Neutron scattering from magnetic materials potx

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 neu

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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

Free neutron decay n → p + e−+ ˜νe

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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

v + dv follows closely the Maxwell–Boltzmann probability distribution,

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

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 k and a spin σ into an outgoing state characterized by a wave vector k

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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

C = NΦ dΩ η

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

V −1/2

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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We now sum over all final states of the sample λ1and final polarization states σ1, average

over all initial states λ0of the sample, which occur with probability p λ0, and over all initial

states of the neutron, which occur with a probability p σ0to obtain

equation in which we make no assumption about the interaction potential V (r) between

the sample and the neutron

5 Nuclear neutron scattering

5.1 Neutron scattering from a single nucleus

The scattering of neutrons by a nucleus is caused by the nuclear forces which have arange of about 10−12–10−13 cm The wavelength of the thermal neutron is of the order

of 10−8cm which is much larger than the range of nuclear forces It is well known from

the theory of scattering that the scattered wave in such cases is spherically symmetric, i.e.,

analyzed in terms of partial waves, has the character of an s wave (l= 0) We take the

ori-gin to be at the nucleus and the z axis to be along the neutron wave vector k If the incident

neutron is represented by the wave function

we have considered the case of elastic scattering (fixed nucleus) and therefore the wave

vector of the neutron k remains the same for the scattered wave The quantity b which has a dimension of length is defined as the scattering length The scattering length b is in

general a complex quantity given by

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10 T Chatterji

where b0is the length associated with the potential scattering and b and ibare the real

and imaginary parts of the resonance scattering which takes place with formation of thecompound nucleus The imaginary term signifies absorption of neutrons by the nucleusand is important for a few strongly absorbing nuclei like103Rh,113Cd,157Gd,176Lu, etc.The scattering length for such nuclei is strongly dependent on the incident neutron energy.However, for most nuclei the imaginary part of the scattering length is small and the scat-tering length can be considered to be real We shall here confine our discussion to suchnuclei

The scattering cross-section dσ /dΩ for scattering of neutrons from a single fixed cleus can be readily calculated If v is the velocity of the neutron which is the same before and after scattering (elastic), then the number of neutrons passing through the area dS per

The value of the neutron scattering length of a nucleus depends on the spin state of the

nucleus–neutron system For a nucleus with spin I the spin of the nucleus–neutron system

is either I + 1/2 or I − 1/2 Each spin state has its own value of b, denoted by b+and b−

corresponding to the spin states I +1/2 and I −1/2, respectively If the spin of the nucleus

is zero, the nucleus–neutron system can only have spin 1/2 and therefore there is only one

value of the scattering length There exists no sufficiently accurate theory of nuclear forces

so far to allow scattering lengths to be calculated Fortunately scattering lengths of mostnuclei have been determined experimentally and have been tabulated in the literature [19].The neutron scattering length depends on the details of the interaction between the neutron

and the components of the nucleus and therefore the sign and the magnitude of b change in

an irregular fashion with the atomic number This irregularity in neutron scattering length

is due to the superposition of resonance scattering with the slowly increasing potentialscattering as a function of the atomic weight This is in contrast to the X-ray scatteringfactor which increases monotonically with the atomic number Figure 2 shows the values

of bound-atom neutron scattering lengths of elements as a function of the atomic number.The values shown are for naturally occurring isotopic compositions The irregular variation

of the neutron scattering length has some important consequences:

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Fig 2 Values of bound-atom neutron scattering lengths of elements as a function of the atomic number The values shown are for the naturally occurring isotopic compositions The solid squares indicate some values for separated isotopes often used in the isotope-substitution experiments (after Price and Sköld [15]).

1 Unlike the case of X-ray scattering, the neutron scattering lengths of light atoms,like hydrogen and oxygen, are quite large and therefore their positions can be easilydetermined in presence of heavier atoms (see Figures 2 and 3)

2 Neutrons can distinguish between atoms of comparable atomic number

3 Neutrons can usually distinguish isotopes of the same element due to their differentscattering lengths These properties sometimes lead to the choice of the neutronsrather than X-rays as the favorable probe for structural investigations

5.2 Coherent and incoherent scattering

Neutron scattering from naturally occurring elements, which are often composed of

differ-ent isotopes with zero and nonzero nuclear spins I , is more complex than that discussed

in the previous section The scattering lengths depend not only on the kind of element, but

also on the kind of isotope and the quantum number of the angular momentum I ± 1/2

of nucleus–neutron system The outgoing partial neutron waves from the individual nucleiattain different phases and amplitudes and do not interfere completely anymore Henceone must distinguish between coherent and isotope and/or spin-incoherent processes The

scattering amplitude operator for a particular isotope α is given by

where σ is the Pauli spin operator of the neutron and the constants A α and B α are tope specific constants The coherent scattering cross-section is given by the square of the

