in the plane. For the neighbouring layers,
connected above and below the page, the commensurate tilt patterns are limited to being exactly in-phase,
or out-of-phase. Below: The Brillouin Zone of the
untilted cubic parent structure. The M and R points
are shown, with which the M3+ and R4+ irreducible
representations governing in- and out-of-phase tilting are associated. The lines, T, joining these points
represent the freedom of phasing of the tilts between
the two commensurate choices. Rx, Ry and Rz represent tilts about the x, y and z-axes associated with
certain wavevectors.
In a free refinement of the geometry of the MA cation in the
ordered phase of MAPbBr3 from neutron data (Fig. 3) it was
found that the MA cation is in almost ideal trans configuration, while the PbBr6/2 octahedra are fairly distorted [14]. This
observation agrees with Raman data [15,16,17] which shows
that the lowest frequency internal mode of MA is higher in
frequency than the highest frequency mode associated with
the octahedral network, implying the cations are far stiffer
than the octahedra. A simplistic view of bond-strength-bond
length rules would imply that if the B-X bond length is shortened the octahedra might be expected to be more rigid, and
one might expect less distorted octahedra. Yet the change
from MAPbBr3 to MAPbCl3 (Fig. 4) shows a strong increase in
distortion of the octahedra. A simple picture was offered of
the relative rigidities of the two ions and the result of placing
the same rigid cation inside a smaller cage as an explanation
for the increased distortion in MAPbCl3 [18]. Another effect on
changing from Br to Cl, is that there is also an increasingly
strong hydrogen bonding interaction between the halide and
MA.
LONE PAIR DISTORTIONS IN OIPS
In MAPbCl3, not only do the X-B-X angles and B-X bond
lengths distort further, but the octahedra also become noncentrosymmetric with Pb moving off the centroid [18]. This is
due to the creation of a stereoactive lone pair on the Pb2+ ion.
Going down the periodic table, Group 14 elements have an
approach (e.g., [9]), is capable of examining non-high-symmetry points of the BZ and shows that rotational freedom
exists associated with the edges of the BZ, linking the M and
R points [9]. This demonstrates that neighbouring planes of
octahedra are not restricted to the two commensurate tilt patterns, but have continuous freedom in the phasing of the tilts.
Knop et al. [10] studied the dynamics of the MA cation and
performed calorimetric measurements of the transition in the
methylammonium lead halides, but at this time the structures of many of the phases was unknown. We have examined the sequence of transitions in methylammonium salts
MAPbBr3, MAPbCl3 and MASnBr3 using neutron and synchrotron powder diffraction. Synchrotron data has proved
very useful in indexing unknown low-temperature phases,
and greatly simplifies the solution of structures, by being
strongly dominated by the metal-halide framework.
Neutron data are essential for refining the geometry of the
cations, and for testing the orientation of these ions in the
cages. Since superlattices can be generated when the amines
orientationally order in the tilt systems, the direct space ab
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Fig 3
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View of the ordered structure of MAPbBr3
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rather than 6-coordinated. In the cubic phases of these compounds dynamical flipping of the 3+3 distortion is associated
with high ionic conductivity [24]. In addition subtle effects
such as the coexistence of two different crystallographic
forms at the same temperature occur; the protonated form of
MAGeCl3 transforms fully to the Pm3m symmetry on
heating, whereas the only 7% of the deuterated form only
transforms into the cubic phase prior to melting, the majority
remaining in the untransformed rhombohedral phase [24].
TMAGeCl3 shows evidence of two phases coexisting over a
large temperature interval on cooling: a monoclinic form, and
an orthorhombic form [25,26]. It has been suggested that the
softness of the material may allow untransformed domain
walls of the orthorhombic phase to exist between monoclinic
domains [24]. Phase coexistence over wide temperature intervals shows that there is very little energy difference certain
states, and the strong isotope effect in MAGeCl3 suggests that
subtle changes in the interaction between amine and GeCl6/2
are at the root. It remains to be seen to what extent similar
disorder effects extend to Sn(II) salts.
INCOMMENSURATE PHASES IN THE OIPS
Fig 4
View of the ordered structure of MAPbCl3
increasing preference to keep their outer s2 electron pair as a
lone pair than for it to participate in bonding, so that Pb is far
more commonly found as a formal 2+ ion than a formal 4+
ion. These lone pairs can become non-spherical, and stereochemically active by hybridization with p-orbitals. The structural effect of this non s-wave lone pair is that the Group 14
element is driven off-centre resulting in alternating long and
short bonds to each pair of halides in trans configuration
across the octahedron. The spontaneous displacements of
ions with d0 electronic configurations, such as Ti4+, are commonly referred to as a second-order Jahn-Teller effect; the
stereoactivity of the lone pair of the Group 14 elements represents another class of this effect [19,20].
We have recently been studying the phases of MASnBr3
below room temperature. We find that the upper structural
transition contains a tilt and a lone pair distortion. This transition is also a semiconductor to insulator transition [21], and
at this point MASnBr3 changes colour from red to yellow [22].
If we extend our view for one moment to non-perovskite
structures such as those containing Sb3+, an ion which has the
same electronic configuration as Sn2+, there have been interesting observations of coupling between hydrogen bonding
and the orientation of the lone-pair inside the octahedron [23].
The polarization of the electrons on the terminal Cl atom by
the amine is coupled to polarization of electrons within the
Sb-Cl bond; i.e. H-bonding interactions may be coupled to
orientations of the stereoactive lone pair. This effect is worth
examining further in the OIPs, since the orientation of the
lone pairs is likely coupled to the ordering of the cations.
The lone pair distortions are usually strong for Ge2+, and
GeCl3-based OIPs are usually described as 3+3 coordinated
(i.e. 3 short bonds opposite 3 long bonds in the octahedron)
As the amines begin to orientationally order on cooling, it is
not uncommon to observe that OIPs pass through incommensurate phases over some temperature interval. These
have been observed in both ABX3 and A2BX4 OIPs. A simple
phenomenological picture of phonon-induced incommensuration arises from the model of Heine and McConnell [27].
This is essentially a strong phonon anti-crossing interaction
in which a strongly temperature-dependent optic or librational mode attempts to cross through a lower frequency
mode, which in practice is usually a transverse acoustic
phonon. Modes with the same symmetry are forbidden from
crossing and repel. If these modes have a different symmetry at high-symmetry points at the centre and boundaries of
BZs but the same in the interior then modes can soften at an
incommensurate position.
One of the most famous cases of incommensurates in insulators, and certainly amongst the OIPs, comes from the propylammonium tetrachlorometallates, where a variety of metals
show similar effects. PA2MnCl4 shows a sequence of transitions α−β−γ−δ−ε−ζ, where α has the I4 / mmm symmetry of an
untilted A2BX4 lattice, and β is a pure tilt phase of space
group Cmca. Phase δ is also Cmca with the same basis (a reentrant of phase β). γ and ε are incommensurately modulated phases, and ζ a low temperature commensurately modulated phase [28,29,30]. The γ, ε and ζ modulated phases are all
associated with instabilities at different points in the Brillouin
Zone. γ−PA2MnCl4 is characterized by transverse modulations running along c with amplitudes parallel to the layer
axis b of the Cmca lattice and the critical wavevector associated with this transition lies on the surface of the BZ along
the H-line of buckling modes (Fig. 5). The amplitude of
the incommensurate distortion decreases to zero at the
γ-δ transition. Thus the Cmca β-phase “reappears” as the
δ-phase. The second incommensurate phase, ε appears
below this as a second order transition with a different
wavevector, k = (1/3 + δ)a* [31], lying along the Σ−direction,
and the lock-in to the commensurately modulated ζ−phase
occurs along a third direction at k = (a* ± b*)/3 [28,29,30,31].
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Using the Rigid Unit Mode (RUM)
approach one can calculate the RUM
spectrum of the tilted layers (Fig 5);
i.e., all possible modes that do not
distort the octahedra. The untilted
α-phase has a potential RUM instability at all wavevectors and does
not give much information as to
the origin of the wavevectors associated with the observed incommensurate phases [33]. Pure tilts exist at
|k|=a*/2, while in the interior of the
BZ buckling modes occur where
the octahedra are both tilted and
displaced perpendicular to the layer.
The buckling modes are “transverse”
waves that buckle the plane of octahedra, rather like buckling a piece of
paper. However, this plane is decorated with corner-bonded rigid octahedra, so a better analogy may be
made to the flexing of chain mail. If
we take the Cmca structure seen in
the β and δ-phases as the parent of
the modulated phases we find only
two orthogonal planes of “buckling”
modes and a single line of tilts at
their intersection [33]. The wavevectors associated with the γ, ε and ζ
phases all lie on these planes. The
related salt PA2CdCl4 shows a commensurately tilted structure at base
temperature in place of the modulat- Fig 5
ed ζ-phase of PA2MnCl4; the
wavevector associated with this
ordering lies at the Y-point, on the
line of pure tilts (Fig. 5).
it is a true ABX3 perovskite, the
modulated structure has been
described in terms of (fused) “layers” [26], and that the major distortion
is a “shear wave”. These planar shear
waves are “frozen” transverse
acoustics.
Hence, although they differ in their
RU and non-RU character, the incommensurate distortions in A2BX4 and
ABX3 have a similar origin in that
they are caused by the onset of
ordering of organic cations interacting
with transverse acoustic modes. The
reasons for the difference in the RU
character lies are the higher flexibility
of layers compared to frameworks,
and the free adjustability of the interlayer distance in the former structure
type.
THE EFFECT OF PRESSURE ON
OIPS
Rigid Unit Mode distribution in the
Brillouin Zone of the commensurately
tilted Cmca-structure of PA2MCl4.
Two planes of buckling modes intersect in a single line of pure tilts along
Γ−Δ
Δ−Y. γ, ε, and ζ represent the
wavevectors associated with the modulated phases of PA2MnCl4, and PACC
represents the wavevector associated
with the final tilted form of PA2CdCl4.
Historically there was much discussion of the interaction between the
PA chains determining the modulations; the PA chains are fairly rigid
and themselves show few degrees
of internal freedom. But it is clear that the layers determine the low-energy choices available for modulation,
since the observed wavevectors all lie in the planes of RU
buckling modes. These RUMs are a special class of acoustic
modes. This implies that the most constrained component
(inorganic layers) governs the behaviour of the whole system.
For the true ABX3 perovskites there are few RUMs, which
exist as solely a line of tilts linking the M and R points
along the edge of the BZ of the untilted phase (Fig. 2); in
principle one could observe incommensurate pure tilt structures associated with this line, but that is not what is seen.
Instead the k-vectors associated with the incommensurate
distortion lie in the interior of the BZ. Acoustic modes
are again involved and the motions can be viewed as buckled layers, but the octahedra are now forced to distort.
Perhaps the best-studied example is δ-TMAGeCl3. Although
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There have been only a limited number of investigations of the response
of these compounds to applied pressure. Lee et al [34] examined the
response of tin-based compounds.
One interesting claim was that FASnI3
was the most compressible perovskite
known with a bulk modulus of 8 GPa
- about three times more compressible
than NaCl. TMAGeCl3 has been
reported as having a bulk modulus of
9 GPa [25]. This softness has been
invoked as an explanation for the
observation of two crystalline forms of
TMAGeCl3 coexisting at low temperatures.
Using a diamond anvil cell in conjunction with synchrotron powder diffraction Lee et al. (2003)
showed transitions between the Pm3m initially to Im3
and finally ending in a reversible amorphization. Very similar behaviour was found for FASnI3, MASnI3 and
(MA1/2FA1/2)SnI3. The space groups Im3 , I4/mmm, and
Immm reported in the study of Lee et al. [34] correspond to the
a+a+a+, a0b+b+ and a+b+c+ tilt systems with in-phase tilting,
associated with differing order parameter directions of the
M3+ irreducible representation [7]. Two interesting observations can be made from this study. First, that the initial phase
entered in the SnI6/2 OIPs is Im3 , independent of the composition of the organic cation; and second that none of the
reported phases is compatible with full orientational ordering
of the organic cations. We have collected additional, unpublished, neutron data that suggest that the preference for
Im3 may extend to being independent of the composition of
the inorganic component as well. Thus, the initial response of
OIPs to pressure appears almost monotonous compared to
the variety of different ordered and incommensurate phases
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seen on cooling at ambient pressure. The early appearance
of Im3 for all compositions strongly suggests that it is coupling to volume that favours its formation. The a+a+a+ tilt
system, in which all three tilts are activated with the same
magnitude, guarantees maximum volume reduction for minimum tilt angle, and keeps the organic cations on high point
symmetries. Our neutron data show that this phase is stable
over a wide temperature interval on cooling, suggesting that
the overall tendency for cation ordering is strongly reduced
under pressure.
It is not clear at this time why in-phase tilting is preferred
over out-of-phase tilting under pressure, to what extent this
is true of OIPs in general, and whether, upon loss of crystallinity, the orientational disorder of the cations changes
from dynamic to a glassy static disorder.
CONCLUSIONS
The organic-inorganic perovskites show a complex set of
transitions, associated with tilt instabilities, ordering of orientationally disordered cations, stereoactive lone pair distortions, and incommensurate instabilities, which are due to
interactions between acoustic modes and the cations. The
existence of so many incommensurate structures in these
organic-inorganic perovskites is evidence of interaction
between ordering of the two components, since similar phases have not been reported in the Cs and Rb analogues of these
salts. Coexistence of more than one crystallographic form
over a wide temperature interval for Ge2+ salts shows that
some structural states have very similar energies. Similar
disorder has yet to be demonstrated in the Sn2+ salts. The
limited data of the response of OIPs to pressure suggests that
coupling to total volume is the most important term, outweighing the orientational ordering of the amines.
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REVEALING THE MICROSTRUCTURE OF POLYMERIC
MATERIALS USING SANS
by Barbara Frisken
O
ne class of soft materials where small angle neutron scatINTRODUCTION TO BLOCK COPOLYMERS
tering is especially useful is polymeric materials built from
Design of nano-structured polymeric materials can be conmulti-component polymers. These polymers phase separate
trolled through the chemical structure of the polymer.
on microscopic length scales to form well-defined microstrucPolymers are built up from individual
tures. In the case of polymers consistunits, called monomers.
If these
ing of two blocks of different
This
article
will
review
monomers
are
all
the
same,
the
polymer
monomers, hexagonal cylinders, closepacked spheres, bi-continuous phases SANS techniques used to is known as a homopolymer. For example,
homopolymers
such
as
and other structures can be achieved
reveal
structure
in
block
poly(styrene)
or
poly(methacrylate)
are
by changing the relative length or the
degree of segregation of the blocks. copolymer materials with composed of multiple units of styrene
and
methacrylate,
respectively.
The tendency of these materials to selfassemble leads to a wide variety of particular examples from Copolymers consist of different
important applications ranging from work done by Canadian monomers which can be organized randomly or in blocks, known as random
high-impact plastics to foams. This
or block copolymers, respectively. The
article will review SANS techniques researchers.
simplest type of block copolymer is the
used to reveal structure in block
diblock, which contains a block of type
copolymer materials with particular examples from work
“A”
monomers
and
a
block of type “B” monomers. Other
done by Canadian researchers.
examples include triblocks, which can either repeat a block
(ABA triblocks) or contain three distinct blocks (ABC triblocks). Figures 1a and 1b show sketches of an AB diblock
and a BAB triblock polymer, respectively. Block copolymers
can also be assembled with branched architecture, such as
star-branched block copolymers.
Mixtures of different polymers tend to phase separate easily,
as the energy of interaction increases with the number of
monomers that make up the polymer. Polymers containing
two or more blocks are only able to phase separate on microscopic length scales because the different blocks are physically joined. This leads to self-assembly into one of a variety of
ordered structures as determined by the fractions of the molecule taken up by the different blocks and the interaction
between the blocks.
Fig. 1
Schematic diagrams of block copolymer architecture
and assemblies. (a) A-B diblock, (b) B-A-B triblock,
(c) disordered phase of diblock polymer and
(d) ordered (lamellar) phase of diblock polymer
(see Ref. 1).
The structural organization of block copolymers in the melt,
i.e. a bulk polymer sample with no solvent, is determined by
the degree of polymerization N, the overall volume fraction
occupied by each component, and the A-B segment-segment
interaction parameter [1]. At high enough temperatures,
entropy dominates and an isotropic or disordered phase with
different blocks interpenetrating each other exists throughout
the sample. As the temperature drops, the interaction parameter increases and fluctuations in the density from the mean
density develop. Eventually the polymer microphase separates at the order-disorder transition. A disordered melt and
an ordered (lamellar) phase are sketched in Figs. 1c and 1d,
Barbara Frisken <[email protected]>, Dept of Physics,
Simon Fraser University, Burnaby BC V5A 1S6.
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respectively. Different ordered structures can be achieved by
controlling the chemical structure of the polymer. For example, if the volume fraction occupied by the A block of a
diblock copolymer is small, the sample will separate into
micelles, small generally spherical aggregates, of A blocks in
a B matrix. At a slightly higher volume fraction of A, the
sample will microphase separate into A cylinders in a B
matrix. At equal volume fractions for the A and B blocks, the
microstructure after separation will be lamellar, or layered.
There are other phases including a bicontinuous phase
known as the gyroid phase or “plumber’s nightmare” that
occurs at volume fractions intermediate between the cylindrical phase and the lamellar phase. Sketches of these four
phases are shown in Fig. 2. For volume fractions of B less
that 0.5, the opposite structures occur. These structures persist as the melt is cooled into the solid phase.
This capacity for structural organization on nanometer
length scales leads to a wide range of applications [2]. By
combining two or more types of monomers in a polymer,
composite materials can be built which combine the properties of different monomers to create unique materials. For
example, combining blocks of poly(butadiene), a rubbery
polymer, and poly(styrene), a glassy polymer, results in a
microphase separated structure of glassy domains in a rubbery matrix that is both rigid and tough, especially if the
copolymer is a triblock with the rubbery block in the middle.
Thermoplastic elasticity due to microphase separation has
application in high-impact plastics and pressure-sensitive
adhesives. Block copolymers are also used as compatibilizers
to stabilize polymer blends. An A-B copolymer can stabilize
a mixture of A and B polymers by segregating at and stabilizing the interface between the two phases, in the same way
that a surfactant stabilizes a dispersion of oil droplets in an
aqueous phase. Surfactant-like behaviour that occurs when
block copolymers are blended with homopolymers has application in foams, oil additives, thickeners and dispersion dispersive agents.
Research on fundamental aspects of these materials focuses
on three main areas: the melt phase at high temperature
where the polymer is fluid, the solid phase that exists below
the glass transition of the polymer, and the behavior of block
copolymer when dispersed in various solvents. The
microphase separation and ordering, both equilibrium [1] and
non-equilibrium [4], is the focus of many studies of both the
melt and solid phase. When copolymer is added to a solvent
at low concentration the molecules disperse completely. But
at higher concentrations, entropy favours aggregation of the
molecules into micelles. Work on copolymer solutions has
focused on the size and structure of these micelles and is
driven by goals of understanding what structures occur in
copolymer samples cast from solution and by the use of
copolymers as compatibilizers.
BLOCK COPOLYMERS AND SANS
Scattering experiments are useful tools for determining the
microstructure and related behavior of these materials. The
length of the structures (10-1000 Å) and the wavelengths of
the radiation available for scattering means that small angle
scattering of x-rays and neutrons is appropriate for the determination of ordering in both melt and dispersion. In both
SAXS and SANS experiments, the intensity of radiation scattered I is measured as a function of scattering angle, which is
related to the scattering wavevector q =
4π ⎛ θ ⎞
sin ⎜ ⎟ . Small
λ
⎝2⎠
angle x-ray scattering (SAXS) and small angle neutron scattering (SANS) are complementary; SAXS provides superior
resolution, while SANS allows use of contrast variation,
achieved mainly by exchange of deuterium for hydrogen
within the polymer, to reveal different parts of the
microstructure. SANS is also useful for samples that are otherwise strongly scattering; the scattering cross section of samples for neutrons is generally very small.
The variety of SANS experiments involving block copolymers is extensive [5-7]. Selective deuteration of some fraction
of polymer molecules has been used to observe the conformation of the polymer chain in the melt. Selective deuteration
of one block or even part of a block has been used to determine intramicellar structure and to enhance contrast between
blocks in the melt. Variation of the deuterium content of the
solvent has also been used to enhance some aspect of the
structure. This can help determine which structural aspects
are responsible for different features observed in the scattering data.
Fig. 2
286
Schematic diagrams of ordered phases of block
copolymer melts. Clockwise from top left (a) micellar, (b) cylindrical, (c) lamellar and (d) bi-continuous
phases. (Adapted from Ref. 3 and used with permission.)
PHYSICS
IN
CANADA
The rest of this article will focus on the work of several
Canadian groups, particularly investigations of micellar
structure, novel nanostructures, and microscopic order in
conducting polymers.
COPOLYMERS IN SOLUTION
The dilute regime of a polymer solution is defined as concentrations below which chains of adjacent polymers begin to
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overlap. Scattering studies below the overlap concentration
yield information about the radius of gyration, the molecular
weight and the packing of the chains. In copolymers, there is
a second concentration that is important at low concentrations: the critical micelle concentration. While the polymer
will always fully disperse at low enough concentration due to
the entropy of mixing, the entropy of the system can be maximized by self-assembly of the polymer into small aggregates
or micelles at concentrations above a critical concentration.
Scattering studies can yield information about the shape and
size of these micelles.
An example of the latter type of experiment which makes
interesting use of selective deuteration can be seen in an
investigation of the coronal structure of star-like block polymer micelles [8]. Adi Eisenberg’s group at McGill is particularly interested in the formation of micelles of ionomers,
lightly charged polymers. In this particular series of experiments, they used the block ionomer micelles as a model system to investigate the behavior of polymer-coated colloidal
particles. The block copolymer consisted of poly(styrene)
(PS) and poly(acrylic acid) (PAA) and the micelles were
formed in tetrahydrofuran (THF), a good solvent for PS, with
the PAA forming the core and the PS the corona. In order to
study the structure of the corona, the PS block was partially
deuterated. The scattering length densities of PS, PAA and
THF are similar; that of poly(d-styrene) (dPPdS) signficiantly
different. By placing the deuterated segment at different distances from the core, the researchers were able to study the
structure of different parts of the corona. To distinguish
between possible structures, the researchers examined the qdependence of the scattering. For a disordered system, scattering is large at small q or large length scales, where fluctuations of all sizes exist. The scattering decreases at larger q,
typically as q-α, where different structures are characterized
by different exponents α. They observed exponents of 5/3
for dPS units far from the core and 1 for dPS units closer to
the core. From this they determined that the chain forms a
flexible “blob” far from the core, but closer to the core the
chain stiffness increases consistent with an increase in the
polymer density and a breakdown of the blob model, as predicted by Gast et al. [9].
NOVEL NANOSTRUCTURES
Amphiphilic copolymers, which have a hydrophobic block
and a hydrophilic block, can be used as surfactants in the
design of new materials. Recently, Guojun Liu and coworkers in the Chemistry Department at Queen’s University
reported the use of poly(acrylic acid)-poly(styrene) (PAA-PS)
diblocks in the synthesis of polymer-coated cobalt nanocrystals for magnetic storage applications [10]. The particles were
made by solution phase reduction of cobalt in the presence of
the PAA-PS diblock. Because of strong binding that occurs
between PAA and Co, the polymer chains remain stuck to the
particles forming a core-shell structure. The presence of the
polymer not only controls particle formation but also enables
film formation; the particles can be solvent-cast to yield bulk
films with magnetic properties.
SANS was used to confirm the core-shell structure of these
particles. Because the particles are metallic, they absorb light
making particle characterization by light scattering unfeasi-
ble. The weak interaction of neutrons with matter makes neutron scattering an ideal choice for studying such optically
dense materials. Liu et al. [10] measured the intensity of scattered neutrons to determine the differential scattering cross
section
dΣ
where
dΩ
I (q) ∝
dΣ
(q) = K P (q) S (q) .
dΩ
(1)
The factor K depends on the contrast and other sample
parameters, P (q) is the particle form factor describing the size
and shape of individual particles and S(q) is the structure factor describing the correlation between the particles. For a
dilute sample, S(q) = 1. Liu et al. analyzed their data using the
Guinier model where at small qRG # 1 the form factor can be
written
⎛ 1
⎞
P ( q ) ∼ exp ⎜ − q 2 RG2 ⎟
⎝ 3
⎠
(2)
where is the radius of gyration of the particles. Using SANS,
they confirmed a 12 nm polymer shell coating 11 nm Co
nanoparticles.
PROTON-CONDUCTING POLYMER FILMS
Recently, we made use of contrast variation techniques to
explore the structure of proton-conducting films cast from
solutions of diblocks consisting of a fluorous block and a sulfonated polystyrene (S-PS) block [11]. Proton-conducting
polymer membranes form the heart of the proton exchange
membrane fuel cell where they provide both a proton-conducting path and mechanical strength. Studies have indicated a relationship between the nanostructure of these materials and their proton conductivity [12]. Block copolymers materials, where a range of architectures is achievable, allow for
systematic investigation of this relationship.
Because these samples absorb water, contrast variation can be
achieved by varying the D2O content in the hydrated membranes. For example, in Fig. 3 we can see the strong effect of
varying the D2O content of the absorbed water on data taken
for one of these samples. There are two main features in the
scattered intensity for these materials; one peak at around
0.01 Å-1 and a second peak at around 0.1 Å-1. As the relative
amounts of H2O and D2O are varied, the magnitudes of these
peaks increase or decrease and a third peak appears around
0.02 – 0.03 Å-1.
The fact that the shape of the spectra changes as the contrast
is varied tells us that the system is not a simple two-component system, but instead consists of at least three components.
Consider scattering from a binary system consisting of a
sphere in a matrix; shown in Fig. 4a. If the contrast is varied,
by increasing the scattering from the matrix for example, the
system still scatters like a sphere in a matrix but with a change
in the overall amplitude, reduced if the contrast is decreased
and augmented if the contrast is increased. If the system is
more complicated, for example it consists of three components, the angular dependence of the scattered intensity can
change. Figure 4b shows a core-shell particle; if the scattering amplitude of the matrix is matched to that of the shell, the
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of partially sulfonated diblocks consisting of a lamellar structure with 3 levels of scattering length density associated with
fluorinated domains, hydrated sulfonated polystyrene
domains and an interface layer of non-sulfonated polystyrene. Contrast variation techniques played an important
role in confirming that our model was consistent with the
data: the structure factor information was removed by dividing three of the scattering spectra by the fourth leaving data
that depended on form factor only. A model function for
core-shell disks was then successfully fit to the data used during these experiments were crucial to our successful modeling of the structure of these films.
SUMMARY
Fig. 3
Fig. 4
Scattering spectra from a series of fluorous-sulfonated poly(styrene) diblock films for four different solvent contrasts: H2O (squares), 50-50 (H2O-D2O) (diamonds), 30-70 (H2O-D2O) (circles) and D2O (triangles). As the D2O content of the solvent is varied, the
shape of the scattering curves changes. (Reproduced
from Ref. 11 with permission.)
(a) When the contrast is varied in a two-component
system, only the amplitude of the scattering changes.
(b) When the contrast is varied in a three-component
system, the angular dependence of the scattering
intensity also changes as it appears that scattering is
from an object of different structure.
particle scatters like a smaller sphere while if it is matched to
the scattering amplitude of the core, the particle scatters like
a spherical shell. As these particles have different form factors, the shape of the spectra will change as the contrast is
varied.
