[1st Edition] 9780444638700, 9780444638717

Table of contents :
Content:
CopyrightPage iv
List of ContributorsPage ix
PrefacePages xi-xiiiEkkes Brück
Contents of Volumes 1–24Pages xv-xxi
Chapter 1 - Physics and Magnetism of Quaternary Heusler AlloysOriginal Research ArticlePages 1-66L. Bainsla, K.G. Suresh
Chapter 2 - Elastic Neutron Diffraction on Magnetic MaterialsOriginal Research ArticlePages 67-143K. Prokeš, F. Yokaichiya
Chapter 3 - Mössbauer Spectroscopy on rare Earth-Based OxidesOriginal Research ArticlePages 145-235P.C.M. Gubbens
Author IndexPages 237-248
Subject IndexPages 249-255
Material IndexPages 257-260

Citation preview

North-Holland is an imprint of Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom First edition 2016 Copyright © 2016 Elsevier B.V. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-444-63871-7 ISSN: 1567-2719 For information on all North-Holland publications visit our website at https://www.elsevier.com

Publisher: Zoe Kruze Acquisition Editor: Poppy Garraway Editorial Project Manager: Shellie Bryant Production Project Manager: Radhakrishnan Lakshmanan Cover Designer: Mark Rogers Typeset by TNQ Books and Journals

List of Contributors L. Bainsla Indian Institute of Technology Bombay, Mumbai, India P.C.M. Gubbens Delft University of Technology, Delft, The Netherlands K. Prokes Helmholtz-Zentrum Berlin fu¨r Materialien und Energie, Berlin, Germany K.G. Suresh Indian Institute of Technology Bombay, Mumbai, India F. Yokaichiya Helmholtz-Zentrum Berlin fu¨r Materialien und Energie, Berlin, Germany

ix

Preface Research on magnetic materials significantly contributes to modern society. Magnetic materials, e.g., help improving the energy efficiency and safety of cars, are used to store the ever growing amounts of data and aid doctors in diagnosing your status of health. New materials are being designed for ever new applications and also new characterization methods are continuously developed. In 1980, Peter Wohlfarth initiated this series of reviews with the aim to have the magnetic research community updated on new developments in ferromagnetic materials. Starting from the fourth volume, Ju¨rgen Buschow joined in as editor and stayed at the helm of this series for 28 years, he enlarged the scope of the series that also was renamed to Handbook of Magnetic Materials. The handbook of which the 25th volume is now lying in front of you, will be continued in the spirit of Ju¨rgen Buschow. It reviews on new materials and developments in experimental techniques. Chapter 1 is an extension of the earlier review on magnetic Heusler alloys, published in volume 21. It reports on substituted quaternary and equiatomic quaternary Heusler alloys (EQHAs) with interesting properties that may be used for applications in spintronic devices. A large number of experimental and theoretical investigations addressing their fundamental and applied interests are reviewed. Combining high Curie temperatures with large spin polarization is the most crucial aspect of these alloys. EQHAs obviously have the advantage of possessing larger spin diffusion length over substituted Heusler alloys such as X2Y1
aY0a Z, due to additional disorder scattering originating from the random distribution of Y and Y0. The structural order and distribution of X, Y, Z atoms in the crystal lattice govern the electronic structure and hence the physical properties of these alloys. The possibility to tune the electronic structure, which in turn affects the magnetic and electrical properties is an advantageous aspect of both types of alloys, making these alloys one of the currently most important material systems that belong to the family of half metallic ferromagnets (HMFs). Within the many HMF materials, spin gapless semiconductors (SGS) recently draw a lot of attention. In the near future many such SGS materials may be developed, both from Heusler and non-Heusler type of alloys.

xi

xii Preface

Various experimental studies indicate that it seems almost impossible, to avoid some degree of antisite disorder. Unfortunately, even a small amount of atomic disorder can cause changes in electronic structure, magnetic and transport properties as well. Therefore the role of disorder in these materials is a crucial factor to be assessed before addressing real-life applications. Elastic neutron diffraction may be the experimental technique of choice to address this, as neutrons can easily distinguish between neighboring atoms in the periodic table, see Chapter 2. Chapter 2 addresses the basic and recent developments in elastic neutron scattering as tool to characterize magnetic materials. This technique has always been intimately connected with studies of magnetic materials, initially in particular applied to large bulky samples. However, as magnetic materials nowadays come along as multilayered films, nanometer sized coreeshell particles, and arrays of quantum dots, this experimental technique has also developed. High-intensity neutron beams, e.g., at modern spallation sources can be used for in situ monitoring of magnetic phase transitions. The small angle neutron scattering with and without polarization analysis has revolutionized the study of magnetic nanoparticles. Exchange bias effects and Skyrmion lattices can be studied in neutron reflectometers. Next to these highlights of modern neutron diffraction, a detailed description is given, how neutron diffraction data can be analyzed to fully exploit the information delivered in a particular experiment. This is done by a step to step elaboration on certain exemplary experiments. Rare earth Mo¨ssbauer spectroscopy is discussed in Chapter 3, in volume 20 the studies on lanthanide intermetallics have been surveyed, here the author concentrates on lanthanide oxides. After a detailed introduction of both experimental and theoretical aspects several rare earthebased oxide materials are discussed. Next to the heavy rare earth sesquioxides such as Gd2O3, Dy2O3, Er2O3, Tm2O3, and Yb2O3 Mo¨ssbauer results on PrO2 are discussed, the spectra are dominated by crystal field effects. The magnetism in perovskite type RMO3 with M, a transition-metal element, can also be explained based on the crystal field results. In the case of RMO4 the crystal field determinations on TmMO4 and YbMO4 are shown in relation to the existence of a JahneTeller effect. Furthermore, the magnetic interplay between rare earth and chromium leads to some interesting features in magnetic and crystallographic behavior. In the superconducting RBa2Cu3O7 compounds studies on magnetic and crystal field behavior leads to quite interesting results. Finally, Gd and Yb Mo¨ssbauer results on the dynamics of the magnetic behavior in pyrochlore and garnet compounds are discussed in combination with mSR results. All three chapters can be used as an introduction to the topic in the particular field of magnetism, without reading the extensive amount of

Preface

xiii

literature. For the researchers in the field, the chapters are intended as source of reference. They are written by experts in the field, who discuss recent developments in experiment and theory. The help from the staff of Elsevier London and Oxford is sincerely acknowledged. Ekkes Bru¨ck TU Delft 2016

Contents of Volumes 1e24 Volume 1 1. Iron, Cobalt and Nickel, by E.P. Wohlfarth 1 2. Dilute Transition Metal Alloys: Spin Glasses, by J.A. Mydosh and 3. 4. 5. 6. 7.

G.J. Nieuwenhuys 71 Rare Earth Metals and Alloys, by S. Legvold 183 Rare Earth Compounds, by K.H.J. Buschow 297 Actinide Elements and Compounds, by W. Trzebiatowski 415 Amorphous Ferromagnets, by F.E. Luborsky 451 Magnetostrictive Rare EartheFe2 Compounds, by A.E. Clark 531

Volume 2 1. 2. 3. 4. 5. 6. 7. 8.

Ferromagnetic Insulators: Garnets, by M.A. Gilleo 1 Soft Magnetic Metallic Materials, by G.Y. Chin and J.H. Wernick 55 Ferrites for Non-Microwave Applications, by P.I. Slick 189 Microwave Ferrites, by J. Nicolas 243 Crystalline Films for Bubbles, by A.H. Eschenfelder 297 Amorphous Films for Bubbles, by A.H. Eschenfelder 345 Recording Materials, by G. Bate 381 Ferromagnetic Liquids, by S.W. Charles and J. Popplewell 509

Volume 3 1. Magnetism and Magnetic Materials: Historical Developments and Present Role in Industry and Technology, by U. Enz

1

2. Permanent Magnets; Theory, by H. Zijlstra 37 3. The Structure and Properties of Alnico Permanent Magnet Alloys, by R.A. McCurrie

107

4. Oxide Spinels, by S. Krupicka and P. Nova´k 189 5. Fundamental Properties of Hexagonal Ferrites with Magnetoplumbite 6. 7. 8. 9.

Structure, by H. Kojima 305 Properties of Ferroxplana-Type Hexagonal Ferrites, by M. Sugimoto 393 Hard Ferrites and Plastoferrites, by H. Sta¨blein 441 Sulphospinels, by R.P. vanStapele 603 Transport Properties of Ferromagnets, by I.A. Campbell and A. Fert 747

xv

xvi Contents of Volumes 1e24

Volume 4 1. Permanent Magnet Materials Based on 3d-rich Ternary Compounds, by K.H.J. Buschow 1

2. Rare EartheCobalt Permanent Magnets, by K.J. Strnat 131 3. Ferromagnetic Transition Metal Intermetallic Compounds, by J.G. Booth

211

4. Intermetallic Compounds of Actinides, by V. Sechovsky´ and L. Havela

309

5. Magneto-Optical Properties of Alloys and Intermetallic Compounds, by K.H.J. Buschow 493

Volume 5 1. Quadrupolar Interactions and Magneto-Elastic Effects in Rare-Earth Intermetallic Compounds, by P. Morin and D. Schmitt 1

2. Magneto-Optical Spectroscopy of f-Electron Systems, by W. Reim and J. Schoenes 133

3. INVAR: Moment-Volume Instabilities in Transition Metals and Alloys, by E.F. Wasserman

237

4. Strongly Enhanced Itinerant Intermetallics and Alloys, by P.E. Brommer and J.J.M. Franse 323

5. First-Order Magnetic Processes, by G. Asti 397 6. Magnetic Superconductors, by Ø Fischer 465

Volume 6 1. Magnetic Properties of Ternary Rare-Earth Transition-Metal Compounds, by H.-S. Li and J.M.D. Coey 1

2. Magnetic Properties of Ternary Intermetallic Rare-Earth Compounds, by A. Szytula

85

3. Compounds of Transition Elements with Nonmetals, by O. Beckman and L. Lundgren 181

4. Magnetic Amorphous Alloys, by P. Hansen 289 5. Magnetism and Quasicrystals, by R.C. O’Handley, R.A. Dunlap and M.E. McHenry

453

6. Magnetism of Hydrides, by G. Wiesinger and G. Hilscher 511

Volume 7 1. Magnetism in Ultrathin Transition Metal Films, by U. Gradmann 1 2. Energy Band Theory of Metallic Magnetism in the Elements, by V.L. Moruzzi and P.M. Marcus

97

3. Density Functional Theory of the Ground State Magnetic Properties of Rare Earths and Actinides, by M.S.S. Brooks and B. Johansson

139

4. Diluted Magnetic Semiconductors, by J. Kossut and W. Dobrowolski 231

Contents of Volumes 1e24 xvii

5. Magnetic Properties of Binary Rare-Earth 3d-Transition-Metal Intermetallic Compounds, by J.J.M. Franse and R.J. Radwa’nski

307

6. Neutron Scattering on Heavy Fermion and Valence Fluctuation 4f-systems, by M. Loewenhaupt and K.H. Fischer 503

Volume 8 1. Magnetism in Artificial Metallic Superlattices of Rare Earth Metals, by J.J. Rhyne and R.W. Erwin

1

2. Thermal Expansion Anomalies and Spontaneous Magnetostriction in RareEarth Intermetallics with Cobalt and Iron, by A.V. Andreev 59

3. Progress in Spinel Ferrite Research, by V.A.M. Brabers 189 4. Anisotropy in Iron-Based Soft Magnetic Materials, by M. Soinski and A.J. Moses

325

5. Magnetic Properties of Rare EartheCu2 Compounds, by Nguyen Hoang Luong and J.J.M. Franse 415

Volume 9 1. Heavy Fermions and Related Compounds, by G.J. Nieuwenhuys 1 2. Magnetic Materials Studied by Muon Spin Rotation Spectroscopy, by A. Schenck and F.N. Gygax 57

3. Interstitially Modified Intermetallics of Rare Earth and 3d Elements, by H. Fujii and H. Sun 303

4. Field Induced Phase Transitions in Ferrimagnets, by A.K. Zvezdin 405 5. Photon Beam Studies of Magnetic Materials, by S.W. Lovesey 545

Volume 10 1. Normal-State Magnetic Properties of Single-Layer Cuprate HighTemperature Superconductors and Related Materials, by D.C. Johnston 1

2. Magnetism of Compounds of Rare Earths with Non-Magnetic Metals, by D. Gignoux and D. Schmitt 239

3. Nanocrystalline Soft Magnetic Alloys, by G. Herzer 415 4. Magnetism and Processing of Permanent Magnet Materials, by K.H.J. Buschow 463

Volume 11 1. Magnetism of Ternary Intermetallic Compounds of Uranium, by V. Sechovsky´ and L. Havela

1

2. Magnetic Recording Hard Disk Thin Film Media, by J.C. Lodder 291 3. Magnetism of Permanent Magnet Materials and Related Compounds as Studied by NMR, By Cz. Kapusta, P.C. Riedi and G.J. Tomka 407

4. Crystal Field Effects in Intermetallic Compounds Studied by Inelastic Neutron Scattering, by O. Moze

493

xviii Contents of Volumes 1e24

Volume 12 1. Giant Magnetoresistance in Magnetic Multilayers, by A. Barthe´le´my, A. Fert and F. Petroff 1

2. NMR of Thin Magnetic Films and Superlattices, by P.C. Riedi, T. Thomson and G.J. Tomka

97

3. Formation of 3d-Moments and Spin Fluctuations in Some RareEartheCobalt Compounds, by N.H. Duc and P.E. Brommer

259

4. Magnetocaloric Effect in the Vicinity of Phase Transitions, by A.M. Tishin

395

Volume 13 1. Interlayer Exchange Coupling in Layered Magnetic Structures, by D.E. Bu¨rgler, P. Gru¨nberg, S.O. Demokritov and M.T. Johnson

1

2. Density Functional Theory Applied to 4f and 5f Elements and Metallic Compounds, by M. Richter

87

3. Magneto-Optical Kerr Spectra, by P.M. Oppeneer 229 4. Geometrical Frustration, by A.P. Ramirez 423

Volume 14 1. III-V Ferromagnetic Semiconductors, by F. Matsukura, H. Ohno and T. Dietl

1

2. Magnetoelasticity in Nanoscale Heterogeneous Magnetic Materials, by N.H. Duc and P.E. Brommer

89

3. Magnetic and Superconducting Properties of Rare Earth Borocarbides of the Type RNi2B2C, by K.-H. Mu¨ller, G. Fuchs, S.-L. Drechsler and V.N. Narozhnyi 199 4. Spontaneous Magnetoelastic Effects in Gadolinium Compounds, by A. Lindbaum and M. Rotter 307

Volume 15 1. Giant Magnetoresistance and Magnetic Interactions in Exchange-Biased Spin-Valves, by R. Coehoorn

1

2. Electronic Structure Calculations of Low-dimensional Transition Metals, by A. Vega, J.C. Parlebas and C.

Demangeat 199

3. IIeVI and IVeVI Diluted Magnetic Semiconductors e New Bulk Materials and Low-Dimensional Quantum Structures, by W. Dobrowolski, J. Kossut and T. Story 289 4. Magnetic Ordering Phenomena and Dynamic Fluctuations in Cuprate Superconductors and Insulating Nickelates, by H.B. Brom and J. Zaanen 379 5. Giant Magnetoimpedance, by M. Knobel, M. Va´zquez and L. Kraus 497

Contents of Volumes 1e24 xix

Volume 16 1. Giant Magnetostrictive Materials, by O. So¨derberg, A. Sozinov, Y. Ge, S.-P. Hannula and V.K. Lindroos 1

2. Micromagnetic Simulation of Magnetic Materials, by D. Suess, J. Fidler and Th. Schrefl 41

3. Ferrofluids, by S. Odenbach 127 4. Magnetic and Electrical Properties of Practical AntiferromagneticMn Alloys, by K. Fukamichi and R.Y. Umetsu, A. Sakuma and C. Mitsumata 209 5. Synthesis, Properties and Biomedical Applications of Magnetic Nanoparticles, by P. Tartaj, and M.P. Morales, S. Veintemillas-Verdaguer, T. Gonzalez-Carren˜o and C.J. Serna 403

Volume 17 1. Spin-Dependent Tunneling in Magnetic Junctions, by H.J.M. Swagten 1 2. Magnetic Nanostructures: Currents and Dynamics, by Gerrit E.W. Bauer, Yaroslav Tserkovnyak, Arne Brataas, Paul J. Kelly

123

3. Theory of Crystal-Field Effects in 3d-4f Intermetallic Compounds, by M.D. Kuz’min, A.M. Tishin

149

4. Magnetocaloric Refrigeration at Ambient Temperature, by Ekkes Bru¨ck 235

5. Magnetism of Hydrides, by Gu¨nter Wiesinger and Gerfried Hilscher 293 6. Magnetic Microelectromechanical Systems: MagMEMS, by M.R.J. Gibbs, E.W. Hill, P. Wright

457

Volume 18 1. Magnetic Properties of Filled Skutterudites, by H. Sato, H. Sugawara, Y. Aoki, H. Harima 1

2. Spin Dynamics in Nanometric Magnetic Systems, by David Schmool 111 3. Magnetic Sensors: Principles and Applications, by Pavel Ripka and Karel Za´veta

347

Volume 19 1. Magnetic Recording Heads, by J. Heidmann and A.M. Taratorin 1 2. Spintronic Devices for Memory and Logic Applications, by B. Dieny, R.C. Sousa, J. He´rault, C. Papusoi, G. Prenat, U. Ebels, D. Houssameddine, B. Rodmacq, S. Auffret, L. Prejbeanu-Buda, M.C. Cyrille, B. Delaet, O. Redon, C. Ducruet, J.P. Nozieres and L. Prejbeanu 107 3. Magnetoelectricity, by L.E. Fuentes-Cobas, J.A. Matutes-Aquino and M.E. Fuentes-Montero 129

xx Contents of Volumes 1e24

4. Magnetic-Field-Induced Effects in Martensitic Heusler-Based Magnetic Shape Memory Alloys, by M. Acet, Ll. Man˜osa and A. Planes 231

5. Structure and Magnetic Properties of L10-Ordered FeePt Alloys and Nanoparticles, by J. Lyubina, B. Rellinghaus, O. Gutfleisch and M. Albrecht 291

Volume 20 1. Microwave Magnetic Materials, by Vincent G. Harris 1 2. Metal Evaporated Media, by Pierre-Olivier Jubert and Seiichi Onodera 65

3. Magnetoelasticity of bcc FeeGa Alloys, by Gabriela Petculescu, Ruqian Wu and Robert McQueeney

123

4. Rare Earth Mo¨ssbauer Spectroscopy Measurements on Lanthanide Intermetallics: A Survey, by P.C.M. Gubbens

227

Volume 21 1. Magnetic Heusler Compounds, by Tanja Graf, Ju¨rgen Winterlik, Lukas Mu¨chler, Gerhard H. Fecher, Claudia Felser, and Stuart S.P. Parkin

1

2. Magnetic Properties of Quasicrystals and Their Approximants, by Zbigniew M. Stadnik

77

3. Bulk Metallic Glasses: Formation, Structure, Properties, and Applications, by Dmitri V. Louzguine-Luzgin and Akihisa Inoue

131

4. Nanocrystalline Soft Magnetic Alloys Two Decades of Progress, by Matthew A. Willard and Maria Daniil 173

Volume 22 1. Magnetic Properties of Perovskite Manganites and Their Modifications by V. Markovich, A. Wisniewski, H. Szymczak 1

2. Magnetocaloric Effect in Intermetallic Compounds and Alloys by Sindhunil Barman Roy

203

3. Future Scaling Potential of Particulate Media in Magnetic Tape Recording by Mark A. Lantz and Evangelos Elefteriou

317

4. Magnetism and Structure in Layered Iron Superconductor Systems by Michael A. McGuire 381

Volume 23 1. Supermagnetism by Subhankar Bedanta, Oleg Petracic and Wolfgang Kleemann

1

2. Non-Fermi Liquid Behavior in Heavy Fermion Systems by Pedro Schlottmann

85

Contents of Volumes 1e24 xxi

3. Magnetic and Physical Properties of Cobalt Perovskites by Bernard Raveau and Md. Motin Seikh

161

4. Ferrite Materials: Nano to Spintronics Regime by R.K. Kotnala and Jyoti Shah 291

Volume 24 1. Spin Glasses by H. Kawamura and T. Taniguchi 1 2. Advances in Giant Magnetoimpedance of Materials by A. Zhukov, M. Ipatov and V. Zhukova

139

3. Advances in Magnetoelectric Materials and Their Application by L.E. Fuentes-Cobas, J.A. Matutes-Aquino, M.E. Botello-Zubiate, A. Gonza´lez-Va´zquez, M.E. Fuentes-Montero and D. Chateigner 237 4. Advances in Magnetic Hysteresis Modeling by Ermanno Cardelli 323

Chapter 1

Physics and Magnetism of Quaternary Heusler Alloys L. Bainslaa and K.G. Suresh1 Indian Institute of Technology Bombay, Mumbai, India 1 Corresponding author: E-mail: [email protected]

Chapter Outline 1. Introduction 1.1 Crystal Structure of Heusler Alloys 1.2 Electronic Structure 1.3 Origin of the Half-Metallic Gap 1.3.1 The Slater-Pauling Rule for HMFs 1.4 Magnetism in Heusler Alloys 1.5 Spin Polarization (PC) 1.6 Electrical Resistivity 1.7 Potential Applications of HMF Materials 2. Results on Substituted Quaternary Heusler Alloys 2.1 Fe2-Based Alloys 2.1.1 Fe2-xCoxMnSi (0
x
2) Alloys 2.1.2 Other Fe2-Based Alloys 2.2 Co2-Based Alloys 2.2.1 Co2Mn1xFexSi (0
x
1) Alloys

2 4 5 6 9 9 10 11 13 13 13 13 19 20

2.2.2 Co2Cr1xVxAl, Co2V1xFexAl, Co2Cr1xFexAl Alloys 2.2.3 Co2FeAl1xSix 2.2.4 Thin Films and Devices 3. Results on Equiatomic Heusler Alloys 3.1 Structural Aspects 3.1.1 CoFeMnZ (Z ¼ Al, Ga, Si, and Ge) Alloys 3.1.2 CoFeCrZ (Z ¼ Al and Ga) Alloys 3.1.3 CoRuFeZ (Z ¼ Si and Ge) Alloys 3.1.4 CoFeTiAl and CoMnVAl Alloys 3.1.5 CoFeCrGe and CoMnCrAl Alloys 3.1.6 NiCoMnZ (Z ¼ Al, Ge, and Sn) Alloys

30 32 32 36 36

36 37 40 41 41 42

20

a

Present address: WPI Advanced Institute for Materials Research, Tohoku University, Katahira 2-1-1, Sendai 980-8577, Japan.

Handbook of Magnetic Materials, Vol. 25. http://dx.doi.org/10.1016/bs.hmm.2016.08.001 Copyright © 2016 Elsevier B.V. All rights reserved.

1

2 Handbook of Magnetic Materials 3.1.7 NiFeMnGa, NiCoMnGa and CuCoMnGa Alloys 3.1.8 MnNiCuSb Alloy 3.2 Magnetic Properties 3.2.1 CoFeMnZ (Z ¼ Al, Si, Ga, and Ge) Alloys 3.2.2 CoFeCrZ (Z ¼ Al and Ga) Alloys 3.2.3 CoRuFeZ (Z ¼ Si and Ge) Alloys 3.2.4 CoFeTiAl and CoMnVAl Alloys 3.2.5 CoFeCrGe and CoMnCrAl Alloys 3.2.6 NiCoMnZ (Z ¼ Al, Ge and Sn) Alloys 3.2.7 NiFeMnGa, NiCoMnGa, and CuCoMnGa Alloys 3.2.8 MnNiCuSb Alloy 3.3 Magneto Transport Properties 3.3.1 CoFeMnZ (Z ¼ Si and Ge) Alloys 3.3.2 CoFeCrZ (Z ¼ Al and Ga) Alloys 3.3.3 CoRuFeZ (Z ¼ Si and Ge) Alloys

43 43 44

44 46 47 47 48 48

49 50 50 50 52

3.4 Spin Polarization Using PCAR 3.4.1 CoFeMnZ (Z ¼ Si and Ge) Alloys 3.4.2 CoFeCrAl Alloy 3.5 Electronic Structure Calculations 3.5.1 CoFeMnZ (Z ¼ Al, Si, Ga, and Ge) Alloys 3.5.2 CoFeCrZ (Z ¼ Al, Ga, Si, and Ge) Alloys 3.5.3 CoMnCrAl, CoFeTiAl, and CoMnVAl Alloys 3.5.4 NiCoMnZ (Z ¼ Al, Ga, Ge, and Sn) Alloys 3.5.5 NiFeMnGa and CuCoMnGa Alloys 3.5.6 CoFeTiZ (Z ¼ Si, Ge, Sn, and Sb) Alloys 3.5.7 Other Equiatomic Quaternary Heusler Alloys 4. Summary and Conclusions References

54 54 55 55

55

57

59

59 60

60

61 61 62

53

1. INTRODUCTION Ever since its discovery in 1903, Heusler alloys starting with Cu2MnAl have attracted the attention of researchers (Bainsla and Suresh, 2016a; Graf et al., 2011; de Groot et al., 1983; Heusler, 1903; Inomata et al., 2008). These are intermetallic compounds, usually consisting of 3d transition elements and some nonmagnetic element from IIIeV group. Majority of the known Heusler alloys are found to be ferromagnetic (FM) in nature, while some are antiferromagnetic. There are mainly two categories of Heusler alloys; ternary metallic or semiconducting materials with a 1:1:1 (represented by XYZ and referred to as half Heusler alloys [HHAs]) or a 2:1:1 (represented by X2YZ and referred to as full Heusler alloys [FHAs]) stoichiometry. When X, Y, or Z

Physics and Magnetism of Quaternary Heusler Alloys Chapter j 1

3

is partially substituted by another 3d or IIIeV group element, it results in the pseudoternary Heusler alloys (given by the formula X2aX0 aYZ or XYZ1bZ0 b with 0 < a < 2 or 0 < b < 1), which are often referred to as quaternary alloys. A special class of quaternary Heusler alloys is the equiatomic Heusler alloys (EQHAs), which show many interesting properties. Recently, these alloys with 1:1:1:1 stoichiometry (XX0 YZ) have also attracted a lot of attention of many researchers (Bainsla and Suresh, 2016a). Heusler alloys, especially the FHAs, are identified as multifunctional materials and draw considerable attention these days. Apart from the halfmetallic ferromagnetic (HMF) nature (de Groot et al., 1983), the multifunctional properties include FM shape memory effect, field-induced strain, giant magnetocaloric effect, giant magnetoresistance (GMR) and anomalous Hall effect. The latter properties are closely associated with the martensitic transition, which is a first-order magnetostructural transition. HMF materials show a fully spin-polarized band structure due to absence of minority spin states at the Fermi level, as shown in Fig. 1, which depicts the difference between a typical metal, a semiconductor, a normal metallic ferromagnet, HMF, and spin gapless semiconductor (SGS). This kind of a band structure leads to many interesting properties, the most prominent being integer magnetic moments, unlike the noninteger moments seen in metallic ferromagnets. The prototype HMF material from the HHA family is NiMnSb. HMF behavior was predicted for the first time in NiMnSb (de Groot et al., 1983), and later this material was studied experimentally (Otto et al., 1989; Soulen et al., 1998). Spin polarization (P) value of 58% was obtained for thin films of NiMnSb (Soulen et al., 1998). In the recent past, several FHAs have been experimentally found to be half metallic. Among these, Co-based alloys have a special place due to their experimentally observed high-spin polarization and high Curie temperatures (Bainsla et al., 2014, 2015a, 2015b; Bainsla and Suresh, 2016a; Varaprasad et al., 2012). HMF Heusler alloys have applications as spin-polarized current sources for GMR [Current perpendicular to plane giant magnetoresistance (CPP-GMR)] devices (Du et al., 2013; Sakuraba et al., 2012), magnetic

FIGURE 1 A cartoon showing the schematic density of states of (A) metals, (B) semiconductors, (C) metallic ferromagnets, (D) half-metallic ferromagnets, and (E) spin gapless semiconductors.

4 Handbook of Magnetic Materials

tunneling junctions, MTJs (Kubota et al., 2009), lateral spin valves (Ikhtiar et al., 2014), and spin injectors to semiconductors (Saito et al., 2013). High values of P were observed at low temperatures for certain Co-based quaternary Heusler alloys such as Co2Fe(Ga0.5Ge0.5) (Varaprasad et al., 2012) and Co2(Fe0.4Mn0.6)Si (Kubota et al., 2009). In the case of Co2Fe(Ga0.5Ge0.5), magnetoresistance (MR) ratios of 183% and 57% were obtained at 10K and at room temperature (RT), respectively (Li et al., 2013). For Co2(Fe0.4Mn0.6)Si, MR ratio of 184% and 58% were obtained at 30K and RT, respectively (Sakuraba et al., 2012). Similarly, many exciting results have been reported in EQHAs (Bainsla and Suresh, 2016a). This article focuses on the half metallic and related properties reported in different types of quaternary Heusler alloys. First, we present the results on systems of the type X2aX0 aYZ, X2Y1bY0 bZ, and X2YZ1bZ0 b with 0
a < 1 and 0
b
1 then on the EQHA systems (a ¼ 1). Various studies have shown that the magnetic properties of Heusler alloys in general are strongly influenced by the structure and the degree of disorder. Moreover, a few crystal structures are preferred as far as half metallicity is concerned. Therefore, we first discuss the general structural aspects, followed by magnetic, transport, spin polarization, and theoretical aspects. After a general discussion on these aspects of FHAs, we present the results reported on various quaternary alloys, thereby illustrating the present status of research of this important class of materials.