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iso-12 T Chatterji

Fig 3 Schematic representation of the scattering cross-sections of elements for X-rays and neutrons The radii of

the circles are proportional to the scattering amplitude b Negative values of b are indicated by the cross-hatched

shading (after Bacon [11]).

average of the scattering lengths

σ i= 4π b2 − b2

Diffusions, crystal-field excitations, Stoner excitations and also nuclear excitations due tohyperfine field splitting are examples of such incoherent processes

6 Magnetic neutron scattering

6.1 Scattering of neutrons from unpaired electrons

We first wish to derive an expression for the interaction potential due to the magnetic

interaction between a neutron in spin state σ and a moving electron of momentum p and spin s The corresponding magnetic moment of the neutron is µn= −γ µNσ and that of the electron is µ = −2µBs, where γ = 1.9132 and µNand µBare the nuclear magneton

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and the Bohr magneton The interaction potential is

is the total magnetization operator The operator D is the Fourier transform of M(r) and

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where α and β stand for x, y, z and δ αβis the Kronecker delta.

It is now necessary to evaluate the matrix elements over the neutron spin states For

unpolarized neutrons the products of the neutron spin operator σ satisfy the equation

where r0= e2/(mec2) = 0.28179 × 10−12 cm is the classical radius of the electron and

Nm is the number of magnetic ions Introducing time dependence equation (45) can berewritten as

the tensor (δ αβ QαQβ ) which picks out the components of the magnetization

perpen-dicular to the momentum transfer Q If the direction of magnetization in the scattering

system is unique and if we define a unit vector ˆη along this direction then the directional

dependence of the magnetic scattering is given by the factor 1 Q· ˆη)2 In fact, in able cases this factor enables us to determine the orientation of the magnetic moments incrystalline materials

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favor-Magnetic neutron scattering 15

6.2 Scattering of neutrons from crystalline magnetic materials

So far we have made no assumption about the spatial distribution of the unpaired electronsand have restricted ourselves to the general case of a many-body system of unpaired elec-trons Now we wish to consider the scattering of neutrons from crystalline magnetic sys-tems At first we will restrict ourselves to a localized electron system in which the unpairedelectrons are situated on the magnetic ions which are arranged in the crystalline lattice If

the magnetic atoms are localized on the positions Riand the electrons have LS coupling,

then the time dependent magnetic interaction operator can be given by

2Si is the magnetic moment associated with the atom at Ri in units

of µBand f i (Q) are the atomic form factors defined by

µ iα (0)µ iβ (t ) e−iQ·Ri (0)eiQ ·Ri (t ) e−iEt/¯h dt. (51)

The above expression still contains the correlation function for the atomic coordinates.Therefore the magnetic scattering is not only influenced by the magnetic behavior of thescattering systems but is also influenced by the crystalline structure and dynamics

6.3 Elastic magnetic scattering from crystals

We have seen in the previous section that the magnetic scattering is dependent on the spin–spin correlation function When the magnetic moments of the atoms order at low tempera-

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16 T Chatterji

ture then the spins are correlated If these correlations persist for infinite time the scattering

is purely elastic and correlations are time independent In such case we get from (46)

where R is the coordinate of the th unit cell, r dis the equilibrium position of the atom

inside the unit cell and ui (t ) is the displacement of the atom from the equilibrium position.

If a, b, c are the direct unit cell vectors we have

The reciprocal lattice vector τ is related to the reciprocal lattice vector H normally used

by crystallographers by τ = 2πH In general the unit cell contains several kinds of

crystallographically distinct atoms located in different sites (see International Tables for

Crystallography [21]) So the index d must be split into d = (j, s), where j is the index of

atom in the site s The number of sites is denoted by nsand for a given site the order of the

site nBgives the number of Bravais lattices for that site

We get from (48) after averaging nuclear and electronic parts independently

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is the magnetic unit cell structure factor The factor e−W d (Q) is the well-known Debye–Waller factor which arises due to the thermal motion of the atoms The differential mag-netic cross-section can be written as

is the magnetic structure factor and τM is the reciprocal lattice vector of the magnetic

structure which can be written as τM= τ ± k The vector k characterizes the magnetic

periodicity and is called the propagation or wave vector of the magnetic structure Just as

in the case of the nuclear Bragg reflection, the magnetic Bragg reflection arises for Q= τM

6.4 Inelastic magnetic scattering

We now consider inelastic magnetic scattering of neutrons Inelastic magnetic neutron tering, like the inelastic nuclear scattering, can originate either from single particle exci-tations like crystal-field excitations or collective excitations like spin waves The startingequation for inelastic magnetic scattering is equation (51) which we repeat here

µ iα (0)µ iβ (t ) e−iQ·Ri (0)eiQ ·Ri (t ) e−iEt/¯h dt. (62)

We can split the two correlation functions into a time-independent part plus the part givingthe time dependence at finite times:

