Table Of ContentMagnetic Small-Angle Neutron
Scattering
OXFORDSERIESONNEUTRONSCATTERINGIN
CONDENSEDMATTER
1. W.G.Williams:Polarizedneutrons
2. E.BalcarandS.W.Lovesey:Theoryofmagneticneutron
andphotonscattering
3. V.F.Sears:Neutronoptics
4. M.F.Collins:Magneticcriticalscattering
5. V.K.Ignatovich:Thephysicsofultracoldneutrons
6. Yu.A.Alexandrov:Fundamentalpropertiesoftheneutron
7. P.A.Egelstaff:Anintroductiontotheliquidstate
8. J.S.HigginsandH.C.Benoˆıt:Polymersandneutronscattering
9. H.Glyde:Excitationsinliquidandsolidhelium
10. V.BalucaniandM.Zoppi:Dynamicsoftheliquidstate
11. T.J.Hicks:Magnetismindisorder
12. H.RauchandS.Werner:Neutroninterferometry
13. R.Hempelmann:Quasielasticneutronscatteringandsolidstatediffusion
14. V.M.NieldandD.A.Kean:Diffuseneutronscattering
fromcrystallinematerials
15. E.H.KisiandC.J.Howard:Applicationsofneutronpowderdiffraction
Magnetic Small-Angle Neutron Scattering
A Probe for Mesoscale Magnetism Analysis
Andreas Michels
University of Luxembourg
1
3
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This book is dedicated to the memory of my parents, and to Anna and Niki.
Preface
One of the very first publications on magnetic small-angle neutron scattering (SANS)
is“DepolarisationundKleinwinkelstreuungvonNeutronendurchGitterfehlerinferro-
magnetischenKristallen”byHelmutKronmu¨ller,AlfredSeeger,andManfredWilkens
from 1963 [1]. The paper is written in German (English translation: “Depolarization
and small-angle scattering of neutrons by lattice imperfections in ferromagnetic crys-
tals”) and was dedicated to Max Born on the occasion of his 80th birthday. In their
theoretical study, the authors have pioneered the use of the continuum theory of mi-
cromagnetics for calculating the magnetic SANS cross section of magnetic materials.
Specifically, Kronmu¨ller, Seeger, and Wilkens studied the spin disorder that is related
tothestrainfieldsofdislocations:inmechanicallydeformedmetals,themagnetization
ishighlyinhomogeneousinthevicinityofdislocations,whichisduetothepresenceof
magnetoelastic coupling. The associated static long-wavelength magnetization fluctu-
ationsrepresentacontrastforelasticmagneticSANS,andthescatteringcrosssection
can be computed by means of micromagnetic theory for samples close to magnetic
saturation. This type of magnetic SANS, denoted as spin-misalignment scattering,
is related to spatial variations in the orientation and magnitude of the magnetiza-
tion. It has been predicted to be about 10−100 times larger than the nuclear SANS
that is related to the volume dilatations of dislocations. With the advent of nuclear
research reactors and the concomitant construction and development of the first ded-
icated SANS instruments, e.g., at Ju¨lich [2] and Grenoble [3], started the exploration
of magnetism and superconductivity on a mesoscopic length scale using neutrons as
a probe. The main ideas of [1] were then verified and extended only later in SANS
experiments on cold-worked single crystals, as summarized in [2,4–6].