1.1 Crystal Structure of Heusler Alloys As mentioned earlier, FHAs are of the form X2YZ and crystallize in the L21 structure with space group Fm-3m (# 225). In general, X and Y atoms are the transition metals, while Z is a nonmagnetic element from the group IIIAeVA. There are four interpenetrating face-centered cubic (FCC) sublattices in the L21 structure as shown in Fig. 2. If each of the four sublattices is occupied by different atoms, a quaternary Heusler structure (XX0 YZ) with a different symmetry (space group no. # 216) is obtained, that is, Y-type (or LiMgPdSn type) (Bainsla and Suresh, 2016a). Depending on the occupancy of the different lattice sites, three different types are possible for the Y-type structure (Alijani et al., 2011a; Dai et al., 2009). The equiatomic quaternary Heusler alloys with 1:1:1:1 stoichiometry exist in the Y-type structure. Inverse Heusler alloy (IHA) structure is another possibility in these classes of materials, where X and Y are the lower and higher transition elements, respectively (Graf et al., 2011). The properties of Heusler alloys are strongly dependent on the atomic arrangements in the crystal structure. A slight disorder in the structure can alter the electronic structure distinctly. The FHAs (X2YZ) show some amount of disorder whereas structural disorder (filling of vacant lattice site) in HHAs rarely occurs. If each of elements (X and Y) resides at their respective sites, the resultant will be a well-ordered cubic structure (L21). Several types of

Physics and Magnetism of Quaternary Heusler Alloys Chapter j 1

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FIGURE 2 Schematic representation of the various structures in Heusler alloys. Reprinted from Galankis, I., Sasio glu, E., Blu¨gel, S., O¨zdogan, K., 2014. Phys. Rev. B 90, 064408 with kind permission from American Physical Society.

disordered structures have been observed in Heusler alloys. Some of the possible disordered structures in FHAs and EQHAs are termed as A2, DO3, and B2 (Graf et al., 2011). The complete disorder in FHA and EQHA structure (X, X0 , Y, and Z distributed randomly) results in the A2 structure with reduced symmetry and body centered cubic (BCC) lattice. On the other hand, the random distribution of X and Y or X and Z leads to the DO3 disorder, which results in a BiF3-like structure. B2 type is another frequently observed structure in which Y and Z sites become equivalent, that is, the disorder is between Y and Z sites only, which leads to a CsCl-like structure. Fig. 3 shows the unit cells associated with different type of structural disorders in Heusler alloys (Table 1).

1.2 Electronic Structure As discussed earlier, half-metallic ferromagnets (HMFs) have two spin bands which show a completely different behavior. While one of them (usually the majority spin band henceforth also referred to as spin up band) shows a typical metallic behavior with a nonzero density of states (DOS) at the Fermi level EF, the minority (spin down) band shows a semiconducting behavior with a gap at EF (as shown in Fig. 1D). Therefore, such materials can be considered as hybrids between metals and semiconductors. The most important manifestation of such a band structure is the observation of integer moments, as

6 Handbook of Magnetic Materials

FIGURE 3 An overview of the different types of crystal structure disorders in Heusler alloys. Here (A), (B), (C) and (D) represents crystal structure for B2, DO3, A2 and B32a disorders. Reprinted from Graf, T., Felser, C., Parkin, S.S.P., 2011. Prog. Sol. Stat. Chem. 39, 1e50 with kind permission of Elsevier.

mentioned later. The band scheme in HMFs allows a completely spin-polarized current. SGS is another important subclass of HMF materials, where a band gap is present for one channel and zero band gap for other channel as shown in Fig. 1E. In some sense, SGS band structure resembles that of a semimetal as far as majority subband is concerned.

1.3 Origin of the Half-Metallic Gap The origin of the half-metallic gap in HHAs and FHAs is the strong hybridization between the d states of the transition metals (Galanakis et al., 2002, 2006a; Graf et al., 2011). Low-lying s-states do not contribute to the DOS near the Fermi level. In the case of HHAs, the gap originates from the strong hybridization between the d states of the two transition metal atoms. For example, in the case of NiMnSb (C1b structure), each Mn ion is surrounded by six Z (]Sb) nearest neighbors. The interaction of Mn with the Z-p states splits the Mn-3d states into a low-lying triplet of t2g states (dxy, dxz, dxz, and a higher lying doublet of eg states (dx2 y2 , d3z2 r2 ). The splitting is mainly due to the different electrostatic repulsion, which is stronger for the eg states. In the

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TABLE 1 Site Occupancy, General Formula, and Structure Type (According to Different Databases) for Different Atomic Orders of Heusler Alloys Site Occupancy

General Formula

Structure Type ICSD

Strukturberichte (SB) Database

Space Group

X, X0 , Y, Z

XX0 YZ

LiMgPdSn

Y

F-43m (# 216)

X ¼ X, Y, Z

X2YZ

Cu2MnAl

L21

Fm-3m (# 225)

X ¼ X0 , Y ¼ Z

X2Y2

CsCl

B2

Pm-3m (# 221)

X ¼ X0 ¼ Y, Z

X3Z

BiF3

DO3

Fm-3m (# 225)

X ¼ X0 ¼ Y ¼ Z

X4

W

A2

Im-3m (# 229)

Space groups corresponding to various structures are also given. Reprinted from Graf, T., Felser, C., Parkin, S.S.P., 2011. Prog. Sol. Stat. Chem. 39, 1e50 with kind permission of Elsevier.

majority band, the Mn 3d states are shifted to lower energies and form a common 3d band with X (X ¼ Ni, Co) 3d states, while in the minority band, the Mn 3d states are shifted to higher energies and are unoccupied, so that a band gap at EF is formed (as shown in Fig. 4). The total magnetic moment (Mt) of HHAs is given by Mt ¼ Zt  18 (Galanakis et al., 2002, 2006a; Graf et al., 2011), where Zt is the total number of valence electrons, which is given by the sum of the up-spin and down-spin electrons while Mt is given by the difference. Z t ¼ N[ þ NY

(1)

Mt ¼ N[  NY ¼ Zt  2N

(2)

FIGURE 4 Schematic illustration of half-metallic band gap (Eg) formation in half Heusler alloys (Galanakis et al., 2006a).

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FIGURE 5 Schematic illustration of the origin of the half-metallic band gap (Eg) in full Heusler alloy Co2MnGe (Galanakis et al., 2006a).

In the case of HHAs, nine minority bands are fully occupied, which results in the relation Mt ¼ Zt  18. The total magnetic moment for half-metallic Heusler alloys should have integer values because both the total number of valence electrons and the number of occupied minority carriers have integer values. This is in sharp contrast to the noninteger moments seen in transition metal ferromagnets such as Fe, Co, and Ni. Similar to the HHAs, the four sp bands lie far below the Fermi level and thus do not contribute to the gap formation in FHAs. For example, in Co2MnGe (Galanakis et al., 2002), for which the hybridization of the 5 d states of the Mn atom and the two Co atoms is important for the gap, as shown in Fig. 5. Co atoms form a simple cubic lattice and Mn atoms (and the Ge atoms) occupy the body-centered site with eight Co atoms as nearest neighbors. The distance between the Co atoms is the second neighbor distance, but the hybridization between these (CoeCo) atoms is very important. The Co 5 d-orbitals are divided into the t2g and eg states, as shown in Fig. 5. The t2g (eg) orbitals of each Co atom can interact only with the t2g (eg) orbitals of the other atom and form bonding and antibonding hybrid orbitals denoted by eg (or t2g) and eu (or t1u). Due to the crystal field, energy of eg > t2g. Considering the hybridization of these CoeCo orbitals with Mn orbitals, eight minority d bands are filled and seven are empty. For these alloys, the minority band contains 12 electrons (as shown in Fig. 5) which results in Mt ¼ Zt  24 rule. Zt is the total number of valence electrons in the primitive cell and for a 2:1:1 based quaternary alloy system can be obtained as the sum of outer s and d electrons of the transition metals and those of s and p electrons of the main group elements. Therefore, the maximum possible magnetic moment in FHAs is 7 mB, which can be achieved only when all the majority d bands are filled. The quaternary Heusler alloys (XX0 YZ) and IHAs (Y2XZ) also follow the Mt ¼ Zt e 24 rule (Galanakis et al., 2014).

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FIGURE 6 Comparison of the experimental magnetic moment per atom with the Slater-Pauling rule in the case of some 3d transition metals and their binary alloys. NiMnSb and Co2MnSi are also included for comparison. Reprinted from Felser, C., Fecher, G.H., Balke, B., 2007. Angew. Chem. Int. Ed. 46, 668 with kind permission of Wiley.

1.3.1 The Slater-Pauling Rule for HMFs Pauling (1938) and Slater (1936) independently found that the magnetic moment per atom (m) for 3d transition metals and their binary alloys could be estimated from their valance electron count (nv). A plot of m versus nv is called Slater-Pauling curve as described by Ku¨bler (2000a). On the basis of Slater-Pauling curve, materials are categorized into two regions as shown in Fig. 6. The first region corresponds to high valence electron concentration (nv > 8) and is the region of itinerant magnetism. The materials with closed packed structures such as FCC or hexagonal close packed are found in this region. The second region corresponds to the low valence electron concentration (nv < 8) and is the region of the localized magnetism. The latter region contains mainly the materials with BCC or its derived structures, and iron is located at the boundary between these two regions, as shown in the figure.

1.4 Magnetism in Heusler Alloys Compared to the HHAs, magnetism in full Huesler alloys is completely different because of the two X atoms occupying the tetrahedral sites, which allow a magnetic interaction between the X atoms and formation of a second, more delocalized magnetic sublattice. Most of the FHAs are FM in nature, while some show antiferromagnetic or ferrimagnetism behavior. A few FHAs have been theoretically predicted to exhibit SGS behavior ¨ zdogan et al., 2013; Skaftouros et al., 2013). However, (Gao et al., 2013; O there are very few among them in which SGS behavior has been confirmed

10 Handbook of Magnetic Materials

experimentally (Bainsla et al., 2015a, 2015c; Quardi et al., 2013; Wang et al., 2008). Mn2CoAl is one such alloy, which has been studied in bulk and thin film forms (Jamer et al., 2013; Xu et al., 2014). Recently, some EQHAs are also identified as SGS materials (Bainsla et al., 2015a, 2015c; Bainsla and Suresh, 2016a).

1.5 Spin Polarization (PC) The spin polarization of a material is defined by P¼

n[ ðEF Þ nY ðEF Þ n[ ðEF Þ þ nY ðEF Þ

(3)

while the transport spin polarization (PC) can be expressed as the imbalance in the majority and minority spin currents (Mazin, 1999); PC ¼

n[ ðEF ÞvF[ nY ðEF ÞvFY n[ ðEF ÞvF[ þ nY ðEF ÞvFY

(4)

where n[(Y)(EF) and vF[(Y) are the DOS at the Fermi level and Fermi velocities for up(down) spin carriers, respectively. The transport spin polarization is the most realistic and relevant parameter from the application point of view. The values of P and PC are same when the Fermi velocities of both the spin currents are equal (Mazin, 1999). Mostly point contact Andreev reflection (PCAR) spectroscopy is used to measure PC. In this measurement, one measures the conductance curves across the FM/superconductor (SC) contact, and the spin polarization for a ballistic contact can be expressed by Eq. (4). Usually, the measured current spin polarization is denoted as P. As most of the spin polarization data reported in this article are based on PCAR, a brief discussion on the working principle of this technique is presented in the following section. Determining spin polarization of an FM material is not easy, as it requires a technique that can discriminate between spin-up and spin-down states near EF. PCAR is a much more practical and convenient technique (Soulen et al., 1998), which requires no field and do not possess any special constraints on sample. It works on the principle of Andreev reflection at a normal metal (N) and SC (S) point contact between the sample and SC by using a mechanical adjustment. The conversion of normal current to supercurrent at metallic interface is called Andreev reflection, which is an important phenomenon in superconductivity. To understand the process of Andreev reflection, consider Fig. 7, which shows an electron from the metal entering the SC in two ways; first by making a Cooper pair with opposite spin electron available at the interface and second by crossing the superconducting band gap (D). In the case of normal metal (P ¼ 0), an electron approaching the interface will find an opposite spin electron at the interface to make the cooper pair, which will lead to the formation of a hole at the interface. This hole has the opposite

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FIGURE 7 PCAR spectra for (A) a normal ferromagnet with P ¼ 0 and (B) a material with P ¼ 100. Andreev reflection process at the metal/superconductor interface for P ¼ 0 and P ¼ 100%. PCAR, point contact Andreev reflection.

momentum to that of the incident electron and propagates away from the interface. Thus, Andreev reflected hole acts like a parallel conducting channel, doubling the normal-state conductance Gn (where G ¼ dI/dV and V is the voltage) of the point contact with applied voltage eV < D. Experimentally, it has been seen that at low voltages, the normalized conductance is indeed twice that of the normal state. For ideal half-metallic systems (i.e., P ¼ 100%), there are no minority carriers available at the interface and so the Andreev reflection is absent there, which means that only single particle excitations are able to contribute to the conductance. These single-particle excitations see the gap in the energy spectrum of SC and suppress the conductance for eV < D. Fig. 7 shows the schematic PCAR data for a normal metallic FM material and an HMF. The normalized conductance curves are fitted to the modified Blonder, Tinkham and Klapwijk (BTK) model using P, D and Z as variables, to estimate the optimum P value. The shape of the conductance curves changes near the superconducting band gap for various Z value curves show a shoulder close to D for high Z values while it became more flat for low Z values (Clowes et al., 2004).

1.6 Electrical Resistivity In addition, information about the spin polarization can be indirectly obtained by a careful analysis of temperature dependence of transport properties, as mentioned in the following section. One of the most commonly used methods to probe for HMF behavior is via electrical resistivity (r) measurements. Electrical resistivity is an intrinsic property of a material that describes the response of a system due to an external electric field. Measurement of r is very effective and useful in probing the magnetic state of the materials because of the strong coupling between the electron mobility and the magnetic state. The origin of electrical resistivity in metals and alloys is due to phonons, magnons, spin fluctuations, etc. Another contribution arises from the electroneelectron

12 Handbook of Magnetic Materials

interaction but mainly at low temperatures. The conduction band in metallic materials is mainly constituted by the s-electrons. Due to their large effective mass, the contribution to the conductivity from the d-electrons is quite negligible. However, d-electrons play an important role in determining the resistivity and its variation due to the large DOS of the d-band. This is due to the fact that scattering probability depends on the DOS into which the electrons are scattered. Therefore, s-d scattering plays an important role for the electrical resistivity behavior in metallic systems. In the high temperature limit, that is, T > qD, the phonon contribution (rph) varies linearly with the temperature. For the low temperature limit, that is, T < qD, rph shows a T5 dependence, where qD is the Debye temperature of the alloy. At low temperatures, the temperature-dependent part of the resistivity mainly arises due to electroneelectron scattering (Balke et al., 2006) and electronemagnon scattering (Blum et al., 2009; Bombor et al., 2013). In the case of HMF materials, electronemagnon scattering is generally found to be absent due to unavailability of the minority carriers at the EF. However, a double magnon scattering is possible in this case. As the temperature increases, the half metallicity trend decreases and the electronemagnon contribution appears. Therefore, the major contributions to the resistivity in a half-metallic ferromagnet arise due to electroneelectron (T2 dependence) and double magnon scattering (T9/2 at low temperatures and T7/2 at high temperatures) (Irkhin et al., 2002; Katsnelson et al., 2008). The change in the resistivity on the application of an external magnetic field is known as MR, which can be used as a probe to study magnetic systems. MR can be estimated from the field dependence of the resistivity data and is given by the relation,   rðT; HÞ rðT; 0Þ MR% ¼  100 (5) rðT; 0Þ Here, r(T,H) and r(T,0) are the resistivity values of the material at temperature T under the application of a magnetic field H and in the absence of the field, respectively. In all metallic systems, a positive contribution to MR arises due to the Lorentz contribution (Sechovsky et al., 1994). In spite of this, many intermetallic compounds show negative MR originating from the suppression of the spin fluctuations and spin disorder by the field. The negative MR arising from the suppression of the spin fluctuations follows quadratic-field dependence in the paramagnetic region (Mallik et al., 1997). Positive MR is obtained in many antiferromagnetic materials, which originates from the enhancement of spin fluctuations by the application of a magnetic field. The change in the magnetic structure of materials, especially in antiferromagnets often leads to the creation of a superzone in its band structure and thus may give rise to positive MR (Das and Rawat, 2000).

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1.7 Potential Applications of HMF Materials As mentioned earlier, Heusler alloys have a special interest due to their high Curie temperatures, theoretically predicted half-metallicity, and tunable electronic properties (Bainsla and Suresh, 2016a; Felser et al., 2007; Graf et al., 2011; Inomata et al., 2008). According to the model by JulliIre (1975), the tunneling magnetoresistance (TMR) of a junction is determined by the spin polarizations of the two FM layers separated by an insulating layer. In addition, GMR ratio in CPP-GMR devices (current perpendicular to the plane) also depends on the bulk spin polarization of the FM electrodes as well as the spin asymmetry at FM/NM interfaces (Nakatani et al., 2010). Therefore, high spin polarization materials will be useful in achieving high MR in both TMR and GMR devices. Therefore, Heuser alloys, in particular have great application potential in many spintronic devices. Since the prediction of half-metallicity in NiMnSb (de Groot et al., 1983), many ternary, pseudoternary, and quaternary Heusler alloys have been found to be half-metallic in nature (Bainsla and Suresh, 2016a; Felser et al., 2007; Graf et al., 2011; Inomata et al., 2008). In the large family of Heusler alloys, Co-based alloys show relatively high Curie temperature and spin polarization (Bainsla and Suresh, 2016a; Felser et al., 2007; Graf et al., 2011; Inomata et al., 2008). Co2FeSi is found to be highest TC among all the half-metallic Heusler alloys, with bulk spin polarization value of 0.45 < P < 0.61 (Makinistian et al., 2013; Yamada et al., 2011). Several attempts are being made to fabricate devices based on these alloys.

2. RESULTS ON SUBSTITUTED QUATERNARY HEUSLER ALLOYS Quaternary Heusler alloys are mainly derived from ternary alloys with substitutions at X, Y, or Z sites. Most of the important quaternary Heusler alloys identified so far belong to either Fe2-based or Co2-based families. In the following section, we first present Fe2-based alloys followed by Co2-based ones.

2.1 Fe2-Based Alloys 2.1.1 Fe2-xCoxMnSi (0
x
2) Alloys 2.1.1.1 Structural Properties One of the well-studied systems from the substituted quaternary Heuasler alloys is the one derived from Fe2MnSi. Partial Co substitution for Fe is reported to affect the magnetic and related properties considerably. Crystal structure analysis for Fe2xCoxMnSi (0 < x < 0.6) alloys was done using RT powder X-ray diffraction (XRD) and 56Fe Mo¨ssbauer data by Bainsla et al.

14 Handbook of Magnetic Materials

(2015d). Analysis of the XRD data revealed that all the alloys existed in single phase with L21 crystal structure. The linear decrease in the lattice parameter value was observed with increase in Co concentration, which is a desirable trend as lattice contraction generally helps in shifting the Fermi level toward the conduction band. This would lead to an increase in the band gap in half metals (Galanakis et al., 2006a). Kondo et al., (2009) also performed structural analysis for Fe2-xCoxMnSi (0 < x < 2.0) alloys and obtained results in agreement with those of Bainsla et al. A rough estimate of the structural ordering was obtained from the I200/I220 and I111/I220 intensity ratios of the XRD data. The S value 1 and a ¼ 0.05 and 0 were obtained for Fe2MnSi and Fe1.4Co0.6MnSi, respectively, by Bainsla et al. (2015d). The obtained S and a values for Fe1.4Co0.6MnSi suggested perfectly ordered structure. Since highly ordered structures are required to obtain high spin polarization, a better knowledge of structural ordering is very important in half-metallic materials. 57 Fe Mo¨ssbauer spectroscopy (MS) has been found to be very useful in probing the structural disorder in some Fe-containing Heusler alloys by Jung et al. (2008, 2009). Therefore, Bainsla et al. (2015d) further studied the crystal structure of these alloys using this technique. In the L21 structure, the parent alloy, Fe2MnSi, has four lattice sites A(1/4, 1/4, 1/4)a, B(1/2, 1/2, 1/2)a, C(3/4, 3/4, 3/4)a, and D(0, 0, 0)a. Here, site A and C belong to Fe atoms, site B belongs to Mn atoms, and D belongs to Si atoms. Mahmood et al. (2004) and Yoon and Booth (1977) have reported a small amount of structural disorder in this compound using the neutron diffraction and Mo¨ssbauer analysis. The results of Bainsla et al. (2015d) on the MS measurements at room temperature (RT) for different x values in Fe2-xCoxMnSi are shown in Fig. 8. As can be seen, the spectra for x ¼ 0.1 are fitted with a singlet which shows that the alloy is paramagnetic at RT. The x ¼ 0.2 spectra are fitted well with a doublet and a sextet, the presence of the former indicating that a fraction of Fe is in the paramagnetic state while the sextet implies a ferromagnetically ordered state. The MS spectra for x ¼ 0.4 are fitted only with a sextet, which hints at a wellordered crystal structure and FM phase at RT. The observed hyperfine field (Hhf) values of 78 and 60 kOe for x ¼ 0.2 and 0.4, respectively, are found to be comparable to the values obtained for other Fe2 base alloys. Bainsla et al. (2015d) estimated the magnetic moment for Fe atoms using the Hhf values. The moment for Fe was found to be 0.5 mB in the case of x ¼ 0.4. Yoon and Booth (1977) had obtained the moment value of 2.3 for Mn using neutron diffraction analysis in the case of Fe2MnSi. From this, the values of Mn, Fe, and Co moments were calculated, which were found to be in agreement with those predicted by Ishida et al. (2008) using electronic structure calculations. 2.1.1.2 Magnetic Properties Kondo et al. (2009) performed thermomagnetic measurements for Fe2-xCoxMnSi (0 < x < 2.0) alloys and found that the materials enter from a higher temperature FM phase to the low temperature antiferromagnetic phase.

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FIGURE 8 57Fe Mo¨ssbauer spectra of Fe2-xCoxMnSi at room temperature. Reprinted from Bainsla, L., Raja, M.M., Nigam, A.K., Varaprasad, B.S.D.Ch. S., Takahashi, Y.K., Suresh, K.G., Hono, K., 2015d. J. Phys. D: Appl. Phys. 48, 125002 with kind permission of IOP.

Hence, the latter transition temperature is termed as the reentry temperature. These authors obtained values of TC ¼ 250K and TR ¼ 61K for (the parent) Fe2MnSi alloy (Pal et al., 2013, 2014), which were in agreement with the reports by Yoon and Booth (1977). Here, TC and TR are, respectively, the Curie temperature and the antiferromagnetic reentry temperature. Kondo et al. (2009) also observed that the reentrant phase gets suppressed with increase in the x, and it completely disappears for x ¼ 0.2. Bainsla et al. (2015d) further investigated the magnetic properties of Fe2-xCoxMnSi (0 < x < 0.6) series and obtained a similar trend for M versus T curves, as obtained by Kondo et al. (2009). TC was found to increase with increase in x; it increases from 220 to 368K as x varies from 0 to 0.2 and it is above 400K for x > 0.2. They attributed the observed magnetization trend to the strong hybridization between the d-states of Fe and Co, which results in the enhancement of the FM nature as reported theoretically by Ishida et al. (2008). The experimental observations of thermomagnetic curves are in good agreement with the calculations by Ishida et al. As per their calculations, one of the assumed antiferromagnetic states is energetically comparable to the FM state for x ¼ 0. This means that these two phases compete with each other. They also found that the FM state becomes stronger with increase in the x value. Based on the estimated Rhodes-Wohlfarth ratio (pc/ps), Bainsla et al. (2015d) have suggested that these alloys show itinerant magnetism. Here, ps and pc correspond

16 Handbook of Magnetic Materials

to the saturation moment at low temperatures and the effective moment per magnetic ion deduced from the Curie constant, respectively. Isothermal magnetization measurements for Fe2-xCoxMnSi (0 < x < 2.0) alloys were also performed by Kondo et al. (2009), and the saturation magnetization (MS) values obtained by them were found to be in good agreement with the Slater-Pauling rule for x  1. The deviation was attributed to the competition between FM and antiferromagnetic transitions. MS values were found to increase with x at temperatures of 3, 150, and 300K, which was attributed to the strong hybridization between the d states of the Fe and Co ions (Ishida et al., 2008). MS value was found to increase from 2.2 to 3.5 mB/ f.u. as x was varied from 0 to 0.4. 2.1.1.3 Magnetotransport Properties Bainsla et al. (2015d) measured the temperature variation of electrical resistivity in the case of x ¼ 0.05, 0.1, 0.2, and 0.4 in different magnetic fields, as shown Fig. 9. There are clear peaks close to the transition temperatures (TC and TR) in the resistivity versus temperature curves. They also found that the slope of these curves changes near the transition temperature with application of a magnetic field, which is due to the reduction in the spin fluctuations. It is

FIGURE 9 Temperature dependence of electrical resistivity in Fe2-xCoxMnSi alloys in different fields. Here (A)e(D) show electrical resistivity curves for x varying from 0.05 to 0.4. Reprinted from Bainsla, L., Raja, M.M., Nigam, A.K., Varaprasad, B.S.D.Ch. S., Takahashi, Y.K., Suresh, K.G., Hono, K., 2015d. J. Phys. D: Appl. Phys. 48, 125002 with kind permission from IOP.

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FIGURE 10 Magnetoresistance (MR) isotherms of Fe1.95Co0.05MnSi and Fe1.8Co0.2MnSi at different temperature regimes. Reprinted from Bainsla, L., Raja, M.M., Nigam, A.K., Varaprasad, B.S.D.Ch. S., Takahashi, Y.K., Suresh, K.G., Hono, K., 2015d. J. Phys. D: Appl. Phys. 48, 125002 with kind permission from IOP.

also evident from these curves that metallic nature changes to nonmetallic as x is increased to 0.4. Electrical resistivity shows different behavior above and below the TC in the case of x ¼ 0.1 and 0.2, the x region where both FM and antiferromagnetic interactions are competing with each other. These alloys (x ¼ 0.1 and 0.2) show metallic behavior up to the TC, while they follow a semiconducting-like behavior above TC. Such a behavior was earlier reported in the case of Fe2Mn1xCrxSi (Pal et al., 2014), Fe2VAl (Nishino et al., 1997), and Mn2CoAl (Quardi et al., 2013). Quardi et al. (2013) studied Mn2CoAl in detail and proved the SGS behavior on the basis of resistivity and Hall effect measurements. Taking into account various factors, Bainsla et al. (2015d) suggested the possibility of SGS behavior in x ¼ 0.4 alloy. A nonmetallic electrical conduction was observed in the case of x ¼ 0.4 (Bainsla et al., 2015d) as shown in Fig. 9D. The results of the fitting showed that the electron-phonon interaction dominates in the high-temperature region for x ¼ 0.05 and 0.1. They also observed the absence of T2 dependence for the low temperature regime, which indicates the absence of electron-magnon interaction, thereby indirectly suggesting the possibility of half-metallic nature. Bainsla et al. (2015d) calculated MR for x ¼ 0.05 and 0.2, as shown in Fig. 10. MR was found to be nominally negative in the TR < T < TC, while it was very small and field independent in the paramagnetic region for x ¼ 0. Below TR, MR is nominally positive and shows field-induced irreversibility which may be due to the spin fluctuations induced by the field on the canted spin structure that causes an increase in the resistivity. The MR trend of the

18 Handbook of Magnetic Materials

parent alloy with x ¼ 0 is retained in the x ¼ 0.05 as well, as can be seen from Fig. 10. Field-induced irreversibility was found to be absent for x ¼ 0.2, and this alloy does not show any reentrant phase. Bainsla et al. (2015d) attributed the presence of field-induced irreversibility to two competing magnetic phases (as evident from the magnetization data). They also mentioned that fieldinduced irreversibility seen in this case is similar to the Nd5Ge3, where the irreversibility is closely related to the occurrence of reentrant antiferromagnetic phase (Maji et al., 2010). 2.1.1.4 Spin Polarization PCAR measurements were performed by Bainsla et al. (2015d) to estimate the current spin polarization for some high TC alloys. The materials with x values of 0.2 and 0.4 show high TC and thus are hence interesting for devices. In addition to high TC, composition with x ¼ 0.4 shows a well-ordered crystal structure as revealed by the XRD and Mo¨ssbauer analysis. The differential conductance curves using PCAR were obtained for x ¼ 0.2 and 0.4 at the sample/Nb interface. Normalized differential conductance curves for these alloys at 4.2K are shown in Fig. 11. The values for P, D, and Z for best fitting are given in Fig. 11. As mentioned earlier, the shape of the conductance curves

FIGURE 11 Normalized differential conductance curves of the Fe2-xCoxMnSi (x ¼ 0.2 and 0.4) alloys. (A) and (C) represents normalized conductance curves, while (B) and (D) show P vs. Z plots for x ¼ 0.4 and 0.2 respectively. Reprinted from Bainsla, L., Raja, M.M., Nigam, A.K., Varaprasad, B.S.D.Ch. S., Takahashi, Y.K., Suresh, K.G., Hono, K., 2015d. J. Phys. D: Appl. Phys. 48, 125002 with kind permission of IOP.