The quantities are in the two square brackets when multiplied give four terms The term

Jαβ ii( ∞)I ii(Q, ∞) gives rise to elastic scattering and has already been discussed The term

Jαβ( ∞)I (Q, t) is elastic in the spin system but inelastic in the phonon system and is

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ii(Q, t) gives scattering which is inelastic in both the spin and phonon

systems Here we are mainly concerned with the term Jαβ

ii (t )I ii(Q, ∞) which gives

in-elastic magnetic scattering For a crystalline solid with long range order the total (in-elasticplus inelastic) magnetic cross-section is given by

by Figure 4 The classical picture of the ground state of a simple ferromagnet has all spins

parallel as shown in Figure 4(a) Consider N spins each of S on a line or a ring with nearest neighbors separated by a distance a and coupled by the Heisenberg interaction given by

H = −2J

i

A possible excitation of the ferromagnet has been illustrated in Figure 4(b) in which an

isolated spin is reversed This is an unlikely situation because it costs an energy 8J S2

which is considerable A low-lying excitation which costs much less energy can be formed

if we let all spins share the reversal as has been illustrated in Figure 4(c) In a classical

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Fig 4 (a) Ground state spin alignment of a simple ferromagnet (b) An isolated spin is reversed which costs a lot

of energy (c) A lower energy is achieved by sharing the disturbance among neighboring atoms (d) A spin wave

that describes their spin direction is clearly seen in the plan view (after Bacon [11]).

picture the spin vectors precess on the surface of cones, with each successive spin advanced

in phase by a constant angle such that the variation of the spin vectors forms a wave.Figure 4(d) shows the spins as viewed from above at a particular instant The excitationillustrated in Figures 4(c) and 4(d) has a wavelike form and is called a spin wave or whenquantized, a magnon There will be a series of such waves and each spin wave will possess awavelength and a quantized energy¯hω This corresponds to the normal modes of a coupled

spin system whose energy E = ¯hω and the wave vector q are given by the dispersion

relation For a ferromagnetic linear chain with nearest neighbor interaction this relation isgiven by

At long wavelength (low energy) qa 12(qa)2 In this limit thedispersion relation for the spin wave becomes

where D is called the spin wave stiffness constant Figure 5(a) shows the typical spin wave

dispersion of a ferromagnet For an antiferromagnetic chain the corresponding dispersionrelation is given by

For the long-wavelength limit this equation is reduced to

(71)

We note the difference between the dispersion of spin waves for a ferromagnet and an

anti-ferromagnet For small q the spin wave energy ¯hω is proportional to q2for a ferromagnet

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20 T Chatterji

Fig 5 (a) Spin wave dispersion of a ferromagnetic chain (b) Spin wave dispersion of an antiferromagnetic chain.

and is proportional to q for an antiferromagnet Figure 5(b) shows the typical dispersion

curve of an antiferromagnet In Figures 5(a) and 5(b) there are no gaps in the dispersion

so that the energy ¯hω is zero at q = 0 In many cases there exist such gaps caused by the

single-ion anisotropy or the anisotropy in the exchange interaction

We have given above the simple examples of one-dimensional linear ferromagnetic andantiferromagnetic chains For a more general case the dispersion of the spin waves can bewritten as

6.6 Scattering from spin waves

For a Bravais lattice the generalized susceptibility can be written as

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given in (65) The inelastic neutron scattering cross-section from the spin waves is given

by the sum of terms for one magnon creation and annihilation:

µ iα (0)µ iβ (t ) = µ iα (0)2δ iiδ αβ= 1

12g i2S i (S i + 1)δ iiδ αβ (82)

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6g i2S i (S i + 1) However, because of the Q dependence of

the magnetic form factor the paramagnetic scattering has the same Q dependence, whereas the nuclear incoherent scattering has no such Q dependence Due to the dipolar interac-

tion between the neutron and the unpaired electronic spin, paramagnetic scattering can beseparated from nuclear incoherent scattering by polarization analysis also

If a magnetic field B is applied along the z direction, a nonzero average spin S zleads to

coherent scattering with an equivalent scattering length of γ r0f i (Q)12g i S z The coherentscattering will be superimposed on the nuclear Bragg scattering The rest of the scatteringwill be just like paramagnetic scattering without a magnetic field but reduced in intensity.For a perfect paramagnet, in which there exist no interactions and no correlations be-tween the atomic spins, the magnetic scattering is spatially isotropic, elastic and is inde-pendent of temperature However, the scattering from exchange-coupled paramagnets isquite different, even in the very high temperature limit for which there exist no short rangecorrelations among the spins For such an exchange-coupled system the neutron scatteringcross-section is directly proportional to the generalized wave-vector-dependent suscepti-bility tensor and thus measures the Fourier components of the microscopic magnetizationfluctuation At a second-order phase transition a strong increase in the susceptibility is ex-pected at the wave vector that characterizes the ultimate ordered phase As the temperature

of the system is decreased to the critical temperature from the high temperature side theinelasticity of the scattering decreases drastically The magnetic structure of the orderedphase and therefore the propagation vector depends on the sign and strength of the ex-change interactions and also the crystal structure For a ferromagnet the ordering wave

vector is k= 0 and therefore corresponds to the nuclear Bragg positions For a simple

anti-ferromagnet the ordering vector is, for example (type-II cubic antianti-ferromagnet), k= 1