Besides being a pioneering study that is full of insights regarding the complex-
ity of the interaction between lattice imperfections and spin structure, the work by
Kronmu¨ller, Seeger, and Wilkens has undoubtly demonstrated the fact that magnetic
SANS is in many respects different than the (nonmagnetic) small-angle scattering by
structural and chemical inhomogeneities. The latter statement may be considered as
the paradigm for this book. It certainly served as the central motivation for the au-
thor to write it. Nuclear SANS and small-angle x-ray scattering are largely based on
the particle-matrixconcept, withparticle formfactors and structure factors being the
basicquantities.MagneticSANSisaboutthemagnetizationdistribution.Theparticle
form factor is obtained as the solution of a volume integral, while the structure-factor
problem, related to the arrangement of and the interaction between the particles,
may be solved by using the methods of statistical mechanics. On the other hand,
thecontinuousvectorialmagnetizationdistributionofamagneticmaterialisobtained
by solving a set of nonlinear partial differential equations (known as Brown’s equa-
tions [7]), which in the context of magnetic SANS—and despite the early work [1]—is
viii Preface
an often overlooked fact. Still, many magnetic SANS studies analyze their data based
ontheparticle-matrixconcept,withtheunderlyingshortcomingassumptionofhomo-
geneously magnetized domains, and neglect the important and often even dominant
magnetic scattering contribution due to misaligned magnetic moments.
As one may anticipate from the previous considerations, the origin of magnetic
SANS is very closely related to the presence of lattice defects in the microstructure of
magnetic materials (e.g., vacancies, dislocations, grain boundaries, pores). This view-
pointhasalsobeenemphasizedintheearlySANSreviewbySpringerandSchmatz[8].
On the mesoscopic length scale that is probed by conventional SANS (∼1−300nm),
the defects are locally decorated by nanoscale spin disorder, which is generated by
(i) spatial variations in the magnetic anisotropy field, and by (ii) spatial variations in
the magnetic materials parameters, most notably the local saturation magnetization.
To be more specific, forces due to the distortion of the crystal lattice in the vicinity of
a microstructural defect tend to rotate the local magnetization vector field along the
main axes of the system of internal stresses (magnetoelastic coupling), while magne-
tocrystallineanisotropytriestopullthemagneticmomentsalongtheprincipalaxesof
the crystal [7]. Likewise, nanoscale spatial variations of the saturation magnetization,
exchange, or anisotropy constants (e.g., at internal interfaces in a magnetic nanocom-
posite or in a nanoporous ferromagnet) give rise to inhomogeneous magnetization
states, which represent a contrast for magnetic SANS. It is of decisive importance to
emphasize that the adjustment of the magnetization along the respective local easy
axes does not occur abruptly, i.e., on a scale of the interatomic spacing, but takes
place over a more extended range. This is a consequence of the quantum-mechanical
exchange interaction, which spreads local perturbations in the magnetization over
larger distances. The size of such spin inhomogeneities is characterized by the mi-
cromagnetic exchange length l , which varies continuously with the applied field and
H
takes on values between about 1−100nm. The ensuing magnetic neutron scattering
appears at scattering angles ψ ∼= λ/l (with λ the neutron wavelength), a regime
H
which is routinely accessible by the SANS technique. We also emphasize that the
observation that lattice-defect-induced magnetization nonuniformities are continuous
functions of the position does not imply the absence of sharp features in the nuclear
grainmicrostructure;forinstance,theremaywellexistsharpparticle-matrixinterfaces
(e.g.,inthechemicalcomposition)inamagneticmaterial,butthecorrespondingspin
distribution (which decorates these interfaces) is continuous over the defects. This is
nothing more than saying that the magnetic microstructure in real space corresponds
to the convolution of the nuclear grain microstructure with micromagnetic response
functionswhichvarywithpositionandfield.Therefore,particleformfactorsmayalso
appear within the micromagnetic description of magnetic SANS, but they naturally
emerge in the course of a calculation, by specifying the geometry of the defect.
The richness and the complex character of magnetic SANS can be grasped by
looking at the figure at the end of this preface, which depicts a selection of computed
spin structures and experimental magnetic SANS cross sections of various polycrys-
tallinemagneticmaterials.Whiletheangularanisotropieswhicharevisibleinsomeof
the cross-section images are exclusively related to the saturated magnetization state,
other patterns have their origin in the nonuniform magnetization distribution of the
Preface ix
material,whichdependsonthemagneticinteractionsandonthecharacteristicsofthe
underlying microstructure. The displayed anisotropies obviously go beyond the well-
knownsin2αdependencyofmagneticneutronscattering,whichepitomizesthehomo-
geneouslymagnetizedsingle-domainstate.Theycanonlybeunderstoodbyanalyzing
thespindistributionofthematerial,i.e.,bycarryingoutmicromagneticcalculationsof
theSANScrosssection.Sincethestandardtextbooksonsmall-anglescattering[9–13]
do not cover the subject of magnetic SANS in sufficient depth, we believe that there
is the necessity to fill this gap with the present book.
