Neutron Scattering from Magnetic Materials part 1 pps

I intentionally chose the title “Neutron Scattering from Magnetic Materials” ratherthan “Magnetic Neutron Scattering” to emphasize that the proposed book was meant to mate-be useful for

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Chapter 1 - Magnetic Neutron Scattering, Pages 1-24

Abstract | Abstract + References | PDF (212 K)

Chapter 2 - Magnetic Structures, Pages 25-91

Chapter 3 - Representation Analysis of Magnetic Structures, Pages 93-151, Rafik Ballou and Bachir

OuladdiafAbstract | Abstract + References | PDF (551 K)

Chapter 4 - Polarized Neutrons and Polarization Analysis, Pages 153-213, J Schweizer

Chapter 5 - Spherical Neutron Polarimetry, Pages 215-244, P.J Brown

Chapter 6 - Magnetic Excitations, Pages 245-331

Chapter 7 - Paramagnetic and Critical Scattering, Pages 333-361

Chapter 8 - Inelastic Neutron Polarization Analysis, Pages 363-395, L.P Regnault

Chapter 9 - Polarized Neutron Reflectometry, Pages 397-471, C.F Majkrzak, K.V O'Donovan and N.F Berk

Chapter 10 - Small Angle Neutron Scattering Investigations of Magnetic Nanostructures, Pages 473-520,

Albrecht WiedenmannAbstract | Abstract + References | PDF (1120 K)

Chapter 11 - Neutron-Spin-Echo Spectroscopy and Magnetism, Pages 521-542, C Pappas, G Ehlers and F

MezeiAbstract | Abstract + References | PDF (322 K)

Author Index, Pages 543-554

PDF (79 K)

Subject Index, Pages 555-559

PDF (47 K)

Neutron Scattering from Magnetic Materials

Edited by: Tapan Chatterji

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The idea of writing a book on neutron scattering from magnetic materials occurred to meabout four years ago I was then acting as a subeditor of the topic Neutron Scattering forthe encyclopedic book “Scattering: Scattering and Inverse Scattering in Pure and AppliedScience” which was to be published by Academic Press, London [1] There I had to coverthe field of neutron scattering in a very limited space and that was a very difficult andfrustrating task indeed I then realized that to cover the whole field of neutron scatteringwas no longer feasible It would be better to concentrate on one of the most useful topics

in neutron scattering, namely, magnetic neutron scattering which happens to be the maintool of my research work Even this field, starting from the well-known work of Shull andSmart [2], has grown to such an extent that it is very difficult, if not impossible, for a singleauthor to cover the whole field successfully Hence I decided to invite eminent researchers

to write separate chapters on different aspects of neutron scattering from magnetic rials I intentionally chose the title “Neutron Scattering from Magnetic Materials” ratherthan “Magnetic Neutron Scattering” to emphasize that the proposed book was meant to

mate-be useful for experimental researchers who intend to study magnetic materials by neutronscattering It is not a book on the principles of magnetic neutron scattering as the alterna-tive title might suggest There was no need to produce such a book because some excellentbooks [3–6], which cover the principles of magnetic neutron scattering, already existed.The fundamental property of the neutron, that it has a spin, and the fact that the neu-tron beams can be polarized and also analyzed rather easily, have led to magnetic neutronscattering being the preferred probe for investigating magnetic materials The potential ap-plication of neutron scattering to magnetism was first recognized by Bloch [7] only fouryears after the discovery of the neutron by Chadwick [8] and the first successful application

of neutron scattering in magnetic materials was, as mentioned before, made by Shull andSmart [2] The next important breakthrough was also made by Shull and coworkers [9] us-ing polarized neutrons Since then polarized neutron scattering has come to be recognized

to be the most versatile probe for the investigation of magnetic materials It is necessary

to remember that neutron scattering probes the magnetic phenomena directly The alized wave vector and energy dependent susceptibility, which contains all the informationthere is to know about the statics and dynamics of a magnetic system, is directly related

gener-to the neutron scattering cross-section; there exists no unknown constant or function inthis relation The claim becomes even more powerful when polarized neutron scatteringtechniques are used No other technique, which probes magnetic properties of condensedmatter, can ever possibly hope to make such a claim The only limitation of neutron scat-tering techniques is the low flux of the available neutron beams, especially when polarized

v

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a total of eleven) are all devoted to the application of polarized neutron scattering.