2 1 2 1

2

which corresponds to Bragg reflections for a lattice whose cell dimensions are twice those

of the nuclear cell Therefore the paramagnetic responses of ferro- and cally coupled systems are very different as one approaches the critical temperature

antiferromagneti-6.8 Crystal-field excitations

Spin wave excitations are collective excitations in the magnetic system Now we considersystems in which single ion effects, rather than the collective excitations, are important

The electrostatic and spin–orbit interactions lift the degeneracy of the unfilled 4f n

con-figuration of, for example, the rare-earth ions and give rise to the J -multiplets Inelastic

neutron scattering is a very powerful technique to determine the positions of energy levelsand the matrix elements of the transitions between them In inelastic neutron scattering

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experiments both energy gain and the energy loss processes can be probed The neutroncan excite the rare-earth ion from a lower state to a higher state with a correspondingloss of the neutron energy or the rare-earth ion is de-excited from a higher energy level to alower energy level and the neutron gains the corresponding energy So the measured energyspectrum exhibits resonance peaks which can be associated with the transitions betweenthe crystal-field-split levels A unique identification of the observed energy levels is onlypossible by comparing the observed intensities of the resonance peaks with the expectedneutron scattering cross-sections of different transitions among the levels The neutronscattering cross-sections of such processes can be derived from (42), (45) and (48),

where Z is the partition function

λ0eE λ0 /(kBT ) The inelastic neutron scattering gations of the crystal field levels is important in cases where optical data are not available,for example, for metallic compounds In neutron scattering investigations one is confrontedwith the problem of distinguishing the crystal-field excitation peaks from, say, phonon

investi-peaks In such cases it is useful to check the temperature and Q dependence of the tensity of the inelastic peak For phonon peaks the intensity is usually proportional to Q2whereas the intensity of the crystal field excitations decreases with Q according to |f (Q)|2.The temperature dependence of the intensity of the inelastic peak is also useful for suchpurpose: phonons obey Bose statistics whereas the population of the crystal-field levels isgoverned by Boltzmann statistics Polarized neutron scattering method is the surest tech-nique for distinguishing phonon scattering from scattering due to the crystal field excita-tions So far we have discussed crystal-field excitations as single particle excitations whichhave no dispersion However, introduction of the interionic exchange interactions results incollective excitations with dispersion These collective excitations are linear combinations

in-of single-ion transitions, called magnetic excitons

7 Concluding remarks

We have in this introductory chapter given only a very short account of the basic principles

of magnetic neutron scattering and its potential applications Some rudimentary account

of the principles of nuclear neutron scattering has also been outlined We have not derivedthe equations but merely listed the important ones The following chapters will consider inmore details many of the points, which are only touched upon here

References

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

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

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24 T Chatterji

[2] W.M Elsasser, C R Acad Sci Paris 202 1029 (1936).

[3] H Halban and P Preiswerk, C R Acad Sci Paris 203 73 (1936).

[4] D.P Mitchell and P.N Powers, Phys Rev 50 486 (1936).

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

[6] L.W Alvarez and F Bloch, Phys Rev 57 111 (1940).

[7] O Halpern and M.H Johnson, Phys Rev 55 898 (1939).

[8] L Néel, Ann Phys (Paris) 17 5 (1932);

L Néel, Ann Phys (Paris) 3 137 (1948).

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

[10] W Marshall and S.W Lovesey, Theory of Thermal Neutron Scattering, Oxford University Press (1971) [11] G.E Bacon, Neutron Diffraction, Third Edition, Clarendon, Oxford (1975).

[12] G.L Squires, Introduction to the Theory of Thermal Neutron Scattering, Cambridge University Press (1978).

[13] S Lovesey, Theory of Neutron Scattering from Condensed Matter, vol 1, Oxford Science Publications (1987).

[14] S Lovesey, Theory of Neutron Scattering from Condensed Matter, vol 2, Oxford Science Publications (1987).

[15] D.L Price and K Sköld, in: Methods of Experimental Physics, vol 23, Part A, p 1, Academic Press, London (1987).

[16] Yu.A Izyumov and R.P Ozerov, Magnetic Neutron Diffraction, Plenum Press, New York (1970) [17] Yu.A Izyumov, V.E Naish and R.P Ozerov, Neutron Diffraction of Magnetic Materials, Consultants Bu- reau, New York (1991).