We were able to study two series of samples, one consisting
of S-PS blocks of different length but full sulfonation, and the
other consisting of a constant S-PS block length but with
varying levels of sulfonation. By varying the contrast
between polymer film and solvent, we were able to study differences in the structure of the two series. We have developed a model that is consistent with all data within the series
288
PHYSICS
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The ability to vary the contrast between solvent and polymer
or between different parts of the polymer molecule by selective deuteration or use of D2O makes SANS a powerful tool
to investigate nanoscale structure in novel materials based on
block copolymers.
REFERENCES
1. For example, please see F.S. Bates and G.H. Fredrickson,
“Block Copolymer Thermodynamics: Theory and
Experiment”, Annu. Rev. Phys. Chem. 41, 525-557 (1990)
and references therein.
2. A.-V. Ruzette and L. Leibler, “Block Copolymers in
Tomorrow’s Plastics”, Nature Materials 4, 19-31 (2005).
3. C. Burger, S. Zhou and B. Chu, “Nanostructures of
Polyelectrolyte-Surfactant
Complexes
and
Their
Applications” in Handbook of Polyelectrolytes and Their
Applications, ed. by S.K. Tripathy, J. Kumar and
H.S. Nalwa (American Scientific Publishers, Stevenson
Ranch, California, 2002), vol. 3, pp. 125-141.
4. For example, please see G.H. Fredrickson and F.S. Bates,
“Dynamics of Block Copolymers: Theory and
Experiment”, Annu. Rev. Mater. Sci. 26, 501-50 (1996) and
references therein.
5. R.W. Richards, “Small Angle Neutron Scattering from
Block Copolymers”, Adv. in Polym. Sci. 71, 1-39 (1985).
6. I.W. Hamley, The Physics of Block Copolymers, Oxford
University Press, Oxford (1998).
7. K. Mortensen, “Block Copolymers Studied with Small
Angle Neutron Scattering”, in Scattering in Polymeric and
Colloidal Systems, ed. W. Brown and K. Mortensen (Gordon
and Breach Science Publishers, 2000).
8. M. Moffitt, Y. Yu, D. Nguyen, V. Graziano, D.K. Schneider
and A. Eisenberg, “Coronal Structure of Star-Like Block
Ionomer Micelles: An investigation by Small-Angle
Neutron Scattering”, Macromolecules 31, 2190-2197 (1998).
9. K.A. Cogan, A.P. Gast and M. Capel, “Stetching and
Scaling in Polymeric Micelles”, Macromolecules 24, 65126520 (1991).
10. G. Liu, X. Yan, Z. Lu, S.A. Curda and J. Lal, “One-Pot
Synthesis of Block Copolymer Coated Cobalt
Nanocrystals”, Chem. Mater. 17, 4985-4991 (2005).
11. L. Rubatat, Z. Shi, O. Diat, S. Holdcroft and B.J. Frisken,
“Structual Study of Proton-Conducting Fluorous Block
Copolymer Membranes”, Macromolecules 39, 720-730
(2006).
12. Y. Yang and S. Holdcroft, “Synthetic Strategies for
Controlling the Morphology of Proton Conducting
Polymer Membranes”, Fuel Cells 5, 171-186 (2005).
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USE OF NEUTRON DIFFRACTION FOR DEVELOPMENT OF
METAL HYDRIDES: CASE OF BCC ALLOYS
by J. Huot, L. Cranswick, I. Swainson
W
ith the growing concern about global warming, a
To reach commercialization, the material should be safe and
replacement to fossil fuels has to be found. Despite many
of low cost. Despite intensive research a metal hydride meettechnological and economical problems, hydrogen is seriousing all these characteristics are yet to be found. Nevertheless,
ly considered as an energy vector, mainly because of its benmetal hydrides could offer a technological solution for onefits in terms of air pollution, energy security, and renewabilboard or stationary applications [6]. More research on the
ity [1,2]. In the perspective of a hydrodevelopment of new metal hydrides
gen economy, a safe, low-cost and high
and the understanding of the fundacapacity hydrogen storage method After a brief introduction of mental aspects of metal-hydrogen systhat could operate at or near room temis still needed. In this task, neutron
systems, tem
perature will be needed. These fea- metal-hydrogen
powder diffraction is an important tool
tures are especially important in the we will review the basic in the development and understanding
case of mobile systems such as the fuel
of metal-hydrogen systems. In this
cell or hydrogen internal combustion facts that make neutron dif- paper we will discuss neutron diffracengine cars. Hydrogen could be stored fraction a unique tool for tion as a tool for studying metal-hydroin pressure tanks or in a liquid form
gen systems. The emphasis will be on
but these two technologies suffer structural determination of crystal structure characterization with
important drawbacks such as high metal hydrides. Two types the BCC system as an example. But
pressure and low volumetric capacity
first, some basic facts of metal hydrides
in the case of pressure tanks, or cryo- of metal hydrides will be should be presented.
genic temperatures (below 20 K) and
discussed: magnesium and
important liquefaction energy in the
METAL HYDRIDES
[3]
case of liquid . The class of materials solid
solution
BodyA simplified view of metal-hydrogen
known as metal hydride seems to be a
interaction is shown in Figure 1. First, it
better candidate for the storage tech- Centred Cubic (BCC).
should be realized that it is the hydronology for mobile systems [4]. Metal
gen atoms which will enter the metal
hydrides (MH) are chemical comlattice
and
not
the
hydrogen
molecule. Therefore, the hydropounds of one or many metals (M) with atomic hydrogen
gen molecule should be dissociated at the metal surface, dis(H) [5]. The advantages of metal hydrides are that they have
solve at interstitial sites of the host metal and form a solid
high volumetric storage capacity (higher than liquid hydrosolution (α phase). When the local hydrogen exceeds a cergen) and they could provide extremely pure hydrogen, which
tain limit (which depends on the metal host), a hydride phase
is essential for fuel cells. For most practical applications the
starts to precipitate (β phase). This is the metal hydride phase
absorption
where the metal and the hydrogen form a chemical bond. The
and desorpheat of formation for elements and intermetallics considered
tion
of
for practical applications varies from 28 to 75 kJmol-1. The
hydrogen has
interested reader could consult review articles on the thermoto be at, or
dynamics of hydrogen with metal and intermetallic
close
to,
alloys [7,8].
room
temperature and
THERMODYNAMICS
at pressure of
around one
The thermodynamics of hydrogenation is easily described by
bar.
The
pressure-composition isotherms (PCT curves) as shown in
m e t a l
Figure 2. The reaction of a metal with hydrogen can be repreh y d r i d e
sented as
should also
have
fast
sorption
J. Huot1 <[email protected]>, L. Cranswick2,
kinetics and
I. Swainson2; 1 Physics Department, Université du
be resistant
Québec à Trois-Rivières, Trois-Rivières, Québec, Canada
Fig. 1 Schematic of hydrogen dissociation at to poisoning
G9A 5H7; 2 National Research Council, Steacie Institute
the interface and solution of atomic
by
trace
for Molecular Sciences, Chalk River Laboratories, Chalk
hydrogen in the bulk. (Adapted
impurities.
River, Ontario, Canada K0J 1J0
from [9])
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between 2 and 3 Å3 per hydrogen atom [10]. This translates to an expansion that could reach 30 vol.% in some
systems and is often anisotropic. A hydrogen sublattice is
formed and the crystal structure may change, typically
with a reduction of symmetry. Nevertheless, the metal
atom substructure does not change appreciably. At low
temperature, the hydrogen sublattice may become
ordered. During hydrogenation, the α and β phases coexists but because these two phases have quite different lattice volumes, a large number of lattice defects will be created. In ductile metals the crystal will be deformed while
brittle metals (or intermetallic compounds) will disintegrate into small grain size powder. Defects will also cause
a broadening of X-ray and neutron diffraction patterns.
Fig. 2.
Schematic of a pressure-composition isotherm. α is the
solid solution of hydrogen and β is the hydride phase.
The right-hand side plot is a van’t Hoff plot giving the
enthalpy of hydride formation ΔH. (Adapted from [7])
x
M + H 2 ⇔ MH x
2
(1)
Region I of Figure 2 is for low hydrogen concentration. At
low concentration (x « 1) hydrogen dissolves in the metal lattice and forms a solid solution phase (α phase). The crystal
structure of the α phase is the same as the metal.
On hydrogenation, hydrogen atoms will occupy specific
interstitial sites. The interstitial sites in three major crystal
structures of hydrides are shown in Figure 3. Only octahedral (O) and tetrahedral (T) sites are shown because
they are the only ones occupied by hydrogen atoms.
Some distinction should be made on the different structures. In the FCC lattice, the T and O sites are respectively enclosed in regular tetrahedral and octahedral formed
by metal atoms. At low or medium H concentration, the
preferred interstitial sites are octahedral. In the HCP
(Hexagonal Close Packed) lattice, the tetrahedral or octahedral sites become distorted as the ratio of lattice parameters c/a deviates from the ideal value of 1.633. Tetrahedral
sites are favoured at low H concentration in HCP metals. In
Hydrogen concentration increases with hydrogen pressure
until the equilibrium pressure of the hydride phase (β) is
reached. The system has now three phases (α, β and hydrogen gas) and two components (metal and hydrogen). The
Gibbs phase rule gives the number of degrees of freedom (f)
as:
f=C-P+2
(2)
where C is the number of components and P is the number of
phases. Therefore, at a given temperature, the hydrogen
pressure is constant in the two-phase region (region II of
Figure 2). The equilibrium pressure Peq at the α6β phase
transition is then given by the van’t Hoff law.
1nPeq =
− ΔH ΔS
+
RT
R
(3)
where ΔH and ΔS are respectively the enthalpy and entropy
of the α6β transition. Experimentally, the transition
enthalpy and entropy are obtained by the van’t Hoff plot of
plateau pressure against reciprocal temperature (Figure 2,
right hand side). Once pure β phase is reached, hydrogen
enters in solid solution in the β phase and the hydrogen pressure again rises with concentration (region III of Figure 2).
CRYSTAL STRUCTURE
With the formation of the β phase, the lattice expands and for
most binary metal compounds the volume expansion is
290
PHYSICS
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Fig. 3
September / October 2006
Interstitial sites of hydrogen occupation (octahedral
and tetrahedral) in fcc, hcp and bcc lattices.
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FCC and HCP lattice, for each metal atom, there is one octahedral site and two tetragonal sites available for hydrogen.
In BCC lattice, the polyhedron is greatly distorted. For the O
sites, two metal atoms are much closer to the interstitial site
than the other four metal atoms. Therefore, the O sites are
subdivided in Ox, Oy, Oz sites according to the direction of
the four-fold symmetry axis. In the same way, the T sites of
BCC lattice are divided according to their symmetry axis and
are noted Tx, Ty, Tz. The hydrogen atom has three octahedral
and six tetrahedral sites available per metal atom. Hydrogen
will preferably occupy the tetrahedral sites in BCC metals.
USE OF NEUTRONS
Neutron scattering is a unique probe for the study of structure and dynamics of condensed matter. Thermal neutrons
(neutrons in thermal equilibrium with a moderator material
near room temperature) have a wavelength comparable in
magnitude with the interatomic distance of matter. Thus,
material structure could be studied by neutron scattering.
For structure determination, one advantage of neutrons over
X-ray scattering is that neutrons are uncharged and thus
interact only weakly in solids. This means they have a better
penetration than X-ray and the bulk of material could
be probed instead of mainly surface properties in the case of
X-ray. The neutron scattering length depends on the nature
of the nucleus and thus varies greatly from one element to
another, contrary to X-ray scattering which is proportional to
the number of electrons. The relative neutron and X-ray scattering for a few elements is presented in Figure 4. It can be
seen that hydrogen and deuterium have a very large neutron
cross section compared to other elements but they are essentially undetectable by X-ray. For hydrogen storage materials
this is a very important feature: hydrogen could be located by
neutron powder diffraction but not by X-ray powder diffraction.
The fact that the cross section depends on the nature of the
nucleus means that the scattered wave will vary between different isotopes. The scattering also depends on the interaction between the two spin states of the neutron-nucleus sys-
Fig. 4
Scattering of X-ray and neutrons for some atoms.
For X-ray the radii of circles are proportional to
the atomic number, for neutron the radii are proportional to the scattering length. Negatives neutron
scattering length are indicated by dark circles.
(Adapted from [11])
tem. Here, we should make the distinction between coherent
and incoherent scattering. As explained in Squires [12], the
coherent scattering depends on the correlation between the
positions of the same nucleus at different times, and on the
correlation between the positions of different nuclei at different times. It gives interference effects and causes Bragg
peaks. In the case of incoherent scattering, it depends only on
the correlations of the same nucleus at different times; it does
not give interference effects.
Bragg scattering is determined by the coherent cross section.
Therefore, for structural studies (neutron diffraction) the
coherent part of the cross section is the significant parameter.
In the case of hydrogen, the incoherent cross section is very
large (80 barn) compare to the coherent one (1.76 barn).
Consequently, the neutron powder diffraction of compounds
with high proportion of hydrogen will show an extremely
high background with only small Bragg peaks thus making
analysis virtually impossible. Fortunately, for deuterium the
coherent cross section is bigger than the incoherent part
(respectively 5.59 and 2.05 barn). Thus, for structural characterization by neutron powder diffraction it is common practice to use deuterium instead of hydrogen.
In most cases, it is difficult to obtain a good quality single
crystal of metal hydride, therefore most of the structural
measurements are performed by neutron powder diffraction.
Sample containers are usually made of quartz or vanadium
(sometimes steel is also used) and the pressure and temperature of the container can be controlled for in-situ experiments.
Almost 60 years ago the first neutron diffraction measurement of metal hydride was performed on sodium hydride by
Shull et al [13]. It is interesting to note that the maximum
intensity on their NaD patter was 25 counts/min! On modern
neutron diffractometers the intensity is many orders on magnitude higher, making measurements much faster and accurate and allowing in-situ experiments during hydrogenation/dehydrogenation.
MAGNESIUM HYDRIDE
Because of their high hydrogen storage capacity magnesium
and magnesium based alloys could be attractive for energy
storage material. However, due to the hydride stability and
slow sorption kinetics, the actual applications are limited.
Important efforts are directed in improving the hydrogen
sorption properties by addition of catalysts, synthesis of new
alloys, new composites, and new ways of synthesis. A powerful way of improving the hydrogen sorption properties is
by energetic ball milling of hydrides [14]. In ball milling elemental powder is mixed with hardened stainless steel balls
and loaded in a crucible. Spinning or shaking the crucible
generates ball-powder-ball and ball-powder-wall collisions.
These repeated high-energy impacts induce fracturing and
cold welding of particles and define the ultimate structure of
the powder [15]. As an example we present here the case of
magnesium. Pure magnesium hydride (MgH2) was ball
milled for 20 hours in a Spex Shaker mill. In Figure 5, X-ray
and neutron powder diffraction patterns of ball milled magnesium hydride are shown. Neutron diffraction pattern was
taken on the high-resolution neutron powder diffractometer
C2 DUALSPEC at Chalk River. The neutron wavelength was
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Such formation of a metastable phase synthesized at room
temperature by ball milling has also been seen in many other
systems [20].
Fig. 5
X-ray and neutron powder diffraction of MgH2 ball
milled 20 hours.
1.3287 Å and the sample holder was a cylindrical can made of
vanadium. For the reasons cited above, this pattern was
taken on a deuterated sample (MgD2). The X-ray pattern was
taken on a commercial diffractometer (Siemens D5000) with
a Cu Kα radiation (wavelength Kα1 = 1.5418 Å).
Crystallographic parameters such as phase abundance, lattice parameters, refinement of atomic positions and occupancies, crystallite size and strain could be determined by
Rietveld refinement. In the Rietveld method the leastsquares refinement is carried out until the best fit is obtained
between the whole observed diffraction pattern and the
assumed crystal structure [21]. This is an extremely powerful
technique and is now extensively used for analysis of X-ray
and neutron powder diffraction patterns. For example,
Rietveld refinement of the powder diffraction patterns
showed in Figure 5 indicated that the crystallite size of
β-MgH2 and γ-MgH2 phases are respectively 11.9±0.1 nm and
17.1±0.7 nm. From neutron pattern, comparison of the Mg-D
bond lengths indicated that in the β phase, the bond
lengths are symmetrically stretched and compressed while in
the γ phase, only one bond is stretched. It was proposed that,
upon ball milling the phase transformation from β to γ is
most likely associated with a micro-stress relaxation
process [22]. This type of study could only be conducted with
neutron diffraction.
The effect of nanocrystallinity on the kinetics of hydrogen
sorption is shown in Figure 6. Both absorption and desorption are much faster in the case of ball milled (nanocrystalline) magnesium hydride than for the unmilled sample. In
desorption, the unmilled sample has an incubation time of
almost 800 seconds while for the nanocrystalline material the
desorption starts immediately with a high rate. The enhancement of hydrogen storage properties with nanocrystallinity
has been observed for a wide variety of elements and intermetallics as well as for nanocomposites [23].
The differences between the X-ray and neutron diffraction
patterns are obvious: the Bragg reflections are shifted
because of the different wavelength of the two radiations but
also the relative intensities are altered because deuterium
atoms scatter neutrons much more than X-rays. Another
striking feature of these
patterns is the broadness
b
of the peaks; this indicates a
the nanocrystallinity of
the material. From the
peaks’ full width at half
maximum the crystallite
size and strain could be
determined.
Usually,
magnesium hydride has
the stoichiometric tetragonal β-MgH2 phase [16,17].
In the case of ball milled
magnesium
hydride,
another phase of magnesium hydride is produced, the orthorhombic
(γ-MgH2) phase [18], as
can be seen on the diffraction patterns of Figure 5.
This orthorhombic (γMgH2) phase was first
synthesized under highpressure and high tem- Fig. 6 Hydrogen sorption curves of unmilled MgH2 (filled marks) and ball milled (hollow
marks) MgH2. (a) absorption at 573K under 1.0 MPa of hydrogen pressure. (b) desorption
perature treatments (2.5-8
[19]
at
623 K under a hydrogen pressure of 0.015MPa.
Gpa and 250-900°C)
.
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further heat treatment
was performed on the
Solid solution BCC alloys (mainly Ti-V-Cr
buttons. The size of the
and Ti-V-Cr based) are promising hydrogen
button
was
15 g.
storage materials because of their relatively
Neutron diffraction of
high storage capacity and their ability to
the arc melted alloy was
absorb and desorb hydrogen in ambient
performed on the highconditions [24]. This type of hydride is also
resolution neutron powconsidered by Toyota Motor Corporation in
der diffractometer C2
hybrid high-pressure/metal hydride storDUALSPEC at Chalk
age tanks for vehicular applications [25]. An
River
(wavelength
important characteristic of this type of
1.3287 Å). The button
alloys is that usually two different strucwas wrapped in a thin
tures are formed during hydrogenation
platinum foil which
process. First, for most BCC solid solution
acted as an internal referalloys when the ratio of hydrogen atoms
ence for lattice parameover the number of metal atoms in the structers calculation.
The
ture (H/M) is around 1 the hydride strucRietveld analysis of the
ture is formed by a BCC metal sublattice,
pattern was carried out
which is sometimes deformed. When the
using the softwares
[30]
H/M ratio is around 2 the hydride has the Fig. 7. Neutron diffraction pattern and Rietveld
and
refinement of TiV0.9Mn1.1. The short vertical EXPGUI
[31]. Refinement of
CaF2 structure where metal atoms form an
GSAS
bars indicate the position of Bragg peaks: the
FCC sublattice and the hydrogen atoms
upper row is for titanium, second row is plat- the neutron pattern of an
occupy tetrahedral (T) sites surrounded by
inum, third row is C14 phase, and fourth row alloy with TiV0.9Mn1.1
four metal atoms. From X-ray and neutron
composition is difficult
is the BCC phase. The bottom curve is the
diffraction Nakamura et al [26,27] found that
difference between experimental and calcubecause, as shown in
for Ti1.0V1.1Mn0.9 solid solution BCC alloy
lated pattern.
Figure 4, the scattering
the mono-hydride Ti1.0V1.1Mn0.9D2.0 has a
cross section of Ti and
pseudo-cubic NaCl structure where the hydrogen atoms
Mn are almost the same, making them hard to distinguish
occupy octahedral (O) sites surrounded by six metal atoms.
with neutron radiation. Moreover, vanadium is difficult to
In an investigation of Ti1.0V1.1Mn0.9Hx and Ti1.0V1.1Mn0.9Dx,
locate because of its small cross section. Despite these disadCho et al [28] found a large difference between the equilibrium
vantages, a good quality neutron diffraction pattern could
pressures of the hydride and deuteride PCT isotherms.
give precise information about the crystal structure. In
However, the crystal structures, lattice parameters, and
Figure 7 we present the Rietveld analysis of the neutron difphase abundance of the isotope hydrides depends only on
fraction pattern of arc melted TiV0.9Mn1.1. The residue curve
the hydrogen content and not on the type of isotope.
is small, indicating a good fit of the pattern.
SOLID SOLUTION BCC
In almost all cases, the highest hydrogen storage capacity is
In Rietveld method various values are used to indicate the
reached when the alloy is single phase BCC solid solution.
goodness of the fit. One of the most meaningful is the ‘RHowever, in one of the first paper on this subject, Iba and
weighted pattern’ Rwp which measures the weighted differAkiba [29] identified a lamellar structure of 10 nm scale conence between the calculated and measured intensities. In the
sisting of two BCC phases with a composition TiMn0.9V1.1.
present case Rwp=5.67%, confirming the quality of the fit. In
They postulated that the high capacity is considered to be
Table 1 the identified phases (beside platinum) are displayed
caused by interactions of these nano-composite phases
long with their respective phase abundance and lattice
through a coherent interface. In metal hydride materials,
parameters. A small amount of pure titanium was identified.
studies of the effect of phase coherency in multiphase alloys
The phase identified as C14 is a member of the so-called
has been limited. Therefore, we decided to investigate the
‘Laves phases’ structures.
effect of coherency in multiphase BCC solid solution. For
this, we selected an alloy close to
the stoichiometry found by Iba
TABLE 1
and Akiba [29]. In their case they
CRYSTAL STRUCTURES OF TIV0.9MN1.1 AS DETERMINED FROM RIETVELD REFINEMENT.
found that the alloy TiMn0.9V1.1
T
HE
VALUES
IN PARENTHESES ARE THREE STANDARD DEVIATIONS AND REFER TO THE LAST DIGIT
has the best storage capacity. For
this study we selected the multiphase alloy TiMn1.1V0.9. Buttons
Phase
Space Group
Abundance
Lattice parameters
of TiMn1.1V0.9 alloys were pre(%)
(D)
pared by arc melting chunks of
C14
P
6
/mmc
32
(4)
a
=4.906(3)
3
pure
metals
(Ti sponge,
c = 8.011(9)
Mn chunk, V chunks) in an
BCC
I
m
-3
m
65
(1)
a =3.018(1)
argon atmosphere. The buttons
were turned over 4 times and
remelted in order to achieve
Titanium
P 63/mmc
3 (1)
a =2.971(5)
homogeneity of the alloy. No
c = 4.63(1)
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FEATURE ARTICLE ( USE OF NEUTRON DIFFRACTION ... )
TABLE 2
5. H. Kohlmann, in Encyclopedia of
Physical Science and Technology,
CRYSTALLOGRAPHIC PARAMETERS OF THE C14 PHASE IN TIV0.9MN1.1 AS DETERMINED
edited by Robert A. Meyers
FROM RIETVELD REFINEMENT OF NEUTRON DIFFRACTION PATTERN
(Academic Press, San Diego,
2002), Vol. 9, pp. 441.
6. D. Chandra, J.J. Reilly, and
Site
Refined
Atoms
Occupancy
R. Chellappa, JOM 56, 26 (2006).
(Wickoff symbol)
Coordinates
7. L. Schlapbach, I. Anderson, and
J.P. Burger, in Electronic and
2a
-Mn
0.65
Magnetic Properties of Metals and
V
0.35
Ceramics Part II, edited by
4f
z = 0.065
Ti
1.0
K.H. Jürgen Buschow (VCH,
Weinheim, 1994), Vol. 3B,
pp. 271.
6h
x = 0.82356
Mn
0.58
8. M. Yamaguchi and E. Akiba, in
V
0.42
Electronic and Magnetic Properties of Metals and Ceramics
Laves phases are intermetallic compounds of AB2 in which,
Part II, edited by K.H. Jürgen Buschow (VCH, Weinheim,
in the case of metal hydrides, a strong hydride former occu1994), Vol. 3B, pp. 333.
pies the A site and a weaker hydride former is on the B site.
9. L. Schlapbach, in Hydrogen in Intermetallic Compounds I, edited by L. Schlapbach (Springer-Verlag, Berlin, 1988), pp.1.
There are three types of laves phases crystal structures; (i)
10. D.G. Westlake, Journal of the Less-Common Metals 90, 251
hexagonal C14 (MgZn2 type)+, (ii) cubic C15 (MgCu2 type),
(1983).
and (iii) hexagonal C36 (MgNi2 type). The difference
11.
D.A. Neumann, Materials Today 9, 34 (2006).
between these structures is explained by different stacking of
12. G.L. Squires, Introduction to the theory of thermal neutron scathexagonal atomic layers [32]. The appearance of this C14
tering. (Dover, Mineola, New York, 1996).
Laves phase is expected because, as pointed out by Akiba
13. C.G. Shull, E.O. Wollan, G.A. Morton et al., Physical Review 73,
[24]
and Iba , these new BCC alloys have a nominal AB2 com842 (1948).
position and are closely related to Laves phases.
14. J. Huot, G. Liang, and R. Schulz, Applied Physics A72, 187
(2001).
In our Rietveld analysis the atomic occupancy and position in
15. C.C. Koch, in Processing of Metals and Alloys, edited by
the C14 phase were refined. Atomic positions are noted by
R.W. Cahn (VCH, Weinheim, Germany, 1991), Vol. 15,
using the Wyckoff notation. The Wyckoff symbol describe
pp. 193.
16. A.S. Pedersen, in Hydrogen Metal Systems I, edited by
the special positions of the space group, beginning with a for
F.A. Lewis and A. Aladjem (Scitec Publications, Zurich,
the highest symmetry. For the C14 structure, position 2a is
1996), pp. 35.
(0,0,0), 4f is (1/3,2/3,z), and 6h is (x,2x,1/4). In Table 2 the
17.
A.A.
Nayeb-Hashemi and J.B. Clark, in Monographs Series on
refined parameters for the C14 phase are shown.
Alloy Phase Diagrams (ASM International, Metals Park, Ohio,
1988), pp. 370.
Titanium atoms are localized exclusively on the 4f site while
18. J. Huot, G. Liang, S. Boily et al., Journal of Alloys and
the manganese and vanadium atoms are distributed on the 2a
Compounds 293-295, 495 (1999).
and 6h sites but with a different abundance. These abun19. J.-P. Bastide, B. Bonnetot, J.-M. Letoffe et al., Materials Research
dances give a stoichiometry of TiV0.8Mn1.2 for the C14 phase.
Bulletin 15, 1215 (1980).
This is an example of the use of neutron diffraction as a tech20. Y. Chen, J.S. Williams, and G.M. Wang, Journal of Applied
nique for structure determination and characterization. A
Physics 78, 3956 (1996).