In Chapter 1 Chatterji introduces some of the basic principles of magnetic neutron tering and gives references to the relevant books and original papers which the readers maywish to consult for further details

scat-In Chapter 2 Chatterji describes some of the magnetic structures which have been mined by neutron diffraction during the past half a century or so After giving only a fewhints for solving magnetic structures from polycrystalline samples or from single crystals,the author describes the most frequently encountered spin arrangements in high symme-try magnetic solids He then introduces the more complex magnetic structures found inrare-earth elements and other magnetic solids Qualitative and phenomenological argu-ments are given in some cases to rationalize such structures The magnetic structures ofimportant electronic materials like high temperature cuprate superconductors and colossalmagnetoresistive manganites are also considered The chapter is intended to be an intro-duction to prepare the reader for more specialized methods of solving magnetic structures

deter-by group theoretical and polarized neutron diffraction described in Chapters 3–5

In Chapter 3 Ballou and Ouladdiaf introduce group theoretical methods for ing possible magnetic structures compatible with the paramagnetic space group symmetryfrom the knowledge of the propagation vector determined by neutron diffraction Thismethod is especially useful for high symmetry paramagnetic space groups for which deter-mination of magnetic structure is less simple and straightforward They have also providedsome pedagogic examples where the group theoretical methods have been used success-fully for solving the magnetic structures

determin-In Chapter 4 Schweizer introduces the method of polarized neutron diffraction Aftergiving useful definitions and some general principles of polarized neutron scattering theauthor discusses two main uses of polarized neutron diffraction In one method the polar-ization of the scattering beam is not analyzed (flipping ratio method) whereas in the otherthe uniaxial polarization analysis is performed

In Chapter 5 Brown goes further and exploits the full potential of the polarized neutrondiffraction in a technique known as spherical neutron polarimetry (SNP) This techniqueneeds a zero-field sample chamber (CRYOPAD) which has been developed at the InstitutLaue–Langevin in Grenoble Spherical neutron polarimetry has proved to be very useful insolving complex magnetic structures The author gives some examples of complex mag-netic structures which could only be solved by this very powerful technique

In Chapter 6 Chatterji describes the experimental methods of inelastic neutron scattering.Triple-axis spectrometry (TAS) is described in some detail including the newly developedmultiplexing technique The time-of-flight (TOF) technique is discussed only briefly Themagnetic excitations in localized ferro-, antiferro- and ferrimagnetic systems for which theHeisenberg model is applicable, have been considered in some detail The spin excitations

in itinerant magnetic systems like Fe and Ni have also been discussed The spin excitations

of CMR manganites are also extensively discussed

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scat-In some magnetic systems the structural and magnetic degrees of freedom give rise tocontributions which are either superposed, or in some cases, strongly interfere, possiblygiving rise to hybrid modes In such cases inelastic polarized neutron scattering with fullpolarization analysis is extremely useful The use of spherical neutron polarimetry allowsone to determine nine coefficients of the polarization matrix which in turn give the variousnuclear–nuclear, magnetic–magnetic and magnetic–nuclear correlation functions.

In Chapter 9 Majkrzak, O’Donovan and Berk describe the principles of polarized tron reflectometry (PNR) Polarized neutron reflectometry is a probe that is particularlywell suited for determining the nanostructures of magnetic thin films and multilayers andtogether with magnetic X-ray scattering provides a unique means of “seeing” the vectormagnetization with extraordinary spatial details well beneath the surface

neu-In Chapter 10 Wiedenmann describes the technique of Small Angle Neutron Scattering(SANS) especially focusing on the newly developed technique of small angle scatteringwith polarized neutrons (SANSPOL) The later technique is a technique of magnetic con-trast variation which allows weak magnetization fluctuation to be analyzed in addition todensity and concentration variations The author then illustrates the use of this techniquefor investigating nanocrystalline microstructures, soft magnetic materials, magnetic col-loids, ferrofluids etc