[18] L Van Hove, Phys Rev 95 249 (1954);

L Van Hove, Phys Rev 95 1374 (1954).

[19] V.F Sears, Methods of Experimental Physics, vol 23, eds K Sköld and D.L Price, Part A, Academic Press, London (1986).

[20] P.J Brown, in: International Tables for Crystallography, vol C, eds A.J.C Wilson and E Prince, Second Edition, p 450, Kluwer Academic Publishers, Dordrecht (1999).

[21] T Hahn, ed., International Tables for Crystallography, vol A, Fourth revised Edition, Kluwer Academic Publishers, Dordrecht (1996).

Trang 32

3 Ferromagnetic and simple antiferromagnetic structures 31 3.1 Cubic Bravais lattice 33 3.2 Hexagonal Bravais lattice 36 3.3 Tetragonal Bravais lattice 36

3.4 Single-k and multiple-k magnetic structures 36

4 Modulated magnetic structures 38 4.1 Sine-wave magnetic structures 39 4.2 Helimagnetic structures 39

5 Complex modulated structures 40 5.1 Magnetic structures of heavy rare-earth elements 40 5.2 Magnetic structures of light rare-earth elements 48 5.3 Spin density wave in chromium 49 5.4 Modulated magnetic structures in CeSb 52 5.5 Modulated magnetic structure of CeAl2 56 5.6 Modulated magnetic structures of EuAs3and Eu(As1−xPx )3 57 5.7 Modulated magnetic structures in MnP: Lifshitz point 62 5.8 Helimagnetic phase in CuO 64 5.9 Modulated magnetic structures in MnSi and FeGe 67 5.10 Microscopic origin of modulated magnetic structures 68

6 Magnetic structures of novel electronic materials 68 6.1 Magnetic structures of cuprates 68 6.2 Magnetic structures of manganites 79

7 Concluding remarks 87 Acknowledgment 88 References 88

NEUTRON SCATTERING FROM MAGNETIC MATERIALS

Edited by Tapan Chatterji

25

Trang 34

Magnetic structures 27

1 Introduction

Elements, alloys and chemical compounds containing atoms with incomplete d- and

f -shells exhibit unique properties characterized by their response to an applied magnetic

field or the magnetic susceptibility They are called magnetic materials Magnetic atomsare mainly situated in three rows of the periodic table, namely the transition elements, the

rare-earth and the actinide elements They have incomplete 3d- (4d-), 4f - and 5f -shells, respectively In Tables 1 and 2 we give 3d- and 4f -shell structures of the iron group and

rare-earth ions For an introduction to the magnetic properties of solids, the reader canconsult, for example, the textbooks of Kittel [1] and Ashcroft and Mermin [2] Due tothe exchange interaction between the magnetic atoms the magnetic moments usually order

at low temperatures Depending on the sign of the exchange the magnetic moments canorder to a ferro- or antiferromagnetic phase In the ferromagnetic phase in a single domainthe magnetic moments are all oriented parallel to each other whereas in a simple antifer-romagnetic phase magnetic moments group together in two sublattices In each sublatticethe moments are oriented parallel to each other but the orientation between the sublatticemoments is antiparallel In a true antiferromagnetic structure the magnetic moments intwo sublattices are equal and compensate each other If the magnetic moments of the twosublattices of the magnetic structure are not equal and therefore a net magnetic moment ex-ists, then the structure is called ferrimagnetic In more complex antiferromagnetic phasesthe magnetic moments can have noncollinear arrangements In another class of magneticmaterials, the magnetic moments are arranged in the form of a spiral or a helix Theseare called modulated magnetic structures In general the periodicities of these modulatedstructures are incommensurate with those of the crystal structures We already mentioned

in Chapter 1 that the first determination of a magnetic structure was done by Shull and

Table 1

Quantum numbers for spin S, orbital L, total angular momentum J , the ground-state terms

of iron group ions with the basic electron configuration 3d n Note that due to the quenching

of orbital angular momentum (L= 0) in a crystal field in the solid state, much better agreement with the experimental values of the magnetic moment is obtained by taking

J = S, than by taking J = |L ± S| appropriate to the free ion according to Hund’s rule

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28 T Chatterji

Table 2

Quantum numbers for spin S, orbital L, total angular momentum J , the ground-state term and the Landé factor g of the ground-state multiplet for rare earth with the basic electron configuration 4f n