21. R.A. Young, in IUCr Monographs on Crystallography-5, edited
complete description of the hydrogen sorption properties
by R.A. Young (Oxford University Press, Oxford, 1993),
and the effect of ball milling on the BCC solid solution alloys
pp. 298.
will be given in a forthcoming paper.
22. J. Huot, I. Swainson, and R. Schulz, Ann. Chim. Sci. Mat. 31,
135 (2006).
In conclusion, neutron diffraction is a unique and powerful
23. J. Huot, in Nanoclusters and Nanocrystals, edited by
tool for the structure characterization and the localisation of
H.S. Nalwa (American Scientific Publishers, Stevenson
hydrogen (deuterium) atoms. In the past it played a crucial
Ranch, California, 2003), pp. 53.
role in the understanding of metal hydrides and will remain
24. E. Akiba and H. Iba, Intermetallics 6, 461 (1998).
a privileged tool for the development of new metal hydrides
25. D. Mori, N. Haraikawa, N. Kobayashi et al., Mater. Res. Soc.
for practical applications.
Symp. Proc. 884E, GG6.4.1 (2005).
26. Y. Nakamura and E. Akiba, Journal of Alloys and Compounds
ACKNOWLEDGEMENT
345, 175 (2002).
27. Y. Nakamura, K.-I. Oikawa, T. Mamiyama et al., Journal of
The authors wish to thank Prof. Louis Marchildon for useful
Alloys and Compounds 316, 284 (2001).
comments and Mr. Sylvain Pednault for drawing figures.
28. S.-W. Cho, H. Enoki, T. Kabutomori et al., Journal of Alloys and
Compounds 319, 196 (2001).
REFERENCES
29. H. Iba and E. Akiba, Journal of Alloys and Compounds 253-254,
21 (1997).
1. G.D. Berry and S.M. Aceves, Journal of Energy Resources
30. B.H. Toby, Journal of Applied Crystallography 34, 210 (2001).
Technology 127, 89 (2005).
31. A.C. Larson and R.B. Von Dreele, Report No. LAUR 86-748,
2. M.L. Wald, Scientific American 290, 66 (2004).
1994.
3. R. Harris, D. Book, P. Anderson et al., The Fuel Cell Review 1,
32. F. Stein, M. Palm, and G. Sauthoff, Intermetallics 12, 713
17 (2004).
(2004).
4. S. Ashley, Scientific American 292, 62 (2005).
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NEUTRONS AND MUONS AS COMPLEMENTARY PROBES
OF EXOTIC MAGNETISM AND SUPERCONDUCTIVITY
by C.R. Wiebe
M
their respective strengths and weaknesses, and close by givany of the “big questions” in science have needed “big
ing examples of recent problems in the field that can be illutools” to answer, such as the Hubble Space Telescope and
minated by using both kinds of probes [3,4].
particle accelerators such as CERN. In the field of condensed
matter physics, some of the “big questions” of the field are
NEUTRON SCATTERING
now being addressed with the creation of scientific centers
such as the National High Magnetic Field Laboratory in
Neutron scattering is a technique whereby neutrons, created
Florida, the Canadian Light Source,
by a source such as a nuclear reactor or
and neutron sources such as NIST in
accelerator (such as a spallation source),
Gaithersburg, Maryland. New neu- In this paper, I will outline
are directed at a material of interest,
tron sources at Oak Ridge (the
and the scattered neutrons are analyzed
how
neutron
scattering
can
Spallation Neutron Source), and the
to learn about how the constituent
proposed facility at Chalk River will be used with muon spin
atoms are located in space (chemical
open new vistas of research in the near
structure), how they are moving with
future. However, there are some prob- relaxation (mSR) to gain a
respect to one another (dynamics), and
lems that neutron science is not well more complete picture of
how magnetic spins order and interact
suited to answer, and other probes of
with one another (magnetic neutron
exotic
magnetism
and
solids are needed. In this paper, I will
scattering). The simplest example of a
outline how neutron scattering can be superconductivity in solids.
neutron scattering experiment is a difused with muon spin relaxation (μSR)
fraction experiment, where neutrons of
to gain a more complete picture of
a particular energy are scattered from a
exotic magnetism and superconductivity in solids. I will dissolid of interest, and the pattern of intensity as a function of
cuss examples of problems in the field that have been
scattering angle is used to learn about how the atoms are
addressed using these two complementary techniques.
arranged within the material. Bragg’s law, 2d sinθ = λ, determines where the intensity of the neutron signal will peak due
INTRODUCTION
to constructive interference of neutrons that reflect from regularly spaced atomic layers. This is called an elastic scattering
The great neutron scientist Bertram Brockhouse (Nobel
experiment since the incoming neutron and the outgoing neuLaureate, 1994) once remarked that if Chadwick hadn’t distron have the same energy, and thus no energy has been lost
covered the neutron, then it would have to be invented [1].
in the solid.
Neutron scattering is an incredibly powerful technique for
investigating different states of matter, and since
If a crystal known as an analyzer is used before the neutron
Brockhouse’s invention of the triple axis spectrometer, the
detector (but after the neutrons have been scattered from the
field has experienced a considerable amount of growth.
sample), the energy of the outgoing neutron beam can be
Neutron scientists around the world are now using neutrons
measured, and an inelastic experiment can be conducted.
to investigate diverse problems from protein crystallography,
Energy can be transferred to the solid in a variety of different
to dynamics in membranes, to spin glass behavior, engineerways, but in solid state physics, inelastic neutron scattering is
ing applications, and of course, a wide variety of correlated
[2]
usually used to measure lattice vibrations (phonons) or magelectron systems .
netic excitations (such as spin waves). An excitation, be it a
phonon or spin wave, can be created or destroyed by an
However, even though there are many problems that neutron
incoming neutron, giving rise to a shift in energy of the outscattering can address, there are some fundamental systems
going neutron beam. The wavevector of the momentum
that have been difficult to characterize with neutrons due to
transfer to the solid, Q, can be measured in addition to the
limitations of the technique. It is rare when a single type of
energy, and information about dispersion relations of the
experiment can answer complex problems such as the nature
vibrations can be obtained.
of high temperature superconductivity in a system of ~ 1023
electrons. An entire arsenal of techniques have, in fact, been
Most physical scientists have been exposed to x-ray diffracused to investigate solids for several decades, and the growtion at some point in their training, and one may ask the quesing consensus is that only a combination of probes can be
used to make progress. In this paper, I will discuss how neutron scattering can be used with muon spin relaxation (μ SR)
C.R. Wiebe <[email protected]>, Florida State
to get a better picture of electronic behavior within solids.
University/NHMFL, Tallahassee, FL, USA 32310-4005
I will give a brief introduction to both techniques, discussing
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tion - why use neutron scattering instead of x-ray scattering?
Neutron sources are far more expensive than conventional
table-top x-ray sources in general – why do we bother with
this technique? There are a number of distinct advantages
that neutron scattering has over x-rays:
(1) Neutrons are charge neutral. The consequence of this is
that neutrons do not interact strongly with the charged
electrons within solids, and they can penetrate to great
depths (on the order of centimeters). The average x-ray
from a conventional source only penetrates on the order
of microns due to the charged nature of electromagnetic
radiation.
(2) Neutrons interact primarily through the strong force that is with the nuclei, compared to x-rays, which interact via the electromagnetic force with charges. Neutrons
can see deep within the constituent atoms, and they are
not affected much by electronic clouds.
(3) Neutrons from a reactor have an average energy which
is on the order of milli electron volts (meV), as compared
to x-rays which have energies on the order of kilo electron volts (1 keV = 106 meV). The meV range is where
many phenomena occur in solids, such as lattice vibrations, as compared to keV, which is a much higher energy scale. This makes inelastic scattering more convenient for neutron scatterers compared to x-ray scientists.
(4) Neutrons have magnetic moments – they have spin.
Due to the fact that neutrons have a quark structure,
even though they have neutral charge they have a spin s
= ½. Each neutron in an experiment can interact with
the spins of a solid, which are usually from the constituent unpaired electrons. Therefore, neutrons can
also give information about the magnetic structure of a
solid (such as a ferromagnet, which has a permanent
magnetic moment below a temperature TC), or about the
magnetic excitations (such as magnons or spin waves).
MUON SPIN RELAXATION (μSR)
Given the many applications of neutron scattering, one may
ask what sort of complementary information can be gained
from the muon spin relaxation technique. To answer this
question, one needs an introduction to μSR [4].
μSR stands for muon spin relaxation, muon spin rotation, or
muon spin research – the name was chosen to draw an analogy to nuclear magnetic resonance, or NMR. NMR is most
often used to look at what are called hyperfine interactions
between the nuclei and the electrons in condensed matter
systems, and thereby gain information about electron-electron interactions within the solid. μSR is a similar technique
– that is, a μSR spectroscopist investigates how muons interact with electrons in a solid. However, the muons have to be
implanted within a material, whereas with NMR, the nuclei
already exist to be probed (or a material can be doped with a
certain nuclei of interest). By monitoring the polarization of
the muon spin as a function to time, the local magnetic field
at each muon site can be obtained, and from this we can gain
complementary information to neutron scattering measurements.
(with a half life of 26 ns) that decay into the longer lived
muons (with a half life of 2.2 μs). The decay of these pions is
governed by the weak interaction, which insures that the
resulting muons are 100% spin polarized. Since muons are
charged (either negative or positive), we can use electric and
magnetic fields to create muon beams which are then directed at our material of interest (see figure 1). The high energy
of the muons is such that these beams can penetrate deep
within solids, and through cryostats, magnets, and pressure
cells which enable measurements at low temperatures, in reasonably high magnetic fields (up to about 7 T) and under
pressure. Once the muons have penetrated within a material of interest, they are rapidly thermalized and reside within
an electrostatically favorable place within the crystalline lattice (at a negatively charged site, for example, for positively
charged muons).
The actual experimental process of μSR involves injecting
muons, one at a time, into a solid of interest, and measuring
the time it takes to decay (see figure 2). The muon will decay
into two neutrinos and a positron with a half-life of about
2.2 μs. Since this is, again, governed by the weak interaction,
the resulting positron is preferentially emitted in the same
direction of the muon spin at the time of the decay. This
enables one to determine the evolution of the muon spin as a
function of time by surrounding the target with a series of
positron detectors, and measuring a large number of muon
decays. The asymmetry, which is a subtraction of either forward – backward, or right - left detector counts, gives a measure of the polarization of the muon spin. Since the initial
polarization of the muon beam is known (the beam is 100%
spin polarized), one can accurately determine the local magnetic field at each muon stopping site within a material.
In a given material, the muon spin precession signal will
change above and below a magnetic transition due to the
development of an internal magnetic field. In the paramagnetic phase, where the spins are rapidly fluctuating, the
muon essentially sees a net field of zero, since the magnetic
Fig. 1
Muons are usually created through high energy collisions of
protons upon a target, which produces short-lived pions
296
PHYSICS
IN
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September / October 2006
A typical schematic of an accelerator-based muon
spin relaxation experiment. Muons are typically produced in a cyclotron, where protons are accelerated to
high speeds and undergo collisions with a target.
The decay products at the target include pions, which
rapidly decay to muons and neutrinos. A crossed
electric and magnetic field separates out unwanted
decay products, and after the muon beam is collimated it is ready for use.
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spins cancel
each
other
out on the
average.
However,
within
the
ordered state,
the
muon
spin precesses. This is
similar
to
Larmor precession
in
NMR – the
muon, possessing
a
spin, will precess with a
frequency
proportional
to the internal magnetic
field generat- Fig. 2 A closer look at muon spin relaxation.
ed from the
The muons are implanted at the left
with a spin polarization in the direco r d e r e d
tion of the beam. After implantation,
moments. A
the muons will feel a local spin envimeasurement
ronment and the orientation will
of this frechange. After decay, the positrons are
quency will
preferentially emitted in the direction
determine
of the muon spin, and are measured in
the size of
a series of detectors, which mark an
this
field.
event on the electronic clock. The
The amplidirection of the magnetic field in the
tude of the
experiment is perpendicular to the
muon spin, indicating a transverse
signal confield configuration (picture courtesy
tains inforof A. Savici).
mation about
what volume
fraction is ordered. So, for example, if only 20% of the sample is ordered and the rest is a paramagnetic, or fluctuating
phase, will we only see a signal which is 20% of the maximum signal (sometimes called the asymmetry). This provides complementary information with neutron diffraction, which
tends to measure only the average ordered
moment within a material. Muons, being a
local probe, give a measurement of the
moment size (through the frequency of the
precession), and the volume fraction.
of experiment that can be done – a longitudinal field (or “LF”)
experiment (where the field is applied along the same axis as
the initial muon polarization), or a transverse field experiment (where the field is applied perpendicular to the initial
muon spin polarization direction). Longitudinal field measurements are used to probe spin dynamics, since one is providing a bias direction for the magnetic spins within a solid
with an applied field. However, the fluctuation rates that can
be probed with μSR are much different than neutron scattering – on the order of 104 to 1012 Hz. This is a time scale which
bridges what NMR and neutron scattering can measure (slow
dynamics and fast dynamics, respectively), and is useful as a
consistency check for these experiments. μSR has also seen a
great deal of success in elucidating the spin dynamics of spin
glasses, which tend to have time scales slower than what neutron scattering can resolve. Transverse field μSR measures a
Knight shift, which is akin to the magnetic susceptibility. TFμSR is also used to measure the penetration depth within
superconductors, which is probably the most successful
application of this technique in condensed matter physics.
The classification of superconductors based upon various
symmetries of the pairing of the electrons (into s-wave, or
d-wave states, for example) has been done through careful
μSR measurements in a series of landmark experiments [4].
COMPARISON OF THE TECHNIQUES
What are the advantages of μSR over neutron scattering? The
table below summarizes and compares these two techniques.
(1) Reciprocal space vs. real space probes. Neutron scattering is a reciprocal space probe; that is, it measures not the real
space distribution of spins and atoms, but the Fourier transform of this. Any neutron data, to be interpreted properly,
needs to be Fourier transformed to real space from the
abstract world of “reciprocal space” to gain information that
we can readily use. Muons, on the other hand, are real space
probes. The muon signal that is obtained is actually measuring the local magnetic field at each muon site. The disadvantage to this is that the muons have resting sites that have to be
elucidated to make sense of the resulting signal in many
cases. Although the muons land randomly within the bulk of
There are three different kinds of μSR
experiments, which are classified according
to the magnitude and direction of the
applied magnetic field. If there is no magnetic field, the experiment is aptly named
“zero field μSR” (of ZF-μSR). These sorts of
experiments are excellent for probing magnetic transitions, and weak magnetic ordering. The muon is an extremely sensitive
probe – fields on the order of a fraction of a
gauss can be measured! If one applies a
magnetic field, there are basically two types
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samples, the actual stopping site within each unit cell is
determined through electrostatics and can be difficult to infer
from the data. There are some cases, such as in the determination of penetration depths within superconductors, where
this is not crucial. However, this is in stark contrast to neutron scattering, where the neutrons can measure the entire
sample at once without the need to determine “stopping
sites”.
peak, which corresponds to an ordered moment of
about 0.03 μB was discovered, but this moment was too
small to account for a large heat capacity anomaly at
17.5 K [7,8].
μSR measurements were among the first to offer a crucial piece of information about the nature of this “hidden order” state. One of the questions which have
plagued experimentalists in this field is the role of impurities. Neutron scattering can detect magnetic impurities through their magnetic transitions, and it was
thought that the small Bragg peak seen was actually due
to a large moment in a small part of the sample, as
opposed to a small moment spread out homogeneously
in a large part of the sample. μSR, on the other hand, can
distinguish between the two situations. A large
moment, from say the ordered electrons on a uranium
site in an impurity, would give rise to a large internal
field developing at the transition temperature. The size
of the volume fraction will determine the amplitude of
the signal. In the case of URu2Si2, muon precession was
observed below 17.5 K, confirming the existence of magnetic order discovered through neutron scattering [9].
However, the size of the oscillating signal was small,
suggesting that only about 10% of the sample was
ordered (see figure 3). Moreover, the internal field was
large, giving a large precession frequency that could not
be accounted for with the small moment predicted from
neutron scattering. It appears, then, that in the case of
URu2Si2, the hidden order phase must be unrelated to
(2) Elastic and inelastic information. With different kinds
of neutron scattering experiments, the elastic and inelastic
information can be separated, through for example the triple
axis spectrometer invented by Brockhouse. The resolution
can approach the micro electron volt (μeV) regime! μSR has
a deficiency in this regard, since the inelastic and elastic data
are often superimposed upon one another. A LF experiment
can sometimes determine the strength of the interactions
between spins, but in general neutron scattering is a far superior technique for the spatial and energetic resolution of magnetic correlations within materials.
(3) Time scales. Neutron scattering typically measures very
fast time scales (108 to 1013 Hz) – after all, the neutrons themselves do not spend much time within solids. Modern techniques have pushed this time window to smaller time scales,
but this still does not cover the dynamic range offered by
μSR – 104 – 1012 Hz (eight orders of magnitude!). When both
techniques are used together, this results in over nine orders
of magnitude of correlation times available to probe.
RECENT EXAMPLES FROM MODERN CONDENSED
MATTER PHYSICS
What questions can the combination of these techniques
answer? Here are some prominent examples from the field
of condensed matter physics:
(1) Heavy fermion materials: Heavy fermion metals belong
to a class of highly correlated electron materials with
extremely high effective carrier mass (as shown through
specific heat measurements) [5]. Most of these systems
are composed of f-electron actinides and transition metals. The physical interpretation of this “heavy mass” for
the constituent electrons is that the carriers themselves
are not moving independently of one another, but they
are strongly interacting with other electrons within the
lattice. The physics behind these systems involves a
competition between magnetic ordering of the f-electrons, through what is called the RKKY interaction, and
a tendency to remain disordered, which is caused by
scattering of conduction electrons with the f-electrons
through the Kondo effect.
URu2Si2 belongs to this class of materials, with an effective electron mass which is about 25 times a free electron
mass [6]. At 17.5 K in this material, there is a “hidden”
order transition, and at about 1 K, there is a superconducting transition. Oddly enough, most of the attention
in this material has been focused on the 17.5 K transition
and not the superconducting one. This is because
despite two decades of research, physicists still do not
know what is ordering. A very tiny magnetic Bragg
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Fig. 3
September / October 2006
Zero field muon spin relaxation spectra in URu2Si2
above and below the transition at 17.5 K [9]. Note the
small precession signal at 2.5 K, which indicates a
small volume fraction of ordered spins (scenario (a)
compared to scenario (b)).
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this small moment, since the entropy loss from the transition would be small if it came from an impurity phase.
NMR measurements, over a decade later, confirmed the
μSR measurements of sample inhomogeneity [10]. The
question of the hidden order phase still remains open.
However, recent neutron scattering results are suggesting that the answer may lie in an analysis of the inelastic spectrum.
(2) Geometrically frustrated materials: Geometrically frustrated oxides usually consist of transition metal or rare
earth magnetic ions that exist in triangular sublattices. If
antiferromagnetic interactions exists between these
species, a conventional Néel ground state cannot arise in
many of these systems, and thus the material is said to
be “frustrated”. A wide variety of unusual ground
states can instead be favored, such as spin glasses in the
absence of chemical disorder, “spin ice” ordering, and
possible “spin liquid” states, where the moments remain
fluctuating down to zero Kelvin [11].
μSR has played an important role in the investigation of
these fascinating materials. Neutron scattering is typically used first as a technique to probe for magnetic
ordering. However, in many of these systems, the time
scale for the fluctuating moments is smaller than the
neutron time window, and thus the characteristic features of frustrated systems are seen as broad lumps of
scattering in the elastic channel. If the moment is too
small, the magnetic signal can be dominated by the
background, and these features can be completely lost.
This was the case in the S=1/2 FCC systems A2BReO6
(A = Sr, B = Ca, Mg) [12,13]. Both of these materials have
networks of face-centered cubic (FCC) Re S=1/2 ions
that have very strong antiferromagnetic interactions.
The FCC lattice can be thought of as a series of edgeshared tetrahedra (complex networks of triangles), so
these systems were prime candidates for geometric frustration. Specific heat and susceptibility measurements
revealed transitions at ~ 14 K and 50 K respectively for
the Ca and Mg systems, but no magnetic Bragg peaks
were detected by neutron scattering. It was μSR that
revealed the nature of the ground state through the
appearance of a characteristic lineshape in the ZF-spectra below the transition temperatures that was typical of
spin glasses – the spins were frozen out into random orientations. Instead of a precession signal, a loss in the
polarization and a signature minimum in the spectrum
is noted (see figure 4).
Despite the successes of these techniques in elucidating
the low temperature ground states of frustrated systems, there are still discrepancies in the literature about
the correct interpretation of the results. For example, the
pyrochlore material Tb2Sn2O7, where the Tb3+ spins lie
on the corners of corner-shared tetrahedra, has been the
subject of continued debate. Neutron scattering clearly
notes magnetic Bragg peaks appearing below 1.2 K [14],
but μSR experiments suggest that this state is not
ordered, but dynamic (ie. there is no precession signal) [15]. Apparently, the spins appear static within the
neutron time window, but dynamic within the μSR time
Fig. 4
(a) Spins with strong antiferromagnetic interactions
which reside in triangular sublattices give rise to the
condition of frustration at low temperatures; (b) Zero
field muon spin relaxation data on the geometrically
frustrated material Sr2CaReO6 [12]. The characteristic
drop in polarization below 14 K is typical of spin
glasses (the fit is to a function used by Uemura et al.
for canonical spin glasses).
window! This situation can arise if there is a large internal field which dampens out the expected precession signal for ordering in a μSR experiment (as expected for
large ordered Tb moments). It is also possible that there
are regions of ordered spins coexisting with dynamic
spins that dominate the signal. The low ordered
moment seen from neutron scattering is consistent with
this result.
(3) Superconductivity: Last, but not least, is the field of
superconductivity, where perhaps μSR has had the
greatest degree of success in determining the symmetry
of the superconducting state, the presence of magnetism
coexisting with superconductivity, and the physics of
vortex dynamics [16]. The literature is replete with examples of the strengths of this method [4]. However, there
are some notable examples of problems within the field
that have needed both neutron scattering and μSR to
address. The existence of magnetism within vortex
cores, for example, remains a controversial question.
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Neutron scattering observes an increase in magnetic
Bragg peak intensity as a function of applied field in La2xSrxCuO4 (x=0.163), eluding to the presence of enhanced
antiferromagnetism within the vortex cores [17].
Some μSR experiments suggest similar behavior in
Pr2-xCexCuO4 through an analysis of the muon lineshape (which is a measure of the local field distribution
at the muon site) [18]. However, the subject is still controversial. What is more certain is the advantage that
μSR has in determining the symmetry of the superconducting state through penetration depth experiments as
a function of temperature. While neutron scattering can
indirectly determine this through inelastic scattering
spectra, μSR has been instrumental in characterizing
many of the known superconductors, and perhaps even
more helpful in drawing upon universal behavior
throughout the high-TCs (through, for example, the
Uemura plot [19]). For as long as the problem of hightemperature superconductivity exists, these techniques
will likely be used together to further our understanding
of this phenomena.
CLOSING REMARKS
A wide variety of problems can be addressed through μSR
and neutron scattering experiments which provide very complementary data due to the range of the interactions and the
respective timescales. It is likely that with the advent of more
powerful spallation sources, μSR will continue to thrive
alongside of neutron scattering, and it is no accident that
sources such as ISIS in the UK have muons and neutrons
available at the same location for experiments. There is no
shortage of important condensed matter systems that are
waiting to be investigated.
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ACKNOWLEDGEMENTS
The author would like to thank the continued support and
interest in the field from many colleagues over the years,
including B.D. Gaulin, W.J.L. Buyers, J.E. Greedan,
J.S. Gardner, G.M. Luke, and Y.J. Uemura.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
September / October 2006
B.N. Brockhouse, Rev. of Mod. Phys. 67, 735 (1995).
See, for example, http://www.neutron.anl.gov/ for a reference on neutron scattering.
G.L. Squires, “Introduction to the Theory of Thermal
Neutron Scattering,” Dover Publications, England (1997).
For an introduction to ìSR and various recent experiments,
see http://cmms.triumf.ca/intro/muSRBrochure.pdf
For an introduction to heavy fermions, please see Z. Fisk et
al., Proc. Natl. Acad. Sci. 92, 6663 (1995).
De Visser et al, Phys. Rev. B34, 8168 (1986).
C. Broholm et al., Phys. Rev. B43, 12809 (1991).
For a review, see V. Tripathi et al., J. Phys. Cond. Matt. 17,
5285 (2005).
G.M. Luke et al., Hyperfine Interactions 64, 517-522 (1990).
K. Matsuda et al., Phys. Rev. Letters 87, 087203 (2001).
For a review of frustration see J.E. Greedan, J. Mat. Chem. 10,
3058 (2001).
C.R. Wiebe et al., Phys. Rev. B65, 155325 (2002).
C.R. Wiebe et al., Phys. Rev. B68, 134410 (2003).
I. Mirebeau et al., Phys. Rev. Letters 94, 246402 (2005).
P. Dalmas de Réotier et al., Phys. Rev. Letters 96, 127202
(2006).
For example, see J.E. Sonier et al., Rev. Modern Physics 72, 769
(2000).
For example, see B. Lake et al., Science 291, 1759 (2001).
J.E. Sonier et al., Phys. Rev. Letters 91, 147002 (2003).
Y.J. Uemura et al., Phys. Rev. Letters 66, 2665 (1991).
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ARTICLE DE FOND ( STATUS OF THE CMCF ... )
STATUS OF THE CANADIAN MACROMOLECULAR
CRYSTALLOGRAPHY FACILITY: DESIGN AND
COMMISSIONING OF THE 08ID-1 BEAMLINE AT THE
CANADIAN LIGHT SOURCE
by Pawel Grochulski, Ingvar Blomqvist, and Louis Delbaere
T
he 08ID-1 beamline is the initial phase of the Canadian
Macromolecular Crystallography Facility (CMCF) [1] located
at the Canadian Light Source (CLS), a 2.9 GeV ring. CLS produced "first light" on December 2003, and the first data collected at CMCF 08ID-1 was in May 2006.
Grenoble, France (6 GeV) and Spring-8 in Nishi Harima,
Japan (8 GeV).