In Chapter 11 Pappas, Ehlers and Mezei describe the principles of Neutron Spin-Echo(NSE) spectroscopy which uses the precession of neutron spins in a magnetic field to di-rectly measure the energy transfer at the sample and decouples the energy resolution fromthe beam characteristics like monochromatization and collimation A very high energyresolution can be achieved by this technique The application of this technique in the field

of magnetism benefits from the unique combination of high energy resolution with larization analysis allowing a direct and unambiguous separation of the weak magneticscattering from all other structural contributions The authors give illustrative examples ofthe use of this technique in spin glasses, superparamagnetic fluctuations in monodomainiron particles and geometrically frustrated magnets

po-It is apparent from the above that we have left out quite a few important topics We havedescribed the triple-axis spectrometric (TAS) technique and its application for the investi-gation of magnetic excitations in some detail But we left out almost completely the equallyimportant time-of-flight (TOF) technique for similar investigations We have left out theimportant topic of neutron depolarization and also small angle neutron scattering investi-gation of the flux lattice in superconductors The topic of nuclear spin ordering has beencompletely left out There are definitely other important topics which are not treated in thisbook But as it is the book has already grown to the limit of a single volume and we muststop somewhere I only hope that the book will be of some use for researchers especiallyfor those who intend to begin their research work in this area

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viii Preface

First of all I must thank all contributors of this book They have written the chapters spite their other heavy duties I wish to thank my M Böhm, P Böni, F Demmel, G Felcher,

de-R Gähler, de-R Ghosh, S Mason, B Roessli, P Thalmeier, F Tasset, C Wilkinson and

A Wills for reading parts of the manuscript and providing many helpful suggestions I alsowish to thank B Aubert for her help in the artwork

Tapan Chatterji

References

[1] R Pike and P Sabatier (eds.), Scattering: Scattering and Inverse Scattering in Pure and Applied Science, Academic Press, London (2002).

[2] C.G Shull and J.S Smart, Phys Rev 76 1256 (1949).

[3] W Marshall and S.W Lovesey, Theory of Thermal Neutron Scattering, Oxford University Press, Oxford (1971).

[4] G.L Squires, Thermal Neutron Scattering, Cambridge University Press, Cambridge (1978).

[5] S.W Lovesey, Thermal Neutron Scattering from Condensed Matter, vol 2, Oxford University Press, Oxford (1984).

[6] Yu.A Izyumov and R.P Ozerov, Magnetic Neutron Diffraction, Plenum Press, New York (1970).

[7] F Bloch, Phys Rev 50 259 (1936).

[8] J Chadwick, Nature (London) 129 312 (1932);

J Chadwick, Proc Roy Soc London Ser A 136 692 (1932).

[9] R Nathans, C.G Shull, G Shirane and A Andresen, J Phys Chem Solids 10 138 (1959).

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List of Contributors

Ballou, R., Laboratoire Louis Néel, CNRS, Grenoble, France (Ch 3)

Berk, N.F., National Institute of Standards and Technology, Gaithersburg, MD, USA

(Ch 9)

Brown, P.J., Institut Laue–Langevin, Grenoble, France and Loughborough University,

Loughborough, UK (Ch 5)

Chatterji, T., Institut Laue–Langevin, Grenoble, France (Chs 1, 2, 6, 7)

Ehlers, G., Institut Laue–Langevin, Grenoble, France and SNS Project, Oak Ridge

National Laboratory, Oak Ridge, TN, USA (Ch 11)

Majkrzak, C.F., National Institute of Standards and Technology, Gaithersburg, MD, USA

(Ch 9)

Mezei, F., Hahn-Meitner-Institut Berlin, Berlin, Germany (Ch 11)

O’Donovan, K.V., National Institute of Standards and Technology, Gaithersburg, MD,

USA, University of Maryland, College Park, MD, USA and University of California, Irvine, CA, USA (Ch 9)

Ouladdiaf, B., Institut Laue–Langevin, Grenoble, France (Ch 3)