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changes near the superconducting band gap for various Z values (can be seen from Fig. 11), and the curves show a shoulder close to D for high Z values while it became more flat for low Z values. The obtained values of D were found to be less than the superconducting band gap (1.5 meV), which was attributed to the multiple contacts between sample/SC (Clowes et al., 2004). These authors estimated the P value by extrapolating the P versus Z curves to Z ¼ 0. The estimated value of intrinsic spin polarization was found to be equal to 0.61  0.1 and 0.66  0.1 for x ¼ 0.2 and x ¼ 0.4, respectively. They also mentioned that the estimated P value for x ¼ 0.4 is higher than that of many ternary or quaternary Heusler alloys. Due to their high TC and high P values, some of the previously mentioned materials are very promising for spintronic devices. Bainsla et al. (2015d) predicted that the material with x ¼ 0.4 would be an SGS. Since these materials are studied only in bulk form, it will be interesting to see the properties of these materials in thin film form. 2.1.1.5 Electronic Structure Calculations Electronic structure calculations were performed for Fe2xCoxMnSi (0
x
2) alloys by Ishida et al. (2008). Based on their calculations, they concluded that these alloys show HMF nature for all compositions. They also mentioned that these alloys may have high TC with increase in x to 1. Their calculations further revealed that partial substitution of Co atoms in place of Fe also helped in stabilizing the FM phase.

2.1.2 Other Fe2-Based Alloys Galanakis (2004) has shown that [Fe1xCox]2MnAl is HMF and obeys the Slater Pauling rule. Nishino et al. (1997) have reported the thermoelectric properties of the Heusler-type Fe2VAl1xGex alloys with compositions 0
x
0.2. They have also reported that unlike Fe2VAl, which shows a semiconductor-like behavior in electrical resistivity, a slight substitution of Ge for Al causes a significant decrease in the low-temperature resistivity and a large enhancement in the Seebeck coefficient. Saha et al. (2009) have studied structural, magnetic, and magnetotransport behaviors of Fe2V1xCrxAl alloys and observed that increasing Cr content promotes site disorder of Fe and Al, which in turn leads to destabilization of L21 superstructure of Fe2VAl. This site disorder was attributed to the enhancement of the magnetic moment through spatially confined magnetic entities and intercluster interactions, giving rise to a magnetically ordered state as x / 1. Low-temperature MR decreases rapidly from w10% for x ¼ 0 to 0.4% for x ¼ 0.8, which is correlated to the change in the magnetic interactions. Also, the MR versus T plots exhibit a maximum in the vicinity of magnetic transition temperature. These authors have suggested that the random anisotropy associated with the cluster glass behavior of Fe2VAl

20 Handbook of Magnetic Materials

weakens in Cr-substituted alloys, resulting in reduced super paramagnetic contribution and low MR values. Vasundhara et al. (2008) investigated the temperature variation of electrical resistivity and Seebeck coefficient S(T) of Fe2VAl1xBx (0
x
1). From the structural analysis, they confirmed that the alloys stabilized in the Heusler type with isostructural L21 phase. They also found as that as x increases from 0 to 1, the system changes from a zero band gap semiconductor to metallic. Among the various dopants, B is found to be the most effective in reducing the electrical resistivity while Si is more effective in increasing the S values. Based on these, the authors claimed that with the optimum substitutions in Fe2Valbased alloys, one could produce materials with low r and high S values, which is essential in thermoelectric devices.

2.2 Co2-Based Alloys As mentioned earlier, Co-based Heusler alloys of the type Co2YZ are of scientific interest because of their predicted high-spin polarization and high Curie temperature, making them attractive for various device applications. Ishida et al. (1995) proposed that ternary alloys of the type Co2MnZ, where Z stands for Si and Ge, are half metals with 100% spin polarization (P ¼ 1). Theoretical calculations on Co2YAl alloys with Y ¼ Ti, V, Cr, Mn, and Fe have also been carried out (Miura et al., 2004). The alloys containing V, Cr, and Mn have been theoretically predicted to be HMF and obey the SlaterePauling curve whereas the Ti- or Fe-containing alloys do not. In particular, Co2CrAl is expected to show a high-spin polarization of 0.84, due to the large enhancement of the Cr moment from 1.54 to 3.12 mB. However, spin polarization of only P ¼ 0.5e0.6 has been experimentally obtained for the alloys. Bainsla and Suresh (2016b) also obtained very high spin polarization for Co2TiX (X ¼ Ge and Sn) alloys using PCAR technique. Mainly the disorder in the structure is found to be the reason for the suppression of the polarization as shown in Co2CrAl, in which it was found that CoeCr-type disorder (Galanakis et al., 2002) drastically reduces the total magnetic moment and the spin polarization. Several investigations carried out over the last few years have shown that HMF properties of many of the previously mentioned ternary alloys can be altered favorably with the help of substitutions, as described in the following section.

2.2.1 Co2Mn1xFexSi (0
x
1) Alloys 2.2.1.1 Structural Properties Kallmayer et al. (2006) studied the crystal structure of Co2Mn1xFexSi (0
x
0.4) alloys by analyzing the RT XRD data. Superlattice reflections (111) and (200) were clearly observed in the XRD up to x ¼ 0.2, while (200) peak was absent for x  0.3. Thus, XRD data indicate that the crystal structure

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of alloys is well ordered up to x ¼ 0.2, while it has some disorder for x  0.3. ˚ was obtained for these alloys independent The lattice parameter value of 5.67 A of the concentration. Balke et al. (2006) studied the crystal structure of Co2Mn1xFexSi (0
x
1) alloys using XRD measurements, and they also used electron spectroscopy for chemical analysis to verify the composition and cleanliness of the samples. No impurities were detected in the samples from electron spectroscopy for chemical analysis analysis. The powder XRD data shows both (111) and (200) peaks with equal intensity for all x values, which indicate that the alloys are in well-ordered crystal structure. They obtained the ˚ for all x values. Due to the nearly equal scattering lattice parameter of 5.64 A amplitudes of all the constituents, it is very difficult to probe structural disorder in this series (if any). To probe the structural disorder in such systems, one needs to have Mo¨ssbauer or extended x-ray absorption fine structure (EXAFS). The observed Mo¨ssbauer patterns for all x values show a sextet, which indicates the magnetically and structurally well-ordered phase of the samples. The Mo¨ssbauer patterns show no quadruple shift, which indicates the cubic symmetry in the crystal structure. Patterns were best fitted using a sextet and a doublet/singlet; however, the intensity of doublet/singlet was quite low (nearly 3.5% for x ¼ 0.5) for all x values. The antisite disorder which causes a small fraction of paramagnetic Fe atoms is the origin of doublet or singlet. The relative contribution of the doublet decreases from 9% to 1.8% as Fe is increased from 0.1 to 1. The average linewidth (over entire series) of the sextet was found to be approximately 0.14  0.01 mm/s, but a relatively higher linewidth of 0.19 mm/s was found for x ¼ 0.7. Comparatively higher value of linewidth indicates the presence of higher disorder. The maximum hyperfine field value at Fe sites was found to be 26.5  106 A/m for x ¼ 0.5 and varies nonlinearly from 25.9  106 to 25  106 A/m for x ¼ 0.1 to x ¼ 1. It was also mentioned by Balke et al. (2006) that the Mo¨ssbauer spectrum recorded at 85K for x ¼ 1 shows a higher hyperfine field of 26.3 A/m without any doublet or a singlet. To find out any high-temperature phase transitions for Co2Mn1xFexSi (0
x
1) series, differential scanning calorimetry was used by Balke et al. (2006). They performed series of DSC measurements with varying heating and cooling rates, but it was not possible to distinguish the magnetic phase transition because it was too close to the structural transition from L21 to B2 phase (TB24L2 t 1). The Fe concentration dependence of L21 4 B2 phase transition is given in Fig. 12. It can be seen that the Curie temperatures for the end members, Co2MnSi (x ¼ 0) and Co2FeSi (x ¼ 1), are slightly below and above the structural phase transition, respectively. The Curie temperature is expected to increase with increase in x value, as TC ¼ 985 and 1100K was reported for x ¼ 0 and 1 (Webster, 1971; Wurmehl et al., 2005). On the other hand, TB24L2 t 1 was found to be almost constant with x. Thus, Balke et al. (2006) found it difficult to unambiguously determine the TC using the DSC measurements. They also mentioned that the Curie temperature of alloys of high Fe concentration was found to be above order-disorder phase transitions.

22 Handbook of Magnetic Materials

FIGURE 12 Structural phase transition in Co2Mn1xFexSi (0
x
1) alloys. Reprinted from Balke, B., Fecher, G.H., Kandpal, H.C., Kobayashi, K., Lkenaga, E., Kim, J., Ueda, S., Felser, C., 2006. Phys. Rev. B 74, 104405 with kind permission of American Physical Society.

2.2.1.2 Magnetic Properties Magnetic properties of Co2Mn1xFexSi (0
x
0.4) alloys were also first studied by Kallmayer et al. (2006). As mentioned previously, the end members (Co2MnSi and Co2FeSi) of this series have high TC ¼ 985 and 1100K, respectively. The saturation magnetization values obtained at 5 and 300K were found to be almost constant. MS value was found to increase with increase in the Fe concentration. Kallmayer et al. (2006) also observed that the experimentally obtained MS value for these alloys roughly matches with those obtained using the Slater-Pauling rule (Galanakis et al., 2002, 2006a, 2014). Balke et al. (2006) further studied the magnetic properties of these alloys using an SQUID magnetometer. On the basis of the remanence and coercivity, they found that these alloys have soft FM nature. Balke et al. (2006) also observed that MS value increases with increase in the Fe concentration. MS values of 4.97  0.05 and 4.97  0.05 mB were obtained for Co2MnSi and Co2FeSi, respectively at 5K, and these values were found to be in agreement with those reported earlier by Wurmehl et al. (2005). MS value increases almost linearly with increase in the Fe concentration and agrees well with the Slater-Pauling rule. Magnetic properties of these alloys were further studied using X-ray absorption spectroscopy by Kallmayer et al. (2006). The X-ray absorption spectra (XAS) and corresponding X-ray magnetic circular dichroism (XMCD) difference spectra obtained at RT for Mn, Fe, and Co-L2,3 edges are shown in Fig. 13. Two resonant peaks L2 and L3 were observed for Co XAS spectra

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FIGURE 13 XAS/XMCD spectra for Co2Mn0.6Fe0.4Si at Mn, Fe, and Co L2,3 edges. Here, solid and dashed lines represent the spectra measured with external field applied parallel and antiparallel to surface normal, while, the bottom panels represent the corresponding XMCD spectra. Reprinted from Kallmayer, M., Elmers, H.J., Balke, B., Wurmehl, S., Emmerling, F., Fecher, G.H., Felser, C., 2006. J. Phys. D: Appl. Phys. 39, 786 with kind permission of IOP.

which correspond to the 2p3/2 / 3d and 2p1/2 / 3d transitions. The multiple peak structure close to L3 was absent for Co spectra, which is similar to the Co2Cr1xFexAl case (Elmers et al., 2003). Kallmayer et al. (2006) mentioned that a multiple peak structure was observed in the L3 region for Co2TiSn. They attributed the observation of different Co spectra (in different alloys) to the difference in the electronic structure of different compounds, while they did not rule out the effect of different sample preparation techniques. XMCD signal, which is the normalized difference in absorption between the left and right polarized X-rays, depends on the exchange splitting and spineorbit coupling of the initial core and final valence states. The strong dichroism signal obtained at Co-L2,3 corresponds to significant magnetic moment on Co atoms. They also obtained the moment for Co using the sum rule analysis and obtained a value of 1.08 mB per Co atom for Co2Mn0.6Fe0.4Si. It is worth noting that the total electron yield is only proportional to the absorption if X-ray penetration (lx) is larger than the probing depth (le) of the emitted electrons. These authors also neglected the effect of lx and le on the reduction of spin magnetic moment. Kallmayer et al. (2006) also obtained a similar absorption edge for Fe as in the case of Co, as shown in Fig. 13. The multiple peak characteristics were

24 Handbook of Magnetic Materials

found to be absent, indicating a metallic band character for Fe d-states too. They claimed that the XMCD spectra obtained for Fe was similar to the spectra obtained for pure Fe. They obtained a total moment of 1.85 mB per Fe atom. The XAS/XMCD spectra obtained for Mn-L2,3 edges for the same alloy are also shown in Fig. 13. XAS spectra for Mn are found to be different as compared to Co and Fe; it shows a doublet at L2 region and L3 peak is followed by two additional peaks at higher photon energy. They mentioned that this kind of XAS spectra were observed earlier in the case of Co2MnSi and Co2MnGe films; however, it was absent in the case of bulk Co2MnGe. These authors also state that this multiplet structure is a clear sign of localized 3d electrons. They found that their XAS spectra match well with the calculated absorption spectra for Mn ground state. They have given various reasons for the multiplet structure, one of them being the oxidation of Mn atoms. Dichroism signal with negative and positive intensity were observed close to L3 and L2 edges, respectively, as shown in Fig. 13. Kallmayer et al. (2006) mentioned that the appearance of the Mn XMCD signal means that the Mn moments are coupled parallel to each other and also parallel to Co and Fe magnetic moments. They obtained total moment of 2.01 mB per Mn atom for Co2Mn0.6Fe0.4Si alloy. The also measured and analyzed XAS/ XMCD spectra for other compositions in Co2Mn1xFexSi (0
x
0.4) series, as given in Fig. 14. The multiplet features were found to vary with change in the composition. The Mn XMCD spectra show a strong variation with different Fe concentrations. The smallest value for XMCD signal was measured for x ¼ 0 and the largest for x ¼ 0.4. The moment values obtained using XMCD sum rules were found to be slightly less than those obtained from magnetization data for x ¼ 0 but the difference decreases for higher x values. 2.2.1.3 Nuclear Magnetic Resonance Wurmehl et al. (2013) used nuclear magnetic resonance (NMR) technique to investigate the local environment of the Co2Mn1xFexSi (0.1
x
0.9) series and, in particular, to find the impact of partial substitution on the 55Mn hyperfine magnetic field. NMR measurements were performed on pieces of polycrystalline samples in an automated, coherent, phase-sensitive spin-echo spectrometer at 4.2K without applying any external magnetic field. The NMR spectra corresponding to all members of Co2Mn1xFexSi (0.1
x
0.9) are given in Fig. 15. The spectra were found to shift toward higher resonance frequencies with increase in the Fe concentration. It can be seen from Fig. 15 that all resonant spectra exhibit a splitting in resonance sublines (satellite lines). They found that the frequency spacing between the satellite lines remained constant for each spectrum; however, it shows a strong dependency on Fe concentration. They mentioned that the linewidth (G) first increases with increase in x but eventually decreases after reaching a maximum value of

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FIGURE 14 (A) Mn 2p / 3d XAS spectra averaged for antiparallel magnetization directions, while (B) shows the XMCD spectra for Co2Mn1xFexSi (0
x
0.4) alloys. XMCD, X-ray magnetic circular dichroism. Reprinted from Kallmayer, M., Elmers, H.J., Balke, B., Wurmehl, S., Emmerling, F., Fecher, G.H., Felser, C., 2006. J. Phys. D: Appl. Phys. 39, 786 with kind permission of IOP.

2 MHz for medium x concentration. Note that the linewidth for Fe-rich samples is smaller than Mn-rich samples. The hyperfine field, which mainly arises from the Fermi contact term (Freeman and Watson, 1960) was found to be the most dominant contribution to the effective field experienced by the nucleus in ferromagnetic materials.

26 Handbook of Magnetic Materials

FIGURE 15 The 55Mn spectra as a function of frequency and function of hyperfine magnetic field for Co2Mn1xFexSi (0.1
x
0.9) series. The inset shows the Heusler crystal lattice in L21 structure. Reprinted from Wurmehl, S., Alfonsov, A., Kohlhepp, J.T., Swagten, H.J.M., Koopmans, B., Wojcik, M., Balke, B., Ksenofontov, V., Blum, C.G.F., Bu¨chner, B., 2013. Phys. Rev. B 88, 134424 with kind permission of American Physical Society.

Keeping this in mind, Wurmehl et al. (2013) attributed the pronounced differences in the resonance frequencies to the differences in the magnetic environment of the NMR-active atoms. They mentioned that the measurement of hyperfine field thus gives the possibility of studying the local magnetic structure and electronic surrounding of the NMR active atom. The effect of antisite disorder (which results in different structures, i.e., A2, B2, or DO3), on local hyperfine field, can be distinguished from the NMR signal as mentioned by Wurmehl et al. (2013). For A2 disorder, a single and very broad NMR line yields, which consists of many unresolved resonance lines corresponding to a large number of possibilities for atomic distribution. 59 Co NMR linewidth of about w100 MHz (w10 T) was observed for A2ordered Co2FeAl (Inomata et al., 2006; Wurmehl and Kohlhepp, 2008). In B2 disorder, resonance lines split into sublines with rather large spacing due to pronounced differences in the magnetic moment of the transition metal and a main group element. The large subline spacing has been observed in some B2ordered Heusler alloys. The reduction of hyperfine fields at Y atoms was seen in the DO3-type Heusler alloys (Jung et al., 2009; Niculescu et al., 1981), and a resonance line splits into nine sublines. On the basis of the observed NMR spectra, Wurmehl et al. (2013) rule out any possibility of any A2, B2, or DO3 disorder in Co2Mn1xFexSi (0.1
x
0.9). They attributed the observed line splitting to the L21 structure to the random distribution of Mn and Fe at 4b position. Fig. 16 shows that the spacing between the adjacent resonance lines

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FIGURE 16 Resonance frequencies of Co2Mn1xFexSi (0.1
x
0.9) alloys as a function of optimized Fe concentration. Here, each symbol corresponds to a particular environment in the third shell of 55Mn. Reprinted from Wurmehl, S., Alfonsov, A., Kohlhepp, J.T., Swagten, H.J.M., Koopmans, B., Wojcik, M., Balke, B., Ksenofontov, V., Blum, C.G.F., Bu¨chner, B., 2013. Phys. Rev. B 88, 134424 with kind permission of American Physical Society.

which can be regarded as almost constant within some experimental uncertainty. In addition to frequency shift, Fig. 16 also demonstrates that the resonance frequencies for a particular third shell environment slightly shift with increase in Fe concentration. Wurmehl et al. (2013) also found that the mean hyperfine field values of 55 Mn nuclei increase with increase in the Fe concentration and also found to be in agreement with their calculations. The linewidth was found to follow an apparent parabolic behavior with Fe concentration as shown in Fig. 17. They mentioned that this behavior is related to the higher shell effects which contribute to the behavior of linewidth as a function of Fe concentration. 2.2.1.4 X-Ray Photoemission Spectroscopy Electronic structure of Co2Mn1xFexSi (0
x
1.0) series was investigated using the high-energy X-ray photoemission spectroscopic studies by Fecher et al. (2007). Note that photoelectron spectroscopy is one of the best techniques to study the occupied electronic structure of materials. The hard X-ray photoelectron spectroscopy (XPS) is different from the common XPS, since Xrays of energy range w8 keV were used for hard XPS as compared to common ˚ XPS w1.2 keV. The increased energies improved the probing depth from 24 A ˚ to 115 A for energy w1.2 keVew8 keV, respectively. The total spin integrated DOS (rtot) and orbital momentum resolved DOS (rl) for alloys with x ¼ 0, 0.5, and 1 are shown in Fig. 18. On the basis of observed DOS as shown

28 Handbook of Magnetic Materials

FIGURE 17 Width of the Gaussian lines obtained from the fit of the Co2Mn1xFexSi series as a function of Fe concentration. Reprinted from Wurmehl, S., Alfonsov, A., Kohlhepp, J.T., Swagten, H.J.M., Koopmans, B., Wojcik, M., Balke, B., Ksenofontov, V., Blum, C.G.F., Bu¨chner, B., 2013. Phys. Rev. B 88, 134424 with kind permission of American Physical Society.

FIGURE 18 (AeC) Total spin averaged density of states (rtot) for x ¼ 0, 0.5 and 1 using hard Xray photoemission spectroscopy. (DeF) represents the corresponding orbital momentum (rl). Reprinted from Fecher, G.H., Balke, B., Ouardi, S., Felser, C., Scho¨nhense, G., Ikenaga, E., Kim, J.eJ., Ueda, S., Kobayashi, K., 2007. J. Phys. D: Appl. Phys. 40, 1576 with kind permission of IOP.

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in Fig. 18, these authors mentioned that the valence band spectra close to the Fermi energy indicate the presence of half-metallic band gap for all the alloys in this series. They also found that the value of effective exchange interaction increases with increase in Fe concentration. 2.2.1.5 Electronic Structure Calculations Balke et al. (2006) calculated the electronic structure of Co2Mn1xFexSi (0
x
1) alloys using the experimentally obtained lattice parameters with Fm3m space group. The results of the calculations for parent Co2MnSi and Co2FeSi are shown in Fig. 19, and the gap in minority band is clearly seen for these alloys. The spin-resolved DOS for alloys with x ¼ 0.25, 0.50, 0.75 are shown in Fig. 20, and all the alloys show HMF behavior with a band gap in the minority band. It was mentioned by Balke et al. (2006) that the Fermi energy falls close to the middle of band gap for intermediate Fe concentrations, which makes the magnetic and electronic properties of these materials very stable against external influences. Thus, they concluded that the Co2Mn0.5Fe0.5Si exhibits the most stable HMF nature in this series as well as those alloys with concentration close to x ¼ 0.5. Electronic structure of Co2Mn1xFexSi (x ¼ 0, 0.1, and 0.2) alloys was also calculated by Galanakis et al. (2006b) and found to preserve HMF nature with Fe substitution.

FIGURE 19 Band structure and density of states (DOS) of Co2MnSi and Co2FeSi. Reprinted from Balke, B., Fecher, G.H., Kandpal, H.C., Kobayashi, K., Lkenaga, E., Kim, J., Ueda, S., Felser, C., 2006. Phys. Rev. B 74, 104405 with kind permission of American Physical Society.

30 Handbook of Magnetic Materials

FIGURE 20 Spin resolved density of states for Co2Mn1xFexSi alloys with x ¼ 0.25, 0.50, and 0.75. Reprinted from Balke, B., Fecher, G.H., Kandpal, H.C., Kobayashi, K., Lkenaga, E., Kim, J., Ueda, S., Felser, C., 2006. Phys. Rev. B 74, 104405 with kind permission of American Physical Society.

2.2.2 Co2Cr1xVxAl, Co2V1xFexAl, Co2Cr1xFexAl Alloys 2.2.2.1 Structural Properties Using 3D elemental mapping, Karthik et al. (2007) have obtained that in Co2Cr0.6Fe0.4Al, the precipitates corresponding to A2 structure are the Cr rich (34% Co; 63% Cr; 1% Fe; 2% Al  2%), while L21 structure is Al rich (50% Co; 8% Cr; 12% Fe; 30% Al  2%) matrix phase. From the results on Co2Cr1xVxAl and Co2Fe1xVxAl alloys, these authors have shown that with increasing V concentration (x > 0.4), the L21 structure becomes more stable. According to the experimental results, as V concentration increases from x ¼ 0

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to x ¼ 1, the lattice parameter increases and consequently the excess strain energy of the disordered phase would increase the thermodynamic driving force for the L21 ordering and, therefore, the V-rich alloy forms in large L21structured crystals, whereas Co2Cr1xVxAl alloy with x ¼ 0.5 and Co2Fe1xVxAl alloy with x ¼ 0.67 form fine L21 domains with antiphase boundaries. 2.2.2.2 Magnetic Properties The concentration dependencies of the saturation magnetization of Co2Cr1xFexAl, Co2Cr1xVxAl and Co2Fe1xVxAl alloys show that for Co2Cr1xVxAl, the Ms at 5K increases from 1.4 to 2.0 mB with increasing vanadium concentration from x ¼ 0 to x ¼ 0.5 and then decreases to w1.4 mB for x ¼ 1. Pendl et al. (1996) also studied Co2Cr1xVxAl explained this variation in terms of the increase in the local moment of Co from w0.78 to w0.95 mB as x increases from 0 to 0.5 and then decreases to w0.92 mB as x ¼ 1. Similarly, in both Co2Cr1xFexAl and Co2Fe1xVxAl alloys, Ms at RT increases from 0.7 and 0.8 to 4.6 mB with increasing Fe concentration from x ¼ 0 to x ¼ 1, respectively, as the local Fe moment increases. The deviation of the saturation magnetization for Co2Cr1xFexAl for an x
0.4 from the SlaterePauling value was attributed to the phase separation which results in the CoeCr disorder (Miura et al., 2004) and the anti-FM coupling between the antisite Cr and its nearest neighbor regular site Cr atom. 2.2.2.3 Spin Polarization Spin polarization measured using PCAR technique showed that it decreases from 0.62 to 0.56 for Co2Cr1xFexAl alloys with increasing x. However, firstprinciple calculations (Miura et al., 2004) have shown much higher values and gradual decrease of the P with x in this alloy both in the ordered L21 structure and in the disordered B2 structure. They have attributed this to the appearance of the Fe 3d minority DOS at the Fermi level. But it is known that CoeCr-type disorder also suppresses P substantially. Therefore, the deviation in the spin polarization of Co2CrAl and Co2Cr0.6Fe0.4Al alloys between the PCAR data and the theoretical calculations could be attributed to both these factors (Karthik et al., 2007). It should also be noted that the A2 phase in these alloys is a Cr-rich phase which drastically reduces the spin polarization, as reported by Miura et al. (2004). The spin polarization of Co2Cr1xVxAl alloys was found to decrease with increasing x from P ¼ 0.62 at x ¼ 0 to P ¼ 0.5 at x ¼ 0.5 and then to P ¼ 0.48 at x ¼ 1. These values are less than the theoretically calculated values, and the difference was attributed to the reduction in the hybridization between Co and V due to the fact that V atom possesses a very small moment and its effect on the Co spinup DOS is only nominal (Galanakis and Mavropoulos, 2003). Karthik et al. (2007) have also discussed the spin polarization

32 Handbook of Magnetic Materials

results of Co2Cr0.5V0.5Al. This alloy with single-phase L21 structure shows P ¼ 0.5, though the total moment is higher than both the end compounds namely Co2CrAl and Co2VAl. The disorder affects the DOS of Co2Cr0.5V0.5Al, resulting in a nonzero population of the minority spin-down band near the Fermi level. This causes a reduction in the spin polarization. These authors have mentioned that factors such as alloying defects, CoeCr disorder, antiphase boundaries, antisite disorder, surface inhomogenities etc. could also reduce the polarization values. In Co2Fe1xVxAl also, the spin polarization decreases with increasing vanadium concentration x from P ¼ 0.62 at x ¼ 0, P ¼ 0.53 at x ¼ 0.67 and eP ¼ 0.48 at x ¼ 1. Another substituted quaternary alloy system that was studied extensively from the point of view of applications is Co2Cr1xFexAl. These alloys mostly formed with B2 disorder and hence half metallicity was poor. Despite this, large TMR was observed in some compositions of this system. Both Co2(Cr1xMnx)Al and Co2Mn(Al1xSnx) are theoretically predicted to be HMF by Galanakis et al. (2004).