Electrons and other charged particles do not radiate while
undergoing linear, uniform motion. However, when accelerated, a rearrangement of their electric
We have designed, constructed and
fields occurs. This field perturbation
are commissioning a beamline which The 08ID-1 beamline is the traveling away from the electrons at the
is illuminated by a small-gap in-vacu- initial
phase
to
the velocity of light is called electromagnetum hybrid undulator (SGU), located
ic radiation. An example of such radiain the upstream half of the straight Canadian Macromolecular tion is a TV antenna where electrons are
section, and chicaned inboard by Crystallography
Facility driven up and down the antenna, in a
0.75 mrad. The downstream half of this
[1] located at the periodic fashion, at a frequency specific
section is reserved for the 08ID-2 (CMCF)
to a particular TV station. In the case of
beamline SGU. The beamline contains Canadian
sources, electrons at
Light Source synchrotron
white beam slits (WBS), a double crysspeeds very close to the speed of light
tal monochromator (DCM) equipped (CLS), a 2.9 GeV ring. CLS are stored in an evacuated circular path.
with an indirectly cryo-cooled first
observed synchrotron radiation is
produced "first light" on The
crystal and a sagittally-focusing seccaused by the transverse acceleration
ond crystal followed by a vertically December 2003,and the first due to magnetic forces. These magnetic
focussing mirror (VFM). The beamline
forces are due to either bending magis terminated by an innovative and data collected at CMCF nets along the ring or by special inservery robust endstation, including a 08ID-1 was in May 2006.
tion device magnets such as, undulators
MarMosaic225 CCD X-ray detector.
and wigglers located in the straight sections of the synchrotron ring. The synFor the most part, the beamline components were manufacchrotron radiation from a bending magnet source is linearly
tured by ACCEL Instruments GmbH (Germany), while the
polarized in the plane of acceleration. In an undulator source,
beamline controls, similarly for the CLS facility, are being
the electron beam is periodically deflected by weak magnetic
developed based on an EPICS platform and complemented
fields, generating a large number of electron beam oscillations
with an innovative user interface. The primary scientific goal
of small amplitude where the electron emits radiation at the
of the 08ID-1 beamline is to interrogate small protein crystals
wavelength of its periodic, sinusoidal motion. Strong interfer(~20 μ) with large unit cells (~1000 Å). The beamline is
ence effects occur in an undulator and the radiation has a high
equipped with a Röntek Spectrometer System (XFLASH
intensity line spectrum over a very restricted frequency
101A) capable of simultaneously carrying out multiwaverange. The spectral resolution of the radiation is proportional
length anomalous diffraction (MAD), X-ray absorption near
to the number of undulator periods - the synchrotron radiaedge structure (XANES), and X-ray fluorescence (XRF) in the
tion wavelength can be altered by varying the magnetic field.
case of protein derivatives containing heavy metal atoms.
For an electron, its wavelength is determined by the undulaThere are ongoing efforts to equip the beamline with a robottor magnetic period divided by γ (due to relativistic Lorentz
ic cryogenic sample changer thus enabling remote access to
contraction - at the CLS γ=5675.3). On the experimental floor
the facility.
of a synchrotron this wavelength appears to the observer to
be further reduced by another factor γ due to the Doppler
effect. For example, an undulator with a period of 0.02 m is
CLS STORAGE RING
The Canadian Light Source (CLS) is a 2.9 GeV "third generation" synchrotron facility in the same class of synchrotrons as
the Swiss Light Source in Villingen, Switzerland and SPEAR3
in Stanford California, USA. The leading edge synchrotron
technologies are the APS in Chicago, USA (7 GeV), ESRF in
Pawel Grochulski <[email protected]> 1,2,
Ingvar Blomqvist 1 and Louis Delbaere 2; 1 Canadian Light
Source, 2 University of Saskatchewan, Saskatoon,
Saskatchewan, Canada
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generating synchrotron radiation in the X-ray regime [~0.02
m/(2(5675.3)2)= 3x10-10 m]. When the magnetic field of an
undulator is increased and the pure sinusoidal transverse
motion of electrons becomes distorted, generating higher
harmonics of the single wavelength undulator radiation, this
device is called a wiggler. The spectrum of a wiggler is similar to the spectrum generated by a bending magnet, but the
photon beam is much more intense and the "critical energy"
(Ec) is shifted towards a higher energy with respect to the
bending magnet radiation. The critical energy is the point in
the spectrum which divides equally the total integrated spectral energy. The CLS critical energy is Ec=7.572 keV and, for
example, the superconducting wiggler of the CLS XAFS
beamline has Ec= 10.7 keV. Both bending magnets and wigglers have a continuous spectrum over a broad range of energies, typically extending from infrared to hard X-rays.
Electrons circulating in the storage ring are compressed into
bunches and, thus the generated radiation is emitted in pulses. However, since the time between bunches is 2 ns, the case
of the CLS, the source can be considered continuous for all
crystallographic applications except nanosecond timeresolved experiments.
The CLS ring currently runs at 200 mA and the size of the
electron beam at the location of the insertion device is
1.09 mm x 0.05 mm full width at half-maximum (FWHM),
horizontal (H) x vertical (V)) and with divergence of 100 μrad
x 21 μrad (FWHM, H x V).
MACROMOLECULAR CRYSTALLOGRAPHY
Macromolecular crystallography is a method for determining
the structures of large biological molecules such as proteins
and nucleic acids. Scientists study the structures of biological
molecules in order to (a) increase their understanding of
structural biology and biochemistry, (b) develop and
improve the design of ligands that bind to macromolecules in
order to use these ligands as pharmaceuticals, and (c) develop a basis for modifying the structures of macromolecules
themselves in order to alter their functions, sometimes with
industrial applications [2].
on an X-ray CCD detector is defined by three properties; the
amplitude, which is measured from the intensity of the spot;
the wavelength, which is defined by the monochromator;
and the phase, which is lost in the X-ray experiment. In order
to determine the atomic positions from a diffraction pattern
we need to know all three properties for each diffracted beam
and then calculate the Fourier transform. Determining phases is the most challenging task in crystallography and today
it is done mostly by the multiwavelength anomalous diffraction method (MAD), realized by tuning the wavelength of the
monochromator to the absorption edge of a "heavy atom"
(e.g. selenium) in the protein.
CANADIAN MACROMOLECULAR
CRYSTALLOGRAPHY FACILITY
The Canadian Macromolecular Crystallography Facility will
have more than 60 protein crystallographers, located across
Canada, and consist of three beamlines, two insertion devices
beamlines and one bending magnet beamline. The first insertion device beamline (08ID-1) was intended to be highly efficient and flexible, capable of satisfying the requirements of
the most challenging and diverse crystallographic experiments, i.e. physically small crystals with large unit cell
dimensions (Figure 1). The recently funded bending magnet
08B1 beamline is designed for high-throughput data collection, capable of being accessed remotely. The third, an undulator based 08ID-2 beamline, is envisioned to have microfocussing capabilities, but with some restrictions in energy.
Since the 08ID-1 beamline is already in a chicaned position,
0.75 mrad inboard (Figure 1), the SGU for the 08ID-2 will be
chicaned 0.75 mrad outboard, so the total separation between
photon beams will be 1.5 mrad. It will allow for the separation between beams to be 30 mm at a distance of 20 m from
the source.
BEAMLINE DESIGN
Insertion device
The CMCF 08ID-1 beamline is illuminated by a hybrid invacuum small-gap undulator (SGU) (80 periods with a period length of 20.0 mm) operating at a minimum gap of 5.5 mm
(Beff=0.923 T). The mechanical supports and vacuum cham-
The first prerequisite for solving the three-dimensional structure of a macromolecule by X-ray crystallography is
a well-ordered crystal that will strongly diffract Xrays. Crystallography depends upon directing a
synchrotron X-ray beam onto a regular array of
identical molecules so that the X-rays are diffracted
in a pattern from which the structures of an individual molecule can be retrieved. The electrons in the
atoms making up the crystal scatter X-rays in all
directions, and only those which constructively
interfere with one another, according to Bragg's
Law, give rise to diffracted beams that can be collected as distinct diffraction spots. Each diffraction
spot is the result of constructive interference from
X-rays with the same diffraction angle emerging
from all atoms of the crystal. For example, diffraction from the 1500 atoms of a myoglobin protein
crystal results in about 25,000 diffracted beams [3].
Each diffracted beam, which is recorded as a spot Fig. 1 Layout of the CMCF "sector"
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ber for the SGU have been manufactured by R.M.P.s.r.l.
(Italy), while the magnetic structure, shimming and controls
have been produced at the CLS (Figure 2). The electron beam
is stabilized by the electron beam position monitors (BPMs)
in the storage ring, however, a four blade X-ray BPM
installed at 8.7 m from the center of the SGU will be used in
the feedback system to maintain the stability of the X-ray
beam to better than 0.5 μrad. The brilliance tuning curves for
the small-gap in-vacuum undulator and a flux spectrum are
shown in Figure 3. The measured filled integrals, RMS phase
angle error and photon energies as functions of undulator
gap are shown in Table 1. It is critical that the RMS phase
errors are less than 2o for the range of energy of 6.5 to 18 keV
since harmonics from 3 to 9 have to be used for those energies.
The 08ID-1 beamline illuminated by the SGU located in the
upstream of the straight section and chicaned inboard by
0.75 mrad. The downstream half of this section is reserved for
the future SGU associated with the 08ID-2 beamline. The
effective size of the source produced by the undulator is practically identical to the size of the electron beam in the undulator whereas, the divergences are different depending on the
wavelength.
the size of the white beam. The central unit of the optical system is a double-crystal Si(111) monochromator manufactured
by ACCEL Instruments GmbH (Germany). The double crystal monochromator is located at 43.5 m from the source. The
first monochromator crystal is cryogenically cooled, while the
second is a sagittally-focussing crystal. All essential components of the monochromator are indirectly water-cooled to
minimize thermal fluctuations. A quadrant of PIN diodes
positioned upstream of a 0.5 μm-thick chromium foil located
between the monochromator and the mirror serve as online
beam position and intensity monitors [4]. Vertical focussing of
the beam is achieved by a 1.1 m long dynamically bendable
ULE (ultra-low expansion titanium silicate) flat mirror manufactured by InSync, Inc. (Albuquerque, NM, USA). It has a
TABLE 1
RMS PHASE ANGLE ERROR (O) AND PHOTON ENERGIES
AS FUNCTIONS OF THE UNDULATOR GAP
(o )
X-ray optics
The narrow bandwidth produced by the SGU cannot be used
directly for macromolecular crystallography. A monochromator is needed to reduce bandwidth and to reject all of the
emission spectrum, except the band centered at the selected
photon energy. Since monochromator crystals let through
certain multiples of the selected fundamental photon energy,
there is a need for a low-pass filter such as, a mirror.
In detail (Figure 4), the optics are comprised of a CVD diamond-based water cooled X-ray beam position monitor
(BPM) followed by white beam slits that are used to define
Fig. 2
SGU in the storage ring
Fig. 3
Tuning curves of the small-gap in-vacuum undultor
(top) and flux spectrum for a 10 mm gap of the SGU
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TABLE 2
SPECIFICATIONS OF THE CMCF 08ID-1 BEAMLINE
WITH 200 MA RING'S CURRENT
Δ
Fig. 4
Layout of the CMCF 081D-1 beamline
measured surface roughness of 1.5 Å RMS and 0.58 μrad
RMS surface figure error without a bender and about 100%
larger with a SESO (France) bender at different bending radii.
The mirror is downward reflecting and fully adjustable in
terms of height, angle, and the radius of curvature can be
changed from infinity to 1.39 km. The active surface of the
mirror has three stripes parallel made of platinum, palladium
and un-coated regions that can be translated into the beam,
depending on the X-ray energy, to remove higher harmonics.
At the normal 2.2 mrad glancing angle the uncoated track is
used for photon energies below 13.5 keV and the Pd-coated
track for 13.5-18 keV. A lateral movement of the mirror substrate inside the vacuum tank allows the appropriate stripe to
be moved into the beam. With a demagnification ratio of 6.8
and 11.8 in the horizontal and vertical directions, respectively, a source size limited focal spot of 160 x 50 μm2 (H x V)
(FWHM) at the sample can be achieved. The mirror tank is
followed by a double safety photon shutter.
Experimental station
The experimental hutch is shown in Figure 5. The monochromatic beam is led through an evacuated filter box and then
through an exposure box. It also contains vertical and horizontal slits, an X-ray beam position monitor and the fast shutter. The protein crystal is mounted on a single axis goniostat
with a fully motorized x-y-z stage. The accumulated mechan-
ical errors amount to less than 6 μm at the sample position.
Diffraction data are recorded using a MarMosaic225 CCD
detector mounted in ACCEL's detector holder that can be
raised and tilted in order to enable ultra-high resolution data
collection. The beamline is equipped with a Röntek
Spectrometer System XFLASH 101A to perform the X-ray
spectroscopy for multiwavelength anomalous diffraction
(MAD) and X-ray absorption near edge structure (XANES)
on the same crystals, and the X-ray fluorescence (XRF) for
metal detection in protein derivative crystals. Specifications
of the beamline are shown in Table 2.
Control system
The beamline's control system is based on the EPICS platform [7], however all commissioning tools have been prototyped using MATLAB (The MathWorks, Inc.). This includes
all scan and sample visualization tools. After the commissioning is finished the user's software will be converted to a
browser based graphical user interface allowing remote
access to the facility. A number of Linux based PCs are available for data transfer, storage and processing.
ACKNOWLEDGEMENTS
We are indebted to the entire CLS staff for their excellent
work during the design, construction and commissioning, in
particular Michel Fodje, Alan Duffy, Russ Berg, Mike
McKibben and Tasha Summers.
REFERENCES
Fig. 5
304
End-station of the CMCF 08ID-1 beamline
PHYSICS
IN
CANADA
1. P. Grochulski, I. Blomqvist, B. Yates, E. Hallin, E. & L. Delbaere,
"Design of the 08ID-1 protein crystallography beamline at the
Canadian Light Source", Acta Physica Polonica A101(5), 589-594
(2002).
2. Third-Generation Hard X-ray Synchrotron radiation Sources, ed.
Dennis M. Mils, John Wiley & Sons, Inc., New York (2002).
3. C. Branden and J. Tooze, Introduction to Protein Structure, Garland
Publishing Inc., New York and London (1991).
4. R.W. Alkire, G. Rosenbaum and G. Evans, "Design of a vacuumcompatible high-precision monochromatic beam-position monitor for use with synchrotron radiation from 5 to 25 keV", J.
Synchrotron Rad. 7, 61-68, (2000).
5. M. Sanchez del Rio, R.J. Dejus, X-ray Oriented Programs 2.0,
http://www.esrf.fr/computing/scientific/xop/
6. C. Welnak, G.J. Chen and F. Cerrina, “SHADOW: a synchrotron
radiation X-ray optics simulation tool”, Nucl. Instr. And Meth.
A347,
344-347
(1994);
SHADOW
VUI,
v 1.0,
http://www.esrf.fr/computing/ scientific/xop/shadowvui/
7. EPICS: http://www.aps.anl.gov/epics/
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ARTICLE DE FOND ( PHONON SPECTROSCOPY ... )
PHONON SPECTROSCOPY AND X-RAY SCATTERING
USING SYNCHROTRON RADIATION
by John S. Tse and Dennis D. Klug
O
INTRODUCTION
wing to advances in the production of high intensity,
coherence and highly monochromatized synchrotron radiaScattering techniques are important tools for research in
tion in recent decade, x-rays scattering has become a highly
material science and condensed matter physics. Through the
desirable technique for the investigation of the dynamical
observation of the distributions and fluctuations in space and
properties of condensed matter. Scattering using x-rays offers
time, the scattering of neutrons and photons can provide
several advantages over neutrons. In
unique insight into the arrangement
this report, the basic theoretical ideas
the dynamics of atoms in the conand recent dramatic developments in Scattering
using x-rays and
densed
(liquid and solid) state.
synchrotron-based inelastic x-ray scatoffers
several
advantages
tering are outlined. Potential applicaThe theoretical framework of using
tions of these techniques are illustrated over neutrons.
In this (neutron) scattering experiments to
through the study of the dynamics of
obtain dynamical information was preclathrate hydrates. The specific appli- report, the basic theoretical sented in the seminal theoretical work
cation of inelastic x-ray scattering for ideas and recent dramatic of van Hove [1] using a space-time corthe characterization of the phonons
relation function formalism. The full
and the role of the guest-host interac- developments in synchro- potential of this technique was fully
tions on the phonon band structure is tron-based inelastic x-ray exploited in the pioneering experiments
described. In addition, the application
of Brockhouse [2]. It was demonstrated
of nuclear resonant inelastic x-ray scat- scattering are outlined.
that the inelastic scattering of thermal
tering is used to reveal the large anharneutrons permit the measurement of
monic nature of the guest host interacsmall energy transfers associated with
tions in the Kr clathrate.
lattice vibrations. In the ensuing years, incoherent inelastic
neutron scattering (IINS) has developed into an indispensable
tool for the investigation of the dynamical properties of crystalline solids, glasses, polymers and liquids.
In contrast, the development of x-ray scattering as a probe of
atomic dynamics have been relatively dormant until the late
1980s. The energy width characteristic of x-rays generated
from conventional cathode tubes have a temporal lifetime of
ca. 0.1 fs which is very short as compare to atom motions in
the order of ps. Moreover, the energy of x-rays is much larger than the vibration energies and very high energy resolution, ca. 10-6 is needed in order to measure vibrational structures. Therefore, x-rays are often used for the determination
of static atomic positions although there were several
attempts to extract dynamical information from the analysis
of diffuse scattering [3]. In the last two decades, the availability of high intensity, spatially coherent and tunable x-rays
from synchrotron radiation has revitalized the interest of
using x-ray in scattering experiments. In particular, advances
in monochromator design have now making routine measurement in a few meV resolution possible [4].
The use of x-ray methods in place of neutrons offers several
practical advantages. Since intense x-rays can be focused into
Fig. 1
ω-Q) region accessible with difEnergy-momentum (ω
ferent probes of inelastic scattering (taken from
ref. 7).
John S. Tse <[email protected]>, Department of
Physics and Engineering Physics, University of
Saskatchewan, Saskatoon, Saskatchewan, Canada
S7N 5E2; Dennis D. Klug, Steacie Institute for Molecular
Sciences, National Research Council of Canada, Ottawa,
Ontario, Canada K1A 0R6
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very small spot size (e.g. microns) only a small volume of
sample is needed for the experiment. This is particularly critical for measurements under extreme pressure and temperature conditions. Excellent examples are recent measurement
of phonon dispersion of single crystals under high pressure
and the dynamical structure of levitated molten liquid
droplets [5]. Another distinct advantage is that x-ray scattering can access a wide range in energy and momentum which
is not accessible to neutrons with conventional triple axis
spectrometers. Figure 1 compares the accessible energymomentum regimes for a number of scattering techniques.
Optical methods, such as Brillouin and Raman scattering are
restricted to small momentum (< 10-3 Å-1) and intermediate
energy transfers (10-6 – 1 eV). Neutron scattering can be used
for momentum transfer larger that 10-2 Å. X-ray scattering
with synchrotron radiation largely overlap with neutrons but
become more accessible at the lower and higher energy and
momentum transfer regimes where neutron scattering is limited by kinematic restrictions.
In this article, the application of x-ray scattering will be illustrated with specific examples. We will focus on the investigation of the lattice dynamics of clathrate hydrates. It will be
demonstrated that valuable dynamical information on the
low energy lattice vibrations and localized motions of the
guests can be obtained from x-ray scattering experiments and
a new site specific nuclear resonant inelastic x-ray scattering
spectroscopy [6].
THEORETICAL AND EXPERIMENTAL
BACKGROUND
Non-resonant x-rays scattering [7]
A schematic of the geometry for a scattering experiment is
shown in Fig. 2. The incident beam has a well defined energy Ei, momentum qi and polarization ei. The scattered beam
at an angle 2θ within an angle element dΩ is
defined by the scattering wave vector qf, energy Ef
and polarization ef . Application of the conservation rules for energy and momentum transfer
gives
E = ω = Ei − E f
and
Q = (qi − q f )
d2σ
⎛ dσ ⎞
=⎜
⎟ S ( Q , ω)
dΩdω f ⎝ dΩ ⎠0
[2]
where (dσ/dΩ)0 is the intrinsic cross-section and in the spacetime correlation formalism
1
e iωt dt Ψ i
∫
2π
S ( Q , ω) =
∑e
− iQ ⋅rj ( t ) iQ ⋅rj ' ( 0 )
e
Ψi
[3]
j , j'
The function 〈…〉 in eqn. (3) describes the fluctuation or correlation of the scattering phase of particles at t=0 and at a different time t at a given state Ψi. Within the one-phonon approximation, eqn (3) can be simplified to
S(Q,ω) = G(Q,q,j)F(ω,T,q,j)
[4]
The dynamical structure factor = G(Q,q,j) is given by
G ( Q , q, j ) =
unitcell
∑
f d (Q )e −Wd [Q ⋅ ed (q, j )] M d−1eiQ⋅d
2
[5]
d
fd(Q) is the atomic form factor of atom d and ed(q,j) is the component of the normalized phonon eigenvector of mode j with
phonon wavevector q for atom d. e-Wd is the Debye-Waller
factor for atom d and Md is the mass. The response function
F(ω,T,q,j) for an undamped phonon is given by
F ( ω, T , q , j ) =
n + 1 /2 ± 1 /2
δ ω ∓ ωq , j
ωq , j
(
)
[6]
where the upper and lower signs are for x-ray energy loss and
energy gain, respectively. The essence of a scattering experiment is the direct measurement of S(Q,ω) which can be
accomplished with a spectrometer with the conventional
triple-axis design. A schematic drawing of a typical beamline
(ID 16 ESRF) is shown in Fig. 3. Typically, the phonon disper-
[1]
For x-rays, the energy of the incident photon is
much larger than the energy transferred (i.e. Ei 〉〉
E), then Q = 2 qi sin θ
The double differential cross section is related to
the scattering function S(Q,ω) by
Fig. 2
306
Geometry for neutron and x-ray scattering.
PHYSICS
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Fig. 3
A schematic drawing of the conventional triple-axis spectrometer
and the layout of a synchrotron x-ray inelastic scattering beamline.
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ARTICLE DE FOND ( PHONON SPECTROSCOPY ... )
sion of an oriented single crystal can be obtained from constant energy scans.
For polycrystalline samples where there is no distinct orientation of the sample with respect to the direction of the incident beam and, as a result of orientation averaging, only the
absolute value of the momentum transfer is well-defined.
However, in order to obtain a dispersion relation of a material, it is necessary to measure the frequency as a function of a
distinct wave vector. In polycrystalline samples, only orientationally averaged longitudinal phonon modes can be measured in the first Brillouin zone. Transverse phonons may
appear in the spectra of the extended zones [7].
Nuclear resonant x-ray inelastic scattering
Nuclear resonant x-ray inelastic scattering (NRIXS) is a new
technique for lattice dynamics measurements [6-8] method,
applicable to specific nuclei with non-zero nuclear moment
analogous to Mossbauer spectroscopy , can provide information on the vibrational density of states. The basic principle
is to tune synchrotron x-ray to match the excitation energy of
a nuclear transition. For a clamped non-vibrating nucleus,
the emitted radiation is very close to recoil free and therefore
has the same energy as the transition between the nuclear
ground and excited state. The emitted photon may then
interact with other nuclei in the crystalline solid and shift the
“recoil” energy (vibration coupled). The measurement of the
resulting time delayed de-excitations contains information
on the vibrations of the nucleus embedded in an Einstein
solid.
The nuclear resonant interaction cross section σ(Q,ω) is related to the nuclear resonance cross section σ0 and scattering
function S(Q,ω) defined above but in this case, can be considered as the probability density for phonon excitation [6]
σ ( Q , ω) = σ 0 ΓS ( Q , ω)
[7]
In effect, apart from a sharp resonance, the NRIXS spectrum
is a convolution with the vibration profile. Within the harmonic approximation, the vibrational density of states of the
nucleus can be obtained [8]. A schematic diagram illustrating
the principle of NRIXS for 83Kr (I=9/2) is shown in Fig. 4.
Since the linewidth of the exited nuclear state is extremely
narrow (3.1 ns for 83Kr), monochromator with very high
energy resolution (ca. meV bandwidth) is required. These
monochromators are often constructed for multiple reflections from high index planes of single crystals. Therefore, the
availability of monochromatic coherence from an intense
incident x-ray source is a prerequisite for the experiment.
The experimental setup for the measurements of nuclear
inelastic scattering is illustrated in Fig. 5.
ILLUSTRATIVE EXAMPLES
In recent years, inelastic scattering employing synchrotron xray has been used to study the phonon dispersion of single
crystals, amorphous solids and liquids. An excellent review
of the state-of-the-art can be found in [7]. We will focus on a
specific topic in this article, namely, the characterization of
the lattice vibrations and low energy guest motions in
clathrate hydrates.
Clathrate hydrates are non-stoichiometric crystalline inclusion compounds with water molecules forming a threedimensional network (host) where small molecules or rare
gas atoms (guests) can be encaged in the empty voids [9].
Fig. 4
Schematic illustration of the principle for
nuclear resonant inelastic scattering (adapted
from ref. 6). (a) The energy level diagram
ω) (resoand the absorption spectrum S(ω
nance) for 83Kr nuclear excitation. (b) The
convolution of the resonance absorption
with vibrational levels. The dashed arrows
indicate the creation and annihilation of
phonons. The dotted arrows indicate multiple-phonon processes.
Fig. 5
Schematic layout of a typical nuclear resonant inelastic scattering beamline. The detector measures the time-delayed fluorescence after nuclear excitation. The intensity of inelastic
scattering can be converted into vibrational density of states.
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FEATURE ARTICLE ( PHONON SPECTROSCOPY ... )
tron scattering. Theoretical lattice
dynamics calculations on a structure I Xe hydrate have shown that
these avoided or anti crossings
occur at very low frequencies (ω)
and at small momentum transfer
(Q) (Fig. 6b). The use of neutrons
to determine the collective excitations is virtually ruled out due to
restrictions in Q-ω space at Q values within the first Brillouin zone,
and due to the incoherent contribution of the hydrogen atoms to the
spectrum. On the other hand, xrays can access the small Q values
needed and provide the necessary
energy resolution. Furthermore,
since polycrystalline samples were
to be used in the experiment, an
unambiguous assignment of the
observed modes is only possible in
the first Brillouin zone: due to the
selection rules in the one-phonon
approximation only the longitudiFig. 6 The three most common structures (type I, cubic Pm3n, type 2, cubic, Fd3m and
nal modes can be observed in the
type H, hexagonal, P6/mmm) of clathrate hydrate stable under ambient pressure(a) first Brillouin zone. Momentum
and (b) calculated phonon dispersion curve along the [111] direction for a Xe
transfers beyond the first Brillouin
hydrate. Note the anti-crossing of the “localized” guest motions with the acoustic
zone will excite several longitudibranches predicted at Q=0.02 and 0.1.
nal and transverse phonons at
once, thus giving information on
Three structures have been observed to be stable under ambithe density of states of these modes. For a cubic system, such
ent pressure (see Fig. 6a). In general, for very small or large
as the low-pressure phase of clathrates, the crystal acoustic
guests, the type II Fd3m structure is adopted. Another very
vibrations transform as T1u symmetry at the Brillouin zone
common structure is the type I Pm3n cubic structure [9]. Even
center. Factor group analysis [13] has shown that guest vibrathough gas hydrates have a well-defined crystalline structions also contain T1u symmetry modes at the zone center. As
ture, the temperature dependence of the thermal conductivithese largely non-dispersive (localized) modes move away
ty is quite complicated but resembles that of a glass [10,11].
from the zone center, avoided crossings with strongly disperFurthermore, the magnitude of the thermal conductivity of
sive acoustic branches having similar symmetry are expected
different hydrates is only weakly dependent on the nature of
to occur within the Brillouin zone.
the guests. Several models have been proposed to account
for this novel property. There is a consensus regarding the
X-rays scattering experiments were performed at the beam
resonant scattering model [10,12] that attributes the scattering
line ID28 at the European Synchrotron Radiation Facility in
of the thermal phonons to the local excitation of the guests.