2 Determination of magnetic structures

Most of the information on the nature of ordered magnetic phases or magnetic structurescomes from neutron diffraction experiments We have already shown that due to their wavenature thermal neutrons show interference phenomena Neutrons are diffracted from thenuclear structure yielding information on the spatial arrangements of the nuclei of theatoms in crystals In Chapter 1 we have shown that neutrons are also diffracted fromthe magnetic moments of the atoms In the present section we shall show how the po-sitions and the intensities of the diffracted neutron beams provide information about themagnetic structure Just as the crystal structure information is absolutely necessary for un-derstanding the physical properties of the crystals, the magnetic structure information is

a prerequisite to understanding of the magnetic properties of the materials Needless tosay to determine the magnetic structure one should know the crystal structure first Crystalstructures are usually determined by X-ray diffraction Only in special cases is neutron dif-

Trang 36

fraction needed But neutron diffraction is necessary to determine the magnetic structure.X-ray magnetic scattering can sometimes provide useful information about the details of amagnetic structure but is seldom employed to determine an unknown magnetic structure

It is useful to stress that the determination of the magnetic structure is fundamentally ferent from conventional crystal structure determination which consists of determining theatomic positions The full determination of the magnetic structure consists of determin-ing both the magnitude and the direction of all magnetic atomic moments in the magneticunit cell The nuclear structure factor is a complex scalar quantity whereas the magneticstructure factor is a complex vector Since in a conventional diffraction experiment onlythe diffraction intensities can be measured the phase problem is aggravated in the mag-netic case However, because of the form of the dipole interaction between the neutronand the magnetization of the crystal, only those components of the magnetic structure fac-tor perpendicular to the scattering vector contribute to the scattered magnetic intensity It

dif-is thdif-is property which allows us to determine the moment direction Experimentally themagnetic structure is determined by using neutron diffraction intensities measured eitherfrom polycrystalline samples or from single crystals New solid state materials first becomeavailable in polycrystalline form and good single crystals become available (if at all) only

at a later stage, so magnetic structure determination is usually attempted first by the tron powder diffraction technique It must be pointed out that unambiguous determination

neu-of magnetic structure is in principle not possible in many cases by neutron diffraction frompolycrystalline samples Even neutron diffraction from a multidomain single crystal doesnot necessarily provide a unique magnetic structure solution In these cases only polarizedneutron diffraction on a single-domain crystal with three-dimensional polarization analysiscan provide a unique solution (see Chapter 5)

In Chapter 1 we have already derived the equations that form the basis of magneticstructure determination We are now concerned with the practical aspects of the determina-tion of the magnetic structure from neutron diffraction experiments This consists of fourimportant steps:

(1) identification of the propagation vector of the magnetic structure,

(2) determination of the coupling between the magnetic moments,

(3) determination of the moment directions, and

(4) determination of the moment values in Bohr magnetons

Before determining the magnetic structure of an ordered magnetic phase it is worthwhile

to characterize the magnetic ordering by bulk magnetic measurements like susceptibility

and magnetization This gives the ordering temperature TC or TN and also preliminaryinformation about the ordered magnetic phase The bulk magnetic measurements can showimmediately whether the ordered magnetic phase is ferro- or antiferromagnetic

2.1 Polycrystalline samples

We have already mentioned that since new magnetic materials are usually synthesized aspolycrystalline materials the magnetic structures are often determined from powder neu-tron diffraction If the ordered magnetic phase is ferromagnetic then we expect magneticreflections superimposed on the nuclear reflections The magnetic cell is the same as the

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30 T Chatterji

nuclear cell and the propagation vector k= 0 The magnetic structure determination is then

reduced to determining the moment direction and the magnitude of the ordered magneticmoment For this one needs to determine the neutron diffraction intensities from the poly-crystalline sample at least at two temperatures, one at the lowest attainable temperature

which must be much lower than the ordering temperature TCand the other at a temperature

higher than TC We assume, of course, that there are no more magnetic phase transitions

below TC The neutron powder diffraction pattern at the higher temperature is used to termine the nuclear contribution to the total intensity which can be subtracted from the lowtemperature intensity to give the magnetic intensity It is, however, desirable to measurethe powder diffraction pattern at several temperatures in order to characterize the ferro-

de-magnetic phase transition and determine the ordering temperature TCfrom neutron fraction If the magnetic moment is small then the powder neutron diffraction technique isnot very helpful because separation of the weak magnetic intensity from the strong nuclearintensity involves large inaccuracy In this case, polarized neutron diffraction from a singlecrystal is recommended (see Chapters 4 and 5) It is possible to have an antiferromagnetic

dif-structure with the propagation vector k= 0 In this case, the magnetic reflections in

gen-eral are superimposed on the nuclear reflections as in the case of ferromagnetic structuresdiscussed above If the magnetic structure is antiferromagnetic with a propagation vector

k = 0 then magnetic Bragg scattering is present at positions which are different from those

of nuclear Bragg reflections and the determination of magnetic intensities by the method

of subtraction becomes more accurate Once the positions of the magnetic intensities areobtained, the next important task is to determine the periodicity of the magnetic unit cell