2.2.3 Co2FeAl1xSix Balke et al. (2007b) have studied this system using several probes and found that L21 structure required for high-spin polarization is obtained for x  0.4. This has been attributed to a better hybridization between Co and Si than between Co and Al. Based on their experimental and theoretical calculations, they have predicted that alloys with x in the range 05e0.7 would be ideal for spintronic applications. 2.2.4 Thin Films and Devices Schmalhorst et al. (2008) studied TMR effect in Co2Mn0.5Fe0.5Si-based magnetic tunnel junctions. The TMR dependence on the barrier thickness is shown in Fig. 21. The maximum value of TMR was observed for barrier thickness of about 1.5 nm. TMR was found to decrease rapidly with decrease in Al thickness. Fully epitaxial CPP-GMR with Co2Mn0.6Fe0.4Si/Ag/Co2Mn0.6Fe0.4Si structures was fabricated and studied by Sato et al. (2011). Using their XRD and scanning tunneling electron microscopy analysis, they found that these devices had high-quality Co2Mn0.6Fe0.4Si electrodes with a good crystallinity and a high level of structural ordering. They also found that Co2Mn0.6Fe0.4Si/ Ag interfaces are very smooth and sharp. MR of 74.8% and 50.1% were obtained at RT for the device with 10 and 3 nm upper Co2Mn0.6Fe0.4Si layers, respectively. Yang et al. (2012) studied the anisotropic magnetoresistance (AMR) effect in epitaxially grown Co2Mn1xFexSi films against varying Fe composition and annealing temperature. They directly deposited Co2Mn1xFexSi films on MgO (001) substrate using an ultrahigh-vacuum (UHV) compatible sputtering system. They confirmed a highly ordered B2 structure

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33

FIGURE 21 Barrier (Al) thickness dependence of the TMR for Co2Mn0.5Fe0.5Si/Al-O/Co70Fe30 junctions measured at room temperature with bias voltage of 10 mV. TMR, tunneling magnetoresistance. Reprinted from Schmalhorst, J., Ebke, D., Weddemann, A., Hu¨tten, A., Thomas, A., Reiss, G., Turchanin, A., Go¨lzha¨user, A., Balke, B., Felser, C., 2008. J. Appl. Phys. 104, 043918 with kind permission of AIP.

for all x values. AMR ratio with respect to f (relative angle between the directions of the magnetic field and the dc current) at 10K for all the films is given in Fig. 22. The sign of AMR ratio was found to change from negative to positive as x is changed from 0.6 to 0.8. They mentioned that the sign of AMR does not change with the temperature (10e300K) for all x compositions. There was no remarkable change in the structural and magnetic properties between x ¼ 0.6 and 0.8. They attributed AMR sign change to the change in intrinsic electronic structure of Co2Mn1xFexSi. They also observed AMR effect for half-metallic Co2MnSi films, and a negative AMR was observed which increased with increase in the annealing temperature. They mentioned that the sign change in AMR indicates a half-metallic to nonehalf-metallic transition. Yang et al. (2012) concluded that the AMR effect can be an indicator of half metalicity, and this information can be obtained without making any device. Later, Sakuraba et al. (2012) studied the fully epitaxial Co2Mn1xFexSi (CMFS)/Ag/Co2Mn1xFexSi (CMFS) CPP-GMR devices with various x values and different top Co2Mn1xFexSi layer thickness (tCMFS). Thin films with tacking structure of MgO(001)/Cr(20)/Ag(40)/CMFS(20)/Ag(5)/CMFS(3-10)/ Ag(2)/Au(5) were grown using an UHV-compatible magnetron sputtering system. The XRD patterns for the 50 nm Co2Mn1xFexSi epitaxial films (on Cr/Ag buffer layers) with various x values revealed that all the films were found to have highly ordered B2 structure. High-resolution transmission electron microscope image for a CMFS/Ag/CMFS (x ¼ 0.4) reveals a (001) oriented fully epitaxial growth with sharp and smooth interface. The highest intrinsic MR ratio was observed for the device with x ¼ 0.4 and tCMFS ¼ 3 nm. MR ratio of 58% and 184% were observed at RT and 30K, respectively, as shown in Fig. 23.

34 Handbook of Magnetic Materials

FIGURE 22 AMR versus f curves for Co2Mn1xFexSi epitaxial films measured at 10K. AMR, anisotropic magnetoresistance. Reprinted from Yang, F.J., Sakuraba, Y., Kokado, S., Kota, Y., Sakuma, A., Takanashi, K., 2012. Phys. Rev. B 86, 020409(R) with kind permission of American Physical Society.

FIGURE 23 Temperature dependence of the observed magnetoresistance (MRobs) for the sample with x ¼ 0.4 and tCMFS ¼ 3 nm. Inset shows the temperature dependence of the normalized MRobs. with x ¼ 0, 0.3, 0.5, and 1.0, while the arrows indicate the highest MR ratio observed for each case. Reprinted from Sakuraba, Y., Ueda, M., Miura, Y., Sato, K., Bosu, S., Saito, K., Shirai, M., Konno, T.J., Takanashi, K., 2012. Appl. Phys. Lett. 101, 252408 with kind permission of AIP.

Physics and Magnetism of Quaternary Heusler Alloys Chapter j 1

35

Very recently, Moges et al. (2016) investigated the factors that affect the half-metallicity of Co2(Mn,Fe)Si thin films by studying the saturation magnetization and TMR ratio of Co2(Mn,Fe)Si/MgO/Co2(Mn,Fe)Si tunnel junctions. They have also investigated the origin of the giant TMR ratio of up to 2610% at 4.2K and 429% at RT observed in the case of these MTJs with Mn-rich, lightly Fe-doped Co2(Mn,Fe)Si electrodes. They suggest that Co antisites at the nominal Mn/Fe sites (CoMn/Fe antisites) can explain the fact that the magnetization for (Mn þ Fe)-deficient Co2(Mn,Fe)Si thin films is lower than the expected half-metallic (Zt  24) value and the TMR ratio for MTJs with (Mn þ Fe)-deficient Co2(Mn,Fe)Si electrodes being lower than that for MTJs with (Mn þ Fe)-rich Co2(Mn,Fe)Si electrodes. It was revealed that the CoMn/Fe antisite is detrimental to the half metallicity of the CMFS quaternary alloy, as in the case of ternary Co2MnSi alloy. It was also shown that (Mn þ Fe)-rich compositions are critical in retaining the half-metallic properties. In addition, a relatively small Fe ratio, rather than a large one, in the total (Mn þ Fe) composition led to a more complete half-metallic electronic state. Furthermore, they found that half metallicity was more strongly enhanced by increasing the Mn composition in Mn-rich, lightly Fe-doped Co2(Mn,Fe)Si than in Mn-rich Co2MnSi. These authors claim that the approach of controlling off stoichiometry and film composition is promising for fully utilizing the half-metallicity of quaternary CMFS thin films as spin source materials. Ko¨hler et al. (2016) have reported on the realization of tuning the intrinsic damping in the half-metallic Heusler compound Co2xIrxMnSi. They studied both bulk and thin films using theoretical calculations and experiments. Control of damping is to remove unwanted magnetization motion, and suppress signal is an important step in the future implementation of this material into devices such as CPPeGMR sensors. Their calculations have shown a stable magnetization and increasing damping parameter with Ir concentration, without affecting the half-metallicity. They also found a good agreement between the calculations and the experimental results from bulk and thin film samples. Gabor et al. (2015) investigated the structural and chemical order in epitaxial Co2FeAl0.5Si0.5 Heusler alloy thin films grown on MgO(001) single crystal substrates and found that these are the most important parameters governing the physical properties of the Heusler compounds. Their XRD measurements indicated that depending on the annealing temperature, the films show B2 or L21 ordering. Longitudinal MR data revealed that for the best L21-ordered film, the magnon scattering is absent at low temperatures, indicating half-metallicity. Anomalous Hall effect measurements indicated that the intrinsic band structure contribution has an opposite sign and that is dominant over the extrinsic skew scattering mechanism. Alfonsov et al. (2015) have studied the role of degree and evolution of structural order in understanding and controlling the properties of highly

36 Handbook of Magnetic Materials

spin-polarized Heusler compounds by using NMR of thin films of Co2FeAl0.5Si0.5 prepared using off-axis sputtering with varying thickness. A detailed analysis of the measured NMR spectra by them shows about 81% L21 ordering along with excess Fe of 8%e13% in place of Al and Si. They also show that, unlike the bulk, the formation of certain types of order depends not only on the thermodynamic phase diagrams but also on the kinetic control in thin films. They could get Co2FeAl0.5Si0.5 films in an almost ideal L21 structure, though with a considerable amount of Fe-Al/Si off stoichiometry. High quality of films obtained by them suggests that the technique of off-axis sputtering method can be of great potential at least in Co2FeAl0.5Si0.5-based spintronic devices.

3. RESULTS ON EQUIATOMIC HEUSLER ALLOYS As mentioned earlier, EQHAs constitute one of the most important constituents of the quaternary Heuser family. A large number of experimental and theoretical investigations have been reported in a whole lot of these alloys. We present some of the most important materials of this family by presenting their physical and magnetic properties in the following section. The results are presented by discussing the structural, magnetic, spintronic, and theoretical aspects, materialwise. Since the composition is 1:1:1:1, usually these alloy names are abbreviated, for example, CoFeMnSi is written as CFMS. Such a scheme is mostly followed in the following section.

3.1 Structural Aspects 3.1.1 CoFeMnZ (Z ¼ Al, Ga, Si, and Ge) Alloys CoFeMnZ (Z ¼ Al, Ga, Si, and Ge) alloys are found to crystallize in the Ytype structure (Alijani et al., 2011a). The RT powder XRD patterns of CoFeMnZ (Z ¼ Al, Si, Ga, and Ge) are given in Fig. 24AeD. Alijani et al. (2011a) found some amount of structural disorder for CoFeMnAl and CoFeMnSi from XRD data, while it is not confirmed for CoFeMnGe and CoFeCrGa due to similar scattering amplitudes of the constituents. To further investigate the crystal structure of CoFeMnZ (Z ¼ Si and Ge) alloys, Bainsla et al. (2014, 2015a, 2015e) performed EXAFS measurements and found large structural disorder in the case of CoFeMnSi, while a well-ordered Y phase was found in CoFeMnGe. RT 57Fe Mo¨ssbauer analysis also revealed a large structural disorder (DO3) for CoFeMnSi (Bainsla et al., 2015a). Some amount of disorder was also obtained for CoFeMnGe from the RT Mo¨ssbauer analysis (Bainsla et al., 2014), which was contradictory to the EXAFS results (Bainsla et al., 2015e). It should be kept in mind that an accurate estimation of the disorder using the Mo¨ssbauer or the EXAFS technique is not possible because of the inherent inaccuracies of the fitting procedures used in these techniques. The structural stability of the CoFeMnZ (Z ¼ Si and Ge) was checked by

Physics and Magnetism of Quaternary Heusler Alloys Chapter j 1

37

FIGURE 24 Powder X-ray diffraction data of CoFeMnZ (Al, Si, Ga, Ge) alloys at room temperature. Reprinted from Alijani, V., Ouardi, S., Fecher, G.H., Winterlik, J., Naghavi, S.S., Kozina, X., G. Stryganyuk, Felser, C., 2011a. Phys. Rev. B 84, 224416 with kind permission of American Physical Society.

Bainsla et al. (2014, 2015a), by performing the differential thermal analysis (DTA) in the temperature range of 400e1450K. They could not see any structural transition in the DTA curves for both the alloys. The absence of (111) peak in the XRD pattern of CoFeMnAl indicates the presence of B2 disorder as shown by Alijani et al. (2011a). Further detailed studies are needed to identify the exact crystal structure of CoFeMnGa as no conclusion could be made by the previously mentioned authors (Alijani et al., 2011a) on the basis of RT XRD data. The results obtained from various measurements are summarized in Table 2.

3.1.2 CoFeCrZ (Z ¼ Al and Ga) Alloys Bainsla et al. (2015b) found that CoFeCrAl exists in B2-type cubic Heusler structure as revealed by the RT powder XRD analysis. Absence of (111) peak indicates that the alloy exists in B2-type structure. Luo et al. (2009) and Nehra et al. (2013) also observed the B2-type structure for CoFeCrAl on the basis of XRD and Mo¨ssbauer analysis. A rough estimate of the degree of B2 ordering was found to be 0.89 by Bainsla et al. (2015b). Such a high value indicates the presence of a highly ordered B2 structure. These results are

Alloy

Structural Ordering

aexp. (A˚)

Ms

exp .

TC (K)

CoFeMnAl

B2

5.79

3.1 (5K)

553

(mB)

P (%)

Physical Nature

References

HMF

Alijani et al. (2011a)

SGS

Alijani et al. (2011a)

2.7 (300K) CoFeMnSi

Y (with DO3 disorder)

5.66

4.1 (5K) 3.7 (300K)

623

64

3.7 (3K)

Bainsla et al. (2015a)

CoFeMnGa

Need more exp.

5.81

3.2 (5K) 2.8 (300K)

567

CoFeMnGe

Y

5.76

4.2 (3K) and 3.8 (at 300K)

711

2.0

460

CoFeCrAl

B2

5.76

70

HMF

Alijani et al. (2011a)

HMF

Alijani et al. (2011a) Bainsla et al. (2014)

67

HMF

Bainsla et al. (2015b) Luo et al. (2009)

CoFeCrGa

Y (with DO3 disorder)

5.79

2.1

>400

CoFeCrGe

Y

5.77

3.0

866

CoFeTiAl

Y

5.85

0

CoMnVAl

Y

5.80

0

CoMnCrAl

L21

5.76

0.9

358

HMF

Enamullah et al. (2015)

CoRuFeSi

B2

5.77

4.8

867

HMF

Bainsla et al. (2015f)

CoRuFeGe

B2

5.87

5.0

833

HMF

SGS

Bainsla et al. (2015c)

HMF

Enamullah et al. (2015)

-

SC

Basit et al. (2011)

-

SC

38 Handbook of Magnetic Materials

TABLE 2 Summary of the Results on Some Equiatomic Quaternary Heusler Alloys (EQHAs)

B2

5.79

4.7

NiCoMnGa

Y

5.80

4.5

NiCoMnGe

Y

5.78

4.1

-

4.8

NiFeMnGa

5.80

4.0

CuCoMnGa

5.85 5.92

NiCoMnSn

MnNiCuSb

Need more exp.

Y

-

HMF

Halder et al. (2015)

HMF

Alijani et al. (2011b)

-

FM

Halder et al. (2015)

-

FM

Halder et al. (2015)

326

HMF

Alijani et al. (2011b)

4.3

631

FM

Alijani et al. (2011b)

-

690

646

Haque et al. (2016)

Type of structural ordering, experimental lattice parameter (Aexp.), experimental saturation magnetization (Msexp ), measured values of curie temperature (TC), current spin polarization (P) at 4.2K, and the magnetic state [half-metallic ferromagnetic (HMF), spin gapless semiconductor (SGS), semiconductor (SC), ferromagnetic (FM)] obtained directly/indirectly are given for comparison figures.

Physics and Magnetism of Quaternary Heusler Alloys Chapter j 1

NiCoMnAl

39

40 Handbook of Magnetic Materials

found to be in good agreement with the reports by Luo et al. (2009) and Nehra et al. (2013). CoFeCrGa alloy is found to exist in the Y-type structure as revealed by the RT XRD and Mo¨ssbauer spectra (Bainsla et al., 2015c). The lattice parameter ˚ . The superlattice reflections (SR) [(111) and (200)] were is found to be 5.79 A absent in the XRD, mostly due to the nearly equal atomic scattering factors of Co, Fe, Cr, and Ga. But absence of SR does not necessary imply structural disorder, as shown previously in the case of Co2FeZ (Z ¼ Al, Si, Ga, Ge) alloys (Balke et al., 2007a). To further investigate the crystal structure of CoFeCrGa, Bainsla et al. (2015c) performed RT 57Fe Mo¨ssbauer spectroscopic measurements. Mo¨ssbauer spectrum was best fitted with two sextets having hyperfine field values of 254 and 142 kOe and relative intensities of 58%, 34%, respectively along with a doublet of relative intensity 8% (paramagnetic). In a well-ordered Y-type structure, there must be a single Hhf due to the presence of only one crystallographic site for Fe. Therefore, presence of the second sextet indicates the structural disorder in the alloy. The authors thus concluded that this alloy crystallizes in the Y-type crystal structure along with some DO3 disorder. They also checked the phase stability of the crystal structure by performing DTA in the temperature range of 500e1600K, and a peak obtained around 1000K was attributed to the phase transition from an ordered Y-type to disordered B2 phase.

3.1.3 CoRuFeZ (Z ¼ Si and Ge) Alloys EQHAs containing a 4d elements like Ru have also been investigated in detail by many researchers. The crystal structure of CoRuFeZ (X ¼ Si and Ge) was investigated by Bainsla et al. (2015f) and Deka and Srinivasan (2015) using XRD and Mo¨ssbauer analysis. (111) and (200) peaks were present in the XRD for both the alloys CoRuFeSi (CRFS) and CoRuFeGe (CRFG) (Bainsla et al., 2015f), as shown in Fig. 25A and B. However, in the case of the former, the intensity of (200) peak was high, while (111) reflection was weak, giving an indication of B2 disorder. However, the superlattice reflections were generally weak in CoRuFeGe, which was attributed to the nearly identical scattering ˚ were factors of the constituents. The lattice parameters of 5.77 and 5.87 A obtained for CoRuFeSi and CoRuFeGe, respectively. Bainsla et al. (2015f) also investigated the crystal structure of these alloys using RT 57Fe MS. The best fit to the Mo¨ssbauer data was obtained with two sextets (S1 and S2) for both the alloys, which indicated that both the alloys are fully FM at room. In the case of CoRuFeSi, Hhf values of 298 and 314 kOe with relative intensities of 76% and 24% were obtained for S1 and S2 respectively, while in the case of CoRuFeGe, they were 307 and 328 kOe with relative intensities of 88% and 12%. The presence of two sextets for both the alloys indicates some amount of chemical disorder in these alloys. The obtained Hhf values suggest that Fe also occupies Z site, which results in the

Physics and Magnetism of Quaternary Heusler Alloys Chapter j 1

41

FIGURE 25 Room temperature powder XRD patterns of CoRuFeZ (Z ¼ Si and Ge) alloys. Here, (A) and (B) correspond to the CoRuFeSi (CRFS) and CoRuFeGe (CRFG) respectively. Reprinted from Bainsla, L., Raja, M.M., Nigam, A.K., Suresh, K.G., 2015f. J. Alloys Comp. 651, 631 with kind permission of Elsevier.

B2-type crystal structure. Therefore, the intensities of S1 and S2 were attributed to the amount of Y and B2 phases, respectively. Both these alloys were found to be reasonably ordered at RT, as revealed by the relative intensities of S1 and S2. Therefore, it was concluded that substitution of Ru for Mn in CoFeMnZ (Z ¼ Si and Ge) improves the chemical ordering in these alloys (Bainsla et al., 2015f). The results of the structural analysis for these alloys are given in Table 2.

3.1.4 CoFeTiAl and CoMnVAl Alloys Structural details of these alloys were obtained by Basit et al. (2011) using the XRD analysis. According to them, characteristic SR (111) and (200) were clearly seen in the XRD patterns of these alloys. The refinement of the XRD data revealed that these alloys exist in Y-type structure with lattice parameters ˚ for CoFeTiAl and CoMnVAl, respectively (given in Table of 5.85 and 5.81 A 2). The difference between the measured and the theoretical intensities for (200) peak indicated the presence of some amount of disorder, which is not quantified. 3.1.5 CoFeCrGe and CoMnCrAl Alloys Structural studies of these alloys were performed by Enamullah et al. (2015) using Rietveld refinement of the RT XRD data. As per this report, these alloys are found to exist in cubic Heusler structure with lattice parameters of 5.77 and ˚ for CoFeCrGe and CoMnCrAl, respectively (as given in Table 2). The 5.76 A intensities of SR (111) and (200) were weak in the case of CoFeCrGe, which may be due to the closeness of the atomic scattering factors of the atoms. According to these authors, the best fit between the experimental and calculated intensities was obtained for the configuration in which Ge and Cr occupy

42 Handbook of Magnetic Materials

4a(0, 0, 0) and 4b(1/2, 1/2, 1/2) octahedral sites, respectively, while Co and Fe occupy 4c(1/4, 1/4, 1/4) and 4d(3/4, 3/4, 3/4) tetrahedral sites, respectively. Ab initio calculations by the same authors were in agreement with the conclusions based on XRD data in this regard. In the case of CoMnCrAl, the (200) peak was found to be stronger than the (111) peak (Enamullah et al., 2015). The observed XRD data clearly suggested a chemical disorder between the octahedral sites. As in the case of CoFeCrGe, there are possibilities of three different configurations for this alloy as well, with the exception that in this case, exchange of atoms between the octahedral sites [i.e., (0, 0, 0) and (1/2, 1/2, 1/2) fcc sublattices] is also to be considered. The configuration in which the octahedral site (1/2, 1/2, 1/2) fcc containing the least electronegative element is found to be energetically the most favorable. Here, Cr and Mn have the least electronegativity, and hence, the two configurations which contain Cr or Mn at octahedral site are favorable. Electronegativity of the Al is also comparable to that of Mn and Cr, and thus, Al tries to occupy (1/2, 1/2, 1/2) fcc sublattice in addition to the (0, 0, 0) sublattice. Due to this behavior of Al, the (111) peak is found to be absent in the XRD pattern. In the light of the previously mentioned observations, Enamullah et al. (2015) have concluded that if two or more atoms possess nearly same electronegativity values, some degree of disorder could be expected in the crystal structure. They gave the example of CoMnCrAl, CoFeCrAl (Bainsla et al., 2015b), and Co2Cr1xFexAl (Graf and Felser, 2013) alloys, where disorder is found between Cr and Al sites. Using the information available in the literature, they proposed an empirical relation between the relative electronegativity values and the occurrence of disorder. This information is important in the material design of new materials.

3.1.6 NiCoMnZ (Z ¼ Al, Ge, and Sn) Alloys Structural analysis of these alloys was done using XRD and neutron diffraction measurements by Halder et al. (2015). The Rietveld refinement of the XRD data revealed that all the alloys formed in single phase with cubic Heusler structure. The SR (111) and (200) were found to be absent for NiCoMnAl and NiCoMnGe, while they were present in NiCoMnSn. The appearance of the (111) and (200) peaks indicates the absence of structural disorder. These authors also mentioned that mere absence of the superlattice reflections in the XRD of NiCoMnAl and NiCoMnGe should not be taken as the sign of disordered crystal structure. The neutron diffraction patterns of the previously mentioned alloys were also analyzed using the Rietveld refinement (using the FULLPROF program). The superlattice reflections (111) and (200) were observed in the neutron diffraction pattern of NiCoMnGe, in contrast to the XRD patterns. In the case of NiCoMnAl, they observed (200) peak in the neutron diffraction pattern, while (111) peak was absent. This shows the importance of the neutron diffraction as a probe in cases where the transition elements and the main group elements

Physics and Magnetism of Quaternary Heusler Alloys Chapter j 1

43

belong to the same period. On the basis of this, they concluded that B2-type structural disorder is present in NiCoMnAl, while NiCoMnGe have well-orderd crystal structure. From the temperature variation of neutron diffraction data, they found that the lattice parameters for both the alloys increased with increase in temperature. The results of structural analysis for NiCoMnZ (Z ¼ Al, Ge, and Sn) are also given in Table 2. In order to get further insight into the order/disorder for NiCoMnAl and NiCoMnGe alloys, Halder et al. (2015) performed model calculations after incorporating varying amounts of disorder. They quantified the disorder with respect to the Mn-Z and Ni-Co sites. Due to the differences in the scattering lengths of neutrons for Ni and Co, they could identify the disorder at Ni and Co sites. Thus, on the basis of the observed neutron diffraction data and the calculations, they concluded that there was a complete disorder at MneAl sites as well as at Ni-Co sites for NiCoMnAl. On the other hand, in the case of NiCoMnGe, they concluded that there was no disorder at Mn-Ge sites, but there was a complete disorder between Ni-Co sites.

3.1.7 NiFeMnGa, NiCoMnGa and CuCoMnGa Alloys Alijani et al. (2011b), obtained the structural details of these alloys by performing the Rietveld refinement of the XRD patterns. All the three alloys were found to crystallize in the cubic Y-type Heusler structure. (111) and (200) peaks were not observed in the XRD for any of these alloys, mostly due to the reason mentioned earlier. Thus, a conclusive structural determination was not possible from the measurements, and a deeper structural analysis is required for these alloys. According to them, the determination of exact crystal structure in such cases is possible from anomalous XRD and EXAFS measurements. The lattice parameters obtained from the refinement of the XRD data are given in Table 2 for all the alloys. They also calculated the bulk moduli of all the alloys and found that the Ni containing alloys are harder as compared to Cu-containing alloy. 3.1.8 MnNiCuSb Alloy The structural properties of MnNiCuSb were investigated by Haque et al. (2016) using the powder XRD measurements. This alloy is found to exist in cubic Heusler crystal structure (Y-type) and F 43m space group. However, a small Cu impurity peak ( a)

(0, 1/2, 0)

(0, 0, 1)

(1/2, 0, 1/2)

(0, 1/2, 1)

(1/2, 1/2, 1/2)

P

(0, 0, 1/2)

(1/2, 1/2, 0)

(1/2, 1/2, 1/2)

C (c > a)

(0, 0, 1/2)

(1, 0, 0)

(1/2, 1/2, 0)

(1, 0, 1/2)

(1/2, 1/2, 1/2)

F (c > a > b)

(0, 0, 1/2)

(1, 0, 0)

(1/2, 1/2, 1/2)

I (c > a > b)

(0, 0, 1)

(1/2, 0, 1/2)

(1/2, 1/2, 0)

(0, 1/2, 1/2)

P

(0, 0, 1/2)

(1/2, 0, 0)

(1/2, 0, 1/2)

(1/2, 1/2, 0)

I (c > a)

(0, 0, 1)

(1/2, 0, 1/2)

(1/2, 1/2, 0)

I (a > c)

(1, 0, 0)

(1/2, 0, 1/2)

(1/2, 1/2, 0)

P

(0, 0, 1/2)

(1/2, 0, 0)

(1/2, 0, 1/2)

R

(0, 0, 1/2)

(1/2, 1/2, 0)

(1/2, 1/2, 1/2)

P

(0, 0, 1/2)

(1/2, 1/2, 0)

(1/2, 1/2, 1/2)

I

(0, 0, 1)

(1/2, 1/2, 0)

F

(0, 0, 1)

(1/2, 1/2, 1/2)

Orthorhombic

Tetragonal

Hexagonal

Cubic

(1/2, 1/2, 1/2)

After Rossat-Mignod, J., 1987. Magnetic structures, in: Simonet, in: Celotta, R., Levine, J. (Eds.), Methods of Experimental Physics, vol. 23C. Academic Press, Inc., London, pp. 69e157 associated with an antiferromagnetic structure.

86 Handbook of Magnetic Materials

TABLE 3 Symmetry Points for k ¼ H/2 of 14 Bravais Lattices

Elastic Neutron Diffraction on Magnetic Materials Chapter j 2

87

moment orientations alternate between two exactly antiparallel directions. The magnetic structure can be viewed as being composed from two interpenetrating ferromagnetic lattices, each having a unit cell twice as large as the crystallographic one (aM ¼ 2a). Correspondingly, in reciprocal space the nodes corresponding to magnetic order do not coincide with the Fourier transform of the crystal structure lattice and lie between them. In other words, the magnetic signal occurs at different reciprocal points than the nuclear one. For Bravais lattices with an antiferromagnetic order, this is always the case. Let us at this point also note that for k ¼ 0 and k ¼ H/2, the Fourier component mj,k is a real number and can be directly identified as the magnetic moment mn,j. For other rational propagation vectors, a more general approach based on Eq. (16) and described below for the case of incommensurate structures has to be used. As we have seen above, in the case of a single magnetic moment in a crystallographic unit cell, one can make a simple correspondence between the type of the propagation vector k and the type of the magnetic ordering. For k ¼ 0 a material with Bravais lattice has to order ferromagnetically. For antiferromagnets there must be k s 0. Such a simple rule does not hold for two or more magnetic moments in the crystallographic unit cell as they can couple within this cell either ferromagnetically or antiferromagnetically. In both cases the propagation vector is k ¼ 0. If in the latter case, these moments are not equal in size, one encounters a ferrimagnetic state as depicted in Fig. 3. Moments in one unit cell point in this case along the same directiondthe magnetic structure is collinear. However, it is clear that different moments in one unit cell can have different orientations preserving the propagation vector value. One has to realize that the description introduced above (Eqs. 15 and 16) describes the translational periodicity of moments and not how they couple within the magnetic unit cell. The information about the mutual orientation is contained in the values of the Fourier components. Since the number of possibilities increases enormously with the number of moments involved, the task of determining the coupling between them can be very complex and tedious. Fortunately, these moments are in many cases symmetry related, and representation theory introduced below offers a possibility to simplify (not in all cases) this task.