Grenoble [14]. By using the silicon [11,11,11] reflection order
Central to the resonant scattering model is the suggestion of
for both the monochromator and the analyzer an overall enersymmetry avoided crossings between acoustic lattice
gy resolution at full width at half maximum of 1.5 meV at
phonons and “localized” guest branches [12]. The existence of
21.747 keV was achieved. Inelastic scans were then recorded
low energy “localized” guest vibrations has been confirmed
at 100 K in the energy region of 220–20 meV with momentum
by several quasi-elastic incoherent inelastic neutron scattertransfers of 1.5 nm-1 〈 ⎢Q ⎢〈11.0 nm-1. These momentum transing (INS) experiments [13]. However, there was no direct evifers were chosen to cover and extend the measurements
dence confirming the crossing of the phonon branches. The
beyond the first Brillouin zone. The size of the first Brillouin
understanding of the nature of the thermal conductivity
zone, determined from the d spacing of the first allowed [110]
where localized oscillators are present is important in a wide
Bragg peak, is ⎢Qmin⎢=3.8 nm-1. A selection of inelastic x-ray
range of materials. For example, efficient thermoelectric
spectra (IXS) of methane hydrate at different momentum
materials will require low thermal conductivities as one of
transfers are shown in Fig. 7.
their essential properties and this will often be determined by
the ‘rattling’ components giving rise to localized vibrations of
The spectra display a well defined dispersive mode. The
the material.
position of the dispersive mode is found to scale linearly with
the momentum transfer in the ⎢Q ⎢range 〈6.0 nm-1 (Fig. 7).
Since most simple clathrate hydrates, such as methane and
This dispersive excitation can be identified with the longiturare-gas hydrates commonly only exist in polycrystalline
dinal acoustic (LA) host-lattice phonon branch. From the
form, it is difficult to grow a large enough single crystal for
slope of the dispersion of the longitudinal acoustic mode, an
the measurement of the phonon dispersion curve using neuorientationally averaged sound velocity of 3950±50 m/s can
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sion, ranging from 6.9±0.2 meV at
⎢Q ⎢=8.0 nm-1 to 9.5±0.3 meV at ⎢Q ⎢=
11.0 nm-1. The spectra in the momentum
transfers beyond the first Brillouin zone,
( ⎢Q ⎢= 5 to 11 nm-1) reflect excitations of
both transverse and longitudinal phonons
and expected to reflect features of the density of states (DOS) of methane hydrate.
This is illustrated in Fig. 8 by comparing
the INS spectrum of methane hydrate in
deuterated water with the IXS spectra
obtained at high momentum transfers.
Fig. 7
Comparison of experimental x-ray inelastic scattering spectra of methane
hydrate with momentum transfers within (a) the first and (b) beyond the
first Brillouin zone (b) with the theoretical spectra computed from lattice
dynamics calculations (c).
be deduced. At ⎢Q ⎢≈3 nm--1, a second non-dispersive peak
at 5 meV becomes visible. Both the position and intensity of
this peak do not show strong variation with increasing
momentum transfer. A plot of the dispersion relation of the
dispersive lattice mode and the nondispersive guest modes is shown in
Fig 8. In comparison to previous
INS spectra, this feature in the IXS is
assigned to the guest vibrations
inside the large cage. It is important
to note that these vibrations appear
in the IXS spectrum after the intersection with the LA lattice mode at
⎢Q ⎢≈2.5 nm-1, which is well within
the first Brillouin zone. This behavior is pointing towards a coupling
between the localized guest vibrations and the acoustic host lattice
modes.
Additional information can be
obtained from spectra at higher
momentum transfers (Fig. 7). Due
to the better contrast, the LA mode
moves out of the energy window, Fig. 8
additional broad features can be
observed at around 7–10 meV. It is
found that the energy position of
this peak displays a slight disper-
The assignment of the IXS is in complete
agreement with the calculated orientationally average scattering function S(Q,ω)
using eqns. 4-6 (Fig. 7). The dynamical
structure factor G(Q,q,j) was calculated
from the eigenvectors generated from a lattice dynamics simulation and averaged
over 239 different directions within the
⎢Q ⎢ range of 0 – 10 nm-1.. The phonon
wave vector q was chosen such that q=GQ, where G was an appropriate lattice vector for Q values beyond the first Brillouin
zone. Thus, the intensities of both the longitudinal and the transverse modes are calculated.
The calculated Q-averaged
phonon-dispersion is also in substantial
agreement with experiment (Fig. 8).
Methane hydrate transforms to two high
pressure phases MH-II and MH-III at moderate pressure of 1 and 2 GPa, respectively [15]. High-energy
resolution inelastic x-ray scattering and diamond anvil cell
(DAC) techniques can be used to determine the orientationally averaged compressional sound velocities. Combining
A comparison of experimental (left side) and calculated (right side) averaged
longitudinal acoustic phonon dispersion curves of methane hydrate. Note the
observed and predicted anti-crossing at Q~2nm-1. The circles and triangles on
the experimental curves are points on the acoustic branch and the localized guest
vibrations, respectively.
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FEATURE ARTICLE ( PHONON SPECTROSCOPY ... )
anvil cell. As the sample is composed of 35% MH-III and 65% ice
VI, the peak at 8.9 meV is assigned
to the LA phonon branches of MHIII and ice VI. At ⎢Q ⎢≥ 6.5 nm-1 an
additional excitation can be
observed at ca. 12 meV. In the
observed ⎢Q ⎢ range this mode displays only a very weak dispersion
and is assumed to have a constant
energy 11.7±0.2 meV within the
experimental error. At the highest
displayed
wave
vector
⎢Q ⎢=8.9 nm-1 a shoulder at
≈18 meV can be observed. Taking
the relatively high momentum
transfer into account, it is reasonable to assume that both are related to transverse phonons in either
ice VI or MH-III.
With the successful separation of
the methane hydrate from the ice
contributions, the phonon dispersion curves for MH-II and ice VI at
17 and 21 kbar, and for MH-III at
21 kbar was obtained. The longiFig. 9 Experimental IXS spectra for high pressure phases of methane hydrate. (a) MH-II
tudinal, or compressional, velociand (b) MH-III. The arrows indicate highly dispersive excitations from the diaties of sound can be derived by fitmond anvil cell.
ting a sine function to the dispersion curves and determining the
these findings with the results from diffraction experiments,
slopes in the ⎢Q ⎢6 0 limit. An orientationally averaged
the elastic properties of the high-pressure phases of methane
compressional sound velocity of ice VI=4700±100 m/s was
hydrate can be derived [16].
determined for ice VI at T=298 K and 17 kbar. For MH-II, an
orientationally averaged compressional sound velocity of
A selection of inelastic x-ray spectra of the MH-II and MH-III
4200±100 m/s was deduced from the phonon dispersion
samples at different wave vector transfers, or ⎢Q ⎢, between
curve. For the MH-III sample, the orientationally averaged
-1
2.0 and 8.9 nm are shown in Figs. 9a and 9b, respectively.
compressional sound velocities at 21 kbar and T=298 K for ice
The IXS data represent the orientationally averaged longituVI are 4950±100 m/s and 4600±100 m/s for methane hydrate,
dinal acoustic dispersion. In the case of the MH-II, the enerMH-III. For a randomly oriented, non-textured powder the
gy positions of the inelastic excitations were determined to be
compressional velocity of sound, which is the same as the ori-1
E=5.9 meV,15.8 meV, and 22.5 meV at ⎢Q ⎢=2.0 nm . The
entationally averaged longitudinal sound velocity, is defined
two high-intensity peaks at E=15.8 meV and E=22.5 meV disas ν p = C / ρ where C is a combination of the elastic conplay strong dispersions and cannot be observed at wave vecstants (effective elastic modulus) and ρ is the density
-1
tor transfers ⎢Q ⎢〉5 nm . From the ⎢Q ⎢ dependence of the
of the material. On the other hand, the compressional
energy positions of these two excitations, sound velocities of
and shear sound velocities can as well be expressed by
≈12 000 m/s and ≈17 000 m/s were determined. These peaks
4
are assigned to the transverse and longitudinal acoustic
ν p = ( 1 /ρ ) ⎛⎜ B + G ⎞⎟ , and ν s = G / ρ where B is the
3 ⎠
⎝
phonons of the diamond anvils. The third dispersive excitais the bulk modulus and G is the shear modulus. The pressure
tion is assigned to the orientationally averaged LA phonon
evolution of B and G were determined separately from x-ray
branches of the sample. As the sample is composed of
diffraction measurements [17]. The bulk modulus and density
61% MH-II and 39% (mole) ice VI, this dispersive excitation
-1
are B(MH-II)=14.4 GPa, r(MH-II)=1.07 g cm-3 and B(MHat 5.9 meV, at ⎢Q ⎢=2.0 nm has contributions from the
III)=23.6
GPa, ρ(MH-III)=1.16 g cm-3. Therefore, for MH-II,
LA phonon branches of both ice VI and MH-II. For
-1
C
=18.9±0.8
GPa, G=3.4±0.6 GPa, and vs =1800±150 m/s and
⎢Q ⎢≥ 6.9 nm an additional non-dispersive excitation is
for
MH-III,
C=24.5±1.0 GPa, G=0.8±0.7 GPa, and
observed at 11.9±0.2 meV and is tentatively assigned to the
vs =800±400 m/s.
TA phonons of either ice VI or MH-II.
In the case of the MH-III at ⎢Q ⎢=2.8 nm-1 three inelastic excitations at E = 8.9, 22.2 and 31.8 meV can be observed. As in
the case of MH-II, the two intense excitations at higher energies are assigned to the acoustic phonons of the diamond
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PHYSICS
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The IXS experiments successfully demonstrated the occurrence of symmetry avoided crossing between the acoustic
phonon branches of the lattice vibrations and the “localized”
motions of the guest. However, the effects of the resonant
scattering cannot be characterized. To resolve this problem a
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Fig. 10 Experimental NRIX spectra (a) and the “harmonic phonon density of states” (b) derived from the corresponding experimental spectra for enclathrated 83Kr in the clathrate hydrate.
site-specific method sensitive only to the guest motions is
required in order to provide an unambiguous characterization of the consequence of the guest–host interactions. The
NRIXS technique can be used to characterize the dynamics of
the guest atoms if it has low-lying nuclear levels that can be
excited by synchrotron radiation. This is the case for 83Kr,
which has a nuclear level of 9.4 keV so that nuclear resonance
can be excited using synchrotron radiation sources.
Therefore, for the Kr clathrate hydrate, nuclear resonant
inelastic scattering is sensitive only to the 83Kr guest atoms in
the clathrate cages and the ice lattice forming the cages is
effectively invisible. Because of this unique property, a
clathrate hydrate of Kr is studied. It is possible to give a
detailed characterization of the localized vibrations of Kr
atoms
The NRIXS measurements were performed at sector 3-ID at
the Advanced Photon Source [18]. A four-silicon-crystal highresolution monochromator based on a weak-link structure
with 1 meV energy bandwidth was used in the experiment.
The high resolution and high throughput of this high-resolution monochromator make it possible to measure the lowenergy phonon modes of 83Kr in the clathrate at its natural
abundance (11.5%) of Kr. The experimental NRIXS spectra
S(ω) (eqn. 7) of the Kr clathrate hydrate at 25, 63 and 158 K
are shown in Fig. 10. The positive energy is the energy gain
(phonon creation) and the negative is the energy loss
(phonon annihilation) spectra.
To extract information concerning the Kr vibration density of
states (VDOS), the observed spectra were subjected to analysis based on harmonic approximation. This procedure has
been used successfully for the determination of the VDOS for
a variety of materials under ambient and extreme conditions.
For example the Kr VDOS of solid Kr under high pressure
obtained from NRIXS experiments are in excellent agreement
with theoretical calculations. In the present case, following
the same procedure, the extracted Kr VDOS yields an unphysical negative density of states (Fig. 10b) at 6–8 meV at the two
lowest temperatures of 25 and 63 K. The failure of the harmonic model indicates a complete breakdown of the harmonic
approximation. Therefore, the vibrations of Kr are clearly
intrinsically anharmonic.
The observable in a NRIXS experiment is related to the
Fourier transform of the self-intermediate scattering function,
L(k0 ,t), where k0 is the incident radiation wave vector and t is
time. For systems with large anharmonicity, the harmonic
assumption is no longer valid and the calculation of L(k0 ,t)
from molecular-dynamics simulations is necessary for a
direct comparison with experiment [6,18]
L(k 0 , t ) =
1
N
∑e
ik ⋅( ri ( t )− ri ( 0 ))
= e ik0 ⋅ri ( t ) e − ik0 ⋅ri ( 0 )
[8]
i
The sum is taken over the N particles in the simulation box
and ri(t) is the position of the ith atom at time t and is obtained
from the trajectory of a molecular-dynamics simulation.
The calculated S(ω) are convolved with the experimental resolution and compared with the background subtracted
NRIXS spectra in Fig. 11a and b. The low-energy feature
(~1–1.5 meV) in the calculated S(ω) is associated with the
vibrations of Kr in the large cages. Owing to limited instrumental resolution, this peak is overwhelmed by the very
strong elastic (zero-energy) peak and cannot be accurately
extracted from the experimental data. Apart from this shortcoming, the higher-energy peak predicted at ~4.2 meV is
clearly observed in the experiment. This peak is mainly
attributed to the localized vibrations of Kr in the small cages.
Molecular-dynamics calculations also reveal several weak
features at ~2 meV and at ~2.8 meV. These features are also
discernible from the NRIXS spectra.
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FEATURE ARTICLE ( PHONON SPECTROSCOPY ... )
ω) for a Kr clathrate. Note the
Fig. 11 A comparison of experimental (a) and theoretical (b) functions S(ω
apparent absence of the predicted peak at ~0.8 meV in the experimental spectra is due to errors in the
background removal of the large central elastic peak.
SUMMARY
In this short article, some applications of inelastic x-ray techniques using synchrotron radiation are illustrated through a
specific example on the characterization of guest-host interaction in clathrate hydrates. It is shown that IXS experiments
provide complementary information to neutron scattering on
the phonon spectra and help to establishes that the avoidedcrossing of the “localized’ guest vibrations with the acoustic
phonon branches of the host lattice with the same symmetry
is responsible for the guest–host interactions. In combination
with site specific NRIXS measurements and theoretical calculations, the unexpected anharmonic nature of the guest vibrations is clearly demonstrated. This large anharmonicity is the
cause of the very low thermal conductivity of the clathrate
hydrates and is a direct consequence of the coupling between
the host and the guest vibration.
3.
4.
5.
6.
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SYNCHROTRON ADVANCES AT THE FRONTIERS OF FOOD
PHYSICS: STUDIES OF EDIBLE FATS SUCH AS CHOCOLATE
UNDER SHEAR
by G. Mazzanti, S.E. Guthrie, A.G. Marangoni and S.H.J. Idziak
C
hocolate and other lipid food materials can form different
ural fats. This peculiar behaviour is very important for the
crystalline structures (polymorphs) depending on the temchocolate industry, yet, despite great advances done by modperature and shear profiles used during their manufacturing.
ern science, many of the reasons for its behaviour are unclear.
These structures determine the quality of the products. Until
Of particular interest is the number of distinct structural
recently, only very empirical knowledge existed on the effects
phases that it forms and the effect that shear flow has on the
of shear. The use of synchrotron x-ray diffraction has allowed
rate and paths of transformation between these different crysfor pioneering advances in our undertalline phases. This is only one example
standing of the mechanisms of struc- The use of synchrotron x- of structure in foodstuffs, a field that
ture formation during the crystallizing
offers very exciting and complex chalof edible multicomponent lipid sys- ray diffraction has allowed lenges in soft condensed matter
tems under shear.
as has been recently highlightfor pioneering advances in physics,
ed in very prestigious publications [1,2].
INTRODUCTION
our understanding of the The food structures result from the
combination of many different compoWhen the western botanists discov- mechanisms of structure
nents in a variety of phases related
ered the cacao trees in the new world,
through several time and length scales.
formation
during
the
crysthey named them “Theobroma”, the
Common food items such as butter,
food of the gods, after the name given tallizing of edible multicommargarines, lard, shortenings, chocolate
by the Mayas. The mystery that still
and spreads rely on microstructured fat
surrounds that ancient civilization ponent lipid systems under
networks to provide excellent mouthsomehow remains also in the very shear.
feel, sensorial properties and physical
peculiar crystalline structure and
attributes [3-6]. The levels of these strucphase behaviour that cacao butter, the
tures are illustrated in Figure 1.
main structural component of chocolate, displays among natThough it may not be obvious to many people who consume
these foods, natural fats form real crystals. These crystals
form clusters, which grow into larger aggregates called flocs,
which eventually produce the fractal solid network that we
can see, spread and chew. The characteristics of the
microstructures depend largely on the type of crystal formed
by the molecular composition of the fat (i.e. lipids) [7] as well
as their crystalline size and spatial distribution [8].
Fig. 1
Schematic representation of different levels of structure in a bulk fat. The crystallite may have one or
more domains of a thickness ξ, composed in turn of
several lamellae of thickness d. Each lamella is
formed by individual fat molecules called triacylglycerides (TAGs) organized with a characteristic
longitudinal stacking and lateral packing.
The type of crystals formed depends on the chemical composition of the material. The principal lipid components of most
edible fats are triacylglycerols (TAG), resulting from the esterification of the long hydrocarbon chain fatty acids upon a
glycerol backbone. Natural fats contain a broad variety of
fatty acids, depending on the kind of fat, its geographic origin
and the techniques used to extract and purify the fat. Thus,
they are multicomponent systems, often with several thousand types of molecules. In addition, minor components,
mostly polar lipids, act as impurities that alter the phase
behaviour of the fats [9]. Some of these minor components,
commonly referred to as surfactants and emulsifiers, are often
employed as natural additives to improve the characteristics
of the materials.
G. Mazzanti[1], S.E. Guthrie[2], A.G. Marangoni[3], S.F.J. Idziak[2],
([email protected]), [1] Dalhousie University, Halifax N.S.,
[2] University of Waterloo, Waterloo, ON, [3] University of Guelph,
Guelph, ON.
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Fig. 2
Cartoon of two main types of longitudinal stacking of
TAGs molecules, corresponding to (a) two (2L) or (b)
three (3L) fatty acid lengths for tripalmitin. The dark
atoms represent carbon, the light ones oxygen. The
resulting lamellae have very different d-spacings.
The complexity is increased by the fact that the same fat can
crystallize in different classes of crystals. This peculiarity of
some materials (that they can be crystallized with different
spatial arrangements of the molecules in the crystals) has
been called polymorphism, while the types of crystals are
referred to as polymorphs, or polymorphic forms. The natural fat systems display a rich and complex crystalline polymorphism, strongly dependent on heat, mass and momentum transfer conditions during crystallization.
TAG molecules in a fat crystal. These structures, and their
crystallization behaviour, bear strong resemblance to those of
alkanes [10-12], since the long hydrocarbon chains are the
dominant portion of the molecules. The metastable α form,
analogous to the rotator phase in alkanes [10], is associated
with a hexagonal unit cell as shown in Figure 3a, where a segment of the hydrocarbon chain is shown represented by two
consecutive carbon atoms (black dots) and their corresponding hydrogen atoms (white dots). The β’ polymorphic
arrangement is consistent with an orthorhombic unit
cell [13,14] as described schematically in Figure 3b, while the β
polymorph has a triclinic unit cell [13], as illustrated in
Figure 3c. Thus knowledge of the lateral packing (e.g. α, β’,
β) and the longitudinal stacking (e.g. 2L) is required to fully
describe the crystalline structure. Margarines and shortenings contain usually a mixture of β’ and β crystals. The crystal size and type is crucial for the structure of margarines,
because an excess of β crystals results in a sandy texture and
poor spread and melting characteristics. The polymorphic
forms for cocoa butter are more complicated, and they have
been traditionally numbered I to VI in roman numerals. The
ideal crystal type for chocolate is in a type of β form known
as form V (or βV) [15,16]. This polymorph is quite stable during storage, and melts in the mouth but not too fast at the
touch of the hand.
For all fats in general, the less stable polymorphs have a
lower free energy barrier of formation, thus can be formed
more readily than the more stable polymorphs [7], if the cooling rate is fast enough. Thus, very often to obtain a given
polymorph one has to crystallize first one of the more unstable (and often undesirable) ones, which then has to be eliminated. This means that the state diagrams of these materials
are time dependent [17], and give rise to processes such as
chocolate tempering. This peculiar characteristic makes the
phase landscape of natural multicomponent lipids extremely
rich and complex, and makes the determination of an equilibrium phase diagram extremely difficult, if not impossible.
Instead, state diagrams are used in lieu of phase diagrams
and do not give the equilibrium behaviour, but rather give
To better understand how this polymorphism happens, we
need to look at the way the molecules are arranged in the
crystals. The TAG molecules
form lamellar structures by
stacking pairs of molecules in
the longitudinal direction,
usually of two (2L) or three
(3L) fatty acid lengths as
shown in Figure 2. This produces crystals that form disclike structures with molecules
arranged perpendicular to the
flat surface. The more common polymorphic forms of
natural fats, analogous to
those formed by pure TAGs,
are usually termed α, β’ and β
in order of their increasing
melting point, packing density
and thermodynamic stability.
This polymorphism is a conse- Fig. 3 Top view of the lateral packing of the fatty acid chains, showing the three characterisquence of the variety of
tic sub-cells: (a) the hexagonal phase α, (b) the orthorhombic phase β’ and (c) and the
arrangements of lateral packtriclinic phase β.
ing of the CH2 groups of the
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smaller repeating spacings of the lateral packing (Figure 3)
produce wide angle x-ray reflections, while the longitudinal
long spacings (Figure 2) produce x-ray reflections in the small
angle region. These small angle reflections provide information on the longitudinal packing of the TAG molecules in the
crystals, i.e. whether they are in the 2L or 3L arrangement.
Very few exact structures have been completely elucidated to
date, but the characteristic signatures of the different polymorphs have been widely documented [19,21].
As rich as the state diagram in Figure 4 may appear, it was
developed under static conditions (no shear applied) and at a
fast cooling rate. However, both cooling rates and flow conditions greatly affect the phase behaviour of natural fats, by
drastically modifying the initial steps of the crystallization
process. These steps depend on the dynamics of structure
formation under different temperature and shear treatments [22,23]. Therefore, to truly understand how these structures are formed, it is necessary to study their development in
time with enough detail.
Fig 4
Time-Temperature state diagram of cocoa butter
cooled very rapidly (Reprinted with permission
from [8], © 2003 American Chemical Society).
phase behaviour along a given timeframe, as illustrated in
Figure 4 for cocoa butter [8,18]. In this paper we will continue
to refer to the individual structures as phases, or forms, as is
conventional in this area of study.
The reader can perform a simple experiment to examine this
phase behaviour by melting some chocolate and pouring
three large drops onto a plate. Upon hardening, the drops
will have a waxy, unpleasant mouth feel and will tend to
melt quite profusely in the hand as well as the mouth (phase
α). If the other drops are left
on the plate at room temperature for several days, they
will develop into the glossy
phase β(V) which tends to
melt less in your hands. After
several months of poor storage on the plate, with temperature fluctuations, the third
drop of chocolate will start to
bloom, or form a white powdery texture on the surface.
The represents the undesirable transition into phase
β(VI).
Melting point determinations
are often used to classify
these polymorphic forms,
since the less stable a polymorph is, the lower its melting point. However, these Fig. 5
structures can only be identified, characterized and differentiated in an unambiguous
manner by their distinctive xray diffraction patterns. The
Synchrotron x-ray diffraction (XRD) allows probing the crystalline structures directly, in-situ and in real time, thus providing irreplaceable information necessary to understand the
genesis and dynamics of structures that depend on the polymorphic state of the material [23]. To study the formation of
these structures we need to capture x-ray diffraction patterns
during the crystallization process, and this is only possible
with the intensity provided by a synchrotron source. The
experiments start from the liquid at high temperature and
progress as the material is observed during the cooldown
and, over time, at the final crystallization temperature.
The use of synchrotron radiation has permitted the development of time resolved studies in the crystallization of fats,
(a) The Couette cell consists of two concentric cylinders made of 0.5mm thick Lexan
walls. The internal cylinder is stationary and has water flowing inside for temperature
control. The outer cylinder rotates. (b) The synchrotron x-rays traverse the sample in a
tangential manner before striking a two dimensional CCD x-ray detector. (c) The
shear profile developed by the velocity gradient between the moving and stationary
cylinders is given by γA = u / δ. (Adapted with permission from [22], © 2003 American
Chemical Society).
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which has opened a new era in the understanding of these
processes. The pioneering static experiments [24] have
evolved to include combined techniques such as DSC [25] and
more recently the application of shear [22,26]. In our research
team this has been accomplished using a Couette shear cell.
Although the challenges due to the geometry of the cell and
the variable conditions of the experiments are difficult, the
results have been extremely rewarding so far, placing our
Canadian group at the world forefront of this line of research.
In this paper, we review our recent work studying the effects
of shear on the crystallization dynamics of fat. Careful analysis of the x-ray diffraction patterns has led us to the discovery
of new phases and transition mechanisms in cocoa butter [27],
and to the mathematical modeling, for the first time, of the
combined phase transition in palm oil and milk fat [9,28] as
well as a description of orientation in fats.
X-RAY SHEAR CELLS
A custom fabricated x-ray compatible Couette shear cell was
used to perform shear measurements in order to develop relatively high shear rates. This cell consists of two concentric
Lexan cylinders, the outer of which rotates at a controlled
rate while the inner remains stationary as shown in Figure 5a.
A specially designed temperature controlled water system
was developed to provide very good temperature control as
well as the carefully controlled temperature ramps necessary
while cooling the fat. The walls of the Lexan cylinders are
sufficiently thin (and relatively high energy 1.1 Å wavelength
x-ray were used) to allow x-ray to pass through and scatter
from the sample as shown in Figure 5b. Shear rates (u/δ in
Figure 5c) up to 2880s-1 were attainable with this system [22].
All measurements reviewed here were conducted at the
ExxonMobil beamline X10A at the National Synchrotron
Light Source in Brookhaven National Laboratory, Upton, NY,
USA. A Bruker 1500 two-dimensional (2D) CCD detector
was used to capture diffraction patterns. One important
advantage of using a 2D detector is the possibility of studying orientation effects which can be seen by the anisotropic
Fig. 6
316
Two-dimensional diffraction patterns of cocoa butter’s phase II (a) crystallized at 17.5 °C. (a) Static
crystallization (b) Under a shear rate of 1440s-1. The
diffraction ring occurs at a scattering angle
θ = 1.27 deg, or q = 0.127 Å-1, which corresponds to a
2θ
d-spacing of 49.47 Å indicating that the molecules are
arranged in a 2L conformation in the layers.
PHYSICS
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scattering illustrated in igure 6b, indicating orientation, as
opposed to the uniform scattering seen in Figure 6a, indicative of scattering by an unoriented, powder-like material.
Two dimensional x-ray images are useful for quickly evaluating the qualitative behavior of the fat. For more detailed
quantitative analysis, the x-ray diffraction intensity from
each 2D diffraction image was radially averaged and plotted
as a function of the reciprocal lattice spacing q, where,
2 π 4π
q=
=
sin θ , d is the interplanar spacing, and 2θ is the
d
λ
Bragg scattering angle as seen in Figure 5b. This radial averaging is accomplished by circular integration of the x-ray
intensity at each value of the radius. Also, azimuthal (or
mosaic) plots were derived from the 2D diffraction patterns,
by plotting the intensity measured along the circumference
(χ angle) of the diffraction rings at a fixed radius. This
allowed for the study of the crystalline orientation in hardening fats [22].