The central aim of the book is to provide an introduction into the theoretical
backgroundthatisrequiredtocomputeSANScrosssectionsandcorrelationfunctions
related to long-wavelength magnetization structures; and to scrutinize these concepts
based on the discussion of experimental unpolarized and polarized neutron data. The
bookisprimarilyaboutthetechniqueofmagneticSANS,notaboutmaterials.Itswrit-
ingstyleanddictionmaybedescribedasamixturebetweenmonograph,textbook,and
reviewarticle(inparticularChapter5).Regardingpriorbackgroundknowledge,some
familiarity with the basic magnetic interactions and phenomena as well as scattering
theoryisdesired.ThetargetaudienceconsistsofPh.D.studentsandpostdoctoraland
seniorresearchersworkinginthefieldofmagnetismandmagneticmaterialswhowish
to make efficient use of the magnetic SANS method. The principles and methods that
are laid out in this book will hopefully enable them to analyze and interpret their
SANS experiments.
Besides exposing the different origins of magnetic SANS (Chapter 1), and furnish-
ing the basics of the magnetic SANS technique (Chapter 2), a large part of the book
is devoted to a comprehensive treatment of the continuum theory of micromagnetics
(Chapter 3), as it is relevant for the study of the elastic magnetic SANS cross sec-
tion. Analytical expressions for the magnetization Fourier components allow one to
highlight the essential features of magnetic SANS and to analyze experimental data
bothinreciprocal(Chapter4)aswellasinrealspace(Chapter6).Chapter5provides
an overview on the magnetic SANS of nanoparticles and so-called complex systems
(e.g., ferrofluids, magnetic steels, spin glasses and amorphous magnets). It is this sub-
field where we expect a major progress to be made in the coming years, mainly via
the increased usage of numerical micromagnetic simulations (Chapter 7), which is a
very promising approach for the understanding of the magnetic SANS from systems
exhibiting nanoscale spin inhomogeneity.
Being the result of more than two decades of research, the book contains the
contributions of many people in one or other form. I would like to express my sincere
gratitude to my former and present master and Ph.D. students, postdocs, and to the
manycollaboratorsandcolleaguesworldwide.Veryspecialthanksaredueto:Michael
Adams, Natalie Baddour, Philipp Bender, Frank Bergner, Dmitry Berkov, Salvino
Ciccariello, Sergey Erokhin, Artem Feoktystov, Luis Fern´andez Barqu´ın, Arsen
Goukassov, Sergey Grigoriev, Patrick Hautle, Dirk Honecker, Joachim Kohlbrecher,
Artem Malyeyev, Jos´e Luis Mart´ınez, Konstantin Metlov, Denis Mettus, Sebastian
Mu¨hlbauer, Yojiro Oba, Ivan Titov, and Andrew Wildes. These researchers took over
the most important task of critically reading various parts of the book, and their
most valuable comments and suggestions have without doubt improved the final
x Preface
result. Stephen Lovesey is thanked for his help in establishing the initial contact
to Oxford University Press and for his continuous interest in the progression of the
book. It is also a pleasure to acknowledge the contribution of Rainer Birringer, who
has relentlessly supported me throughout my scientific career. Last, but not least, I
would like to thank my Ph.D. adviser, J¨org Weißmu¨ller, who introduced me to the
magnetic SANS technique.
Luxembourg AndreasMichels
February 2021
SelectionofcomputedspinstructuresandexperimentalmagneticSANScrosssections.