5.2 Incommensurate Magnetic Structures For incommensurate structures, one cannot define a magnetic unit cell that would be an integer multiple of the crystallographic unit cell. There are many different types of magnetic structures that fall into this category. The simplest incommensurate magnetic structure is a sine waveemodulated structure (see Fig. 7), where the magnetic moments are modulated in size in direct space as one moves along the k direction according to   mn;j ¼ mj u cos k$Rn þ fj ; (18)

88 Handbook of Magnetic Materials

FIGURE 7 Schematic representation of a sine wave (A) modulated structures: longitudinally modulated collinear magnetic structure (B), transversally modulated collinear magnetic structure (C), a cycloidal magnetic structure (D), and a spiral magnetic structure (E) together with an example of a commensurate collinear magnetic structure (F).

where the vector u describes the direction of the propagating magnetic moment and its phase fj. If u and mj are parallel to each other, one speaks about a longitudinally modulated structure, if these vectors are perpendicular, a transversally modulated structure is realized (a simple example of such structures are shown in Fig. 7B and C). Since the magnetic moment is a real quantity, its spatial distribution is described by two terms, one associated with k and the other with k. These two terms, which are conjugated complexes, read as mj,k ¼ mj/2uexp(ifj) and mj,k ¼ mj/2uexp(ifj), respectively. The magnetic structure factor (Eq. 17) is then of the form     1 X FM ðQÞ ¼ p fj ðQÞmj $u exp ifj exp iQ$rj ; (19) 2 j where the factor 1/2 expresses the fact that one sums over k and k. One can immediately see that such a formula is valid also for commensurate structures. The difference is that in the latter case the moment values are after a period corresponding to the propagation the same while for truly incommensurate

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magnetic structures all the moments within the crystal are different (either in value or in direction). That is why such a structure cannot be called antiferromagnetic, although this term is frequently used in literature. Sine waveemodulated collinear magnetic structures shown in Fig. 7B and C are in fact a special case of a more general structure, in which moments are not collinear, and having two or three nonzero Cartesian components. If only one propagation vector is present, they are simultaneously modulated along this same propagation vector. Restricting ourselves to the case with two Cartesian components having the third one zero, the moments lie within one plane. Another degree of freedom, available for this structure, is the phase shift between modulations of both components. If the phase shift equals to zero, the magnetic structure is again collinear, with the moments aligned along a single direction within the plane defined by both Cartesian components. If the phase shift equals to p/2 (the term cos (k$Rn þ fn þ p/2) in Eq. (18) becomes sin (k$Rn þ fn)), one encounters a helical magnetic structure. Here are again two possibilities as the propagation vector k can be perpendicular to the plane in which the moments are modulated (spiral structure) or lies within this plane (cycloidal structure). The former magnetic structure can be found, for instance, in pure elements as Tb, Ho, and Dy at low temperatures (Koehler, 1972). The latter type of magnetic order is found frequently in multiferroic materials, for instance, in manganites (Mochizuki and Furukawa, 2009) and MnWO4-related material (Urcelay-Olabarria et al., 2012). Depending on the magnitudes of the two Cartesian components, such a helix structure is either elliptically (m1,j s m2,j) or circularly modulated (m1,j ¼ m2,j). In the more general elliptical case, the magnetic moments are modulated in direct space according to     mn;j ¼ m1;j u cos k$Rn þ fj þ m2;j v sin k$Rn þ fj (20) and the corresponding magnetic structure factor is

     1 X FM ðQÞ ¼ p fj ðQÞ m1;j $u þ m2;j $v exp ifj exp iQ$rj ; 2 j

(21)

We see that for the case m2,j ¼ 0 the formula reduces to the case of a sine waveemodulated structure. Simplified examples of such cycloidal and helical structures are depicted in Fig. 7D and E. For comparison we show in Fig. 7F a simple commensurate magnetic structure having a sequence of “þ þ þ þ    .”

5.3 Multi-k Structures All the magnetic structures mentioned above were fully described by a single propagation vector k even if one also needs the associated k vector (except for special cases mentioned at the section of commensurate structures). Such magnetic structures are called correspondingly single-k structures. This is in contrast to the situation where the moments in direct space are modulated in

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more directions at the same time (this requires more than one k vector). The Fourier expansion (16) contains several propagation vectors that might not be collinear. In this case are the magnetic structures called multi-k, and one needs to consider combinations of different ordering types. Obviously, there are many combinations and possibilities. For instance, a canted arrangement of moments in direct space consisting of a set of ferromagnetically aligned moments and moments alternating in opposite directions as one moves along the a-axis leads to a combination of a ferromagnetic kF ¼ (0, 0, 0) vector with an antiferromagnetic kAF ¼ (1/2, 0, 0). To obtain the correct moment values, one needs to sum the two Fourier components appropriately according to Eq. (16). Another example, which we would like to mention briefly, is a conical type of structure that combines a ferromagnetic or an antiferromagnetic component and a helical spiral one. Such a magnetic structure is realized, for instance, in several rare earth elements at low temperatures [Ho and Er (Koehler, 1972)]. All moments rotate at a surface of a cone around the axis defined by the ferromagnetic or antiferromagnetic component. Let us at this moment mention a somewhat special situation frequently encountered in incommensurate magnetic structures. An incommensurate sine wave modulation means that magnetic moments are unequal in size. Such a structure is not very stable as it is not clear (without entering peculiarities of the symmetry of the particular problem eventually leading to frustration effects) why some of the moments should be negligible whereas some others fully developed, although they reside on the same crystallographic site. The moments therefore tend to be equal in size, and sine waveemodulated magnetic structures are unstable at low temperature (in contrast to helical structures). This imposes its squaring-up (if the modulation wavelength does not become commensurate with the nuclear lattice) and a necessity to introduce new, higher harmonic Fourier components in Eq. (16) that are odd multiples of the original periodicity. Corresponding propagation vectors have the same direction. So, although the magnetic structure seems to be of a multi-k type, the higher-order propagation vectors are directly related to the primary propagation vector. Their appearance in Eq. (16) is needed because more components are necessary to describe a square profile in terms of Fourier series. The magnetic structure is still a simple arrangement of collinear magnetic moments equal in size. The situation in truly multi-k structures is very different as the individual components are not related to each other and may describe a different type of moment couplings along different directions. A special class of multi-k structures represents commensurate magnetic structures in materials crystallizing in highly symmetry paramagnetic space groups. In Fig. 8 we show a relation between a single-k magnetic structure [k ¼ (0, 0, 1/2)] in a primitive cubic material with respect to a double-k and triple-k structures (see also the appropriate information in Table 3). Although these magnetic structures are very different (even the direction of moments and the size of the magnetic unit cells

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FIGURE 8 Schematic representation of a single-k, double-k, and triple-k antiferromagnetic structure associated with a primitive cubic lattice and propagation vector k ¼ (0, 0, 1/2). After Ressouche, E., 2014. Reminder: magnetic structures description and determination by neutron diffraction, in: Simonet, V., Canals, B., Robert, J., Petit, S., Mutka, H. (Eds.), Neutrons et Magnetisme, EDP Sciences, p. 02001, 1e22.

are different), they are very difficult to distinguish in experiments. The reason is that in the case of a single-k structure, different parts of a sample may adopt symmetrically equivalent but different propagation vectors. Due to high symmetry (here cubic), magnetic Bragg reflections appear at the same positions as they appear if the magnetic structure would be of double- or triple-k type.

5.4 K and S Magnetic Domains There are two main types of magnetic domains, the so-called K domains exist in symmetrical systems where two or more propagation vectors are equivalent with respect to the symmetry of the underlying crystal lattice. The volume of the sample divides in large macroscopic areas where moments order with different propagation vectors. These vectors are called sometimes members of a K-star. This effect, called problem of magnetic domains, is in the simplest case responsible for nearly zero macroscopic magnetic polarization of a ferromagnetic material in the absence of a magnetic field. Let us assume a tetragonal system with propagation vector k1 ¼ (1/2, 0, 0). In the simplest case the moments are arranged in a way as depicted in Fig. 2, and magnetic peaks appear half-way along the a* direction. No signal is found along the b* direction (missing reflections in Fig. 2 represent merely the fact that for such reflections the projection on the plane perpendicular to Q is zero). However, it is obvious that k1 is symmetrically equivalent to a vector k2 ¼ (0, 1/2, 0)dthe two propagation vectors are fully equivalent and the Kstar has two members. In the absence of any external perturbation and other effects, the sample’s volume splits in two nearly equally populated spatially separated parts, one ordered according to k1 and the other according to k2. The diffraction pattern recorded in the case when two different propagation vectors are present in two different macroscopic regions separately (single-k structures

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in different regions called domains) is indistinguishable from the case where the two propagation vectors define the magnetic structure in the whole sample’s volume at the same time (multi-k structures). The distinction can only be done, after applying a perturbation that is able to significantly modify the volume ratios between the two single-k domains. This can be done, for instance, by application of a magnetic field or by a uniaxial stress (Burlet et al., 1985; Rossat-Mignod, 1987). Another type of domains can be realized within a single K domain, in which several equivalent directions of magnetic moments with respect to k exist. Such domains are called S-domains. Such a situation appears, for instance, in cases when magnetic moments make an angle with the propagation vector. In Fig. 9 we show the situation of a simple cubic system.

6. IDENTIFICATION OF MAGNETIC SIGNAL IN PRACTICE From the above it is clear that the diffracted neutron signal contains information on both nuclear and magnetic orders present in the system under study. Although the intensities of diffracted Bragg reflections originating from the two processes are comparable in magnitude (number of neutrons recorded at a given reciprocal space point), their origin is very different. It may seem at first sight that it is difficult to separate nuclear and magnetic signal, but, at least in the case of elastic diffraction on materials with large magnetic moments neglecting all other processes (as magnetostriction leading to variations in extinction and/or crystal structure lattice distortions), it is relatively straightforward. The first point to make is that nuclear scattering occurs at all temperatures and is essentially independent on Q. Magnetic scattering appears below the magnetic ordering temperature of the studied material and decreases sharply as Q increases due to a reduction in magnetic form factor f(Q) (see Eqs. (13) and (14)). By subtracting the low (T < TC,N) from the high (T > TC,N) temperature data, an estimate of the magnetic scattering can be made. In the case of antiferromagnetic ordering, yielding the magnetic unit cell larger than the crystallographic unit cell, extra Bragg peaks appear at positions different from those of nuclear origin. For systems with commensurate magnetic and crystallographic unit cells, the magnetic reflections are situated often at the same positions as nuclear peaks. It is clear that in the latter case the situation is more difficult because the magnetic signal can be comparable with statistical uncertainties of the measurement.

6.1 Determination of Magnetic Structures To fully determine the magnetic structure, we need to determine four specifics: First, the temperature at which the moments develop some type of longrange order (for antiferromagnets below the Ne´el temperature TN, for

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FIGURE 9 Schematic representation of different kind of domains. In the first row, K-domains present in a simple cubic system are shown, in the second row the S-domains existing for each Kdomain are illustrated (only one K-domain is shown). The third row shows two so-called 180degree domains. After Ressouche, E., 2014. Reminder: magnetic structures description and determination by neutron diffraction, in: Simonet, V., Canals, B., Robert, J., Petit, S., Mutka, H. (Eds.), Neutrons et Magnetisme, EDP Sciences, p. 02001, 1e22.

ferromagnets the ordering temperature is called Curie temperature TC) and compare it with the phase transition temperature determined from another method. This is necessary because (1) there are always experimental uncertainties in the determination of the temperature and (2) near the ordering temperature critical phenomena occur where the exact value of the ordering temperature is indispensable. Second, the wave vector k of the propagation of the magnetic structure. This can be done either by mapping the reciprocal space using a single crystal sample or by trying to index a powder diffractogram using a suitable

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propagation vector or combination of vectors to account for all the observed magnetic reflections. Apparently, both methods have advantages and disadvantages. The first requires a (large) single crystal of a suitable quality and the possibility to cover a large portion of reciprocal space. While the latter requirement can be fulfilled by using the appropriate diffractometer, the former one cannot be always achieveddfor instance, when no single crystals of the material can be prepared. On the contrary, the powder diffraction does not require single crystalline samples, and also the mapping is easy as the random orientation of crystallites automatically brings in the reflection condition for all Bragg reflections in the available range. The problem is that the signal to noise ratio (especially in the case of weak ferromagnets) is significantly worse than in the case of a single crystal. Nevertheless, powder diffraction is often used as the first step in an attempt to identify the magnetic signal and to determine the propagation vector. One can use try-and-fail method. However, this usually fails for propagation vectors with nonrational components or if two or more propagation vectors define the magnetic structure. Automatic indexing routines are also available that simplify this task enormously. However, this step could be the most difficult part in the magnetic structure determination. Third, determine the coupling and direction of the Fourier components. Normally this is established by inspection of systematic extinction rules and magnetic intensities. We recall that when the scattering vector, Q, is parallel to the magnetic moment, the magnetic scattering intensity is zero. Thus, for a simple collinear magnetic structure, it is usually a simple matter to find the series of (hkl) reflections that are absent because they arise from scattering vectors parallel to the magnetic moments. However, in the case of noncollinear and multi-k magnetic structures, such a simple way cannot be used. Instead, one compares observed intensities (derived magnetic structure factors) with calculated ones for a series of magnetic structure models derived in some suitable systematic way. The point is to generate and test all possible models. For magnetic structures with large magnetic unit cells built from different magnetic moments in the presence of several propagation vectors, it may seem to be impossible to test all the combinations. However, the complexity of this task can be enormously reduced by taking into account symmetry constrains of the problem. In this respect, one can utilize the magnetic space group approach (called also Shubnikov groups) (Izyumov et al., 1991) and group representation theory (Bertaut, 1968). Both methods rely on the fact that the symmetry of the magnetic structure cannot be higher than that of the underlying crystal structure and is derived from its space group. The former approach is very suitable for commensurate magnetic structures and limits itself usually on the invariance symmetry properties for this kind of configurations. It introduces a new symmetry elementdtime reversal. By multiplying the 230 space groups, one obtains 1421 magnetic space groups. The latter approach that is more general works with irreducible representations that are derived from the

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original representation of the crystallographic lattice after considering the symmetry of the propagation vector. We will introduce this method below. Let us only note at this point that recently another, more general approach (related to the Shubnikov approach) has been introducedda superspace formalism extending the invariance concept to incommensurate magnetic structures and allowing to discover “hidden” symmetries that are not obvious using standard descriptions (Perez-Mato et al., 2012). In the last, fourth, step the magnitude of the magnetic moments on the different sites has to be determined from the determined Fourier components. This requires a careful consideration taking into account macroscopic physical properties of the system. In the course of the reduction the knowledge of magnetic form factors, f(Q), is needed. Standard values are tabulated in available literature (Descleaux and Freeman, 1978; Freeman and Descleaux, 1979). Apparently, for different phase shifts between individual Fourier components, different magnetic structure models and consequently different magnetic moment magnitudes are determined.

6.2 Limitations of the Neutron Technique The first difficulty for determining the magnetic structure of a material follows from a fundamental limitation of scattering diffraction techniquesdthe impossibility to determine phase factors in Eq. (18). The Fourier components mj,k ¼ mj/2uexp (ifj) (and their conjugated complexes) defining the distribution of magnetic moments in the direct space are in general complex vectors. The problem in the magnetic diffraction experiments is the fact that one detects intensity proportional to the square of the modulus of the structure factor defined by Eq. (19). The information regarding the phase fj is lost because the origin of the phase shift can be defined arbitrarily. The only information remaining is the phase difference between two distributions originating from moments residing at two different sites in the case of a nonBravais lattice. Let us document this fact on a very simple example of a material with a propagation vector k ¼ 1/3 containing two magnetic atoms in the unit cell. The magnetic moments become according to Eq. (18) m1,1 ¼ m1$ucos (f1), m1,2 ¼ m1$ucos (2p/3 þ f1), and m1,3 ¼ m1$ucos(4p/ 3 þ f1) for moments on the first site in three consecutive unit cells and m2,1 ¼ m2$ucos (f2), m2,2 ¼ m2$ucos (2p/3 þ f2), and m2,3 ¼ m2$ucos (4p/ 3 þ f2) for the second site, where mi and fi are the moment amplitudes and phases at the two (i ¼ 1,2) sites. Considering only the first site, one gets for a choice f1 ¼ 0 a sequence m1, m1/2, m1/2,. for the moments in the first, second, and the third unit cell. This sequence þ   consists of one moment pointing along the direction that is antiparallel to the direction of the two remaining moments that are of half size. The net magnetization is zero. Choosing f1 ¼ p/6 the sequence becomes 0.866 m1, 0.866 m1, 0 (sequence is þ  0). Two moments of an equal size are oriented in antiparallel directions

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and the third one is zero. Also in this case the net magnetic moment is zero. The two moment configurations give the same diffraction result and cannot be distinguished from the diffraction experiment. Other physical arguments or techniques are then needed to select one or the other magnetic structure. The same procedure can be done for moments residing on the other site. It leads also to a degree of freedom in choosing the origin of the phase. However, once one defines such an origin, the phase difference between the two waves becomes important. The easiest way to document this fact is to consider two possibilities. In one we choose f1 ¼ 0 and f2 ¼ p that leads to antiparallel orientation of m1 and m2 moments and, if they would be equal in size to cancellation of the net magnetic moment in each of the unit cells. Such a magnetic order is truly antiferromagnetic with a specific extinction rules. For any other difference between f1 and f2, the net magnetic moment in each unit cell is nonzero leading to another set of extinction rules and hence also to different diffracted intensities.

7. SHORT INTRODUCTION TO REPRESENTATION GROUP THEORY Here we give a short introduction to the group representation theory, which is used to generate possible configurations for a given crystal structure and propagation vector considering symmetry of both. Although the idea of this method is quite straightforward, the subject requires complicated mathematics. We therefore advice the reader to consult specialized literature sources (Ballou and Ouladdiaf, 2006; Bertaut, 1968; Izyumov et al., 1991; Rodriguez-Carvajal and Boure´e, 2012). Here we summarize only the main ideas of the method that are documented below on very few examples. This method utilizes, in addition to the symmetry operations on moments as it is done in the case of Shubnikov groups also, the associated irreducible representations. In the group representation approach, one decomposes the original group of the paramagnetic state into a set of irreducible representations. This can be done because the set of symmetry operations that leave the propagation vector invariant is at maximum the same as in the case of the paramagnetic group. Usually, the symmetry of the magnetic structure (defined by k) is lower, leading to a subset of such operations that form the so-called little group. Note that this procedure is quite general and is used for other physical properties of ordered systems as well. The more general character of representation theory results in some advantages over the Shubnikov space group analysis. First, it can deal with multidimensional irreducible representations (Shubnikov space group analysis can deal only with one-dimensional irreducible representations). It can deal with any propagation vector of the magnetic structure, not only with commensurate ones, and it can even simplify the analysis of systems with very low symmetry.

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Let us recall that in mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element of the group. If a mapping between these elements and square matrices that preserves the multiplication rules of the group can be established, we say that the set of such matrices forms a representation of the group. There is an infinite number of equivalent ways to represent a given group. It is clear that the simplest way to represent a group is to use a description within the lowest possible dimension. In general, an ndimensional representation is reducible if there exists a subspace with dimension m < n, which is invariant under all transformations of the original group. The representation is said to be irreducible, if there is no subspace with smaller dimension invariant under all transformations of the original group. The representation is fully reducible if there exist a set of basis vectors for which all the matrices of the representation have block diagonal form. In literature devoted to group theory (Ballou and Ouladdiaf, 2006; Bertaut, 1968; Izyumov et al., 1991; Rodriguez-Carvajal and Boure´e, 2012), it is shown that in the finite-group representations (which is the case of magnetic structures), one can always choose a unitary representation which is fully reducible. Then, the reducible representation G can be decomposed into a direct sum of irreducible representations Gn in a unique way. Some of the lower-dimensional invariant subspaces which are represented by irreducible representations can be contained in the original representation several times. This fact is expressed by the multiplicity nn of the irreducible representation Gn. The decomposition of a reducible representation can be written X G¼ n v Gv ; (22) 4v

Irreducible representations satisfy several important relations. Among them, the great orthogonality theorem is a very useful tool to actually find them. It also allows to determine how many times a particular irreducible representation appears in the decomposition. One can also show that traces of respective matrices are orthogonal. The following important relation that allows quickly to discard some of the possibilities reads as X n2v ¼ h; (23) v

where nn is the dimension of the n-th irreducible representation and h is the order of the original group which is the same number as the number of the symmetry elements of the group. The symmetry operations of the paramagnetic group act of course on both propagation vector k and the atomic positions. While the atomic positions are mapped by definition on the same equivalent positions, this may not be the case for propagation vectors.

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Having the transformation relations of the magnetic moment components, which are obtained by application of symmetry elements that leave the propagation vector of the magnetic structure invariant or which transform it to equivalent ones on the Cartesian components of all magnetic sites and a set of irreducible representations, one can construct all possible magnetic moment ðmÞ configurations (determined by basis vectors Gi ), by using the projection operator. ðvÞ

ðmÞ

PIm Gi

ðmÞ

¼ Gl dmi dmv ;

where the projection operator P reads as nm X ðmÞ ðmÞ PIm ¼ R ðSÞOs ; g s lm

(24)

(25)

where OS denotes group element. Eqs. 24 and 25 provide us with basis vectors belonging to a particular irreducible representation, from which one can construct the possible magnetic moment configurations. In the case that the crystal structure consists of several decoupled (i.e., independent) sites or crystallographically inequivalent sites (non-Bravais lattices), one has to combine the resultant moment arrangements from models belonging to the same irreducible representation. At this point we would like to mention that all the mathematical descriptions given above assume that the crystal lattice has a translational symmetry in three directions. This property has been for a long time considered as property defining a crystal. However, an increasing number of materials violate translational symmetries. As examples, one can name quasicrystals (Steurer, 2004) and many other compounds as Na2CO3 (Dusek et al., 2003), Cr2P2O7 (Li et al., 2010), or CeRuSn (Prokes et al., 2014). Many other materials exhibit some kind of aperiodicity that is present along with commonly understood translationally symmetric lattices. In these cases a more general description is needed. One of the simplest cases of incommensurate crystal structures is represented by a crystal where atomic positions are shifted from periodic positions (that are in accord with one of the space groups) as a function of a distance from some reference point. The additional modulation can be described by different functions (e.g., crenel, sawtooth), a sine wave modulation being one of the simplest. The modulation can be realized either along a single axis, but cases where two or three different modulations are present are also existing [Ca2MgSiO7, Ba2TiSi2O8 (Bindi et al., 2006)]. To describe the additional “aperiodicity” (actually periodicity that is not commensurate with the underlying average structure), one needs to introduce additional dimensions. The resulting (3 þ d)-dimensional supercrystal has then translational symmetry in these (3 þ d) dimensions. It then is not surprising that such a generalization allows to formulate equations analogous to all formulas for the structure

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factors listed above, allowing also to determine all the parameters as if the crystal would be of the ordinary type. The generalization of the representation group theory, called superspace symmetry formalism, enables to treat theoretically also such (3 þ d)-dimensional materials. Even more, it helps to understand hidden symmetry relations in incommensurate magnetic structures that are not obvious if the normal representation group formalism is utilized. For further reading we advice the reader to consult several excellent reviews (Jannssen et al., 2007; Perez-Mato et al., 2012; Toudic et al., 2008).

8. INSTRUMENTATION 8.1 Neutron Sources Neutrons are produced in sufficient quantity only either at nuclear reactors or at spallation sources (Beckurts and Wirtz, 1964; Furrer et al., 2008; Manning, 1978) The former method require a fission reaction of suitable material (usually 235U) in which nuclei become excited into a high energy level and decay into a few fission fragments with an average of 2e3 emitted fast neutrons. One of the emitted neutrons is required (after lowering its energy) for the next fission, leaving 1e2 usable neutrons. Produced neutrons have erratically distributed energies and have to be thermalized by a moderator. The probability of neutrons having a velocity between v and v þ dv follows the MaxwelleBoltzmann probability distribution. Inside the moderator, the velocities and thus also energies of neutrons are thermally averaged by many scattering processes with the moderator’s molecules and brought to thermal equilibrium with the moderator. Consequently, the energy distribution depends only on the temperature of the moderator. In most cases, for thermal neutrons with an energy between 5 and 100 meV ˚ ), heavy water is used, which provides pro(wavelengths between 1 and 4 A tection for the environment as well. Such a type of moderator is called a thermal source. For smaller energies (lower temperature), liquid hydrogen, nitrogen, or neon are used. Such a type of moderator is referred to as cold source, and energies of neutrons lie between 0.1 and 10 meV (wavelengths ˚ ). In hot sources, usually graphite heated to 2000e4000K, higher4e30 A energy neutrons (energies between 100 and 400e500 meV and wavelengths ˚ ) are produced. Further neutrons are extracted through between 0.4 and 1 A tubes that are evacuated to high vacuum, geometrically limited by slits and brought to either a monochromator or a system of fast rotating disksdchoppers which “filter” out neutrons with desired properties. The available flux is restricted mainly by the power density and irradiation damages of the reactor hot zone. The most powerful reactor sources are found at Institut Laue-Langevin, Grenoble, France; Heinz-Maier Leibnitz, Garching, Germany; the Oak Ridge National Laboratory ORNL, USA; NIST Center for Neutron

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Research, Gaithersburg, USA; China Advanced Research Reactor, Beijing, China; Japan Research Reactor 3, Ibaraki, Japan; and Laboratoire Le`on Brillouin, Saclay, France. There are also other, smaller reactor sources that play, however, an important role since they serve the community as development places and enable many experiments that would not get beam time otherwise. Another way to produce neutrons reliably and in large quantities is the spallation process that produces pulsed or quasicontinuous neutron beams. Accelerated protons hit the target made of a heavy nuclei material (Hg, Pb, W). In contrast to the fission of heavy nuclides, one spallation reaction leads to about 20e30 emitted neutrons. These, of course, have also energies not suitable for use and must be moderated. However, the moderation is not complete due to the necessity to preserve a pulse character of the source that is used in the scattering experiments with the so-called time-of-flight method (TOF). Neutrons of desired properties are defined, selected, and analyzed by a system of choppers that are carefully designed to enable a large range of flexibility of the instrument but keeps the background (for instance, due to the unmoderated fast neutrons) minimal. The ISIS neutron source, Didcot, UK; Spallation Neutron Source SNS, Oak Ridge, USA; J-PARC, Tokai, Japan; and the Swiss spallation neutron sources SINQ, Villingen are among the most powerful spallation devices worldwide. Currently the European Spallation Source (ESS), a project of European countries, is being constructed in Lund, Sweden (Hall-Wilton and Theroine, 2014). With its 5 MW long pulse, it is expected to become by 2019 the world’s brightest neutron spallation source. TOF diffractometers and spectrometers are also built at reactor sources. As an example, we show the layout of the HFM-EXED facility installed at the Helmholtz-Zentrum Berlin (HZB) in Fig. 10 (Peters et al., 2006; Prokhnenko et al., 2015).

8.2 Data Collection Methods Expressions (15)e(17) represent the fundamental equations for a magnetic structure determination, relating magnetic moments to the differential magnetic cross sections, the magnetic structure factors, and Fourier components. However, the Fourier components and magnetic structure factors are not the quantity measured in a diffraction experiment. One records the Bragg peak intensity I(Q) equal to the number of detected neutrons entering the detector when the diffraction condition according to the Bragg’s law 2dsin q ¼ l is satisfied. This can be achieved for a given d spacing and diffraction angle q either by varying the wavelength l or, if the wavelength is fixed, by changing the diffraction angle. The first method is used in the Laue technique. In the second method, one rotates a single crystal bringing the Bragg reflection in a constructive reflection condition to a detector positioned at the expected diffraction position or by covering different

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FIGURE 10 Instrument layout of the time-of-flight instrument HFM-EXED installed at Helmholtz-Zentrum Berlin. The instrument is optimized for restricted geometry imposed by the horizontal 26 T hybrid solenoid offering the world’s highest steady magnetic field for neutron diffraction. After http://www.helmholtzberlin.de/pubbin/igama_output?modus¼einzel&sprache¼en&gid¼1939. ©HZB/E. Strickert/K. Prokes.

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diffraction angles using a large-area detector in the case of a powder sample. In the case of a TOF method, the detection is coded using the time. The recorded intensity corresponds to an integration of the differential cross sections and is proportional to the square of the relevant structure factors. In the case of a nuclear diffraction, it is jFN(Q)j2 and in the case of magnetic scattering, it is to the square of the projected magnetic structure factor jFMt(Q)j2. One has to correct the detected intensity I(Q) for various geometrical factors that are introduced below. The final coefficient of proportionality between experimentally determined and calculated jFN(Q)j2 leads to a scaling factor that is the same also for the magnetic part and offers the possibility to put the magnetic moment magnitudes determined from measured jFMt(Q)j2 on an absolute scale.