Time dependent crystallization studies can yield tens of
thousands of two dimensional diffraction patterns, making
data analysis a complex task. We designed a custom plug-in
program for the ImageJ software, typically used by the optical microscopy community, which was used to perform radial averaging as well as the other data transformations done
on the 2D images [22].
CRYSTALLINE ORIENTATION
Figure 6a shows a typical small angle diffraction pattern of
cocoa butter’s phase II (or α) crystallized statically, while
Figure 6b shows the same phase crystallized under a shear
rate of 1440s-1. The anisotropy of the scattering intensity
around the ring clearly indicates crystallite orientation in the
sheared sample, whereas the sample crystallized statically
presents uniform intensity around the diffraction ring. The
orientation of the particles is also evident from the x-ray scattering at small angles seen near the beamstop. Azimuthal
Fig. 7
September / October 2006
Azimuthal intensity profiles from phase a of palm oil
and cocoa butter, crystallized under a shear rate of
1440s-1, as a function of the azimuthal angle around
the diffraction ring indicating the increased orientational ordering seen in cocoa butter.
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formation. The initial nucleation is characterized by the
appearance of nuclei far apart from each other. These
platelet-like nuclei [29] do not interact with each other. As
they start to grow, the system becomes a disperse suspension
of rapidly growing crystals. Eventually the crystals start
forming clusters as they impinge upon each other. This produces aggregates crystallized under static conditions at a fast
cooling rate, which are not spherulites, but rather clusters of
randomly oriented crystallites as seen in Figure 1.
Fig. 8
Two-dimensional diffraction patterns of cocoa butter’s phase V (a β polymorph) crystallized at 17.5 °C
under a shear rate of 360s-1. (a) Pure cocoa butter.
(b) Commercially available dark chocolate. (Adapted
with permission from [22], © 2003 American
Chemical Society).
plots derived from the 2D images are shown in Figure 7 for
palm oil and cocoa butter under a shear rate of 1440s-1. As
can be seen, the degree of orientation, as determined from the
width of the peak seen in the azimuthal scan (narrower is better oriented), is dependent on each particular material. All
four fat systems studied displayed orientational ordering
under shear, strongly suggesting that the shear-induced orientation effect is universal to all fats. In most cases, this orientation effects were evident from the onset of crystallization
but did depend on shear rate [22].
This orientation was also observed in real chocolate, and not
only in pure cocoa butter. Figure 8a shows a typical diffraction pattern for cocoa butter crystallized under shear after the
phase transition to phase β(V). A commercially available
dark chocolate cooled under a shear rate of 360 s-1 produced
the diffraction pattern shown in
Figure 8b demonstrating that the
orientation effect remains after the
transition to phase β(V). Recall that
phase β(V) gives chocolate its ideal
sensory attributes. Current studies
indicate that this preferred orientation remains for long periods of
time. Also seen in Figure 8b is the
greatly enhanced smallangle scattering at the center of the chocolate
scattering pattern which is due to
the large amount of fine particulate
matter present in real chocolate,
such as cocoa powder and sugar.
To understand how the orientation
of fat crystallites happens, the different stages of the crystallization Fig. 9
process of these multicomponent
lipid materials can be described.
During its crystallization from the
liquid state, the fat material goes
through different steps, beginning
with nucleation, and followed by
unhindered growth and structure
The presence of a moderate shear field hinders the formation
of these clusters, but the exact response depends on the composition of the system and processing conditions, as illustrated by the different response of palm oil and cocoa butter in
Figure 7. The orientation of suspended particles in a flowing
system depends on the interaction of shear, inter-particle, and
Brownian forces [30]. In low volume fraction particle suspensions, like the ones present at nucleation and the early stages
of crystallization, inter-particle forces are negligible.
Therefore, the distribution of particle orientations results
from the interplay between ordering induced by shear forces
and disordering induced by Brownian forces. If shear forces
are prevalent, a particle rotates slowly when nearly parallel to
the direction of flow, and much faster when perpendicular to
it, resulting in a time-averaged distribution preferentially parallel to the direction of flow. Triacylglycerol systems tend to
crystallize initially forming small disc-like platelets [29], so it is
reasonable to assume that the nuclei are platelet-like shaped.
The time dependent x-ray diffraction pattern analysis is therefore consistent with the formation of small asymmetric crystals, which in the presence of a shear field adopt a non-random distribution around an average preferred orientation
(Figure 9a). Weak or no orientation was observed at low
shear rates either due to a random distribution of anisotropic
crystals (Figure 9b) or the formation of spherical particles
upon platelet aggregation (Figure 9c). This process can be
Idealized schematic of the behaviour of platelet crystallites. a) Under high shear
rates, the crystallites adopt a preferred orientation parallel to the walls of the
cell, as long as the shear forces prevail over the segregation and adhesion forces.
b) At very low shear the crystallites may just tumble in the flow, since the segregation forces prevail over shear and adhesion forces. c) If the adhesion forces
between crystallites prevail over the shear and Brownian forces, the crystallites
will form spherical clusters. (Adapted with permission from [22], © 2003
American Chemical Society).
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formally described using the concept of the Peclet number [22].
PHASE TRANSITION ACCELERATION
A very important effect of shear forces is the acceleration of
the transformation of crystalline fats to more stable polymorphs [22,26,31-33]. For example, the times required for the a
to b’ transition in milk fat, milk fat TAGs and palm oil were
reduced by up to an order of magnitude in sheared samples
relative to statically crystallized samples (Figure 10 a,b,c).
Other interesting findings include the comparison between
milk fat TAGs and native milk fat, that suggests that polar
lipids present in the native milk fat hinder crystallization
events, stabilize the α phase, and delay the formation of the
more stable β’ phase.
Cocoa butter displays a very rich and complex set of polymorphic transformations. As illustrated in Figure 10d, under
static conditions, phase α (II) persists for approximately
75 minutes while phase β’(IV) starts to emerge after 50 minutes. The desirable β (V) form of cocoa butter, present in high
quality chocolate, requires at least an additional 15 hours to
form through the tempering process [18,34]. However, when
cocoa butter was crystallized under a shear rate of 1440s-1,
the formation of the β’(IV) phase was not observed at lower
temperatures, whereas the β(V) phase formed in less than
20 minutes [16,22,26,27]. The persistence time and the relative
amount of phase α (II) remaining after the beginning of the
phase transition was reduced as shear was increased, and at
1440 s-1 no evidence of the presence of phase α (II) was
observed after the phase transition. This is especially important for the confectionery industry where a reduction in the
time required to induce the formation of desirable crystalline
structures is extremely valuable. At higher temperatures, the
relative concentration of phases depends on the combination
of shear rate, cooling rate, final crystallization temperature
and time [27].
MATHEMATICAL MODELING OF THE PHASE
TRANSITIONS
Integrated peak intensities derived from the 2D x-ray diffraction patterns can be used to describe quantitatively the phase
transition between α and β’ in
palm oil, milk fat and similar systems (margarines and shortenings, for example) using a model
based on the competition of three
events [28]. The first event is the
crystallization of phase α from a
fraction of the liquid, that will be
denoted A. This assumes that
only a certain fraction of the original material can crystallize in
phase α, mostly composed of the
molecules with the highest melting temperature. The second
event, after the onset of phase β’,
is the crystallization of phase β’
from the liquid fraction. Phase β’
can be formed from the molecular composition found in the liquid fraction “A” as well as from
an additional liquid fraction B.
The third event is the direct
transformation of existing phase
α into phase β’. The concept of
the solid fat content (SFC), which
defines the relative fraction of
solid material in the fat is useful
to describe the amount of crystallized material, as a significant
fraction of a “solid” fat such as
butter is actually in the liquid
state. The SFC was approximatFig. 10 Relative crystalline content represented by X-ray integrated intensity of the diffrac- ed to be proportional to the total
tion peaks as a function of time. Phase α (circles) and phase β’ (squares) are for (a) integrated intensity under the xmilk fat, (b) milk fat triacylglycerols, and (c) palm oil. (d) Phase α (II) (circles),
ray diffraction peak, even under
phase β’(IV) (squares), and phase β (V) (triangles) are for cocoa butter. The results
conditions of weak orientation.
from static experiments are represented by open symbols, whereas the results from The driving force is the supersatthe experiments under shear are represented by solid symbols. It can be clearly
uration σ defined as the ratio
appreciated that the transition between phases happens much earlier and faster
between the mass of the untrans-1
when the materials are crystallized under a shear rate of 1440 s . (Adapted with
formed material (liquid or cryspermission from [22], © 2003 American Chemical Society).
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The initial conditions are
A t=0 = A*, B t=0 = B* and
αSFC t=0 = 0. The system of
differential equations was
integrated numerically in
MATLAB® and fit to the high
resolution x-ray diffraction
data. The fits were remarkably good for the experiments
at different temperatures and
shear rates, as seen in
Figure 11 where the solid line
represents the fit. The effort
represents the first time that
the multiple phase transitions
in a real fat system have been
successfully modeled.
talline α) and the total mass of the material
potentially crystallizable. The equilibrium
value that the SFC tends to is called SFC*. The
supersaturation is defined by Eq. ( 1).
σ≈
SFC * −SFC
SFC
= 1−
SFC *
SFC *
( 1)
The supersaturation σ is considered to follow
the differential form of the Avrami equation [35]
(also known as the Johnson-Mehl-AvramiErofeev-Kolmogorov (JMAEK) model [36])
given by:
ni − 1
∂σi
= −ni ⋅ ki ⋅ σi ⋅ ⎡⎣ − ln ( σi ) ⎤⎦ ni
∂t
(2)
Where k is a time scaling constant and n is a Fig. 11 Solid fat content (SFC,%) of-1palm oil
crystallized at 22°C and 90 s as a
growth mode exponent for the ith phase that is
function of time, estimated from the CONCLUSION AND FURcrystallizing (e.g. α, β’). The phase behavior of
integrated intensities. The open
THER DIRECTIONS
the crystallizing fats was postulated to proceed
squares represent phase α and the
as follows: Phase α nucleates from the melt and
The discovery of new areas in
open triangles phase β’. The solid
grows from a fraction of the liquid called A*. At
line is the model estimate SFC for
the fascinating world of the
some later time, which depends on the shear
phase α and for phase β’.
soft condensed matter physics
rate, the α crystals act as nucleation sites for the
of edible materials has been
formation of β’ crystals. The β’ crystals can
made possible by the use of
grow from the liquid fraction A*, from another liquid fraction
synchrotron x-ray diffraction. In the field of multicomponent
B* (of an average lower melting point), and at the expense of
bulk lipid systems the finding of orientation under shear and
α crystallites. The two liquid fractions, A* and A*+B*, repreits correlation to the reduction of the phase transition characsent the maximum crystallizable liquid that can go, at a given
teristic times are particularly important. The possibility of
temperature, into each one of the phases α and β’.
quantitatively modelling these processes also offers the possibility to improve the design of industrial equipment and proThus the process can be modeled with three differential
cedures. The relevance of these findings is therefore of interequations and one balance equation [28] as shown below.
est both for the scientist and the engineer.
∂A
⎡
⎛ A ⎞⎤
= −na ⋅ ka ⋅ A ⋅ ⎢ − ln ⎜
⎟
∂t
⎝ A * ⎠ ⎥⎦
⎣
na − 1
na
−
nb − 1
⎧
⎫
+
A
B
A ⎪
⎡
⎤
⎪
⎞ nb
⎛
⎨nb ⋅ kb ⋅ ( A + B) ⋅ ⎢ − ln ⎜
⎬
⎟⎥
A + B⎪
⎝ A * +B * ⎠⎦
⎣
⎪⎩
⎭t > t o β '
Many aspects still await explanation, and it is the task of soft
condensed matter physicists to seek them. The development
of correlations between shear, temperature, rheology, thermology, structural characteristics and mechanical properties
is now under way in our group thanks to the development of
new tools such as the split Couette cell [37] and a micromechanical analyzer.
REFERENCES
∂α SFC
⎡
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⎣
na − 1
na
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⎧
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⎡
⎪
⎛ α SFC ⎞ ⎤ nc ⎪
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⎨ c c SFC ⎢
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⎬
⎟
∂t
A + B⎪
⎝ A * + B * ⎠ ⎥⎦
⎣
⎪⎩
⎭t > t o β '
β 'SFC = ( A * + B*) − ( A + B) − α SFC
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A.G. Marangoni and S.E. McGauley, Crystal Growth &
Design 3, 95 (2003).
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S.H.J. Idziak, Crystal Growth & Design 4, 1303 (2004).
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in fats and lipid systems, edited by N. Garti and K. Sato
(Marcel Dekker, Inc., New York, 2001), p. 118.
P. Fryer and K. Pinschower, MRS Bulletin 25, 25 (2000).
S.T. Beckett, The Science of Chocolate (Royal Society of
Chemistry, Cambridge, UK, 2000).
J.H. Los, W.J.P. van Enckevort, E. Vlieg, and E. Floter,
Journal of Physical Chemistry B106, 7321 (2002).
K.F. van Malssen, R. van Langevelde, R. Peschar, and
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S.H.J. Idziak, Crystal Growth & Design 3, 721 (2003).
G. Mazzanti, S.E. Guthrie, E.B. Sirota, A.G. Marangoni, and
S.H.J. Idziak, in Soft Materials - Structure and Dynamics, edited by J.R. Dutcher and A.G. Marangoni (Marcel Dekker,
Inc., N.Y., 2004).
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Chemistry and Physics of Lipids 58, 131 (1991).
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S.H.J. Idziak, Crystal Growth & Design 4, 409 (2004).
G. Mazzanti, A.G. Marangoni, and S.H.J. Idziak, Physical
Review E71, 041607 (2005).
T. Unruh, K. Westesen, P. Bösecke, P. Lindner, and
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LIVRES REÇUS
BOOK REVIEW POLICY
Books may be requested from the Book Review Editor, Andrej Tenne-Sens, by using the online book request form at http://www.cap.ca.
CAP members are given the first opportunity to request books. Requests from non-members will only be considered one month after the distribution
date of the issue of Physics in Canada in which the book was published as being available (e.g. a book listed in the January/February issue of Physics
in Canada will be made available to non-members at the end of March).
The Book Review Editor reserves the right to limit the number of books provided to reviewers each year. He also reserves the right to modify any submitted review for style and clarity. When rewording is required, the Book Review Editor will endeavour to preserve the intended meaning and, in so
doing, may find it necessary to consult the reviewer.
LA POLITIQUE
POUR LA CRITIQUE DE LIVRES
Si vous voulez faire l’évaluation critique d’un ouvrage, veuillez entrer en contact avec le responsable de la critique de livres, Andrej Tenne-Sens, en
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Les membres de l'ACP auront priorité pour les demandes de livres. Les demandes des non-membres ne seront examinées qu'un mois après la date
de distribution du numéro de la Physique au Canada dans lequel le livre aura été déclaré disponible (p. ex., un livre figurant dans le numéro de janvier-février de la Physique au Canada sera mis à la disposition des non-membres à la fin de mars).
Le Directeur de la critique de livres se réserve le droit de limiter le nombre de livres confiés chaque année aux examinateurs. Il se réserve, en outre,
le droit de modifier toute critique présentée afin d'en améliorer le style et la clarité. S'il lui faut reformuler une critique, il s'efforcera de conserver le
sens voulu par l'auteur de la critique et, à cette fin, il pourra juger nécessaire de le consulter.
BOOKS RECEIVED / LIVRES REÇUS
The following books have been received for review. Readers are invited to write reviews, in English or French, of books of interest to them.
Books may be requested from the book review editor, Andrej TenneSens by using the online request form at http://www.cap.ca.
A list of ALL books available for review, books out for review, and
copies of book reviews published since 2000 are available on-line -see the PiC Online section of the CAP's website : http://www.cap.ca.
Les livres suivants nous sont disponible pour une évaluation critique.
Celle-ci peut être faite en anglais ou en français. Si vous êtes
intéressé(e)s à nous communiquer une revue critique sur un ouvrage
en particulier, veuillez vous mettre en rapport avec le responsable de
la critique des livres, Andrej Tenne-Sens par internet à
http://www.cap.ca.
Il est possible de trouver électroniquement une liste de livres disponibles pour la revue critique, une liste de livres en voie de révision,
ainsi que des exemplaires de critiques de livres publiés depuis l'an
2000, en consultant la rubrique "PiC Électronique" de la page Web de
l'ACP : www.cap.ca.
GRADUATE TEXTS
AND PROCEEDINGS
GENERAL INTEREST
DARWINISM AND ITS DISCONTENTS, MICHAEL RUSE, Cambridge
University Press, 2006, pp8: 316; ISBN: 0-521-82947-X (hc); Price:
$30.00.
THE TROUBLE WITH PHYSICS: THE RISE OF STRING THEORY, THE FALL OF A
SCIENCE AND WHAT COMES NEXT, Lee Smolin, Thomas Allen
Publishers, 2006, pp: 343; ISBN: 978-0-618-55105-7; Price: .
UNDERGRADUATE TEXTS
A SHORT INTRODUCTION TO QUANTUM INFORMATION AND QUANTUM
COMPUTATION, Michel Le Bellac, Cambridge University Press, 2006, pp:
167; ISBN: 0-521-86056-3 (hc); Price: $60.00.
AN INTRODUCTION TO GENERAL RELATIVITY AND COSMOLOGY, Jerzy
Plebanski and Andrzej Krasinski, Cambridge University Press, 2006,
pp: 534; ISBN: 0-521-85623-X (hc); Price: $80.00.
CLASSICAL MECHANICS, R. Douglas Gregory, Cambridge University
Press, 2006, pp: 596; ISBN: 0-521-82678-0 (hc); 0-521-53409-7 (pbk);
Price: $120/60.00.
SPACE-TIME, RELATIVITY AND COSMOLOGY, Jose Wudka, Cambridge
University Press, 2006, pp: 320; ISBN: 0-521-82280-7; Price: $55.00 -hc.
THE IDEAS OF PARTICLE PHYSICS: AN INTRODUCTION FOR SCIENTISTS,
G.D. Coughlan, J.E. Dodd, B.M. Gripalos, Cambridge University Press,
2006, pp: 254; ISBN: 0-521-67775-0 (pbk); 0-521-84728-1 (hc); Price:
$50/100.00.
AN INTRODUCTION TO UNCERTAINTY IN MEASUREMENT, L. Kirkup,
R.B. Frenkel, Cambridge University Press, 2006, pp: 233; ISBN: 0-52184428-2 (hc); 0-521-60579-2 (pbk); Price: $80/$34.49.
CHAOS AND COMPLEXITY IN ASTROPHYSICS, Oded Regev, Cambridge
University Press, 2006, pp: 455; ISBN: 0-521-85534-9 (hc); Price: $80.00.
CONDENSED MATTER FIELD THEORY, A. Altland, B. Simons, Cambridge
University Press, 2006, pp: 624; ISBN: 0-521-84508-4 (hc); Price: $85.00.
DISCRETE INVERSE AND STATE ESTIMATION PROBLEMS WITH
GEOPHYSICAL FLUID APPLICATIONS, Carl Wunsch, Cambridge
University Press, 2006, pp: 371; ISBN: 0-521-85424-5 (hc); Price: $125.00.
FINITE-TEMPERATURE FIELD THEORY PRINCIPLES AND APPLICATIONS,
Joseph I. Kapusta and Charles Gale, Cambridge University Press, 2006,
pp: 428; ISBN: 0-521-82082-0 (hc); Price: $140.00.
LIQUID CRYSTALLINE POLYMERS - SECOND EDITION, A. Donald, A.
Windle, S. Hanna, Cambridge University Press, 2006, pp: 589; ISBN: 0521-58001-3 (hc); Price: $90.00.
PATH INTEGRALS AND ANOMALIES IN CURVED SPACE, Florenzo
Bastianelli and Peter van Nieuwenhuizen, Cambridge University
Press, 2006, pp: 379; ISBN: 0-521-84761-3 (hc); Price: $120.00.
STEPS TOWARDS AN EVOLUTIONARY PHYSICS, Enzo Tiezzi, WIT Press,
2006, pp: 157; ISBN: 1-84564-035-7 (hc); Price: $95.00.
WEAK SCALE SUPERSYMMETRY FROM SUPERFIELDS TO SCATTERING
EVENTS, H. Baer and X. Tata, Cambridge University Press, 2006, pp:
537; ISBN: 0-521-85786-4 (hc); Price: $80.00.
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BOOK REVIEWS
BOOK REVIEWS / CRITIQUES DE LIVRES
A GUIDED TOUR OF MATHEMATICAL METHODS FOR THE
PHYSICAL SCIENCES, SECOND EDITION, Roel Snieder,
Cambridge University Press, 2004, pp: 507, ISBN 0521834929
(hc); Price: US$60
Reviewing this text was an enjoyable experience. As the name
suggests, it gives the reader a tour of the mathematical methods
used in physical sciences. But this tour is a very practical one, as
it engages the reader in an active manner by describing the
methods and then posing challenging problems.
Most of the mathematical methods needed to solve problems in
physical sciences are introduced here. All the methods are given
due attention according to their complexity and usefulness. The
advanced concepts, such as Cartesian tensors, perturbation theory, asymptotic evaluation of integrals, and variational calculus,
are discussed in a very easy-to-understand manner. The author
relates the mathematical methods to appropriate physical problems, which makes the text highly engaging.
With plenty of problems and to-the-point approach, this book is
an absolute essential for students of physics, chemistry, and
other physical sciences as well as professionals working in related areas.
Syed Naeem Ahmed
Sudbury Neutrino Observatory/Queen’s University
Ontario, Canada
Book Review Editor’s Note: For a more detailed review of the first (2001) edition of this book, see the Book Reviews section of the March/April 2002
issue of Physics in Canada. The review is available on the Web at:
http://www.cap.ca/BRMS/Reviews/Math-Sneider-Buckmaster.html
ALL YOU WANTED TO KNOW ABOUT MATHEMATICS BUT
WERE AFRAID TO ASK: MATHEMATICS FOR SCIENCE
STUDENTS, VOLUME 2, L. Lyons, Cambridge University Press,
1998, pp: 382, ISBN 052143601X (pbk); Price: US$28
All You Wanted to Know About Mathematics but Were Afraid to Ask:
Mathematics For Science Students, Volume 2 est un livre qui
s’adresse aux étudiants effectuant un baccalauréat en physique
ou en génie. On y retrouve plusieurs concepts mathématiques
qui seront utiles aux futurs bacheliers. Il s’agit d’un petit livre
contrastant avec les grands livres habituels de mathématiques
de l’ingénieur. De plus, ces notions mathématiques sont toujours abordées avec des exemples physiques concrets. Cette
méthode est efficace pour conserver l’intérêt et l’attention d’un
lecteur qui pourrait rechercher l’utilité de ce qu’il apprend. On
développe également de cette manière la capacité d’analyse de
systèmes physiques avec des outils mathématiques. L’auteur
fait preuve d’une certaine originalité pour nous captiver. Par
exemple, au chapitre 15, on verra qu’il y a un lien entre la multiplication de matrices et le lavage et le séchage de nos mains! Par
ailleurs, plusieurs schémas illustrent les différentes notions
expliquées facilitant ainsi la lecture. De plus, le livre n’est pas
une suite d’équations difficile à suivre. Il y a en fait beaucoup de
texte pour expliquer les équations écrites. En somme, il s’agit
d’un bon livre d’ordre général, pédagogique et permettant de
comprendre les bases de la mathématique et non d’un livre spécialisé permettant d’approfondir certaines notions déjà acquises.
322
PHYSICS
IN
CANADA
Ce deuxième volume contient les chapitres 9 à 16 et l’annexe C
de la série. Il n’est pas nécessaire d’avoir lu le premier volume
pour l’entamer en supposant que l’on dispose déjà de bonnes
connaissances. Les notions abordées dans le premier livre
étaient les équations simultanées, la géométrie tridimensionnelle, les vecteurs, les nombres complexes, les équations différentielles ordinaires, les dérivées partielles, les séries de Taylor
et les multiplicateurs de Lagrange. Quelques problèmes à la fin
de chaque chapitre permettent de s’assurer de la compréhension
de la matière. Selon l’auteur, il est nécessaire que l’étudiant
essaie chacun de ces problèmes puisqu’ils sont peu nombreux.
Les réponses à ces problèmes ne sont cependant pas données.
De plus, il n’y a malheureusement aucune référence vers
d’autres livres ou articles concernant les sujets traités.
Le chapitre 9 débute avec le calcul des intégrales de ligne. On
introduit ensuite les intégrales multiples en divisant la carte de
l’Angleterre en rectangles pour calculer la population totale,
connaissant la densité de population de chacun des rectangles.
On y voit un exemple original d’aborder un concept mathématique. La généralisation pour des intégrales à n dimensions est
ensuite effectuée avant de s’attarder aux limites d’intégration et
aux changements de variables. Au chapitre 10, les opérateurs
gradient, divergence et rotationnel sont expliqués en détail. On
exprime aussi ces opérateurs en coordonnées cylindriques et
sphériques. Ensuite, le théorème de la divergence, le théorème
de Stokes et le théorème de Green sont tour à tour décrits.
Quelques exemples mathématiques et physiques terminent le
chapitre.
Les équations différentielles partielles font l’objet du chapitre 11.
Il est intéressant que l’équation de conduction de chaleur serve
de point de départ à l’explication. Graduellement, on introduit
les notions de conditions frontières, la séparation des variables
pour la résolution de problèmes ou encore la méthode
d’Alembert. Les exemples importants de l’équation d’onde, des
équations de Poisson et de Laplace ainsi que de l’équation de
Schrödinger sont entre autres mentionnés.
Les transformées de Fourier constituent un outil bien important
dans plusieurs domaines de la physique et le chapitre 12 leur est
consacré. De façon bien structurée, l’auteur donne les étapes que
l’on doit prendre pour déterminer les coefficients de Fourier
d’une fonction donnée et pour vérifier que la réponse calculée
est plausible. Cette approche est particulièrement intéressante.
Ensuite les applications physiques données, par exemple les circuits électriques, permettent de comprendre l’importance de
bien étudier ces séries.
Les deux chapitres suivants se consacrent à des notions de
physique. Le chapitre 13 porte sur les modes normaux. Les deux
pendules couplés identiques et différents sont le point d’ancrage
du chapitre. Pour être général, l’auteur aborde également les
modes non-normaux. À la fin du chapitre, l’auteur a eu le souci
de faire un résumé des étapes à suivre pour résoudre ce genre de
problèmes. Il y a également un résumé de la notation utilisée
pour que tout soit le plus clair possible. Pour sa part, le chapitre
14 traite des ondes. L’équation d’onde, la vitesse de groupe et la
vitesse de phase, l’énergie des ondes, la réflexion, la polarisation,
les ondes longitudinales et l’interférence sont autant de notions
qui seront développées.
September / October 2006
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CRITIQUES DE LIVRES
Les matrices et les opérations possibles avec ces matrices sont
expliquées en détail au chapitre 15. La partie sur les propriétés
des matrices est particulièrement intéressante comme aidemémoire. Finalement, le chapitre 16 montre comment obtenir les
vecteurs et valeurs propres d’une matrice et leur utilité. Un
retour sur les modes normaux est aussi effectué. L’annexe C ne
fait que la liste des principales équations de chaque chapitre,
mais peut constituer un bon aide-mémoire qui évite de relire
tout le livre inutilement.