It is convenient to use the nuclear or paramagnetic unit cell and express the magnetic

pe-riodicity in terms of the propagation vector k If k is commensurate with the nuclear cell, i.e., when for example, k= (1

2, 0, 0) (the magnetic cell is 2a × b × c), or k = (1

2,12, 0) (the magnetic cell is 2a × 2b × c) or k = (1

2,12,12) (the magnetic cell is 2a × 2b × 2c),

then the determination of the propagation vector from the powder diffraction diagram isnot a very difficult task A simple graphical method described by Rossat-Mignod [4] is

adequate to determine k But if the magnetic cell is incommensurate with the nuclear cell,

i.e., the components of the propagation vector cannot be expressed by simple fractions, the

determination of k becomes more difficult In such case, neutron diffraction investigation

with a single crystal, if available, is recommended Otherwise a computer program oped by Wilkinson et al [5] can be used to index the magnetic reflections of the neutronpowder diffraction pattern Wilkinson et al [5] illustrate their method by giving several ex-amples Once the propagation vector of the magnetic structure (step 1) is determined onetries to determine the coupling (step 2) between the Fourier components of the magneticmoments Sometimes group theoretical methods are employed for this purpose Chapter 3gives the details of this method In addition the readers are referred to the relevant litera-ture [4,6,7] The spin density Patterson function, which is analogous to the crystallographicPatterson function, is also employed to determine the magnetic coupling from the magneticintensity data [8–10] This method works best with a good single crystal data set, but canalso be employed by the calculation of the magnetic ion radial distribution function usingthe magnetic intensity obtained from a powder diffraction diagram [11] The third step

devel-of the magnetic structure determination is to find the moment directions The intensities

of the magnetic reflections, especially the absent reflections, provide useful information

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on the spin directions Otherwise one uses a least-squares method in which the momentdirections are continuously varied to get the best agreement between the observed and cal-culated magnetic structure factors [12] If the magnetic structure model is correct then theleast-squares refinement of the magnetic structure factors provides the information on themoment directions (step 3) and moment values in Bohr magnetons (step 4) Lastly oneshould point out that a complete structure determination is often not possible with powderdiffraction data [13,14]

2.2 Single crystals

If a single crystal is available then the determination of the magnetic structure is easier,first of all because the propagation vector can be determined unambiguously Secondly,the intensities of the magnetic reflections can be determined without the problem of over-lapping reflections For the calculation of magnetic structure factors and refinement of themagnetic structure using single neutron diffraction data the reader is referred to the Cam-bridge Crystallography Subroutine Library developed by Brown and Matthewman [15].However, one must always remember that the intensities are affected more by the extinc-tion effect in the single crystal case This is especially so for crystals which tend to growwith a high degree of perfection Also the correction for the absorption effects is morecomplex Historically magnetic single crystal experiments have normally been performedwith a single detector and there is always a danger of missing some information, espe-cially in the cases where several propagation vectors coexist So even if single crystalsare available it is always worthwhile to carry out a powder diffraction experiment first, or

to perform the single crystal experiment using a large multidetector As we have alreadypointed out the measurement of neutron diffraction intensities from a single crystal is notalways enough for the unique determination of the magnetic structure especially in the highsymmetry cases One needs to produce a single domain by applying symmetry breakingperturbations like magnetic field or uniaxial stress Sometimes unpolarized neutron dif-fraction from a monodomain single crystal is also not enough to determine the magneticstructure unambiguously In this case, it is necessary to use polarized neutrons with three-dimensional polarization analysis for a unique determination of the magnetic structure.This will be discussed in Chapter 5

3 Ferromagnetic and simple antiferromagnetic structures

In the last section we discussed the methods of magnetic structure determination In thepresent section we will describe the different types of magnetic ordering that have beenfound by neutron diffraction As previously mentioned most of our knowledge of theordered magnetic phases and their structures originate directly from neutron diffractiondata We will not attempt to describe exhaustively all types of magnetic structures deter-mined Interested readers can consult [16] and [17] for magnetic structures However, manymagnetic structures have been solved since those publications Rossat-Mignod [4] has de-scribed some selected interesting magnetic structures Examples of magnetic structures are

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32 T Chatterji

given in the book by Bacon [18] For modulated magnetic structures readers are referred

to [19,20] Here we will discuss some of the most common magnetic structures tered in practice The most frequently encountered structure of the ordered magnetic phase

encoun-is a simple ferromagnetic or an antiferromagnetic structure A ferromagnetic structure can

be described by a propagation vector k= 0 which means that the magnetic unit cell is

identical to the chemical unit cell Magnetic reflections are observed in the same tions as nuclear reflections in neutron diffraction experiments Examples of ferromagnetic

posi-structures among 3d elements are body centered cubic Fe, hexagonal Co, face centered

cubic Ni The low temperature magnetic structures of several hexagonal rare-earth metalsare also ferromagnetic There are many examples of ferromagnetic structures in transitionmetal, rare-earth and actinide compounds A simple antiferromagnetic structure is defined

by a propagation vector which corresponds to a symmetry point of the Brillouin zone, i.e.,

k= H/2, where H is a reciprocal lattice vector In Table 3 we give the symmetry points

of the Brillouin zones of the 14 Bravais lattices The propagation vectors corresponding

to each of these symmetry points define distinct antiferromagnetic structures The gation vector corresponding to the symmetry points which are marked by a star are those

propa-which keep the full symmetry of the paramagnetic group Gp, i.e., Gk = Gp The

corre-sponding magnetic structures do not have S or rotational domains.