8.2.1 Laue Technique This technique is the simplest one to detect Bragg reflections. The singlecrystalline sample is static and surrounded by a static large area detector. It requires a relatively broad range of available incident wavelengths of a known distribution. This means that neutron Laue instruments can be built as a standalone instrument or at neutron guides where no other instrument has filteredout some wavelength. Bragg reflections occur at directions where the Bragg’s law is satisfied. This leads to a large coverage of the reciprocal space and the possibility to map various propagation vectors at once. However, due to the large range of wavelengths, it may happen that at the same position more than one Bragg reflection contributes, since the reflection condition is satisfied also for higher harmonic (equivalency with diffraction of the same d-spacing but with l/2, l/3, .). These individual contributions are not easy to separate. Together with the fact that geometrical conditions along with associated extinction and absorption corrections are not easily taken into account makes this method difficult in terms of quantitative analyses. Examples of dedicated neutron Laue instruments include CYCLOPS at the Institut Laue-Langevin (ILL), FALCON (HZB), and Koala at Australian Nuclear Science and Technology Organization (ANSTO). 8.2.2 Powder Diffraction In a powder experiment, due to the random orientation of many singlecrystalline particles, the nodes in reciprocal space form spheres with radii corresponding to the inverse separation of planes in direct space. The same holds for “magnetic planes” made of magnetically active atoms. Note that now the term “magnetic planes” is not necessarily connected with physical crystallographic planes of the crystal structure as the magnetic structure can have incommensurate character. Consequently, in reciprocal space, two sets of spheres exist. The first set corresponds to the crystal structure and the second

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corresponds to the magnetic structure. They are both observed in the experiment as Bragg reflections at diffraction angles q given by the Bragg’s law 2dsin q ¼ l. The relation between these angles and the modulus of the scattering vector Q ¼ H þ k, where H is a reciprocal lattice vector follows directly from the Bragg’s law reads as Q ¼ jQj ¼ jk0  k1 j ¼ 4p

sin q l

(26)

For powder samples, since the diffracted intensity lies on a cone, the information about the intensity of such a cone can be obtained by detecting only a portion of the cone, since it is assumed that powder grains are randomly oriented in space. In other words, being at the cone surface, there is always a portion of grains that fulfill the reflection condition. The detected intensity measures therefore a representative statistical portion of the sample. The signal strength is proportional to the covered cone part. It is therefore useful to cover as much spherical angle around the sample as possible. This is done either with linear detectors covering a certain angular range around the horizontal plane or with position-sensitive detectors (PSDs). The latter detectors have an advantage that it is possible to correct for the curvature of cones intersecting the active zone of the PSD. This curvature leads in the case of a linear detector to instrumental broadening of detected reflections, especially at low diffraction angles. Powder diffraction experiment thus leads to a one-dimensional diffraction pattern, in which one gets the angular dependence (or equivalent onedfor instance, in terms of d spacing) of the detected intensity. This dependence contains all information about the substance under study but is also affected by other factors, for instance, by instrumental conditions. To the former (sample) group belong, for instance (apart from structural parameters as the atomic positions, stoichiometry and Debye-Waller factors), absorption, extinction, microabsorption, and preferential orientation. To the latter, instrumental part belong, for instance, systematic shift of reflections, peak shape, and width (these are, however, to a certain extent dictated also by the sample’s physical form and state) and background. It is outside the scope of this chapter to give the complete list of phenomena that possibly take place in the sample influencing the recorded intensities. The reader is advised to consult the available textbooks. One can see that if the material consists indeed from a large number of statistically randomly oriented crystallites, all various coherent constructive signals are detected. In terms of the magnetic diffraction on a material with unknown propagation vector k, it is a big advantage since one covers for sure all possible orientations and the signal (if sufficiently strong) cannot be missed. However, it is immediately seen that there are two problems connected with powder neutron diffraction. The first is that two Bragg reflections may

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have accidentally the same d spacing and appear in the diffraction pattern at the same angle. Fortunately, this so-called peak overlapping can be overcome successfully by the Rietveld method (Rietveld, 1967). In this technique, the whole diffraction pattern is least-square fit at the same time using parameters defining the sample’s structure, shape and width of Bragg reflections, background, and all other parameters defining the sample and experimental conditions. The number of numerical operations is large. However, nowadays there are several very good computer codes accessible for the community for free that master this task very well. Among the most used ones belong Fullprof, GSAS, and Jana2006 (Larson and VonDreele, 2000; Petricek et al., 2006; Rodriguez-Carvajal, 1993; Roisnel and Rodriguez-Carvajal, 2001). The quality of the fit is judged by several sophisticated parameters. However, the user can judge immediately using the integrated graphical interface where the observed and calculated profiles are shown. The second problem arises from the fact that not always the signal is strong enough to be detected although from geometrical point of view it enters the detector. This can have several reasonsdfor instance, magnetic moments involved are too small or they appear at the top of the nuclear Bragg reflections where the signal to noise ratio is for magnetic contribution much worse. In the former case, one may wish to use larger samples (might not be always possible); in the latter, however, one will always run into problems as the nuclear structure parameters may inhibit the proper determination of magnetic parameters. In that case, only neutron single crystal diffraction, possibly combined with a polarized neutron beam (see below), may lead to a successful determination of the magnetic structure. Let us now assume that two readily observable sets of Bragg reflections were detected. The distinction whether a particular reflection belongs to the nuclear or magnetic set can be done by subtracting the paramagnetic diffraction pattern from the state where the material is supposed to order magnetically. While the former set is accounted for by the nuclear crystal structure of the material, it is not clear what periodicity the magnetic structure has. The task is therefore to find propagation vector(s) k that would explain all the observed magnetic Bragg reflections. There are several approaches to solve this problem. The first one, called graphical method, is based on drawing a particular section of the reciprocal space together with circles according to Eq. (26) and circles with radius jkj. Possible positions of magnetic reflections are defined by intersections of the corresponding circles. This method which is below documented for the case of UPdSi (shown in Fig. 11) could be very time-consuming as it is not clear which section of the reciprocal space of the single crystal should be used. Fortunately, automatic indexing computer codes, such as K-search and DICVOL (Boultif and Louer, 2004; Roisnel and Rodriguez-Carvajal, 2001), can try to match observed reflections (data either in diffraction angle or d-spacing values) with calculated values. This is done in a systematic way for a whole series of propagation vectors, generated within

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FIGURE 11 Schematic representation of the graphical method to determine a propagation vector demonstrated on the case of low-temperature antiferromagnetic structure of UPdSi with propagation vector k ¼ (1/4, 0, 1/4). The correspondence of the reciprocal space of this material where observed magnetic reflections are highlighted by arrows with the recorded diffraction pattern is shown at the bottom right.

given boundary conditions imposed by the symmetry of the crystal structure. Possible propagation vectors are then tested during the indexing of the observed pattern. However, this method does not lead necessarily to a successful indexing scheme, especially in the case of incommensurate structures. The number of propagation vectors, systematic absences, and the influence of possible magnetic impurities also play an important role. For a given wave vector k, there are several other equivalent propagation vectors that are given by symmetry operations of the crystallographic space group. This set of equivalent propagation vectors is called the star of k and has the same number of members as the number of symmetry operations. This number is usually even as þk and k are always associated (except for vectors ending at the Brillouin zone boundaries listed in Table 3). Therefore, in each Brillouin zone there exists a set of magnetic peaks, each of these being associated with a given propagation vector from the star of k (Rossat-Mignod, 1987). They all contribute to the same diffraction angle but since magnetic moments have different projections for different k (and hence different partial contribution in the diffraction pattern), only the average intensity can be

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determined. As a consequence, a part of the information is lost. The only information available is the angle a between the magnetic moment and the propagation vector k. This follows immediately from the expression for the magnetic structure factor that relates its projection on the plane perpendicular to the scattering vector which reads as FMt ðQÞ ¼ sinðaÞFM ðQÞ;

(27)

The recorded intensity is then proportional to the multiplicity of the reflection (containing the information regarding how many equivalent propagation vectors contribute) and the mean value of sin2 (a). It appears (Shirane, 1959) that for highly symmetrical structures this leads to the impossibility to determine the moment direction (in cubic systems) and only the angle with a unique axis can be determined (in hexagonal, trigonal, and tetragonal systems with respect to the c-axis). This fact often leads to the question if the ordering is described by one or several of the equivalent vectors. Therefore only experiments on single crystals while imposing an external perturbation (uniaxial pressure, magnetic field) can give an answer regarding this problem. The problem is solved naturally if a crystallographic distortion is observed at the phase transition, giving an indication of lowering of symmetry. There are many powder diffractometers around the world differing in detailed specifications (flux at the sample position, available neutron wavelengths, collimation, shielding, ability to accept different sample environments, detector system used, angular and Q-range coverage, etc.). All of them, however, use more or less the same principles of detecting diffracted neutrons that were before prepared to have known properties and directed on the sample. The instrumental layout is therefore rather generic. In Fig. 12, we show an example of the double-axis high intensity diffractometer D20 (Hansen et al., 2008) installed at ILL. The biggest advantage of this diffractometer is its very high incident flux that enables stroboscopic measurements and monitoring of chemical reactions.

8.2.3 Single Crystal Experiments At constant-wavelength neutron sources, the necessity to bring a single crystalline sample in some reflection condition requires that at least one component changes its value or orientation. There are several possibilities to do so. However, in practice two basic designs are used. In the first one, called a four-cycle geometry, the crystal is mounted on a Eulerian cradle that enables to orient the crystal in any direction so that any scattering vector can be brought to the horizontal plane. The advantage is that with a reasonably short incident wavelength a large number of independent reflections can be measured, an advantage used when details of the crystal/magnetic structure are needed. Since the space available on a Eulerian cradle is limited, usually only close

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FIGURE 12 Instrument layout of the high-intensity double-axis diffractometer D20 installed at Institut Laue-Langevin (ILL). After https://www.ill.eu/instruments-support/instruments-groups/ instruments/d20/.

cycle refrigerators and furnaces are used. For a more complicated sample environment as helium- and nitrogen-cooled cryostats, dilution refrigerators, cryomagnets (that cannot be put up side down anyway), pressure cells, and a combination of those are to be used, the so-called normal-beam geometry with a lifting detector is preferred. The sample in this case rotates around a predefined axis, usually around the vertical axis of the diffractometer, called u. The detector is able to be positioned out of the horizontal scattering plane by rotating it by an angle n around the sample position. This imposes of course some limitations on the reachable reciprocal space. For instance, if the single crystal is oriented in such a way that its c-axis is vertical, all (00l) reflections are inaccessible. However, the advantage over powder diffraction is the fact that it is possible to inspect all reachable points of the reciprocal space individually. In other words, if all the members of some K-star are reachable, one can collect all Fourier components belonging to such a propagation vector separately. This requires the knowledge of the crystal’s orientation with

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respect to the laboratory coordinate systemda task that is done with the help of sufficiently known strong nuclear reflections. The determination of the so-called UB-matrix enables then to position the crystal in any allowed orientation using reciprocal space indices. Since at each point of the reciprocal space the orientation of the crystal is known and the magnetic intensity is proportional to the projection of the magnetization on the plane perpendicular to the corresponding scattering vector k, the direction of moments can be determined (here we assume that no S-domains exist; if they do exist, they are averaged). This is the main reason why single crystal diffraction is so useful. It is more versatile, and in principle many other sample-related parameters can be determined. However, considering our primary subject of magnetic neutron diffraction, these parameters constitute, however, complications. At first, in the case of a single crystal, effects of extinction that lead in general to lower intensities than calculated using a kinematic theory have to be evaluated. The effect of extinction rests in the fact that the diffracted beam also is in a scattering condition and is rescattered. This holds exactly for a monochromatic beam scattered on a perfect crystal and leads to the necessity to calculate the diffracted intensities using a more complicated dynamical theory. However, in practice there is always a certain distribution of incoming wavelengths, and crystals are also not always perfect in a mathematical sense. Dislocations, vacancies, thermal vibrations, and internal stresses in general (to name a few) apart from the existence of grains having a finite angular distribution lower the quality of a crystal and the effect of extinction. If the effect of extinction is small, corrections to the kinematic approach are possible (Becker and Coppens, 1974; Zachariasen, 1967). Different models to approximate real crystals exist. In the most common one the crystal is modeled using regions (blocks) of a certain size that are supposed to be perfect. These regions are then supposed to be misaligned by a small angle with respect to each other. The distribution of the misalignment angle can be either Gaussian or Lorentzian. The width of the distribution and the typical size of the perfect crystal blocks are the two most decisive parameters to be determined. Another problem that effects both nuclear and magnetic structure determination is absorption that depends on the averaged total length of the neutron trajectory inside the sample (and therefore depends on the shape of the crystal). However, modern computer codes contain the possibility to account for these effects, and after proper scaling corrections to nuclear reflections, this leads to reliable magnetic moment magnitudes. Examples of typical single crystal diffractometer installed at reactors include D3, D9, D10, and D23 diffractometers installed at ILL; E2 (Graf, 2003), E4, and E5 installed at HZB; HEiDi at MLZ Mu¨nchen (Heinz Maier-Leibnitz Zentrum et al., 2015); and 6T2 at LLB Saclay. The layout of the exemplary D10 instrument is shown in Fig. 13. This instrument is very versatile and enables, in addition to the four-circle geometry, also the use of a flat-cone option and energy analysis that is useful in suppressing the

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FIGURE 13 Instrument layout of the single-crystal diffractometer D10 installed at Institut LaueLangevin. After https://www.ill.eu/instruments-support/instruments-groups/instruments/d10/ description/instrument-layout/.

background signal. A small PSD ensures that the signal is not lost even if it moves a bit due to thermal expansion of the sample’s lattice and/or due to a lattice distortion and similar effects. The latter two options allow also for a more complex and heavier sample environment.

8.2.4 Use of Polarized Neutrons As mentioned above, neutrons carry magnetic moments. With help of polarizing monochromators or benders, it is possible to extract from the unpolarized beam neutrons with a particular orientation. The polarization P isP given by the average of the polarization vectors of N individual neutrons: P ¼ Pi/N. For a fully polarized beam jPj ¼ 1, and for unpolarized jPj ¼ 0. In the case of a partially polarized beam that is usually attainable in reality, we have 0 < jPj < 1. Neutrons prepared in such a way are brought to the sample using neutron guides that keep their orientation and undergo interaction with both nuclei and unpaired electrons just like in the case of unpolarized diffraction. Without giving details we only state that the difference with respect to the case of unpolarized neutrons is that now one has to consider also interference between nuclear and magnetic structure factors. In other words, the number of diffracted neutrons for different initial polarizations is not the same. In the simplest polarized neutron experiment, the analysis of the final spin state of the scattered beam is not performed, and the neutrons are detected irrespective of their spin. This method is called the flipping ratio method. It

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consists in measuring the intensities Iþ and I of a Bragg reflection, for an incident beam polarized parallel (superscript þ) and antiparallel (superscript ) to the initial polarization and calculating the ratio between them. The advantage of this method is that both intensities are measured at the same position and intensities are subject of the same corrections. The respective intensities read as I þ zjFN þ FMt ðQÞj2 and I  zjFN FMt ðQÞj2

(28)

Assuming now, for instance, FN ¼ 1.0 and FM ¼ 0.2, one gets for detected intensity in the case of unpolarized beam I z jFNj2 þ jFMj2 ¼ 1.04, whereas in the case of fully polarized neutron beam Iþ z jF2N þ FMj2 ¼ 1.44 and I z jF2N þ FMj2 ¼ 0.64. This certainly is an enormous improvement in sensitivity as far as magnetism is concerned!

8.2.5 Neutrons and Nanoobjects In the last 30 years, the study of magnetic nanoparticles and nanostructure (Shi et al., 1996) has increased, and the application of neutron techniques has become important and uniquely suited to characterize these materials. Concerning these, techniques such as neutron reflectivity [mainly polarized neutron reflectivity (PNR)] and small-angle neutron scattering (SANS) (also with neutron polarization: SANSPOL and grazing incident small-angle neutron scattering: GISANS) are very useful tools in this field of knowledge (Fitzsimmons and Schuller, 2014; Ott, 2014a,b). Due to the increasing number of new magnetic structures for which the nanometer scale plays a key role in their physical properties, it is possible to classify them in some categories according to the neutron techniques to be applied (Ott, 2014a,b). First category is related to three-dimensional materials such as (1) magnetic nanoparticles (ferrofluids) (Rajnak et al., 2015), (2) percolating phases as observed in manganites (Castellano et al., 2012; Craus et al., 2014; De Teresa et al., 2006; Saurel et al., 2010), (3) magnetic domains and domain wall (Mudivarthi et al., 2010; Turtelli et al., 2004), (4) vortex lattices in ferromagnetic systems and superconductors (Mazzone et al., 2014; Metlov and Michels, 2016; Ramazanaglu et al., 2014), and (5) long-range helical structures (Cameron et al., 2016; Hamann et al., 2011; Lamago et al., 2006; Okorokov et al., 2005). In these cases, SANS is an ideal tool to investigate the shape of individual object and their range of organization. Second category is related to two-dimensional systems such as objects organized on surface. The production of these systems can be made via lithography techniques or self-organization process. For these systems it is possible to use either SANS or GISANS (Klokkenburg et al., 2007). The third category includes magnetic thin films (for example, metal thin films, oxide thins films, magnetic semiconductors), and they can be

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characterized using the neutron reflectivity (or polarized neutrons reflectivity) technique. Here we will briefly describe the basics of these two techniques: SANS (including GISANS and SANSPOL) and neutron reflectivity (and PNR) techniques. Moreover some examples of recent studies involving magnetic nanostructures, nanoparticles, and thin films will be presented as well.

8.2.6 Small-Angle Neutron Scattering Known as a nondestructive nanoanalytic method for crystalline, amorphous, or liquid materials, SANS is a conventional technique used in neutron scattering and is mainly applied to polymer science and soft matter studies due to the contrast variation possibilities. Probing nanometric properties of magnetic materials is also possible because of the coupling between the spin of the neutron and the magnetic moment of the particles. For a good formalism of SANS theory, we advice to consult the excellent monographs about small angle scattering (Feigin and Svergun, 1987; Guinier and Fournet, 1955; Kostorz, 1979). As explained in the section Essentials of Elastic Neutron Diffraction, here we will condense the important and basic terms that allow us to understand the subject. In Fig. 14 we display a schematic of the experimental SANS setup. Some examples of SANS instruments include D11 (Lieutenant et al., 2007), D16, D22, and D33 (Dewhurst et al., 2016) at ILL; V4 (Keiderling et al., 2008) and

FIGURE 14 Schematic representation of the small-angle neutron diffraction setup. Four steps used in any scattering experiment are illustrated: monochromatization, collimation, scattering, and detection.

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V16 (Vogtt et al., 2014) at HZB; KWS-1 (Feoktystov et al., 2015) and KWS-2 (Radulescu et al., 2012) at MLZ; PA20 (Chaboussant et al., 2012), PAXY, PACE, and TPA at Laboratoire Le´on Brillouin (LLB); SANS-I, SANS-II (Strunz et al., 2004), and USANS at Paul Scherrer Institut (PSI); and QUOKKA (Gilbert et al., 2006) at ANSTO. A white beam from a reactor is “monochromatized” (usually of order of 10% is filtered out) with the use of a velocity selector, a monochromator, or a set of choppers. Then the neutron beam is tightly collimated and directed onto the sample. The scattered beam from the sample is registered in an area detector, simultaneously a beam stop is placed in front of the detector to absorb the direct beam that crosses the sample and would saturate or even destroy the detector. In principle, the diffraction theory for nanoparticles is the same as above. However, here we are dealing with large objects (1 e1000 nm objects) and ˚ ), leads to a this, when compared to normal diffraction (in the range of A significantly different Q range. Typically, the Q range in the SANS regime is between 102 and 5 nm1 which corresponds in real space to values in the range of 0.5e500 nm. In SANS measurements we are not sensitive to the atomic details of the sample, and it is possible to apply an optical approximation. Differently from the definition of a discrete atomic scattering b of an atom, we can use a locally averaged scattering length density h(r), due to the low spatial resolution for the SANS case. We can define a nuclear scattering length density X ci bi hN ¼ (29) Ui where bi is the nuclear scattering length, ci is the concentration, and Ui is the atomic volume of the speciment i. For instance, for water H2O, one gets h(r) ¼ 0.56  1010 cm2 and for heavy water D2O h(r) ¼ 0.56  1010 cm2. Similar to Eq. (29) we can define the magnetic scattering amplitude, the interaction between the neutron magnetic moment and a set of magnetic moments reads as X ci M t i hM f (30) Ui At this place we should note again that the magnetic scattering amplitude is a vector, and it depends on the orientation of the moment with respect to the scattering vector Q. Considering Fig. 15, where we schematically show a nanoobject in a matrix, SANS technique can determine the form factor of the object which is the Fourier transformation of the scattering length density difference between the nanoobject (hp) and the matrix (hmatrix) around. Dh ¼ hp  hmatrix

(31)

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FIGURE 15 Nanoobject of scattering length hp in a matrix or solvent with scattering length hmatrix. After Ott, F., 2014b. Neutron scattering on magnetic nano-objects, in: Simonet, V., Canals, B., Robert, J., Petit, S., Mutka, H. (Eds.), Neutrons et Magnetisme. EDP Sciences, p. 02005, 1e16.

The form factor in the SANS case can be conveniently defined as: Z FðQÞ ¼ dr3 DheiQrJ

(32)

In the case of magnetic diffraction, the interaction is limited again to the component of the magnetization perpendicular to the scattering vector Q. A brief description of this fact was explained previously. When we have an unpolarized monochromatic incoming neutron beam, i.e., the neutron spins are randomly distributed, we speak about a conventional SANS experiment. For this configuration, the total scattering intensity is equal to the sum of the squared amplitudes from individual magnetic and nuclear contrasts. Suppose we have a material containing magnetically active nanoobjects (e.g., precipitates) immersed in a magnetic field H that orients moments in these nanoobjects or the nanoobjects themselves along the field direction in a certain way. The scattering cross section can be written in analogy to Eq. (8) as dsðQÞ ¼ AðQÞ þ BðQÞsin2 a dU

(33)

We observe that this scattering cross section is anisotropic due to the vector nature of the magnetic form factor. A and B are the isotropic and anisotropic terms, respectively, and the angle a is the azimuthal angle between the direction of the field H and the scattering vector Q. If all magnetic moments in the sample are fully aligned along the applied magnetic field, then in a SANS measurement it is possible to separate the contribution of the nuclear origin A(Q) and the magnetic contribution B(Q). In spite of the fact that SANS is a powerful technique for measuring the magnetic contribution from magnetic nanoobject systems, it shows two limitations: first, the magnetic contrast is weak when compared to the nuclear one; and second, it is not possible to obtain at absolute scales the composition,

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densities, and magnetization of the matrix and the particles. This is a direct consequence of the loss of the phase information between individual contributions. Let us now consider nanospheres with a particular radius R in a solvent as shown in Fig. 16. Although the scattering length density of the particles is different to the solvent (or the matrix), the form factor depends only on the shape of particles. In some cases it can be calculated analytically. As an example we show in Fig. 16B the form factor for a spherical particle of radius R, which is given by the formula f ðQRÞ ¼ 3

sinðQRÞ QR cosðQRÞ ðQRÞ3

(34)

In this simulated case, we consider that the size distribution of the particles is narrow. In a real case, when the nanoobject exhibits a significant distribution

FIGURE 16 Schematic representation of a colloidal suspension of diluted spheres (A) and simulated form factor of a sphere of R ¼ 30 nm (B). After Ott, F., 2014b. Neutron scattering on magnetic nano-objects, in: Simonet, V., Canals, B., Robert, J., Petit, S., Mutka, H. (Eds.), Neutrons et Magnetisme. EDP Sciences, p. 02005, 1e16.

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in sizes, the oscillations observed in Fig. 16B are smoothed. For the isotropic case, the two-dimensional signal from the detector can be integrated circularly to provide a better signal to noise ratio. In the examples mentioned until now, we just considered diluted systems in which the distance between the nanoobjects is large. However, it is possible to have also a dense packing of nanoobjects, where the scattering starts to become sensitive on the correlation between the particles. In this case the scattered intensity I(Q) follows the expression IðQÞ ¼ jFðQÞj2 $SðQÞ;

(35)

where S(Q) is the structure factor which characterizes the correlation between the particle positions. It can be observed that the nanoobjects can be more or less well packed leading to variations of the diffraction pattern as documented in Fig. 17. Modeling of such patterns leads to detailed knowledge of the degree of packing. The SANS technique has been used in many different studies. At this place we list only a few representative examples. 1. Information linked to magnetic transformations at large scales that are related to percolation setting can be obtained successfully using SANS

FIGURE 17 Influence of a packing of particles on the small-angle neutron scattering (SANS) pattern. Observe that the system can be more or less packed, influencing in the scattering pattern detected by the two-dimensional position-sensitive detector. After Ott, F., 2014b. Neutron scattering on magnetic nano-objects, in: Simonet, V., Canals, B., Robert, J., Petit, S., Mutka, H. (Eds.), Neutrons et Magnetisme. EDP Sciences, p. 02005, 1e16.

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techniques. Recently, several manganite materials have been subject of this type of study, for instance, Pr0.7Ca0.3MnO3. Saurel et al. (2010), have used the SANS instrument D22 at ILL to investigate the evolution of the conducting phase at the percolation threshold in this material. 2. To investigate vortex lattices in a mixed state of stannide superconductor Yb3Rh4Sn13, Mazzone et al. (2014) have used SANS technique at SANS-I and SANS-II, PSI. By applying magnetic fields (in the range of 35e1850 mT) at low temperatures (in the range of 0.05e6.5K), they observe a single-domain vortex lattice with a slightly distorted hexagonal symmetry. 3. To investigate a long-range helical magnetic structure, SANS has been used to determine the magnetic phase diagram of the helimagnetic spinel compound ZnCr2Se4 (Cameron et al., 2016). Temperature- and magnetic fieldedependent measurements were made, from 2 up to 16K, and in fields up to 11 T. This sample presents an incommensurate screw like ˚ making SANS a suitable magnetic structure with a pitch of 22.4 A technique. A special case of SANS is called grazing incident small angle neutron scattering, GISANS. Here the incident angle with respect to the surface of the studied sample is so small that the propagating neutron wave is confined to a small surface region. For good reviews of several grazing incidence techniques, see the monographs of Hexemer and Mu¨ller-Buschbaum (Hexemer and Mu¨llerBuschbaum, 2014; Mu¨ller-Buschbaum, 2013). There are three different geometries as shown in Fig. 18: (1) specular reflection; (2) scattering in the incidence plane (also known as off-specular scattering); and (3) scattering perpendicular to the incident plane. Each scattering geometry has a different length scale x and direction within the sample surface. For the specular reflectivity, for instance, the structure along the depth in the film (3 < x < 100 nm) is probed. To probe surface features at a micrometric scale, 600 nm< x < 60 mm, the off-specular geometry is the appropriate technique. In the case of the last method, the surface characteristics are in the range of 3 < x < 100 nm. Typically, GISANS may be used to investigate small particles on a surface in the order of 20e100 nm, arrays of nanowires (20e100 nm), magnetic domains self-organized in a regular structure (w100 nm), and magnetostructural surface correlations (10e20 nm). Usually, GISANS experiments can be performed on general SANS instruments due to good collimation in the plane of incidence and in the perpendicular plane. However, it is also possible to find instruments dedicated to this technique, such as PAPYRUS and the new PA20 (Chaboussant, 2012) at LLB; FIGARO (Campbell et al., 2011) at ILL; and REFSANS (Kampmann et al., 2004) at FRMII. As an example of the application of GISANS on magnetic systems, we refer to the study of magnetic stripes in FePd and FePt layers (Pannetier et al., 2003).

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FIGURE 18 Three different geometries for scattering involving small angles. Specular reflectivity geometry (red line); off-specular scattering plane corresponding to the incidence plane (red plane); grazing incident small-angle neutron scattering plane perpendicular to the incidence plane (green plane). After Ott, F., 2014a. Neutron surface scattering. Application to magnetic thin films, in: Simonet, V., Canals, B., Robert, J., Petit, S., Mutka, H. (Eds.), Neutrons et Magnetisme. EDP Sciences, p. 02004, 1e21.

Using polarized neutrons in combination with the SANS technique, a new technique was developed SANSPOL, to improve and increase the magnetic contrast. This has been utilized, for example, in systems that have weak magnetization fluctuations such as Co ferrofluids (Wiedenmann, 2001; Wiedenmann et al., 2003), in soft magnetic Fe-Si-B-(Nb,Cu) and Fe-Nb-B alloys (Wiedenmann, 2001), or in polydisperse systems that present different constituents and where the conventional SANS cannot differentiate between them [for example, in ferrofluids based on magnetite, Fe3O4 (Kammel et al., 2001, 2002)]. For polarized SANS measurements, there are two possible geometries to be used: (1) apply the magnetic field parallel to the neutron propagation direction; and (2) apply a magnetic field perpendicular to the neutron propagation vector. In the first case, the scattering will be isotropic, and in the second case the scattering will be modulated by a factor sin2 a. These two setups are illustrated in Fig. 19. The procedure to get information with this technique is then identical to the ordinary polarized neutron diffraction mentioned briefly above under

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FIGURE 19 Two different configurations of SANSPOL, where the magnetic field can be applied longitudinally (A) or transversally (B) to the incoming neutron beam. SANSPOL, small-angle neutron scattering with neutron polarization. After Ott, F., 2014b. Neutron scattering on magnetic nano-objects, in: Simonet, V., Canals, B., Robert, J., Petit, S., Mutka, H. (Eds.), Neutrons et Magnetisme. EDP Sciences, p. 02005, 1e16.

considerations of a special character of SANS. Examples for this application of SANSPOL include, for instance, investigations of Co ferrofluids to observe how the magnetic core behaves. In this case, conventional SANS technique could not provide the information about the systems. Authors of this study noticed a superparamagnetic behavior of the diluted Co ferrofluids. The other example we want to mention at this place is a study of a magnetite ferrofluid diluted in H2O and D2O with different compositions (Kammel et al., 2002). Using SANSPOL it was possible to determine three different components of the particles: the magnetic coreeshell particles, free organic-shell molecules, and magnetic aggregates.