Pappas, C., Hahn-Meitner-Institut Berlin, Berlin, Germany (Ch 11)

Regnault, L.P., SPSMS/MDN, CEA-Grenoble, Grenoble, France (Ch 8)

Schweizer, J., DRFMC/MDN, CEA-Grenoble, Grenoble, France (Ch 4)

Wiedenmann, A., Hahn-Meitner-Institut Berlin, Berlin, Germany (Ch 10)

ix

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CHAPTER 1

Magnetic Neutron Scattering

Tapan Chatterji

Institut Laue–Langevin, B.P 156X, 38042 Grenoble cedex, France

E-mail: chatt@ill.fr

Contents

1 Introduction 3

2 Basic properties of the neutron 3

3 Neutron source 5

4 Neutron scattering 5

4.1 Definitions of scattering cross-section 6

4.2 The master equation 7

5 Nuclear neutron scattering 9

5.1 Neutron scattering from a single nucleus 9

5.2 Coherent and incoherent scattering 11

6 Magnetic neutron scattering 12

6.1 Scattering of neutrons from unpaired electrons 12

6.2 Scattering of neutrons from crystalline magnetic materials 15

6.3 Elastic magnetic scattering from crystals 15

6.4 Inelastic magnetic scattering 17

6.5 Spin waves 18

6.6 Scattering from spin waves 20

6.7 Paramagnetic scattering 21

6.8 Crystal-field excitations 22

7 Concluding remarks 23

References 23

NEUTRON SCATTERING FROM MAGNETIC MATERIALS

Edited by Tapan Chatterji

1

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Magnetic neutron scattering 3

1 Introduction

The discovery of the neutron in 1932 by Chadwick [1] certainly had the most profoundconsequences The era of nuclear physics began culminating into nuclear technology andthe particle physics was born Elsasser [2] was the first to suggest that the motion of neu-trons would be determined by wave mechanics and thus would be diffracted by crystallinematerials The first demonstration of the diffraction of neutrons was done by Halban andPreiswerk [3] and also by Mitchell and Powers [4] These experiments were done using

a radium–beryllium neutron source The scattering process was of nuclear origin, i.e., theneutrons were scattered by nuclei The idea of magnetic neutron scattering originated from

Bloch [5] He wrote a two-page letter to the editor of the Physical Review in which he

suggested that if the value of the magnetic moment of the neutron was of the same order asthe known measured magnetic moment of the proton, then neutron scattering by the spinand orbital moments of magnetic atoms should be observable Later Alvarez and Bloch [6]showed experimentally that the neutron magnetic moment was about 0.7 of the protonvalue A detailed discussion of the magnitude of magnetic neutron scattering was given

by Halpern and Johnson [7] Following the prediction of antiferromagnetism by Néel [8],Shull and Smart [9] provided the first experimental evidence of this phenomenon in MnO

by neutron diffraction Starting with these developments, decades of research in magneticneutron scattering followed and is still contributing enormously to the microscopic un-derstanding of condensed matter The principles of magnetic neutron scattering have beentreated in several excellent books [10–17] and readers are advised to consult them Here

we attempt to summarize the essential aspects of magnetic neutron scattering

2 Basic properties of the neutron

The scattering of slow neutrons is a very powerful technique to investigate the structureand dynamics of condensed matter The usefulness of this technique stems from the funda-mental properties of the neutron summarized in Table 1

The value of the mass of the neutron 1.674928(1)× 10−24g leads to a de Broglie

wave-length of thermal neutrons of about 1.8 Å which is of the order of the interatomic

dis-tances in condensed matter making interference effects possible Thus neutron scatteringcan yield structural information about condensed matter The energies of thermal and cold

Table 1

Basic properties of the neutron

β-decay life time 885.8 ± 0.9 s

Free neutron decay n → p + e−+ ˜νe

Magnetic moment, µn −1.9130427(5)µN

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For the purpose of neutron scattering investigations there exist two types of neutronsources, viz reactor and spallation neutron source The neutrons emerging from thesesources have very high energies (epithermal neutrons) and are therefore moderated to haveuseful energy ranges The neutrons are called thermal, cold or hot depending on the tem-

perature T of the moderator The probability of neutrons having a velocity between v and

where m is the mass of the neutron and kBis the Boltzmann constant The maximum of the

function P (v) occurs at a velocity v that corresponds to the kinetic energy of the neutron E