Bref, ce deuxième volume présente de façon originale et stimulante plusieurs notions des mathématiques utiles en faisant constamment des liens avec des problèmes physiques concrets. Il
s’agit donc d’un bon outil pour tout étudiant entreprenant des
études de baccalauréat en physique ou en génie. Quelques problèmes à la fin des chapitres permettent aussi de s’assurer de sa
compréhension. Peu volumineux, il est probablement idéal
pour une personne ne disposant que de peu de temps pour comprendre certains des sujets présentés. Évidemment, cette concision n’en fait pas un livre spécialisé, mais bien un livre général.
Léo Barriault
Université Laval
Québec, QC, Canada
BAYESIAN LOGICAL DATA ANALYSIS FOR THE PHYSICAL
SCIENCES: A COMPARATIVE APPROACH WITH MATHEMATICA
SUPPORT, Phil Gregory, Cambridge University Press, 2005, pp:
460, ISBN: 052184150X; Price: US$70
I became intrigued by Bayesian statistics several years ago when
I saw some applications of the methods to problems in physics
and the success that they had. A graduate student worked with
me to apply the methods in some new areas. In order to get up
to speed on the subject matter I audited a graduate course in
Statistics that focused on Bayesian statistics. This led to an
awareness of the works of the physicist E.T. Jaynes who, in addition to publishing some applications, commenced to write a
book on the subject. The chapters were available on the web,
and so I downloaded everything and became excited about the
potential of this field. Dr. Jaynes died before finishing the book,
but with the help of Larry Bretthorst and others, the book,
Probability Theory: The Logic of Science, was published in 2003
(also by Cambridge).
Hence, it was very interesting to read the preface to this work by
Phil Gregory, indicating how he also was stimulated by the
work of Jaynes. Much of the work in this text is based on material from Jaynes, Bretthorst and others, who were pioneers in
many of the applications of Bayes statistics to the sciences.
The book contains 14 chapters and 5 appendices. The author
begins with an introduction to the concept of scientific inference,
its relation to probability theory, and the role of Bayes’ theorem.
The concept of marginalization and the advantages of this
approach over the standard statistical methods are introduced in
the early chapters.
Some of the basics of logic are introduced next, with truth tables
and Boolean algebra. This leads into the operations required for
plausible inference and some of the rules associated with this.
The author then gets into the meat of the matter by discussing
parameter estimation, selection of priors, assigning probabilities
and building likelihood functions. Along the way he employs
examples from physics and astronomy to show the relevance of
the developments and their applications.
In order to help readers appreciate the role of Bayesian statistics,
three chapters of the book are devoted to reviewing the standard
frequentist approach. Sampling theory, probability distributions
and related topics such as pseudorandom numbers are presented. Calculating statistics, chi-squared values and confidence
intervals are presented in Chapter 6. Hypothesis testing forms
the basis of the following chapter.
The central topics of the book include the maximum entropy
principle and its role in generating probability distributions. Its
application to image reconstruction is one example that is included. Computing means and standard deviations using Bayes’ theorem are also covered: How do different samples compare?
Parameter estimation is another primary application. This is
applied to both linear and nonlinear models.
The Markov-chain Monte Carlo method has a chapter to itself.
This is also a valuable technique for parameter estimation. An
example included in the text is the analysis of astronomical data
that was used to discover an extra-solar planet in 2003. Another
very well-known application is in spectral analysis. Here the
work of Jaynes and Bretthorst and others was revolutionary in
the ability to obtain spectral information from noisy data.
The appendices provide the required mathematical material for
singular-value decomposition, the discrete Fourier transform, as
well as elaborating on some of the math in earlier chapters.
The author includes Mathematica support in some sections by
including the required commands in the text. In addition, he has
made available a tutorial, to support the book, on a website.
Each chapter also has problems to test the reader’s understanding. There is also a very comprehensive list of reference material
at the end of the book.
Overall, this is an excellent text to introduce readers to the many
applications of Bayesian logic.
Richard Hodgson
University of Ottawa
Ottawa, Ontario, Canada
BIOHAZARD, Ken Alibek with Stephen Handelman, Random House,
1999, pp: 319, ISBN 0375502319 (hc); Price US$25
Biohazard is a real-life account of Ken Alibek’s experience in the
Soviet Union’s biological weapons program from 1975 until the
collapse of the Soviet Union. His story is more ominous in
today’s climate of international suspicion than it was in 1999.
The authors’ clinical descriptions of Frankensteinian science are
worded with such serious clarity that what could normally be
disregarded as fiction feels authentic and real. I opened the front
cover skeptical of the insert’s claims and closed the back cover
accepting every word. It is the lack of emotion in the authors’
writing which convinced me this is not a heroic drama but a factual insider’s account of a well-established program.
Alibek describes the mass manufacture and weaponization of
lethal viruses, bacteria and toxins created for the sole purpose of
immobilizing and killing entire populations. From the exotic
(Marburg and Ebola) to the traditional (smallpox and plague),
Alibek discloses the former Soviet Union’s collection of killer
bugs. There is even mention of a Chimera virus, a hybrid
designed to trigger multiple diseases in concert.
LA PHYSIQUE AU CANADA
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The majority of this book focuses on the politics and agencies
involved in the mass manufacture of biological weapons. It outlines how the Soviet Union maintained a trail of deception for
two decades. Millions of dollars were funneled into a secret program to design viruses and bacteria meant to kill millions of people. A central theme emerges from the pages – secret government policies, left unchecked, can avoid social responsibility and
steal potentially life-saving research away from its citizens.
Absent from this book is an open discussion about the personal
goals, values and principles of the scientists involved in this program. Personal beliefs remain justifiably silent in Alibek’s
Soviet-era world. There is, however, a description of the psychological toll the secrets and lies have on the lives of those
involved. At first, the authors’ lack of emotion and observational style portrays scientists detached from their work, detached
from the moral consequences. Reading further, glimpses of frustration, guilt and denial unfold into disillusionment and horror.
A memorable moment is when Alibek introduces the reader to a
man involved in the accidental release of anthrax into the city of
Sverdlovsk: “His face was riveted on an invisible spot in front of
him, and his hands began to shake so violently that he had to put
his teacup down. He looked as if he was about to burst into
tears”.
The authors’ words are just as relevant now as they were in 1999.
The book actually made accurate predictions about our world
that explain why the Russians will continue to invent biological
weapons: “…the weakened Russian military machine confronts
a greater variety of challenges…These include armed separatist
movements in the Caucasus, civil wars in central Asia, the
spread of Muslim fundamentalism from Iran and
Afghanistan…Biological weapons can play an important part in
such conflicts, often compensating for the weakness or ineffectiveness of conventional forces”. Today, we live in a world
where terrorists in Afghanistan are reportedly pursuing biological weapons, Chechen rebels have terrorized Russian citizens,
and in 2002 Russian security forces used a secret narcotic gas to
save hostages in a Moscow theatre. The realization of some of
Alibek’s general predictions place more weight upon his dire
warnings.
This story is the only one of its kind, as the principal author is
the only man in the world who has written about the secret life
of a biological warfare scientist. If Ken Alibek ever reads this, I
ask him, please pick up a pen and write another book.
Graeme Drysdale
University of Regina
Regina, Saskatchewan, Canada
BOSONIZATION AND STRONGLY CORRELATED SYSTEMS,
A.O. Gogolin, A.A. Nersesyan and A.M. Tsvelik, Cambridge
University Press, 1999, pp: xxii+423, ISBN 0521590310 (hc);
Price: US$100
Bosonization and Strongly Correlated Systems by Gogolin,
Nersesyan and Tsvelik provides an extensive overview of
important non-perturbative techniques for the study of manybody systems. The book surveys the many technical details of
bosonization and provides important references for further
details. The book itself would serve as a good reference to such
topics; however, in its attempts to cover such a huge wealth of
methods and results, many details are skipped and many steps
are left unjustified. Thankfully, the authors present references
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to more elaborate treatments in their discussions. For use as a
reference, their terse style would suffice to remind the reader of
a particular method or result if the material didn’t necessarily
rely so heavily upon previously presented work.
In the first part of their book, they present the many technical
details associated with bosonization and the results of noninteracting fermions and bosons, including topics such as onedimensional fermions, the Gaussian model (of bosons), the
structure of Hilbert space in conformal theories, Bose-Einstein
condensation in two dimensions, non-Abelian bosonization,
and various spin-chain models.
In the second part, the authors apply the techniques developed
in the first part to a variety of many-body systems, including
interacting fermions with spin, a Tomonaga-Luttinger liquid, a
variety of spin models, superconductivity in a doped spin liquid, and edge states in the quantum Hall effect.
In the third part, the effects of impurities are examined. Topics
include potential scattering, the x-ray edge problem, impurities
in a Tomonaga-Luttinger liquid, and the multichannel Kondo
problem.
The book has a wealth of information and is a convenient reference to someone already working in the field. For newcomers,
the authors direct the reader to good sources of further details,
while trying to present the essential ideas and results for a
coherent, if brief, presentation.
Lara Thompson
University of British Columbia
Vancouver, BC, Canada
DATA
AND ERROR ANALYSIS, SECOND EDITION, William
Lichten, Prentice Hall, 1999, pp: 188, ISBN 0133685802 (pbk, CDROM included); Price: US$27
The ability to accurately interpret data and error is a fundamental skill for scientists and engineers. This relatively short text
(188 pp.) and accompanying CD-ROM (Windows and
Macintosh compatible) provides an introduction to data and
error analysis with “simple, handy rules for estimating errors,
both by graphical and analytic methods”. It has a learn-by-doing
approach as opposed to a rigorous theoretical one. The author
addresses this in his introduction by noting that it helps “science
students process their data without lengthy and boring computations” that entail long discussions and involved derivations.
There are five chapters: “Measurement and Errors”, “Error
Analysis for One Variable”, “Error Analysis for More than One
Variable”,” Finding Relations Between Variables” (according to
the Table of Contents, but entitled “Linear Regression: Fitting a
Straight Line to a Set of Points” within the text), and “Using
Trigonometric and Exponential Functions in the Laboratory”.
Each chapter contains practice problems (with answers), examples, and figures which demonstrate methods.
Chapter 1 covers introductory measurement topics and definitions such as the distinction between exact and approximate
statements, possible sources of error, significant digits, relative
error, and percentage error. Chapter 2 introduces statistical
terms, common frequency distributions, grouping, and properties that can be analyzed for a given data set. Chapter 3 extends
on previous concepts by considering more than one variable.
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Topics include error propagation, correlation, independence,
and methods for error analysis. In Chapter 4, step-by-step
instructions and examples are provided for linearly fitting data,
manually with graph paper and computationally with the
method of least squares. Chapter 5 explores trigonometric, exponential and power-law functions because they are commonly
used to describe relationships between variables. A mathematical review, data-analysis methods, plotting (semilog and log-log
graphs), and example calculations are presented. Power-series
approximations for small angles are listed for trigonometric
functions as an alternative method of calculation, although the
derivation is not included. Demonstrations of how linear regression can determine coefficients for linear, log, exponential, exponential with counts, and power-law fits are particularly useful.
Detailed calculations are provided for applications such as
measuring the time constant of an RC circuit and measuring
radioactive-isotope lifetimes.
There is a strong focus on the included programs and guides for
calculators and computers. The lengthy “Appendix B” (85 pp.)
gives procedures for specific calculators. Unfortunately, I could
not test these procedures since my scientific calculator is not on
the list. The CD-ROM contains Excel programs to practice calculations and create plots, a cT program for error analysis, and
BASIC and Pascal programs.
Excel is a good choice of platform since it is highly familiar and
widely available. The included files are user-friendly and provide a valuable medium for learning, but the size of the data sets
is limited to less than 40 data points. Thus, these files are best
suited for students who simply want to understand concepts,
since they can input small data sets and not have to program in
all the calculation details.
As I was unfamiliar with cT, an Internet search informed me that
the cT programming language was developed at Carnegie
Mellon University. Its main niche is “the creation of programs
for education, and many prize-winning educational programs
were written in cT, especially in the areas of physics”.l Although
the language is no longer supported, these files do not require
the user to know how to program in cT – it is a graphical user
interface for learning text material interactively. These files are
not meant for practical research purposes, for which there are
many programming languages and software applications capable of computing and displaying large data sets more efficiently
and adaptively. I would have liked to see additional references
to Matlab or other software applications.
A strength of this text is its multidimensional learning environment. There is a good emphasis on using computers to make the
computational part of the methods easier and faster, even
though it does not give the user the knowledge to program their
own applications and, in my opinion, does not make sufficient
use of the vast array of software and programming languages
available.
It is suited toward undergraduate scientists and engineers who
are not yet comfortable with programming and/or mathematical derivations, but who need to perform data and error analysis
for their coursework. Students with strong math and computer
science skills may require a text with more theory and programming material. Due to its limited scope, I would treat this text
only as an introduction. Other resources should be consulted for
mathematical derivations, more complex functions, and a com-
prehensive source of statistical methods, distributions and formulas.
Reyna Jenkyns
University of Victoria
Victoria, British Columbia, Canada
1. See: http://vpython.org/cTtsource/cToverview.html
INTRODUCTION TO CHAOS: PHYSICS AND MATHEMATICS OF
CHAOTIC PHENOMENA, H. Nagashima and Y. Baba, Institute of
Physics Publishing, 1999, pp: 166, ISBN 0750305088 (pbk); Price:
US$27
This book attempts to introduce some of the ideas of nonlinear
dynamics, largely through the mathematics of one-dimensional
iterated maps. Although the authors do take the time to explain
how 1D maps come about from specific physical phenomena, it
is my opinion that this approach is not the best way to come to
appreciate the physical meaning of chaos. But, as far as expositions on chaos that employ this approach, I would highly recommend the book. Despite the title of the book under review, it is
unfortunately quite slim on physics. Keeping in mind this is a
review in a journal for physicists, I think it is necessary to comment on whether this book would be a good book for physicists,
or even more generally, who this book would be good for. I
think it is useful for self-study if you want to do numerical experiments with 1D maps and want to get a deeper understanding of
that topic. It would also be useful as a course text for a short
undergraduate applied math course in discrete dynamical systems. However, for physicists who would like to learn about
chaos, I think there are much better starting points. I would
rather recommend reading something like the collection of articles in the book Exploring Chaos, edited by Nina Hall. For formal
undergraduate courses, I would highly recommend Baker and
Gollub’s excellent junior undergrad text Chaotic Dynamics, or
Francis Moon’s Chaotic and Fractal Dynamics for a senior undergrad physics course.
An initial qualm I had with the book was that it did not cover in
detail a significant range of topics that I think are essential in any
“non-popular” introduction to the subject. In other words, I
thought the book was ‘too thin’. But, as I read on it became clear
that what the authors did cover, they did with clarity and precision. It is an elegant book. It is well thought out, contains many
helpful figures, and the writing is excellent. So, it is perhaps
more fair to judge the book on what is actually there instead of
on what is not.
The book consists of four chapters, a long (indispensable) set of
appendices, a reference list, and a set of solutions to problems
given in the chapters. Chapter 1 gives an introductory discussion. Chapter 2 covers the Li-Yorke theorem, Sharkovski’s theorem, the topological entropy, and the Lyapunov number. This
chapter is (necessarily) quite rigorous and technical. The derivations are clean, and there are sufficient examples and problems,
but this material is pretty hard going and not easy to get through
without some mathematical experience. Beware that there is not
a shred of physics to be found in this chapter!
Chapter 3 covers various routes to chaos, windows, and intermittency using the standard (and no doubt fine) examples of the tent
and logistic maps. The analysis is all laid out and it is well written, and again there are many elegant figures. Of course, anyone
who has already studied these topics in some depth can proba-
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bly appreciate many of the fine tricks the authors perform, but
someone who has not studied the subject before, after going
through this chapter, might wonder if they are “getting it”? They
might ask: “so, what is chaos anyway?” It is not that there are
any secrets here; for example, the authors do explain (albeit
tersely) where the chaos resides in the window of the logistic
map, but there is a lot of mathematical subtlety here. The
impression one might get from this type of presentation is that
chaos means “complex structure”, which is of course really misleading. In Section 3.2, the authors pursue the problem of trying
to find a condition that a 1D map must satisfy in order that an
infinite sequence of pitchfork bifurcations exists. No doubt this
is an interesting mathematical problem, but I wonder if knowing
the answer is that the Schwarz derivative of the map is negative
(a fabulous mathematical result) helps the physicist to better
understand what chaos is. Perhaps this is a heretical remark. As
for genuine physics, all the physics you get in this chapter is two
sentences about a laser experiment in connection with the period-doubling route to chaos. Undoubtedly, for anyone new to
chaos, it would take quite some effort to get through Chapters 2
and 3. The payoff for the physicist seems negligible. This is
where I would again question the 1D iterated maps approach
(see first paragraph).
Chapter 4 is entitled “Chaos in realistic systems”. However, the
first section of this chapter is the only place where the reader will
see any physics really discussed in detail. Here, conservative
versus dissipative chaos is distinguished using the examples of
the simple harmonic and damped oscillators, respectively. (Of
course, there is no chaos to speak of for the simple harmonic
oscillator.) The rest of the chapter contains material on limit
cycles, strange attractors, Poincaré sections and maps, Lyapunov
numbers, fractal dimensions, and scaling indices. In particular, I
liked the nice simple discussion of the Hausdorf dimension and
how it leads naturally to the capacity dimension. On the whole,
the chapter is well written, but it is just not educational. Let me
be more specific. The material on the Rossler and Lorenz attractors is very sketchy. The Henon map is just barely mentioned but
nevertheless one finds a figure of the Henon attractor. There is
also a figure that shows a strange attractor obtained from
“magnon chaos”, but the authors never explain what this is.
“Magnon chaos” also creeps into the section discussing the correlation dimension and in the section on scaling indices. The latter section is another case in point. The last section in Chapter 4
is about scaling indices and their spectra. There is standard
material here such as the relation between the generalized boxcounting dimension, the q-parameter, the index, and the spectrum (although this result is admittedly derived in an unconventional way), and there are the standard examples like the
“weighted Cantor set”. As before, the derivations are very clean,
but the discussion is simply too terse. The authors cover this
topic without explaining what is really being done and why. The
notion of a multifractal is entirely absent. A more pedagogical
and more meaningful discussion (for example, like that in
Hilborn’s text Chaos and Nonlinear Dynamics) is necessary in
order to make what is there at all intelligible. The authors do try
and bring in some physics. In addition to the “magnon chaos”
experiments mentioned before, the authors very briefly discuss
data from a Benard convection experiment reported in a beautiful Letter, but the reader cannot really get too much out of it
since it is all wrapped up in two or three sentences.
Undoubtedly, the reader would have to look up the original article to understand what is going on. I think the figure from the
original paper is insufficient; perhaps it would have been better
to just mention the paper and only give the reference.
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The appendices elaborate on some of the finer points and get
into some of the technicalities. One finds a lot of pure classical
analysis in the appendices to Chapter 2 such as countable and
uncountable sets, Lebesgue measure, and so on. I think this
material is definitely helpful and the authors cover this material
in a way that would be appealing to non-purist mathematicians.
Personally, I found the appendix on “normal numbers” and
periodic orbits with finite-fraction initial conditions very interesting. The appendices to Chapter 3 are mostly supplementary
to Chapter 3, and it would have made more sense to include
them in Chapter 3 itself. Appendix 3D, for instance, gives examples of invariant measure and it would have been helpful to
include that stuff in the main text. Generally speaking, the
appendices are quite beneficial. I did, however, have some specific qualms with the last three appendices. One of these is about
chaos in a double pendulum. Here would have been a nice
opportunity to discuss chaos in a conservative system. The contents of this section are disappointing. All the authors do is
obtain the equations of motion using the Lagrangian formalism
(through the Euler-Lagrange equations). So, they write these
down, give a Mathematica program which “solves these equations”, and then finally display a single Poincaré section. It is
simply not enough. I think it would have been illuminating to
give a full discussion of this example, something more along the
lines of what is done in Chapter 11 of Hand and Finch’s
Analytical Mechanics. The last two appendices are about the “singular points” of the van der Pol equation and the Roessler
model. In these two sections, the authors embark upon a quickand-dirty analysis of the linearized dynamics in the vicinity of
the equilibrium points of these two systems (although they don’t
say it, this is what they are really doing). Honestly, I am
impressed at how much ground they cover in a few pages. But,
I would seriously question the benefit of embarking upon this
material in the final pages of the text. This stuff constitutes a subject in its own right and any meaningful exposition demands a
systematic study. For example, the same material spans more
than three full chapters in the classic Strogatz text Nonlinear
Dynamics and Chaos.
The book ends with a page of references and solutions to problems. There are very few references given (most of these are original papers). This is not really a shortcoming, but rather reflects
the limited scope of the text. There are about 45 problems in the
book and full solutions are given at the end of the book. The
solutions are nicely laid out and detailed.
In conclusion, this book offers some nice systematic discussions
on certain properties of one-dimensional iterated maps and very
sketchy discussions of many other topics and ideas encountered
in the subject of nonlinear dynamics. I would recommend this
book as a starting point for understanding many of the mathematical features of 1D maps, but for physicists who would like
to learn about chaos, there are many better alternatives.
Jamal Sakhr
Harvard University
Cambridge, Massachusetts, USA
QUANTUM FINANCE: PATH INTEGRALS AND HAMILTONIANS
FOR OPTIONS AND INTEREST RATES, Belal E. Baaquie,
Cambridge University Press, 2004, pp: 314, ISBN 0521840457
(hc); Price: US$70
In this book, the author shows how to approach problems related to financial markets with mathematical techniques that are
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traditionally used in quantum field theory. This approach is in
contrast to the usual stochastic methods that are generally
applied to handle such problems. In particular, the author shows
that path-integral formalism can be applied to understand the
dynamics of options and interest rates.
The author devotes the first chapter to introduce the reader to
the terminologies and basic mathematics of financial markets.
This useful chapter makes the book appealing to readers who
are not very familiar with the financial sector. It gives a very
good, brief, and to-the-point introduction of the subject. The
author then moves on to introducing and applying quantummechanical and field-theoretical methods to stock options and
interest rates. With plenty of appendices at the end of each chapter, the author has done a fairly good job in defining complex
concepts and mathematical methods. The author touches on all
the relevant segments of options and interest rates, such as stochastic volatility, moments, forward rates, and hedging.
For non-specialists, that is, readers who are not already familiar
with quantum field theoretical methods, this book may be fairly
challenging. However, to physicists and mathematicians, it provides an excellent introduction to this novel approach. Since
most quantitative analysts in the financial sector have backgrounds in physics or mathematics, this book would be highly
beneficial for them in analyzing the markets in a non-traditional
way.
The author has done a marvelous job in showing that the quantum field theoretical methods can be usefully applied to financial market analysis. It is not intended to be a practical guide to
the subject and therefore lacks examples of actual data analysis.
As the author notes, the book is a research tool that can be used
by physicists and mathematicians interested in the financial sector. I recommend it highly for such a readership.
Syed Naeem Ahmed
Sudbury Neutrino Observatory/Queen’s University
Ontario, Canada
SOLAR SYSTEM DYNAMICS, C.D. Murray and S.F. Dermott,
Cambridge University Press, 2000, pp: 575, ISBN 0521575974
(pbk) / 0521572959 (hc); Price: US$40/$90
This textbook comprehensively outlines the main techniques,
both old and new, and mathematical tools of planetary and solar
system dynamics, demonstrating how these apply to a wide
array of interesting, modern problems. This widely acclaimed,
authoritative book succeeds in bridging the gap between the
classical and modern methods of celestial mechanics.
The work is distinguished by the great care taken to convey
understanding and by the emphasis placed on the phenomena
of resonance. By laying out the basics of the two- and three-body
problems and some perturbation theory, it goes on to explain the
dynamical origin, evolution, and stability of the bodies in the
solar system.
Programs written in the computer algebra system Mathematica
(available as a free download on the author’s website) were used
to help produce many plots of data and graphics in the text; several of these (and more) are available online and greatly enhance
the educational value of the book. Notable topics omitted are
mentioned in the preface: lunar theory, geophysics, and Cassini
states. Altogether, this is a great textbook from which to learn
and it is very suitable for a graduate-level course on celestial
mechanics and for self-study by either the professional or the
technically-inclined amateur astronomer.
My personal interests in astronomy typically fall within the confines of “practical” astronomy and related ephemerides. Several
excellent books have been published on these subjects, notably
Montenbruck and Pfleger’s Astronomy on the Personal Computer
and the venerable standard reference Explanatory Supplement to
the Astronomical Almanac. There are many other, however, less
“practical” subjects that are not covered therein despite their
intrinsic appeal. For instance, it is known that the phenomenon
of resonance underlies much of the dynamical structure in the
solar system (e.g., planetary ring structure) and that most dynamical systems are not deterministic (i.e., chaos rules). Murray and
Dermott’s text covers both the theory of deterministic and nondeterministic motion. For example, Chapter 8 on “Resonant
Perturbation” develops a theory-based (Hamiltonian approach)
investigation of orbit-orbit resonance and Chapter 9 covers
“Chaos and Long-Term Evolution” of orbits. As such, it provides
comprehensive coverage.
The problem sets are well-crafted in their own right and some
successfully integrate interesting elements from some recent
high-profile astronomical media events. Problems which pique
the interest of the reader can serve as the basis for further investigation and this is facilitated by the inclusion of a fairly extensive list of (mostly modern) references. The list of references
could be further aided by the addition of an author index. There
are no colour plates or colour figures (although this not meant to
be a “coffee-table book”). The figures consist mainly of diagrams,
graphs, and only a few photos.
Solar System Dynamics is thoughtfully laid out, is a joy to read,
and is a thought-provoking, enthralling journey. The authors
have accomplished their goal with great skill and sensitivity.
Grant I. Nixon
MDS Nordion
Ottawa, Ontario, Canada
SYMMETRIES IN PHYSICS: PHILOSOPHICAL REFLECTIONS, Edited
by K. Brading and E. Castellani, Cambridge University Press, 2003,
pp: 445, ISBN 0521821371 (hc); Price: US$100
This book is a collection of papers written by physicists, mathematicians, and philosophers on the subject of symmetries in
physics. The contributions span a wide range of topics related to
symmetries and symmetry breaking. It seems that the editors
have spent considerable time and energy in selecting the articles
as each one of them gives the reader a new flavour of the subject.
Some of the articles have been written by prominent philosophers, physicists, and mathematicians, such as Hermann Weyl,
Leibniz, Kant, Black, Curie, and Wigner. They introduce the
reader to the initial thought processes that eventually led to the
current philosophical foundations of the symmetries in physics.
These classic articles are followed by new articles and commentaries on the same subject, keeping the reader engaged in the
learning process.
The editors have ensured that all of the fundamental symmetries
in physics receive due exposure in the book. The introduction of
complex ideas, such as gauge theories and spontaneous symmetry breaking, takes the reader to a higher level of understanding
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in a philosophical sense. Most of the articles in the book deal
with the philosophy of symmetries without going into complex
mathematics. This makes the book appealing to non-specialists
with little background in physics and mathematics. For philosophy students, this book provides an excellent introduction to the
subject.
Throughout the rest of the book, new pieces of code and relevant
algorithms are presented which add more functionality to the
basic program discussed in the previous chapters. Chapter 3
presents methods of calculating interatomic forces, two algorithms for integrating equations of motion, and code for setting
up the initial state of the simulation.
Highly recommended for students and researchers working in
the fields of physics and philosophy of science.
Measuring equilibrium properties of simple fluids is the topic of
the next chapter. It is mostly concerned with structural properties of the system. Therefore, methods for calculating radial distribution functions as well as for studying packing arrangements
through the Voronoi algorithm are discussed. Cluster analysis is
also described.