8.2.7 Neutron Reflectivity Different new artificial materials, comprising a stacking of different materials in thin sandwiches (heterostructures), started to be produced with the advanced techniques for deposition of ultrathin films in the early 1980s. Thereafter, new physical phenomena began to emerge like magnetic exchange-coupling in rare earth and metallic superlattices, exchange-bias coupling at antiferromagneticeferromagnetic interfaces, enhanced magnetism in ultrathin films, giant magnetoresistance in metallic spin valves, or tunnel magnetoresistance (Ott, 2008). Facing this new technological revolution, new methods of characterization needed to be implemented. Thus, consequently the neutron reflectometry technique was developed at the same time as the fabrication of these new devices. Knowing the magnetic coupling between neutrons and the magnetic moments of the sample, it was expected that neutron reflectometry would arise as a perfect tool to acquire information about the magnetic configuration in all these new systems. Initially, the neutron reflectivity technique was employed

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to investigate ferromagnetism in metallic layers. However, to probe multilayers with different materials and obtain information on the amplitude and the direction of the magnetization of the different layers, one needs to combine polarized neutrons with the neutron reflectivity measurements. In neutron reflectivity experiments it is convenient to introduce an equivalent neutron “optical index” n, according to the expression n z1  dN  dM ;

(36)

where dN is the nuclear contribution and dM is the magnetic contribution to the optical index. The sign of the magnetic contribution depends on the orientation of the neutron spin with respect to the magnetization (parallel or antiparallel). According to relation (36) the measured reflectivity is different for incident neutrons with spin parallel or antiparallel to the magnetization. Examples of experiments involving polarized neutron reflectometry include studies of the recently highly debated skyrmion lattices. They are observed in a hybrid structure of asymmetric Co nanodot arrays grown on Co/Pd thin film with perpendicular magnetic anisotropy (Gilbert et al., 2015), confirming the chiral texture in the vortex state of Co dots imprinted in the Co/Pd underlayer. These measurements were performed at MAGIK reflectometer at NIST. In another interesting study it has been shown that the interface of the Pd-doped FeRh thin films exhibits an anomalous magnetization (Bennett et al., 2015). The results prove the existence of unusual thin interfacial regions with strong magnetization that have unique thermomagnetic properties when compared to the rest of the system. For these studies the Magnetism Reflectometer at the Spallation Neutron Source at Oak Ridge National Laboratory has been employed.

8.2.8 Sample Environment Although powder diffraction leads to indispensable knowledge regarding, for instance, the length of the propagation vector, in many cases it cannot solve the magnetic structure to details. On the contrary, single crystal diffraction has a potential to disclose even the secrets of K and S domains. To do this one has to have an opportunity to alternate the population of different domains distribution using some suitable external perturbation. This can be either uniaxial stress or magnetic or electric fields and leads to a necessity to build and use additional machinery that surrounds and acts on a sample called sample environment. It can be a simple cryostat or a complicated magnet combined with application of electrical field and pressure. In a broader sense, under the term “sample environment” one understands any auxiliary apparatus creating strongly nonambient conditions under which the sample is studied including low and high temperatures. Temperatures down to 30 mK using a dilution refrigerators or temperatures up to 2000K are routinely available. Pressure cells offer pressures ranging from near-ambient pressures of a few bar or kilobars that are possible to be changed in situ up to

120 Handbook of Magnetic Materials

several GPa in the case of bulk PariseEdinburgh cells that weight several tens of kilos. There exist a clear trade off between the maximum pressure and the volume space available for the sample (Guthrie, 2015). As far as magnetism is concerned, a special role plays the possibility to alternate the magnetic state of the sample using a magnetic field. Two basic geometries are then recognized. In the vertical geometry the field is directed along a particular crystallographic direction that is parallel to the rotational axis of the diffractometer. The magnetic field is produced by a split-pair coil that has a certain splitting (typically 20e50 mm) and opening angle (typically 5e15 degrees). In such a setup the reciprocal space experimentally accessible is limited to directions close to a plane perpendicular to the field direction. The angular range available for diffraction can be expanded by increasing the opening angle. This, however, leads to a lower field acting on the sample. The steady magnetic fields available for this geometry are typically limited to 15e17 T (Prokes et al., 2001). The other geometry is horizontal, where the sample is rotated around an axis that is perpendicular to the field direction. This geometry enables, for instance, studies where the field has to be aligned along the scattering vector. As examples we want to mention here studies of flux lattices in high-temperature superconductors or in materials where magnetic reflections appear close to the scattering vector. In this case the maximum field is limited to 16 T (Holmes et al., 2012) (superconducting technology) and to 26 T (hybrid magnet at HZB (Smeibidl et al., 2010) that combines both superconducting and resistive coilsdsee Fig. 10). In the latter case, the total energy consumption is much higher than in the case of a superconducting coil.

9. EXAMPLES OF MAGNETIC STRUCTURE DETERMINATIONS 9.1 Example I: Powder Sample of UPdSi Let us document the whole procedure of the magnetic structure determination on one, relatively simple example of UPdSi (Prokes et al., 1997a, 1998) that was investigated in the form of a powder. UPdSi is one member of a large group of isostructural equiatomic UTX (T-late transition metal, X ¼ Si or Ge) compounds (de Boer et al., 1991; Sechovsky and Havela, 1998; Troc and Tran, 1988). Their magnetic properties are strongly dependent on the constituent transition metal elements although the only magnetic moment is of 5f-electron origin residing on uranium. UPdSi adopts the TiNiSi-type (space group Pnma) of structure, which is derived from the CeCu2 type (space group Imma) (Prokes et al., 1997b). Uranium atoms are coordinated on zigzag chains running along the a-axis. The nearest UeU distance is 351 pm. In Hill’s classification (Hill, 1970), such a value indicates that the compound is in the critical region between localized and itinerant

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5f-electron behavior. Magnetic bulk measurements document that this material orders magnetically around 33K with an additional magnetic phase transition at 27K (Prokes et al., 1997a). From the magnetization curves exhibiting metamagnetic behavior at low temperatures, one expects that the ground state of UPdSi is antiferromagnetic (de Boer et al., 1991). The neutron-diffraction data were obtained some time ago at HZB using a 400-channel multidetector ˚. powder diffractometer. The incident-neutron wavelength was l ¼ 2.4 A To be able to determine the magnetic structure, details of the crystal structure have to be known. In the case of UPdSi, this task was of enormous importance since two different crystal structures were published in literature (Sechovsky and Havela, 1998). It appears that the Bragg reflections observed at 62K, well above the anticipated magnetic phase transition, can only be fully indexed assuming the TiNiSi type of structure. The space group Pnma allows due to its lower symmetry (and hence lower number of extinction rules) for reflections with h þ k þ l ¼ 2n þ 1 which would be absent for the CeCu2 type (space group Imma). The strongest reflection of this type is the reflection at ˚, 52.5 degrees (see Fig. 20). The refined cell parameters are a ¼ 7.026 (6) A ˚ ˚ b ¼ 4.205 (5) A, and c ¼ 7.670 (8) A. There are four U atoms in the unit cell, which occupy 4c sites (x, 0.25, z) with the local symmetry $m$, where the dot means an absence of any symmetry operator and the m denotes a mirror plane perpendicular to the b-axis. The positions of the U atoms read as U1: (x, 0.25, z), U2: (x þ 0.5, 0.75, z þ 0.5), U3: (x, 0.75, z), and U4: (x þ 0.5, 0.25,

FIGURE 20 Powder neutron diffraction pattern of UPdSi adopting the TiNiSi type of structure (space group Pnma) taken at 62K, above the magnetic phase transition. The experimental points are the filled points. The full line through the points is the best fit. The vertical lines at the bottom denote the position of nuclear Bragg reflections that are indicated near the vertical lines. Those denoted by a star would be absent in the case UPdSi would adopt the space group Imma. In the inset the crystal structure is schematically shown.

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FIGURE 21 Powder neutron diffraction pattern of UPdSi taken at 62, 31, and 3.5K. Bragg reflections that appeared at 31 and 3.5K are denoted by an arrow and stars, respectively.

z þ 0.5) with x ¼ 0.003(1) and z ¼ 0.184(2). The Pd and Si atoms occupy the same 4c Wyckoff position but with different x and z parameters. A schematic representation of the structure is shown in the inset of Fig. 20. In a first step of the magnetic structure determination, one has to verify the magnetic phase transition temperature by observing the additional diffracted signal that is possible to be associated with long-range magnetic order. As shown in Fig. 21, with lowering the temperature, indeed, new Bragg reflections appear (shaded in Fig. 21). While at 31K we observe a single reflection of rather low intensity (marked by an arrow), the pattern taken at 3.5K contains several such reflections (marked by stars). As all these reflections appear at positions where no nuclear reflections are situated, one can immediately conclude that the magnetic structure has lower symmetry than the nuclear structure. The detailed temperature dependence of the magnetic intensity shown below indicates two magnetic phase transitions (around 34 and 26K, respectively) in accordance with bulk measurements. Second, we need to identify the propagation vector of such a magnetic structure. If we assume that there is no structural transition between 62 and 3.5K, the magnetic signal can be extracted in first approximation by subtracting the high temperature from the pattern taken at the relevant temperature. By doing so, one obtains a set of diffraction angles that define the projection of a three-dimensional lattice onto a single axisdthe diffraction angle q. In the case of UPdSi it appeared that the propagation vector of the lowtemperature magnetic structure is k0 ¼ (0.25, 0, 0.25). Then the visible magnetic reflections can be indexed as (000)þ, (101), (101), and (200)/(200)þ/(102). Of course, there is also an equivalent vector k00 ¼ ð0:25; 0; 0:25Þ, forming together with k0 and k00 a K-star having

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four members. With k00 one can fully index the reflections in the same manner. In this case, however, the indices would read as (000)þ, (101), (101), and (200)þ/(200)/(102). In the refinement it is necessary to consider, however, only one pair since for powders K-domains are not distinguishable. In the next step, one needs to determine the coupling and direction of the magnetic moments. To do this in a systematic way (not to miss any possible configuration) we adopt the group representation theory introduced above. The space group Pnma contains eight symmetry operations, which form a group G. There are only two symmetry operations within G which leave the propagation vector k0 ¼ (0.25, 0, 0.25) invariant, or which transform it into the equivalent propagation vector k0: the identity E and the mirror plane perpendicular to the b-axis m(x, 1/4, z). As there are four members of the K-star and the original space group contains eight symmetry elements, these two elements form a little group Gk of k of order two. In this case, there is no possibility for two-dimensional irreducible representations, as 22 ¼ 4 exceeds the order of the little group (¼ 2, Eq. 23). Consequently, there are two one-dimensional irreducible representations denoted as G1 and G2. The effects of the symmetry elements on the moment components are listed in Table 4, and it follows that all four atomic positions are decoupled. So, there is no relation between the magnetic moments residing at the four sites that were equivalent in the paramagnetic state, but became nonequivalent in the magnetic state. Still, the number of possible magnetic structures is significantly reduced with respect to the general case since the existence of the mirror plane restricts the possible directions to be either along the b-axis or perpendicular to it. The most probable magnetic structure has been obtained by fitting our data to all the models, at first with no restriction on the orientation of the U magnetic moments (along the b-axis or perpendicular to it but belonging to the same irreducible representation, all U moments independent). It appears that all the moments clearly prefer to align ferromagnetically within one unit cell along the b-axis. The best agreement (see Fig. 22) between the model and the observed spectrum is found for the model belonging to representation G2. This step, which is the last step in the determination, leads to the knowledge of the Fourier components describing the distribution of the four U magnetic moments in direct space. Since the magnetic structure is commensurate but not one of the special cases listed in Table 3, one has to sum according to Eq. (16) over k0 and k0. It follows that at 3.5K (after normalizing to the scale factor obtained from the best fit of 62K data to the nuclear structure) the magnetic moments at all U sites amount to 1.40 (1) mB/U. They are coupled within one unit cell ferromagnetically and point along the b-axis. The antiferromagnetic structure, which is shown in the inset of Fig. 22, consists of collinear U moments in a sequence þ þ þ þ     as one moves along the [1 0 1] direction. Such a structure is encountered very often in literature for different materials.

mx1

mx2

mx3

mx4

my1

my2

my3

my4

mz1

mz2

mz3

mz4

Identity E

mx1

mx2

mx3

mx4

my1

my2

my3

my4

mz1

mz2

mz3

mz4

m(x, 1/4, z)

mx1

mx2

mx3

mx4

my1

my2

my3

my4

mz1

mz2

mz3

mz4

G

mx1

mx2

mx3

mx4

0

0

0

0

mz1

mz2

mz3

mz4

G

0

0

0

0

my1

my2

my3

my4

0

0

0

0

1 2

The positions of U atoms read as U1: (x, 0.25, z), U2: (x þ 0.5, 0.75, z þ 0.5), U3: (x, 0.75, z), and U4: (x þ 0.5, 0.25, z þ 0.5) with x ¼ 0.003 and z ¼ 0.184(2). The lowest two lines describe possible U magnetic moment configurations as derived by using group theory. mx1 denotes the x cartesian component at the site 1.

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TABLE 4 Transformation Rules for Components of U magnetic Moments (Wyckoff Position 4c(x, 0.25, z) in the Orthorhombic UPdSi (Space Group Pnma))

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FIGURE 22 Powder neutron diffraction pattern of UPdSi taken at 3.5K from which pattern taken at 62K has been subtracted along with the best fit to a model belonging to the irreducible representation G2 listed in Table 4. Magnetic Bragg reflections indexed by the propagation vector k0 ¼ (0.25, 0, 0.25) appear at positions marked by vertical lines. The resulting antiferromagnetic structure is shown in the inset. It consists from U moments oriented either parallel or antiparallel to the b-axis.

With increasing temperature, the intensity of the strongest magnetic reflection smoothly decreases and vanishes around Tm ¼ 26K. Between this temperature and TN, one observes, however, another magnetic reflection that is at even lower diffraction angle than the lowest-lying (and the strongest) ground-state magnetic reflection. This reflection disappears around 34K, a temperature that is reasonably close to TN ¼ 33K derived from bulk measurements. This suggests that the modulus of the propagation vector is smaller than that of k0. Since within statistical uncertainties only one reflection is observed, no unambiguous indexation is possible. However, as mentioned above, one encounters very often sine wave incommensurate structures that precede the low-temperature magnetic ground state. This is the reason why we assume that the intermediate structure between Tm and TN is related to the lowtemperature ground state by a lock-in transition. The propagation vector k1 is assumed to take the form k0 ¼ (d, 0, d). Then, d amounts to about 0.215. Its temperature dependence together with the intensity of both the commensurate and the incommensurate reflections is shown in Fig. 23. The magnetic structure is in principle very similar to the ground state magnetic structure. The only difference is that it is not commensurate and the U moments are modulated according to a transversal sine wave (see also Fig. 7C). The example just given is a relatively simple case where powder diffraction was enough to solve under certain assumptions the problem completely (except for a problem of the K-domains). This might be not always the case as propagation vectors may be accidentally projected to the same diffraction

126 Handbook of Magnetic Materials

angle or the signal is very small. In this case, single crystal diffraction is indispensable. To be able to interpret the measured quantities in an unambiguous manner, single crystals are equally required for diffraction experiments in magnetic fields or under uniaxial pressure. Let us therefore also describe one single crystal neutron diffraction case.

9.2 Example II: Single Crystal of UNiGa UNiGa is one of the most frequently studied UTX compounds crystallizing in the noncentrosymmetric hexagonal ZrNiAl-type of structure. This structure consists of two basal planes alternating along the hexagonal c-axis: one containing the U atoms and one-third of the T atoms and the other plane contains the rest of the T atoms together, with the X atoms. While the three U atoms occupy the 3g(xU, 0, 1/2) position (xU w 0.58) with local symmetry m2m and

FIGURE 23 Temperature dependence of the integrated intensity of the lowest-lying magnetic reflection seen in UPdSi below the magnetic phase transition temperature of 33K (A). While at low temperatures it appears at a commensurate position (blue filled points), above 26K it can be indexed by an incommensurate propagation vector of the form k0 ¼ (d, 0, d), with d dependent on temperature (B). This description is dependent on an assumption of a close relation between both magnetic structures.

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all p-atoms the 3f(xp, 0, 0) positions (xp w 0.24) with the same local symmetry, but with different position parameter, T atoms are distributed over two inequivalent positions. One of them is the 2c(1/3, 2/3, 1/2) position, and remaining T atoms take the 1b(0, 0, 1/2) position. There are more than 22 compounds in this group. All of them exhibit very strong magnetocrystalline anisotropy fixing the U moments along the hexagonal axis (Sechovsky and Havela, 1998). UNiGa has been previously reported to order ferromagnetically (Andreev et al., 1984), antiferromagnetically (Palstra et al., 1987) or classified even as a spin-glass system (Zeleny and Zounova, 1989). These variety of contradictory claims exist because the ground state of UNiGa is very sensitive to the exact stoichiometry and heat treatment (Andreev et al., 1995). There is today no doubt that it orders after a proper heat treatment antiferromagnetically (Havela et al., 1991). The temperature dependence of bulk magnetic properties clearly demonstrates the presence of a strong uniaxial anisotropy. Magnetic ordering in UNiGa is manifested by a faint anomaly in the electrical resistivity, specific heat, and low-field magnetization around TN ¼ 39K (Sechovsky and Havela, 1998). There are three additional magnetic phase transitions at T1 ¼ 34.7K, T2 ¼ 36.0K, and T3 ¼ 37.4K in a small temperature range below TN ¼ 39.0K. While the transition at TN is of the second-order type, the first three transitions are of the first-order type. Although the symmetry analysis can be done only in the case that the underlying crystal structure remains intact in the course of magnetic phase transitions (i.e., in the case of the second-order transition), no crystal structure distortion has been detected in UNiGa down to the lowest temperatures. This enables us to use the group representation theory also in the case of UNiGa. The existence of four-phase transitions that are not of a crystal structure type suggests an existence of four different magnetic phases in UNiGa. Let us denote the magnetic state below T1 as AF1, between T1 and T2 as AF2, between T2 and T3 as AF3, and between T3 and TN as INC. As we will show, magnetic structures AF1, AF2, and AF3 are commensurate with the crystal structure, while the last one is incommensurate. The magnetic response of UNiGa to an applied magnetic field is very anisotropic. While only very little effect is seen for fields applied perpendicular, the application of a field along the c-axis effects the magnetic state of UNiGa dramatically. When a field of approximately 1 T is applied along the hexagonal axis at T ¼ 4.2K, it exhibits a metamagnetic-like transition (MT) that is reflected in all bulk properties. The saturation magnetization attains 1.4 mB/U (Jirman et al., 1992; Sechovsky and Havela, 1998). Compiling all results, a complicated magnetic phase diagram has been established comprising, apart from the ferromagnetic phase that appears above the MT, six different phases (Sechovsky et al., 1995). Here, however, we concentrate only on the zero-field magnetic structures.

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Neutron diffraction studies performed on two UNiGa single crystals in zero field clearly distinguished four different magnetic phases with transition temperatures (TN ¼ 39.5K) agreeing with bulk data (Sechovsky and Havela, 1998; Sechovsky et al., 1995). In Fig. 24, we show the reciprocal space of UNiGa mapped using single crystal neutron diffraction at 41.8K using the flat cone diffractometer E2 installed at HZB. The incident neutron wavelength was ˚ , and the crystal was oriented with its (hhl) plane perpendicular to the 2.38 A vertical axis of the instrument. It was rotated around this axis by 180 degrees with a step size of 0.2 degree. As can be seen from Fig. 24, all the observed Bragg reflections can be indexed. As the temperature is lowered, new Bragg reflections due to the magnetic order appear. These extra reflections document that UNiGa indeed orders antiferromagnetically and appear in the diffraction pattern collected at 1.6K (see Fig. 25) at places indexable by three propagation vectors: k1 ¼ (0, 0, 1/6), k2 ¼ (0, 0, 1/3), and k3 ¼ (0, 0, 1/2). It is seen that the vector k3 is the third harmonics of the vector k1. Generally, reflections indexed by k2 appear to be stronger than the other two groups. No additional intensity is detected at the top of nuclear reflections suggesting that the propagation vector (0, 0, 0) is not realized in UNiGa in this phase. The identification of these propagation vectors agrees well with previous powder work (Maletta et al., 1992).

FIGURE 24 A portion of the reciprocal space of UNiGa mapped using single crystal neutron diffraction at 41.8K. The section shown is the (hhl) plane.

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FIGURE 25 A portion of the reciprocal space [the (hhl) plane of UNiGa mapped using single crystal neutron diffraction at 1.6K]. New Bragg reflections of magnetic origin can be indexed by three individual propagation vectors k1 ¼ (0, 0, 1/6), k2 ¼ (0, 0, 1/3), and k3 ¼ (0, 0, 1/2) (highlighted in the inset).

At this stage we immediately recognize the power of single crystal diffraction. When the sample is appropriately oriented, enabling the effective mapping of the reciprocal space, one obtains all the propagation vectors directly. In the case of UNiGa it is important that no other propagation vectors were detected in the flat-cone geometry that enables mapping in layers above the shown horizontal scattering plane, in our case with components along the [100] direction. In the next step we need to determine the coupling and direction of the U magnetic moments. We utilize again the group representation theory and consider a magnetic propagation vector in the form k ¼ (0, 0, kz), where kz s 1/2. The space group P-62m contains 12 symmetry operations, which form a group G. There are six symmetry operations which leave the propagation vector k ¼ (0, 0, kz), where kz s 1/2 invariant, or which transform it to an equivalent propagation vector k: the identity E; rotoinversion axes 3þ(0, 0, z) and 3(0, 0, z); and three mirror planes m(x, x, z), m(x, 0, z), and m(0, y, z). It should be noted that some of the symmetry operations mutually relate the three uranium sites U1: (x, 0, 1/2), U2:(0, x, 1/2), and U3: (x, x, 1/2). For instance, the rotoinversion axis 3þ(0, 0, z) permutes the sites in the following

130 Handbook of Magnetic Materials

sequence: U1 / U2 / U3 / U1. This means that if the moment residing at the first 3g(xU, 0, 1/2) position (xU w 0.58) has a certain moment orientation, the orientation on the two remaining positions is restricted by such a symmetry operation. At the same time, the translation symmetry has to be preserved. As there are two members of the K-star (k and k) and the original space group contains 12 symmetry elements, the symmetry elements listed above form a little group Gk of k of order six. In this case, there is the possibility to have two-dimensional irreducible representations. It appears that there are according to Eq. (22) two one-dimensional irreducible representations denoted as G1 and G2 and one two-dimensional denoted as G3 appears in the decomposition. This is in agreement with Eq. (23), according to which 12 þ 12 þ 22 ¼ 6 (order of the Gk). Although the derivation of possible magnetic structure models can be performed “by hand,” there are several suitable computer codes (e.g., MODY, BasIrep) available (Roisnel and Rodriguez-Carvajal, 2001; Sikora et al., 2004). The analysis of the UNiGa case has been performed using BasIreps which is a part of a larger distribution package associated with Winplotr (Roisnel and Rodriguez-Carvajal, 2001). Examples that are included with the distribution make the necessary calculations easy and understandable. In Table 5 we give basis vectors calculated according to Eq. (24) which show how the Cartesian components of the magnetic moments residing on the three uranium sites relate to each other. These can be used in the fitting procedure directly for both k1 and k2. As one can see, magnetic moments associated with the irreducible representation G1 are confined to the basal plane. The G2 irreducible representation, on the other hand, allows moments to be either along the a-axis (we note that in the hexagonal system there are three equivalent a-axes), along the c-axis, or confined within the aec plane. Models associated with the third, two-dimensional irreducible representation are at the first sight rather complex. However, closer inspection reveals that it is rather similar to the case of the G2. The difference is that the relation between moment components at different sites is more complex allowing for a more general direction of moments, not confining them to a particular, highly symmetric crystallographic direction or plane. Now we turn to models associated with the propagation vector k3 ¼ (0, 0, 1/2). The situation for the propagation vector k3 ¼ (0, 0, 1/2) is somewhat different as it belongs to a special case listed in Table 3. In this special case, all of the 12 symmetry operations keep this vector invariant (or transform it to an equivalent vector). The little group Gk has order 12. There are six irreducible representations, four one-dimensional and two two-dimensional. The relation (23) reads as 12 þ 12 þ 12 þ 12 þ 22 þ 22 ¼ 12. Models allowed by symmetry for k3 ¼ (0, 0, 1/2) follow from the basis vectors listed in Table 6. As can be seen, the situation is different to the previous case of k1 and k2. No combinations between basal plane and c-axis components are allowed. Moreover, one of the irreducible representations (G4) requires moments to be zero.

U1

U2

U3

Mx

My

Mz

Mx

My

Mz

Mx

My

Mz

G

a

2a

0

2a

a

0

a

a

0

G

a

0

b

0

a

b

a

a

b

G

a þ 0.500 (c  d) þ i0.866 (c  d)

b 0.500d  i0.866d

e þ 0.500f þ i0.866f

0.500 (b  d) þ i0.866b

0.500 (b  a  c) þ i0.866 (b  a)

0.500 (f  e)  i0.866e

0.500(a  b c)  i0.866 (a  b  c)

0.500(a  c þ d)  i0.866 (a  c þ d)

0.500(f  e)  i0.866(f  e)

1 2 3

The positions of U atoms read as U1: (x, 0, 1/2), U2: (0, x, 1/2), U3: (x, x, 1/2). Mx denotes the x-cartesian component and a, b, c, d, e, and f are in general complex numbers

Elastic Neutron Diffraction on Magnetic Materials Chapter j 2

TABLE 5 Basis vectors for U magnetic moments (Wyckoff position 3g(x, 0, 1/2), x ¼ 0.58 in the Hexagonal UNiGa (Space Group P-62m) for the Propagation vector k ¼ (0, 0, kz)), Where kz s 1/2

131

U1

U2

U3

Mx

My

Mz

Mx

My

Mz

Mx

My

Mz

G

a

2a

0

2a

a

0

a

a

0

G

0

0

b

0

0

b

0

0

b

G

a

0

0

0

a

0

a

a

0

G

0

0

0

0

0

0

0

0

0

G

0 þ i0

0 þ i0

a  0.500b  i0.866b

0 þ i0

0 þ i0

b  0.500a  i0.866a

0 þ i0

0 þ i0

0.500 (a þ b) þ i0.866 (a þ b)

G6

a þ 0.500 (e  c) þ i0.866 (e  c)

b  0.500c  i0.866c

0 þ i0

0.500 (b  c) þ i0.866b

e þ 0.500 (b  a) þ i0.866 (b  a)

0þi0

0.500(a  b  e)  i0.866 (a  b  e)

0.500(a  e þ c)  i0.866 (a  e þ c)

0 þ i0

1 2 3 4 5

The positions of U atoms read as U1: (x, 0, 1/2), U2: (0, x, 1/2), U3: (x, x, 1/2). Mx denotes the x-cartesian component and a, b, c, d, e, and f are in general complex numbers.