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Magnetic neutron scattering 5 Table 2

Values of some physical constants

Physical constant Value

at the Institut Laue–Langevin in Grenoble is the most powerful and has contributed much

to the neutron scattering investigation of condensed matter in general and of the magneticproperties of condensed matter in particular

4 Neutron scattering

Neutrons are scattered by the nuclei and also by the unpaired electrons of the magneticatoms in condensed matter The corresponding neutron scattering is called nuclear neutronscattering and magnetic neutron scattering, respectively In the present book we will mainlyconsider magnetic neutron scattering However, neutron scattering intensity from magneticmaterials is a superposition of both types of scattering In order to be able to separatemagnetic scattering from nuclear scattering and to extract information about the magneticstructure and spin dynamics, it is important to understand the basic principles of bothprocesses In the present section we will give definitions and describe some basic principles

of neutron scattering in general which are valid both for nuclear and magnetic neutronscattering In the following sections we will describe nuclear neutron scattering brieflyfollowed by a more detailed treatment of magnetic neutron scattering

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6 T Chatterji

4.1 Definitions of scattering cross-section

The double differential scattering cross-section is defined by the equation

of identical scatterers N is introduced such that the scattering cross-sections are expressed

per scatterer (per atom) In subsequent formulae for the neutron scattering cross-section

we have divided the right-hand side by the number N or Nm (the subscript “m” standsfor “magnetic”) so that the scattering cross-section is expressed per atom or per magneticatom This number depends on the summation carried out on the right-hand side (totalnumber of scatterers, total number of magnetic atoms or total number of magnetic atomsper unit cell)

If we do not analyze the energy but simply count all the neutrons scattered into a solid

angle dΩ in the direction θ, φ, then the corresponding cross-section, known as the

differ-ential cross-section, is defined by the equation (see Figure 1)

If the scattering is axially symmetric, i.e., the scattering depends only on θ and not on φ,

the above equation becomes

The scattering of neutrons by condensed matter from an incoming state characterized by

a wave vector k0and a spin σ0 into an outgoing state characterized by a wave vector k1

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Magnetic neutron scattering 7

Fig 1 Geometry for scattering experiment (after Squires [12]).

and a spin σ1can be represented by the differential scattering cross-section dσ /dΩ In a neutron scattering experiment the count rate C in a detector that makes a solid angle dΩ and has an efficiency η is given by

dσ dΩ

k0,σ0→k1,σ1

4.2 The master equation

The theory of neutron scattering has been treated in several textbooks [10,12–14,16] andarticles [15,18] We recall here some basic results We consider scattering of neutrons by

a sample consisting of condensed matter which undergoes a change from a state λ0 to a

state λ1while the state of the neutron changes from (k0, σ0) to (k1, σ1) The corresponding

differential scattering cross-section is given by

where Wk0,σ0,λ0→k1,σ1,λ1 is the number of transitions per second from the state k0, σ0, λ0

to the state k1, σ0, λ1 and Φ is the flux of incident neutrons The summation is over all

values of k1 that lie in the small solid angle dΩ in the direction θ, φ, the values k0, λ0

and λ1remaining constant The right-hand side of the above equation is evaluated by usingFermi’s golden rule,

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8 T Chatterji

on the validity of first-order perturbation theory It is certainly valid for nuclear neutron

scattering since the nuclear potential is short range and only s-wave scattering is possible.

The magnetic scattering potential is not short range but it is weak and therefore the goldenrule is still valid To calculate the matrix element in (17) we consider the neutron and the

sample in a large box of volume V0, the incident and scattered neutron functions being

0 eik0 ·r|σ0 and V0−1/2eik1 ·r|σ1, respectively The number of states of the scattered

neutron in the energy interval dE1is

where|k0, |k1 denote the plane waves V0−1/2eik0·r, V −1/2

0 eik1·r From the law of

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