Syed Naeem Ahmed
Sudbury Neutrino Observatory/Queen’s University
Ontario, Canada
THE ART OF MOLECULAR DYNAMICS SIMULATION, SECOND
EDITION, Dennis C. Rapaport, Cambridge University Press, 2004,
The next three chapters discuss how to calculate dynamical
properties of a fluid, its transport coefficients, and how to perform a correlation analysis.
Most physics students know that molecular dynamics (MD) simulation consists of numerically solving equations of motion for a
system of interacting particles. In other words, their positions
and velocities are calculated at a series of time steps. Some of
those students would probably be able to write code simulating
elastic collisions of billiard balls. If that is where your programming knowledge ends, and you always wondered about the
details of MD code or your career choice calls for writing one,
then this book is for you.
Up to this point, all example simulations are performed in constant-energy and volume ensemble (NVE or microcanonical
ensemble). Extending the functionality of the code so that it covers constant-temperature (NVT or canonical) and constant-pressure (NPT or isothermal-isobaric) ensembles is covered next, followed by studies of nonequilibrium dynamics, rigid and flexible
molecules, as well as molecules with internal degrees of freedom
and geometric constraints. As case studies, water, surfactants,
and alkane chains are chosen, respectively. Chapter 12 is entirely devoted to the subject of three-body and many-body interactions and Chapter 13 discusses long-range interactions where
the Ewald sums and multiple-moments expansion methods are
described and implemented. Chapters 14 through 16 cover step
potentials, time-dependent phenomena, and granular dynamics.
pp: 549, ISBN 0521825687 (hc); Price: US$60
Dennis C. Rapaport is a professor at the Department of Physics
of Bar-Ilan University in Israel. His research interests include not
only performing MD simulations to study properties of a variety
of physical systems but also designing simulation algorithms for
parallel computers. He is also interested in interactive simulations. I think that The Art of Molecular Dynamics Simulation,
Second Edition can be considered a classic in the field, and as part
of a canon of books about application of computer simulations in
physical sciences that includes Understanding Molecular
Simulation by D. Frenkel and B. Smit, and Computer Simulation of
Liquids by M.P. Allen and D. J. Tildesley.
The Art of Molecular Dynamics Simulation is claimed by the author
to be a practical guide to writing MD simulation code and I fully
agree. In fact, an entire MD code is in the book. In that sense, it
is unique because I do not think that there is any other book on
that topic which includes working molecular dynamics source
code. Due to its availability and popularity among potential
readers, the author decided to use C as the programming language. Although an entire source code can be downloaded from
the Internet, typing it in as you read could serve as a valuable
programming exercise, especially for an inexperienced programmer. Personally, I found the author’s programming style a little
difficult to follow, but this should not be a problem for a more
experienced programmer. On the other hand, a complete lack of
comments in the code is, in my opinion, a more serious drawback.
The book covers a lot of material and it is difficult to list all the
topics without actually rewriting its table of contents here. It
starts with some introductory remarks in Chapter 1, where the
author describes the very basics of MD code for studying
monatomic systems interacting through Lenard-Johns potential.
The main emphasis of this section is on the general methodology of MD simulation and the programming style used throughout the book. However, after working through those first 43
pages, the reader will be able to perform his or her first, very
simple simulations of a soft-disk fluid.
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Considering the rapid development of computer technology,
and the increasing availability of multiprocessor computers at
many universities, it was certainly a good choice on the author’s
part to include a chapter about making your code run in parallel. Implementation of message passing, threads, and vector processing approaches are discussed in Chapter 17. Small tidbits of
information regarding general functionality of the code are
described in the next chapter. Those include generation of random numbers, sorting, some utility functions such as solving a
system of linear equations, memory allocation, and sorting.
The author’s choice of a standard structure for each chapter
helps the reader to fully benefit from the material covered. He
always starts with a presentation of necessary formulae and theoretical concepts. Detailed implementation remarks with the
listing of actual code follows. A measurement section, which
tells the reader how to set up a simulation run and presents
some results which are useful for comparison purposes, concludes most of the chapters. It is also worth mentioning that each
chapter ends with a selection of exercises for the reader. Their
purpose is twofold. Some of them will ask him or her to perform
simulation runs with different initial conditions than those
described in the text. Those very often include references to original papers on a given topic which again allow for easy checking
of validity of the results. Others are coding exercises which
result in improving and extending the functionality of the code.
In my opinion, a serious reader will have to work through those
exercises.
I have never used or written simulation code of any kind, neither
have I a need to do so right now; however, I always wanted to
know how such calculations are performed. The Art of Molecular
Dynamics Simulation contains all information needed to satisfy
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my curiosity and a lot more. I regard it as a very good book for
anybody who wants or needs to write molecular dynamics simulation code. Source availability, choice of case studies, and a
cookbook approach are its three major assets. A large number of
relevant references to original journal articles as well as books on
the topics of molecular dynamics simulation, properties of liquids, and statistical mechanics are included at the end of the
book. Using them will certainly enhance and solidify the reader’s knowledge of the subject.
been devoted to wind-wave investigation and this monograph by Peter
Janssen is an important contribution.
However, there is no such thing as a perfect book and The Art of
Molecular Dynamics Simulation is certainly no exception
(although in my opinion the author did a good job writing it). I
have to admit that I found parts of the book difficult to follow —
at least during the first read. This is probably a result of my lack
of a thorough background in theoretical physics of fluids in particular and in computational physics in general. However,
returning to a given section after browsing through some of the
references allowed me to grasp a few more details rather quickly. Obviously, the level of difficulty Rapaport’s book will present to a reader will vary from person to person and will depend
on a level of familiarity with classical and statistical mechanics,
computer programming, and numerical methods. However, I do
feel that a few concepts resurfacing throughout the book could
be explained a little more thoroughly. For example, the concept
of linked list and cluster analysis algorithm were new to me and
I had to resort to browsing through some of the references given
to understand them a little better. In my opinion, their importance in computer simulations warrants a more thorough discussion; perhaps a few simple diagrams would be very helpful.
“Peter Janssen pays attention to the problem of interaction of atmospheric boundary layer and sea waves. Winds generate ocean waves but,
at the same time, airflow is modified due to loss of energy and momentum to the waves; thus, momentum from the atmosphere to the ocean
depends on the state of the waves. Wind-wave numerical simulation is
made on the basis of the wave action balance equation.”
One more negative aspect caught my attention, namely, a noticeable number of graphs, with multiple curves differing in a value
of some parameter, which are presented without a suitable
description. Although it is sometimes possible to find relevant
information from the subsequent discussion in the text, it would
improve the book if the author included appropriate legends.
On the other hand, a serious reader will try to reproduce all
those graphs from his own simulation runs, therefore he or she
will be able to sort out that issue rather quickly. Perhaps that
was the author’s intention.
In summary, I can testify that Dennis C. Rapaport’s The Art of
Molecular Dynamics Simulation, Second Edition will provide a
source of countless hours of enjoyment for interested readers,
studying a plethora of properties of liquids by means of molecular dynamics simulation. It will certainly improve the reader’s
understanding of computer simulation methodology and classical and statistical mechanics. Through an excellent selection of
exercises and references it will also serve as a starting point for
studies at a much more advanced level than that covered by the
book.
Marek Bromberek
Memorial University of Newfoundland
St. John’s, NL, Canada
THE INTERACTION OF OCEAN WAVES AND WIND, Peter
Janssen, Cambridge University Press, 2004, pp: 292, ISBN
0521465400 (hc); Price: US$120
Je débuterai cette revue de littérature par l’introduction tiré du
World Meteorological Organisation Bulletin, July 2005:
“Surface gravity waves are well known, complicated phenomena,
which have always been the subject of great interest. They are easily
observed but difficult to describe mathematically. Many works have
“About 20 years ago, the author was a member of the International
Wave Modelling Group (WAMDI). At present, the resulting WAM
model is being improved, tested and widely used both at the global scale
and in local water areas. The operational WAM model variant assimilates satellite information for updating the wave forecast. Nowadays, it
is one of the most popular wind-wave models used in many countries.
Cette introduction de I.G. Lavrenov présente bien ce que constitue le chapitre 1, l’introduction, de l’ouvrage de P. Janssen.
Le chapitre 2 sur le bilan d’énergie des vagues océaniques en eau
profonde, présente la dérivation et la formulation des ondes
linéaires, des ondes groupées et le bilan d’énergie. Le bilan d’énergie fait intervenir des termes sources comme le forçage du
vent, l’interaction non-linéaire vagues-vagues et la dissipation
par le déferlement. Ici on introduit succinctement ces termes
sources; ils seront approfondis dans les chapitres suivants.
Le chapitre 3 sur la génération des vagues par le vent, décrit principalement l’effet de la turbulence sur la génération, et l’effet à
double sens du vent sur les vagues, et la cambrure des vagues
sur l’écoulement du vent. Ici, il aurait été intéressant de présenter graphiquement cet effet à double sens à l’aide des profils du
vent et la fréquence des vagues au fur et à mesure que s’installe
cette interaction.
Au chapitre 4, l’auteur présente plus spécifiquement les termes
sources d’interaction non-linéaire vagues-vagues et de dissipation de ceux-ci. L’interaction vagues-vagues est un processus qui
fait intervenir une représentation mathématique sophistiquée et
encore une fois j’aurais aimé que l’on présente de façon conceptuelle la physique se rattachant à ce phénomène de façon
moins abstraite. La dissipation des vagues fait intervenir l’équation de V. Zakharov.
Enfin le chapitre 5 présente l’aspect de la prévision des vagues à
l’aide du modèle numérique du European Centre for MediumRange Weather Forecasts (ECMWF) et l’analyse résultant des
observations conventionnelles des bouées en mer (plus spécifiquement distribuées sur la côte est de l’Amérique du Nord) et
les observations de hauteur de vague provenant de l’altimètre à
bord du satellite ERS-2. C’est un chapitre très intéressant, mais il
ne constitue pas le pôle principal de l’ouvrage. Ce chapitre mentionne aussi les recherches en cours qui pourront devenir dans
un futur rapproché les points majeurs d’avancées dans le
domaine, comme par exemple le rôle des vagues dans la circulation océanique et les courants.
Personnellement, j’ai lu cet ouvrage avec beaucoup d’intérêt et il
pourrait être utile aux étudiants, ingénieurs et scientifiques qui
sont intéressés par la génération des vagues et les problèmes
reliés à leurs prévisions et leurs formations.
André April
Institut des sciences de la mer,
Université du Québec à Rimouski, Rimouski, QC, Canada
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BOOK REVIEWS
WHISTLER-MODE WAVES
IN A HOT PLASMA, Sergei Sazhin,
Cambridge University Press, 1993, pp: 257, ISBN 0521401658
(hc); Price: US$70
An idea of the content and theme of this book can be gained
from a list of the chapters, including their length. Chapters: 1.
Basic equations (35 pp.); 2. Propagation in a cold plasma (25 pp.);
3. Parallel propagation (weakly relativistic approximation) (15
pp.); 4. Parallel propagation (non-relativistic approximation) (45
pp.); 5. Quasi-longitudinal approximation (27 pp.); 6. Quasi-electrostatic approximation (23 pp.); 7. Growth and damping of the
waves (41 pp.); 8. Non-linear effects (15 pp.); 9. Application to
the Earth’s magnetosphere (26 pp.).
This review is structured in the form of a response to the publisher’s description (which is given below in italics):
The book provides an extensive theoretical treatment of whistler-mode
propagation, instabilities and damping in a collisionless plasma. This
is true enough, in that there is presented a massive development
of whistler wave behaviour in a uniform Vlasov plasma and the
results of making various simplifying approximations.
Unfortunately, in spite of the publicity outline (see below), there
is little beyond that. The reader interested in deciding whether it
would be worthwhile to wade through this large mass of
detailed analysis is given little guidance as to what will be
gained thereby. In particular, no connection is made to any actual data except for the first part of the discussion of whistler sonogram “noses” such as shown (one supposes to whet the reader’s
appetite) on p. 1 in Fig. 1. The first part of the discussion of data
at the very end of the book (in Sec. 9.1, pp. 210-214) uses very elementary cold-plasma analysis, based on whistler behaviour near
the magnetosphere equator. The actual application of the
approximations developed in the previous 200 pages is to be
seen in the subsequent part of Chapter 9 (Sec. 9.1(b, c) pp. 214224). There the conclusion is reached that “before the method
can be recommended for practical applications we need to be
able to specify more accurately the model of the electron density and temperature distribution in the ionosphere, have a better
estimate for the effect of ducted ray paths and increase the precision of determining the whistler parameters.” Put in the opposite way, it seems that the generally modest corrections that this
analysis gives in the context of the magnetosphere are usually
less than the other uncertainties. In effect, we have a painstakingly developed methodology which appears to be rather far in
advance of the state of knowledge of the magnetosphere. (By the
way, considering the importance of whistler wave ducting to
observed whistlers, the total lack of discussion of ducting with a
cold plasma and of the effects on ducting and ducting analysis
of the temperature terms the author discusses at such length
make the work much less useful than it might be, even for devoted whistler specialists.)
This book fills a gap between oversimplified analytical studies of these
waves, based on the cold plasma approximation, and studies based on
numerical methods. The principal preoccupation of this work is
the derivation of many variations of the equations for wave
propagation in uniform plasmas, usually quickly specialized to
the whistler wave context. The link to numerical work is not
made with useful clarity. Also, whether the adjective “oversimplified” is justified is unclear. In other words, it is up to the
author to make the case that the whistler enthusiast needs to follow all this work and that the more complete theory is necessary
for adequate understanding of whistler waves. My impression is
that it would take very high-quality whistler data indeed to
necessitate even first-order-temperature theory.
330
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Although the book is primarily addressed to space plasma physicists
and radio physicists, it will also prove useful to laboratory plasma
physicists. “Laboratory plasma physics” would appear to mean
electron-ion laboratory near-Vlasov plasma physics (i.e., excluding pure-electron plasmas). This should not include magnetic
confinement fusion physics (already well developed by the specialists in this field) and apart from the industrial plasma applications where collisions usually cause dominant warm or Vlasov
plasma effects (apart from sheaths). This leaves pretty well the
usually low-density plasma experiments addressing issues of
space plasmas; however, the author does not discuss this aspect
at all. The experimental results and conclusions of such “stars”
in the field as Reiner Stenzel or Walter Gekkelmann (let alone
the earlier work of Roy Gould) pass completely without mention. The statement on laboratory utility would thus seem to be
more of a pious hope than a realistic opinion.
Mathematical methods described in the book can be applied in a
straightforward way to the analysis of other types of plasma waves.
With no actual examples given, it is not clear just what the
author has in mind here, beyond the presentation of the various
approximations.
Problems included in this book, along with their solutions, allow it to
be used as a textbook for postgraduate students. If a course were to
be taught along the lines of the material given here, this book
could well be useful as an associated reference. However, as
implied above, the application of this work is so limited that
such a use by anyone other than the author would seem to be an
unlikely occurrence. In another textbook aspect, that of accessibility, a serious drawback is that the author seems either unwilling or unable to indicate or highlight the plasma physics significance of any of the work. In particular, the lack of any use of
constant-frequency contours in wavevector space to illuminate
the basic aspects of anisotropic propagation seems to border on
the perverse. In addition, the reader is constantly forced to refer
back to equations given in earlier chapters; if the intention is to
use the book for a course, it is much better to repeat the relevant
equations for convenience as needed.
To sum up, while the author has succeeded in getting into print,
in this very specialized monograph, much of the work to which
he has devoted his life, the specialization is such that it is hard to
think of any customers for the work as presented. There may
well be an interesting and exciting book on whistler waves
including their kinetics to be written, but this is not it.
In addition to the general comments just given, while no actual
errors leapt off the page, some few points which are at best confusing and at worst misleading are given below.
(1) On p. 20 at the end of Sec. 1.4, the notation is introduced for
a normalized parallel component of phase velocity pph =
ωmα /kz , (“z” here means component parallel to the magnetic field, the symbol “parallel” not being available for this
review). The point is that since the (n = 0) resonance is ω –
kAv = 0, the quantity ωmα /kz is to be compared with mανz
and not with pz = mανz(1 - (ν 2/c 2))-1/2 = Mανz, the z-component mα momentum corresponding to the z-directed phase
velocity ω/kz. (For instance, in a phase space (z, pz) plot, it is
pzph = Mανz that is needed to make sense of the orbits, and
not mανzph.) My preference would be to use mανzph for the
quantity now termed pph.
(2) On p. 48, with reference to the Storey angle (in any case
more easily seen as arcsin(1/3)), here is a case where any
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snapshot of the constant-frequency contour in wavevector
space (similar to that shown by Helliwell) would show
clearly what is being discussed. To be more specific, the
group velocity (Mw/Mk) is in the direction normal to the frequency contour in k-space. The whistler contours (with electrostatic effects ignored) are bell- or Gaussian-shaped, with
axis along the magnetic field. The largest angle from parallel to the magnetic field the normal makes is at the inflection
site of the contours and this is the Storey angle, giving the
propagation zone angular width. This is readily seen graphically. Similarly, when the effect of electrostatics is included
the curves become W-shaped (the contour edges tending
now to slope away from a right angle). The so-called
Gendrin angle (Sec. 2.2, Eq. (2.45) p. 45) is then simply associated with the bottoms of the W’s.
(3) On p. 95 there is a particularly flagrant example of complicated referencing to follow a change of notation. One sees,
after Eq. (5.5) “ … and P = 1 – νY 2 is the same as in (1.79).”
Now on p. 21, after Eq. (1.79), we were given “P = 1 –X”. It
is after this until after Eq. (1.89) on p. 22, that one learns that
X = Π 2/ω2, where Π α was earlier defined at the top of p.20.
Also on p.22 one is told that Y = Ω/ω. So far so good. What
about ν? After a bit of a search one finds in Chapter 2, on
p. 38, after Eq. (2.10) that (as one might hope) ν = Π 2/Y 2, so
that νY 2 = X, and all is well. The reader should not have to
follow such a tortuous path. This sort of opacity is not what
one would recommend in a text.
CANADIAN INSTITUTE FOR THEORETICAL ASTROPHYSICS/
INSTITUT CANADIEN D’ASTROPHYSIQUE THÉORIQUE
SENIOR RESEARCH ASSOCIATE POSITIONS
CITA is a national centre for theoretical astrophysics located at the University of
Toronto. The Institute expects to offer one or more senior research associate positions of three to five years duration. The starting date will be 1 September, 2007.
Applicants should have an excellent research record in astrophysics and postdoctoral experience. Funds will be available for travel and other research expenses.
The primary duty is to carry out original research in theoretical astrophysics, but
senior research associates are also expected to work with postdoctoral fellows and
to assist with administration of the Institute.
All applicants for senior research associate positions will also be considered
automatically for postdoctoral fellowships.
HOW TO APPLY:
We would prefer electronic submissions.
Please check http://www.cita.utoronto.ca under “Working at CITA” for instructions.
Applicants unable to access the web should mail: a curriculum vitae; statement
of research interests; and arrange for three letters of recommendation to be sent
to:
Professor N. Murray, Director
Canadian Institute for Theoretical Astrophysics
University of Toronto, 60 St. George Street
Toronto, Ontario
Canada M5S 3H8
DEADLINE FOR APPLICATIONS AND ALL LETTERS OF RECOMMENDATION IS 1, DECEMBER 2006.
All qualified candidates are encouraged to apply; however, Canadians and permanent residents will be given priority. In accordance with its Employment Equity
Policy, the University of Toronto encourages applications from qualified women and
men, members of visible minorities, aboriginal peoples and persons with disabilities.
Tudor Wyatt Johnston
INRS-EMT (Université du Québec)
Varennes, Québec, Canada
Faculty Positions, Department of Physics
Astrophysics, Condensed Matter Theory, Particle
Physics and Geophysics
The Department of Physics, University of Alberta
(www.phys.ualberta.ca ) invites applications for five tenure-track faculty
positions. We plan to hire in each of the following areas: observational
astrophysics, condensed matter theory, experimental particle physics,
theoretical particle physics and geophysics. We primarily seek
candidates at the Assistant Professor level, but exceptional candidates
at a more senior level will be considered. The start date for these
positions is July 1, 2007.
Applicants must possess a PhD, have outstanding promise in
research and be committed to teaching. The successful candidates
will be expected to build strong research programs, supervise
graduate students and teach at the undergraduate and graduate
levels.
The Department of Physics has approximately 35 faculty and
115 graduate students, with research interests in astrophysics,
subatomic physics, condensed matter physics, geophysics and
medical physics. The Department has excellent electronics,
machine shop and computational facilities and staff, as well as
access to high performance computational infrastructure (see
www.westgrid.ca ).
Initiatives by the Governments of Alberta and Canada provide
exceptional opportunities for additional funding to establish new
research programs at the University of Alberta. See, for example,
www.albertaingenuity.ca, www.gov.ab.ca/sra, www.icore.ca, and
www.innovation.ca for further information.
The application should include a curriculum vitae, a research
plan, and a description of teaching experience and interests. The
applicant must also arrange to have at least three confidential
letters of reference sent to the relevant selection committee. For
details about the positions and application procedures please see
www.careers.ualberta.ca or www.phys.ualberta.ca/jobs/; or contact
the Department of Physics by e-mail at: [email protected].
All qualified candidates are encouraged to apply; however, Canadians and permanent residents will be given priority. If suitable Canadian citizens and
permanent residents cannot be found, other individuals will be considered. The University of Alberta hires on the basis of merit. We are committed to the
principle of equity in employment. We welcome diversity and encourage applications from all qualified women and men, including persons with disabilities,
members of visible minorities, and Aboriginal persons.
LA PHYSIQUE AU CANADA
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OPPORTUNITIES
TENURE TRACK FACULTY POSITION (S)
Department of Physics and Astronomy
The University of British Columbia
Competition 2006 - 05 — Planetary Astronomy
The Department of Physics & Astronomy at the University of British Columbia
seeks applications for one or more tenure track faculty positions in Planetary
Astronomy / Science. These positions are primarily intended to be at the
Assistant Professor level, but applications from senior candidates will also be
considered. Applicants must have a PhD. Degree or equivalent, relevant
postdoctoral experience, an outstanding research record and a strong interest in
teaching at the undergraduate and graduate level.
UBC is expanding into the area of planetary sciences, led by the Physics and
Astronomy and Earth and Ocean Sciences departments. Candidates are sought
whose research addresses issues related to the formation and evolution of
planetary systems. The Department of Physics and Astronomy has expertise in
solar system dynamics, small-body observations, searches for and studies of
planets in globular clusters, and transit studies of extrasolar planets.
The
deadline
for
receipt
of
all
application
materials
is
November 15, 2006. Applicants should complete the online application form at
http://www.physics.ubc.ca/cgi-bin/Job_Appl_Info.cgi, making sure to select the
Planetary Astronomy competition. A CV, publications list, and statements of
research and teaching interests are required and can be uploaded directly.
Three letters of reference may be submitted electronically to
[email protected] , or sent by mail to:
Chair, Planetary Astronomy Search Committee
Department of Physics and Astronomy
University of British Columbia
6224 Agricultural Road
Vancouver, B.C. V6T 1Z1 Canada
The University of British Columbia hires on the basis of merit and is committed
to employment equity. We encourage all qualified persons to apply - however,
Canadian citizens and permanent residents will be given priority.
PHYSICS UNDERGRADUATE
LABORATORY TECHNOLOGIST
Classification:
Salary:
Education:
Continuing non-academic
starting at $52,000 per annum plus benefits
Bachelor degree in Physics or closely related fields;
Masters/PhD degree with experience in experimental
physics is desirable.
Qualifications:
Teaching:
- Demonstrated excellence in teaching undergraduate labs, including the ability
to assess students’ performance.
- Excellent writing and communication skills and ability to interact with graduate
and undergraduate students from diverse backgrounds.
- Ability to instruct and supervise graduate teaching assistants (TA’s).
- Experience with web development, spreadsheets and other software for laboratory teaching.
Technology:
- Strong background in experimental physics.
- Experience in a range of experimental techniques, such as optics, electronics,
instrumentation, and computer based data acquisition.
Duties:
- Provide technical support for senior labs: setup, troubleshooting, teaching, TA
training, consultation and supervision.
- Develop and produce lab manuals for senior labs.
- Develop and maintain new experiments for senior labs.
- Coordinate and teach laboratories for junior level courses: equipment setup,
TA meetings, develop manuals, administer marks.
Deadline for applications:
Expected starting date:
30 September, 2006
01 December, 2006
Send a cover letter and a CV including the names and contact information
of three references via email to: [email protected]
For information regarding the Department of Physics at the University of Alberta
go to www.phys.ualberta.ca
The University of Alberta hires on the basis of merit. We are committed to the principle
of equity in employment. We welcome diversity and encourage applications from all
qualified women and men, including persons with disabilities, members of visible minorities, and Aboriginal persons.
TENURE TRACK FACULTY POSITION (S)
Department of Physics and Astronomy
The University of British Columbia
Competition 2006 - 06 — Biological Physics
The Department of Physics & Astronomy at the University of British Columbia
seeks applications for one or more tenure track faculty positions in Biological
Physics. These positions are primarily intended to be at the Assistant Professor
level, but applications from senior candidates will also be considered. Applicants
must have a PhD. degree or equivalent, relevant postdoctoral experience,
an outstanding research record and a strong interest in teaching at the
undergraduate and graduate level.
Candidates are sought who will lead a vigorous research program in theoretical,
experimental or computational biophysics, inspired by fundamental questions
in biology. The University of British Columbia offers a rich collaborative research
environment that in biophysics includes members of the Faculties of Science,
Applied Science and Medicine, many of whom are affiliated with the Michael
Smith Laboratories and the British Columbia Cancer Agency.
The University of British Columbia hires on the basis of merit and is committed to
employment equity. We encourage all qualified persons to apply - however,
Canadian citizens and permanent residents will be given priority.
Applicants should complete the online application form at
http://www.physics.ubc.ca/cgi-bin/Job_Appl_Info.cgi, making sure
to select the Biological Physics competition. A CV, publications list,
and statements of research and teaching interests are required and can
be uploaded directly. Three letters of reference may be submitted
electronically to [email protected], or sent by mail to:
Chair, Biological Physics Search Committee
Department of Physics and Astronomy
University of British Columbia
6224 Agricultural Road
Vancouver, B.C. V6T 1Z1
Canada
The deadline for receipt of all application materials is December 15, 2006.
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L’ART DE LA PHYSIQUE
ART
OF
PHYSICS EXHIBITION
There are currently 38 photographs and captions in the CAP's Art of Physics Exhibition. It is
available for loan to any group or organization
wishing to display it. The cost for insurance of
the exhibition is $50 per loan. The CAP will
arrange for shipping to you; you will be responsible for the shipping costs to either return the
exhibit to the CAP or else forward it on to the
next exhibitor.
Interested?
Simply download the booking form found at
https://www.cap.ca/art/artex.html and return it
with your fee to the CAP office.
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ALL UNDELIVERABLE
COPIES IN
CANADA /
TOUTE CORRESPONDANCE NE
POUVANT
ETRE
LIVREE AU
CANADA
should be
returned to /
devra être
retournée à :
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Association of
Physicists/l’Association canadienne
des physiciens et
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Suite/bur. 112
Imm. McDonald Bldg.
Univ. of/d’Ottawa,
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Ottawa, Ontario
K1N 6N5