Table 3

Symmetry points of the Brillouin zones of the 14 Bravais lattices (after Rossat-Mignod [4]) The symmetry

points, k= H/2, are associated with an antiferromagnetic structure Those marked by an asterisk (∗) keep the

full symmetry of the group Gk= Gp

Bravais lattice Symmetry points, k= H/2

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3.1 Cubic Bravais lattice

From Table 3 we notice that for the face centered cubic (f.c.c.) lattice there are two try points and two possible propagation vectors for simple antiferromagnetic ordering with

symme-k= H/2: (0, 0, 1) and (1

2,12,12) corresponding to the so-called type-I and type-II

antifer-romagnetic ordering, respectively Note that the type-III ordering with k= (1,1

2, 0) should

not be called antiferromagnetic ordering since k = H/2 Rossat-Mignod [4] preferred

to classify this magnetic structure as a commensurate structure For type-III structures

k= (1,1

2, 0)=1

4(4, 2, 0) = H/4 Similarly the commonly encountered magnetic structure

type-IA with the wave vector k= (0, 0,1

2) = H/4 is a commensurate structure rather than

an antiferromagnetic structure The stability condition of the three types of ordering in

the f.c.c lattice has been given by Villain [21] on the basis of nearest neighbor J1 and

next nearest neighbor J2Heisenberg exchange interaction The stability condition for the

type-I structure is J1< 0 < J2and those for type-II and type-III are J2<12J1< −J2and

α-MnS and MnSe have type-II antiferromagnetic structures Rare-earth monopnictides

and monochalcogenides RX (R= rare-earth, X = N, P, As, Sb, Bi, S, Se, Te) also have

NaCl-type crystal structures These compounds order usually with type-II netic structures, the exceptions being cerium and neodymium monopnictides which havetype-I magnetic structures Type IA ordering has been found in CeSb and CeBi at lowtemperatures [4] Type-III ordering has been found in MnS2 which crystallizes with the

antiferromag-pyrite-type crystal structure (space group Pa¯3) [18] Although the Mn atoms are situated

on an f.c.c Bravais lattice, strictly speaking there are four Bravais lattices This type of

ordering has also been found [18] in β-MnS, which crystallizes with the cubic zincblende

structure The Mn atoms are situated on an f.c.c sublattice It is to be noted that the f.c.c.lattice is inherently frustrated with respect to antiferromagnetic ordering Topological frus-tration can be explained most simply by considering an equilateral triangle of magneticmoments If any of the magnetic moments are aligned antiparallel, then the third momentcannot be simultaneously aligned antiparallel to both, so the third magnetic moment isfrustrated and is unable to decide where to point An f.c.c lattice has many such equilat-eral triangles (two nearest neighbors of a given atom can themselves be nearest neighbors)and therefore many magnetic moments are frustrated Because of this frustration the f.c.c.lattice leads to a wealth of observed magnetic phases A different magnetic structure ob-

served in CrN is called a type-IV structure with the wave vector k= (1

2,12, 0) Again this

structure should be considered as a commensurate structure rather than an netic structure Another type of magnetic structure been found in MnSe2 [18] in which

antiferromag-Mn atoms lie on an f.c.c Bravais lattice but the crystal structure is primitive cubic (Pa¯3)

and therefore strictly speaking there are actually four Bravais lattices The magnetic

struc-ture is a commensurate square-wave type with wave vector k= 1

3, 0, 1  and 3k = 0, 1, 0.

The four common modes of the magnetic ordering of the f.c.c lattice have been illustrated

in Figure 1

of nuclear Bragg reflections and the determination of magnetic intensities by the... determine the magnetic coupling from the magneticintensity data [8–10] This method works best with a good single crystal data set, but canalso be employed by the calculation of the magnetic ion... breakingperturbations like magnetic field or uniaxial stress Sometimes unpolarized neutron dif-fraction from a monodomain single crystal is also not enough to determine the magneticstructure unambiguously

Tiêu đề	Neutron Scattering from Magnetic Materials
Trường học	University of Science and Technology of China
Chuyên ngành	Physics
Thể loại	Book
Năm xuất bản	2006
Thành phố	Hefei

Định dạng
Số trang	559
Dung lượng	8,05 MB