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TABLE 6 Basis vectors for U magnetic moments (Wyckoff position 3g(x, 0, 1/2), x ¼ 0.58 in the Hexagonal UNiGa (Space Group P-62m) for the Propagation Vector k ¼ (0, 0, 1/2))

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The deduction of the real magnetic structure in UNiGa is subject of fitting of experimentally determined magnetic structure factors to calculated ones with the help of models listed in Tables 5 and 6, keeping the same irreducible representation. One has to keep in mind that one has to include in the calculations structure factors associated with k1 and k2 vectors, considering in general a free phase shift between the three k1, k2, and k3 contributions and that the refined values represent for the first two propagation vectors merely Fourier components (in Bohr magnetons). In the case of the Fourier components belonging to the propagation vector k3, the refined values are directly interpretable as magnetic moment. This vector ends at the Brillouin zone boundary, and the magnetic moment associated is always real. The best agreement has been obtained for the model associated with the (G2) irreducible representation, keeping moments exclusively along the c-axis. This is in agreement with all bulk magnetic measurements that indicate strong uniaxial magnetocrystalline anisotropy and the fact that all the magnetic Bragg reflections of the (00l) type are absent in the diffraction pattern shown in Fig. 25. Refined Fourier components belonging to the k1, k2, and k3 propagation vectors amount to 0.96(1), 1.58(1), and 0.45(1) mB with associated phase shifts between the k1 and k2 and k1 and k3 components of 0.55(2)p and 1.80(2)p, respectively. A nonzero phase shift between k1 and k3 suggests that k3 is a fundamental propagation vector and is not caused by a squaring-up of the magnetic structure defined by k1. Resulting U moment values at different sites are to be calculated using Eq. (16). The graphical construction using the refined Fourier components and phase shifts introducing an arbitrary general shift is shown in Fig. 26. As there is no dependence on the x and y positional parameters, the only variable is the phase for a given propagation vector as one moves along the c-axis, all three moment values within one crystallographic unit cell are identical and ferromagnetically coupled. The magnitude is 1.42(5) mB that agrees very well with magnetization data (Jirman et al., 1992). As one moves along the c-axis, the moments form a sequence “þ þ  þ  .” As the temperature is raised, reflections indexed using k1 and k3 disappear above T1, and new sets of magnetic reflections indexable by k4 ¼ (0, 0, 1/8) and k5 ¼ (0, 0, 3/8) appear in reciprocal space, marking a change from the AF1 phase to another, AF2. The latter reflections are stronger. The experimentally determined map of reciprocal space of UNiGa at 35.4K in the same orientation as at low temperature is shown in Fig. 27. Closer inspection, however, shows that a remainder of reflections associated with the propagation vector k2 ¼ (0, 0, 1/3) are still visible (marked by stars in Fig. 27). This is in accord with the first-order character of the transition from AF1 to AF2. Considering only k4 ¼ (0, 0, 1/8) and k5 ¼ (0, 0, 3/8) propagation vectors and performing the same analysis as done above for k1 ¼ (0, 0, 1/6) and k2 ¼ (0, 0, 1/3) (one can immediately use Table 5), the magnetic structure of the AF2 state between T1 ¼ 34.7K and T2 ¼ 36.0K can be determined. It appears that

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FIGURE 26 A graphical representation of Eq. (16) in the case of the ground state magnetic structure of UNiGa. The “þ þ  þ  ” sequence of U moments oriented along the hexagonal axis is determined by three Fourier components resulting from propagation vectors k1 ¼ (0, 0, 1/ 6), k2 ¼ (0, 0, 1/3), and k3 ¼ (0, 0, 1/2) components (dashed, dotted, and dash-dotted lines, respectively). The sum describes the experimentally determined values of U moments at the discrete atomic positions.

while the magnitude and confinement of the U moments that are aligned along the hexagonal axis remains the same, the coupling is alternated. The sequence as one moves along the c-axis changes to “þ þ  þ   þ .” As it is documented in Fig. 28, with further increasing the temperature between T2 ¼ 36.0K and T3 ¼ 37.4K, reflections indexable by k4 ¼ (0, 0, 1/8) and k5 ¼ (0, 0, 3/8) disappear and reflections described by k2 ¼ (0, 0, 1/3) reappear. The analysis for this propagation vector has been done above. The difference between this phase, called AF3, and the phase AF1 is the fact that only one propagation vector is present. No combination of different Fourier components belonging to different propagation vectors is needed. However, one has to still combine k and k components and calculate the real magnetic moment according to Eq. (16). Let us note that also for this magnetic phase we do not have any evidence for propagation vector (0, 0, 0). Due to principal features regarding the inaccessibility of the phase information, it is not possible to decide between a sequence “þ  0” or “þ þ ,” of U moments oriented along the c-axis, where in the latter case “þ” is half of the “.” Both magnetic moment arrangements lead to the same distribution of intensities. Bulk magnetic properties do not help either as both structures impose a zero macroscopic magnetization.

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FIGURE 27 A portion of the reciprocal space [the (hhl) plane of UNiGa mapped using single crystal neutron diffraction at 35.4K, in the AF2 state]. New Bragg reflections of magnetic origin can be indexed by two individual propagation vectors k4 ¼ (0, 0, 1/8) and k5 ¼ (0, 0, 3/8) (inset). The remaining propagation vector belongs to the ground state phase AF1 and documents the firstorder character of the transition.

In Fig. 29 we show a portion of the UNiGa reciprocal space map taken in the same orientation as above at 38.5K. The propagation vector seen before at lower temperature changes slightly its length from k2 ¼ (0, 0, 1/3). Although not shown here, it is temperature dependent (Sechovsky and Havela, 1998) and at this temperature one can define the new vector k6 ¼ (0, 0, 0.36). Such a value indicates a magnetic periodicity incommensurate with the crystal structure. We have denoted this structure above as INC. The group symmetry analysis is identical to the case of phase AF3. In the simplest way, the INC structure can be viewed as being longitudinally sine wave modulated with moments pointing along the c-axis (similar to the situation in Fig. 7B). Interestingly, the moment magnitude decreases with temperature only marginally with respect to the ground state value. Only 1K below the TN it still amounts to 90% of its low-temperature value. The example of UNiGa clearly demonstrates that single crystal neutron diffraction, although more demanding from the point of view of diffraction technique (e.g., the necessity to fix the crystal in one particular orientation), can disclose information that would be difficult to obtain from powder diffraction.

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FIGURE 28 A portion of the reciprocal space [the (hhl) plane of UNiGa mapped using single crystal neutron diffraction at 36.5K, in the AF3 state]. All magnetic Bragg reflections can be indexed by a single propagation vector k2 ¼ (0, 0, 1/3) (see the inset).

In the case that the magnetic structure orders with several propagation vectors, the diffraction pattern recorder would lead to an excessive Bragg peak overlap and difficulties to index and resolve the pattern unambiguously. Other problems occur if moments involved in the ordering are small or if they appear at the top of nuclear reflections. In the latter case the use of polarized neutrons introduced above can be helpful.

10. CONCLUDING REMARKS In this chapter we have tried to summarize the main ideas, methods, instrumentations, and practical ways concerning elastic neutron diffraction in magnetic materials. As a result, microscopic picture regarding the spatial arrangement of elementary moment is usually obtained. This information is needed as the detailed knowledge of mutual coupling between various magnetically active atoms in the structure is indispensable in understanding how the material’s magnetic properties can be explained and modeled with the aim to prepare materials suitable for technological applications. This concerns, for instance, permanent magnets where the transition metal elements increase the magnetic phase transition and 4f or 5f atoms increase the anisotropy.

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FIGURE 29 A portion of the reciprocal space [the (hhl) plane of UNiGa mapped using single crystal neutron diffraction at 38.5K, in the INC state]. All magnetic Bragg reflections can be indexed by a single propagation vector k6 ¼ (0, 0, w0.36) (see the inset).

It concerns also multilayers, multiferroics, spintronic materials, and much more. The area is very broad and expanding. A vast amount of various materials have been studied using neutrons, and it is not possible to cover all different types of materials, magnetic orderings, experimental techniques, all notions of theoretical approaches or instrumentation. Nevertheless, we hope the reader gets inspired by the information contained.

ACKNOWLEDGMENTS We would like to acknowledge many stimulating discussions with our colleagues from the Helmholtz Zentrum Berlin for Materials and Energy and the support and understanding of our families.

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Chapter 3

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides P.C.M. Gubbens1 Delft University of Technology, Delft, The Netherlands 1 Corresponding author: E-mail: [email protected]

Chapter Outline 1. Introduction 2. Rare Earth Mo¨ssbauer Spectroscopy and Methodology 2.1 The Recoilless Fraction 2.2 Nuclear Energy Levels 2.2.1 Isomer Shift 2.2.2 Electric Quadrupole Interaction 2.2.3 Magnetic Hyperfine Interaction 2.2.4 Line Positions and Intensities 2.3 Methodology of Rare Earth Mo¨ssbauer Spectroscopy 3. Theoretical Aspects 3.1 Introduction 3.2 Crystal Fields 3.3 Magnetic Interaction

145

147 147 147 147 148 148 149 149 153 153 153 156

3.4 Relation to Mo¨ssbauer Parameters 3.5 Magnetic Relaxation 3.6 Analysis Procedure: Examples 3.6.1 TmBa2Cu3O7
x 3.6.2 Yb2Ti2O7 4. Overview of Rare Earth-Based Oxides 4.1 Introduction 4.2 R2O3 Compounds 4.3 RMO3 Compounds 4.4 RMO4 Compounds 4.5 RBa2Cu3O7 Compounds 4.6 R2BaMO5 Compounds 4.7 R2M2O7 Compounds 4.8 R3M5O12 Compounds 5. Conclusions, Justification, and Acknowledgment References

160 163 165 165 169 171 171 171 175 182 194 206 215 225 229 231

1. INTRODUCTION The Mo¨ssbauer effect, discovered in 1958 by R.L. Mo¨ssbauer, has become a powerful nuclear measuring technique in different branches of physics, chemistry, biology, geology, and materials science. He discovered that nuclei Handbook of Magnetic Materials, Vol. 25. http://dx.doi.org/10.1016/bs.hmm.2016.10.001 Copyright © 2016 Elsevier B.V. All rights reserved.

145

146 Handbook of Magnetic Materials

imbedded in solids can show recoilless absorption and emission of radiation. The bond of the nuclei with the solid results in quantization of the recoil energy and therefore a part of the nuclei, the recoilless fraction, shows a zero recoil energy. This phenomenon made resonant absorption of nuclear radiation possible and small variations in the nuclear levels could be studied. Detailed descriptions of the technique are published in a lot of standard works (Frauenfelder, 1962; Goldanskii and Herber, 1968; May, 1971; Wegener, 1964; Wertheim, 1964). First rare earth studies were already performed in the sixties and seventies of the 20th century. The Mo¨ssbauer technique gives information about crystal field effects and magnetic ordering, indirectly, by measuring the electric quadrupole-splitting and the magnetic hyperfine field. The studies are relevant for magnetic materials, superconductors, minerals, and chemical rare earth complexes. This paper will survey the results of rare earth-based oxides studied with rare earth Mo¨ssbauer spectroscopy. The emphasis will be mainly on the nuclei 141 Pr, 155Gd, 161Dy, 166Er, 169Tm, and 170Yb. In this chapter of the Hand book of Magnetic Materials a survey of the different magnetic aspects of the rare earth insulator compounds will be given. Only in the case of Gd, Tm, and Yb Mo¨ssbauer spectroscopy a clear quadrupole-splitting above the magnetic ordering temperature is found. Below the magnetic ordering a clear quadrupole-splitting is also found in Dy and Er Mo¨ssbauer spectroscopy. An overview of all these Mo¨ssbauer nuclei will be given in Chapter 2. Mo¨ssbauer spectroscopy is a very sensitive tool and selective for crystallographically different rare earth atoms. If the total angular momentum of a rare earth is an integer, the ground state of the rare earth atom shows a larger variety in ground-state level than in the case of a half integer total angular momentum, which consists only of Kramers doublets. Then, below and above the magnetic ordering temperature crystal field levels consisting of singlets, doublets give a large variety of physical behavior. This depends mainly on the local symmetry of the rare earth atom. In general, one derives from the hyperfine field and the electric quadrupole-splitting the physical behavior of the ground state at low temperature. At higher temperatures the magnitude of the hyperfine field and the quadrupolesplitting are proportional to the Boltzman distribution over the energy levels of the crystal field. So far, more extended publications on this topic were published as invited papers by Gubbens et al. (1985a, 1985b, 1990), Stewart (1994), Gubbens and Mulders (1998) and Stewart (2010). An extended overview on rare earth Mo¨ssbauer spectroscopy on intermetallic rare earth compounds is published by Gubbens (2012). Two papers concerning a theoretical fundamental explanation and interpretation of 169Tm Mo¨ssbauer spectra in terms of crystal fields were published by Stewart (1985) and Stewart and Gubbens (1999).

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

147

¨ SSBAUER SPECTROSCOPY AND 2. RARE EARTH MO METHODOLOGY 2.1 The Recoilless Fraction The Mo¨ssbauer effect is based on resonant absorption of gamma radiation by atomic nuclei. In the process of emission and absorption of gamma radiation, energy as well momentum conservation have to be obeyed. The line width G of the Mo¨ssbauer resonance is equal to G ¼ Z=s where s is the half-life of the excited nuclear level. s can take values from 1 to 100 ns, corresponding to G values between 0.01 and 1 meV. When a nucleus of a single atom initially at rest emits a photon, it will obtain the same momentum of opposite sign. Then there is a recoil energy ðR ¼ E02 =2mc2 Þ of the order of 10
2 eV, which is at least four orders of magnitude larger than an average line width of the nuclear energy levels involved. However, when the nucleus is embedded in a solid, the momentum to the solid is quantized via the phonon excitations. Phonon quantizations in solids can amount to 10
2 eV, comparable with the recoil energy. Therefore, for a certain fraction of photons, the emission and absorption will occur without energy loss due to the fact that the momentum is transferred to the whole crystal with a large mass and the recoil energy is zero.

2.2 Nuclear Energy Levels The hyperfine splitting of the energy levels of the nuclei is determined by the following Hamiltonian H hf ¼ H e þ H m of which H e represents the electrostatic interaction and H m the magnetic interaction of the nucleus. The electrostatic part of the Hamiltonian H e consists of a “dipole” and a “quadrupole” part giving rise to the Isomer Shift and to the Electric Quadrupole Interaction, respectively.

2.2.1 Isomer Shift The Isomer Shift (IS), the first parameter, is unique for Mo¨ssbauer spectroscopy and is sensitive to the electronic charge density around the nucleus, that is, it gives a value that is mainly proportional to the valency of the atom. The Isomer Shift is expressed in mm/s and it has the form:   
 2  D 2 E 2pcZe2
 re
rg IS ¼ jA ð0Þ2 
jS ð0Þ2  (1) 3Eg In Eq. (1), c is the speed of light, Ze the nuclear charge, Eg the energy of 2 i the averaged nuclear radii of the excited and the Mo¨ssbauer resonance, hre;g ground state, respectively, and ejjA,S(0)2j the electronic charge density at the nucleus of the absorber (A) and the source (S).

148 Handbook of Magnetic Materials

2.2.2 Electric Quadrupole Interaction The second contribution to the electrostatic part of the Hamiltonian is the electric quadrupole interaction, which is associated with the nonspherical charge distribution around the nucleus. For a level of nuclear spin I, the interaction between the quadrupole moment of the nucleus (Q) and the electric field gradient tensor (Vii with i ¼ x, y and z) is given by the Hamiltonian
i eQVzz h 2 H e;q ¼ (2) 3Iz
IðI
1Þ þ h I2x
I2y 4Ið2I
1Þ   with the asymmetry parameter h ¼ Vxx
Vyy =Vzz. In this equation Vxx, Vyy and Vzz are the nonzero electric field gradient tensor elements expressed with respect to the principal x, y and z axes and h is the electric field gradient tensor asymmetry parameter (h ¼ 0 for axial asymmetry along the z axis). In the case of 169Tm Mo¨ssbauer spectroscopy, only the excited state I ¼ 3/2 level has a nonzero quadrupole moment. In the absence of a magnetic hyperfine field, this level is split into two levels resulting in a simple doublet spectrum.   1 1 2 1=2 (3) DEQ ¼ eQVzz 1 þ h 2 3

2.2.3 Magnetic Hyperfine Interaction The magnetic part of the Hamiltonian characterizes the magnetic hyperfine field at the nuclear site. The degeneracy of the nuclear levels is lifted by the so-called Zeeman splitting. The magnetic hyperfine field interaction is then given by Hm ¼
gN mN I$Heff

(4)

where gN is the nuclear g-factor, mN the nuclear magneton, and I the nuclear spin operator. For 169Tm the nuclear ground-state moment mg ¼ 0.231 mN and the moment ratio between excited and ground states me/mg ¼
2.223 as found by Wit and Niesen (1976). For the ground nuclear spin operator, one has Ig ¼ 1/2 and for the excited nuclear spin operator Ie ¼ 3/2. The magnetic hyperfine field is composed of several contributions: Heff ¼ HJ þ HFC þ HD þ HTR þ Happ :

(5)

HJ is the total angular moment and by far the largest contribution, HFC is the Fermi contact interaction term, HD the dipolar field, HTR the transferred hyperfine field, and Happ the external field. HD and HTR are for rare earth very small and can be neglected.

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

The 4f total angular magnetic moment gives rise to a field

1 HJ ¼
2mB 3 hJi r

149

(6)

A much less important contribution is the Fermi contact interaction, which originates from the spin polarization of the electrons at the nuclear site: HFC ¼


8p m hSz i; 3 B

(7)

where mB is the Bohr magneton and hSzi the spin polarization around the nucleus. This spin polarization can be divided into two contributions originating from the core s-electrons and from the valence electrons. 2.2.4 Line Positions and Intensities By combining the electric quadrupolar and magnetic interactions of the hyperfine Hamiltonian H hf ¼ H e þ H m , one composes a total Mo¨ssbauer spectrum. H hf has specific eigenvalues and energy levels, which lead to a number of transitions between the nuclear levels and the transition probabilities between these levels, given by the ClebscheGordan coefficients. As shown in Tables 1 and 2, the multipolarity of the Mo¨ssbauer nucleus gives the rules for the allowed transitions. In the case of M1 and E1, it is DM or DE ¼ 0, 1. In the case of E2, it is DE ¼ 0, 1, 2.

2.3 Methodology of Rare Earth Mo¨ssbauer Spectroscopy In Tables 1 and 2, eight rare earth Mo¨ssbauer resonances are listed, which are amendable to material research. In Table 1 the sources are listed, which can be used at room temperature. The resonance energy is then low enough to have a recoilless fraction. A material under study with these sources can be measured at least up to room temperature, in the case of 169Tm due to its low energy even up to 2000K. The sources tabulated in Table 2 can only be used at low temperatures due to their high energy. In this case to achieve a recoilless fraction, both source and absorber have to be cooled down to helium temperatures. In the case of 166Er (80.56 keV), shown in Table 2, the highest temperature, where the nuclear resonance is observable, is about 100K. With exception of 149Sm, all the sources for these nuclei can be made in a reactor via neutron irradiation. The source (149Eu) for 149Sm has to be made with proton irradiation (150Sm þ pþ ¼ 149Eu þ 2n). The source for 151Eu, which has a very long halflife time of 87 year, is commercially available. As an example the energy spectrum of the 169ErAl9 as source of 169Tm is given in the overview paper of Gubbens (2012). The main applications of the Mo¨ssbauer effect in these nuclei is also given in Tables 1 and 2. In Fig. 1, a schematic representation is given of the pure electric quadrupole-splitting and this parameter as a perturbation on

Isotope

149

Ie and Ig

5/2 and 7/2

7/2 and 5/2

5/2 and 5/2

3/2 and 1/2

Multipolarity

M1

M1

E1

M1

Energy (keV)

22.49

21.53

25.66

8.40

Qe and Qg (barn)

0.51 and 0.058

1.48 and 1.14

2.35 and 2.35


1.20 and 0

me and mg (mN)


0.67 and
0.72

3.47 and 4.64


0.48 and 0.39


0.23 and 0.10

Parent activity

149

151

161

169

Half-life

106d

87y

6.9d

9.4d

Raw source material

150

150

160

Activation

(p,2n)

(n,g)

(n,g)

(n,g)

Natural line width (mm/s)

1.71

1.30

0.38

8.33

Application

IS and Heff

IS and Heff

IS, QS and Heff

QS and Heff

Sm

Eu

Sm2O3

151

Eu

Sm

SmF3

161

169

Dy

Tb

Gd162 0.5

Dy0.5F3

168

Tm

Er

ErAl3-Al

Ie and Ig are the nuclar spin operators of excited and ground states. Multipolarity indicates the character of the transitions between the nuclear levels. Qe and Qg are the quadrupole moments and me and mg are the magnetic moments of the excited and ground states.

150 Handbook of Magnetic Materials

TABLE 1 Relevant Parameters of Rare Earth Nuclei With Mo¨ssbauer Effect Possible at Room Temperature

Isotope

141

Ie and Ig

7/2 and 5/2

5/2 and 3/2

2 and 0

2 and 0

Multipolarity

M1

E1

E2

E2

Energy (keV)

145.4

86.54

80.56

84.25

Qe and Qg (barn)

0.28? and
0.059

z01.30


1.59 and 0


2.14 and 0

Pr

155

Gd

166

Er

170

Yb

me and mg (mN)

2.80 and 4.28


0.515 and
0.254

0.63 and 0

0.67 and 0

Parent activity

141

155

166

170

Half-life

32.5d

1.81y

27h

130d

Raw source material

CeF3

154

HoPd3/(Ho,Y)H2

TmB12

Activation

(n,g)

(n,g)

(n,g)

(n,g)

Natural line width (mm/s)

1.02

0.5

1.89

2.03

Application

IS and Heff

IS, QS and Heff

QS and Heff

QS and Heff

Ce

Eu

SmPd3

Ho

Tm

Ie and Ig are the nuclear spin operators of excited and ground states. Multipolarity indicates the character of the transitions between the nuclear levels. Qe and Qg are the quadrupole moments and me and mg are the magnetic moments of the excited and ground states.

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

TABLE 2 Relevant Parameters of Rare Earth Nuclei With Mo¨ssbauer Effect, for Which Helium Temperatures are Required

151

152 Handbook of Magnetic Materials

FIGURE 1 Mo¨ssbauer spectra for the pure electric quadrupole interaction (lower spectrum of each pair) and for coaxial quadrupole and magnetic hyperfine interactions (upper spectrum of each pair). Unless indicated otherwise, the spectra are for the trivalent ions. They are based on values close to the free ion values. This figure has been taken from Stewart, G.A., 1994. Mater. Forum 18, 177.

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

153

the hyperfine field. More details of this paragraph can be found in the publication of Gubbens (2012).

3. THEORETICAL ASPECTS 3.1 Introduction In the rare elements the incomplete 4f shell, which becomes more and more filled going from La to Lu, is responsible for their magnetic properties. Most rare earths have three conduction electrons. These electrons have either 5d or 6s character and are mainly delocalized. On the other hand, the 4f electrons remain localized deep in the electron shell and there is in general no overlap between the 4f wave functions of two neighboring atoms. The “normal” electronic configuration of rare earths in the metallic state is then4fn5d16s2. In Tm, n ¼ 12. Since 4f electrons are deeply embedded in the electron cloud, electrical fields are rather small and the spin orbit coupling is not broken up as in the case of the 3d metals. Spin, orbital, and total angular moments, present in rare earths, are determined by Hund’s rules. For trivalent rare pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi earths the magnetic moment is proportional to gJ mB JðJ þ 1Þ in the paramagnetic and to gJJmB in the magnetically ordered state. Tm has a 3H6 multiplet ground state. This means that S ¼ 1, L ¼ 5 and J ¼ 6. Since gJ ¼ 7/ 6, the free-ion value of the magnetic moment in the paramagnetic phase is 7.57 mB and in the magnetically ordered state 6 mB. The relevant parameters of the other rare earths are tabulated in Table 3. The dominant terms in the Hamiltonian, which describes the magnetic properties of rare earth atoms is usually presented as H ¼H

cf

þH

(8)

mag

where the term H cf represents the crystal field interaction and H scribes the magnetic exchange interaction.

mag

de-

3.2 Crystal Fields A rare earth atom in a crystal is situated in a potential field (electric field gradient). The crystalline field arises from the electric charges on the neighboring atoms. This potential partially lifts the (2J þ 1) degeneracy of the ground state multiplet levels of the 4f electrons of the rare earths. Since the 4f electrons are deeply embedded in the electron cloud, in general, the splittings due to the potential are much smaller than the multiplet separations. This means that there is no mixing of multiplets. The number of singlets and doublets is depending on the symmetry of the crystal field. Rare earths with a

154 Handbook of Magnetic Materials

TABLE 3 Selected Ionic Properties of Trivalent Rare Earth R

Ground Term

S

L

J

gJ

gJ[J(J þ 1)]1/2

gJ J

S0

0

0

0

e

0

0

F3/2

1/2

3

5/2

6/7

2.54

2.14

H4

1

5

4

4/5

3.58

3.20

I9/2

3/2

6

9/2

8/11

3.62

3.28

H5/2

5/2

5

5/2

2/7

0.84

0.72

F0

3

3

0

0

0

0

S7/2

7/2

0

7/2

2

7.94

7

F6

3

3

6

3/2

9.72

9

H15/2

5/2

5

15/2

4/3

10.63

10

I8

2

6

8

5/4

10.60

10

I15/2

3/2

6

15/2

6/5

9.59

9

H6

1

5

6

7/6

7.57

7

F7/2

1/2

3

7/2

8/7

4.54

4

S0

0

0

0

e

0

0

La

1

Ce

2

Pr

3

Nd

4

Sm

6

Eu

7

Gd

8

Tb

7

Dy

6

Ho

5

Er

4

Tm

3

Yb

2

Lu

1

For symbols refer to the text.

higher symmetry have more unsplit levels. Since the charges are distributed in a nonspherical way, there will be a preferential orientation of the 4f shell resulting in an electric field gradient at the 4f site. In this description of the socalled crystal field effects an interaction Hamitonian is introduced, describing the electric characteristics of the 4f shell by Stevens (1952). The crystal field Hamiltonian of a rare earth atom can be described by the Eq. (9) X X m m H cf ¼ Bm qn hr n iAm (9) n On ¼ n On m;n

where Om n erators Jz,

m;n

are operator equivalents (polynomials of the angular moment opn m J2, Jþ and J
) and Bm n ¼ qn hr iAn . In this equation, qn represents the Stevens constants aJ, bJ and gJ for n ¼ 2, 4, and 6, respectively. The symbol hrni represents the nth power of the radial integrals of the 4f shell as calculated by Freeman and Desclaux (1979). Am n are the crystal field potentials. In Table 4, Stevens constants of all the rare earth are tabulated. The aJ is the most dominant Stevens constant. The sign and magnitude of aJ determines for

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

155

TABLE 4 The Total Angular Momentum Parameters of Trivalent Rare Earth R

J

aJ hr 2 ið10
3 a20 Þ

bJ hr 4 ið10
4 a40 Þ

gJ hr 6 ið10
6 a60 Þ

Ce

5/2


74.8

þ251.7

0

Pr

4


25.54


24.95

þ1141.66

Nd

9/2


7.161


8.471


570.96

Sm

5/2

þ40.209

þ56.527

0

Gd

7/2

0

0

0

Tb

6


8.303

þ2.101


7.683

Dy

15/2


5.020


0.891

þ6.260

Ho

8


1.676


0.459


6.959

Er

15/2

þ1.831

þ0.564

þ9.969

Tm

6

þ6.976

þ1.916


24.33

Yb

7/2

þ21.04


18.857


581.94

The Stevens factors and its multiplication with the radial integrals of the 4f shell are given. The last
2 values were taken from Freeman and Desclaux (1979). Note that a
2 ¼ 3:571  A ,
4


6


6   a
4 0 ¼ 12:751 A , and a0 ¼ 45:541 A Bnm and the crystal field potentials Am n.

0

. These values interconnect the crystal field parameters

the second order the shape of the potential of the 4f shell: for aJ > 0, it has a cigar shape and for aJ < 0, it has a disc shape. For Sm, Er, Tm, and Yb, aJ is positive, while for Pr, Nd, Tb, Dy, and Ho, it is negative. For Gd, aJ ¼ 0. If there is a center of inversion for the local symmetry, m and n are even. For rare earth atoms, there are no Bm n with m and n larger than 6. In the Hamiltonian for cubic symmetry with z chosen in the (001) direction:



(10) H cf ¼ B4 O04 þ 5O44 þ B6 O06
21O46 For hexagonal local point symmetry: H

cf

0 0 0 6 6 ¼ B02 O02 þ BO 4 O4 þ B6 O6 þ B6 O6

(11)

For trigonal local point symmetry: H

cf

¼ B02 O02 þ B04 O04 þ B34 O34 þ B06 O06 þ B36 O36 þ B66 O66

(12)

For tetragonal local point symmetry: H

cf

¼ B02 O02 þ B04 O04 þ B44 O44 þ B06 O06 þ B46 O46

(13)

In all these cases the asymmetry parameter of the Mo¨ssbauer effect h ¼ B22 =B02 ¼ 0, while the principal axis of the electric field gradient is parallel to the crystallographic c axis as main axis.

156 Handbook of Magnetic Materials

For orthorhombic symmetry: H

cf

¼ B02 O02 þ B22 O22 þ B04 O04 þ B24 O24 þ B44 O44 þ B06 O06 þ B26 O26 þ B46 O46 þ B66 O66

(14)

In this case the asymmetry parameter h s 0. The principal axes are orthogonal. They can potentially be relabeled. Also for lower symmetries like monoclinic and triclinic the asymmetry parameter h s 0, but the main crystallographic axes are not orthogonal. For a locally monoclinic symmetry the crystal field Hamiltonian has the form: H

cf


2 0 0 2 2
2
2 ¼ B02 O02 þ B22 O22 þ B
2 2 O2 ðrank 2Þ þ B4 O4 þ B4 O4 þ B4 O4
4 0 0 2 2
2
2 4 4 þ B44 O44 þ B
4 4 O4 ðrank 4Þ þ B6 O6 þ B6 O6 þ B6 O6 þ B6 O6 (15)
4 6 6
6
6 þ B
4 6 O6 þ B6 O6 þ B6 O6 ðrank 6Þ

For the above shown Hamiltonian for monoclinic symmetry the large number of unknown rank 4 and rank 6 crystal field parameters for each site can 0 be reduced to just B04 and B06 by calculating within-rank ratios, rnm ¼ Bm n =Bm , using simple crystal field models. For the purpose of such computations, only the nearest-neighbor oxygen atoms can be considered here, an approximation which is expected to be reasonable for the shorter-range rank 4 and rank 6 crystal field components. The three alternative sets of ratios can be computed using the superposition model of Bradbury and Newman (1967) with the following different radial dependencies: (1) the ideal point charge model, (2) a model proposed by Nekvasil (1979), and (3) a model as derived theoretically by Garcia and Faucher (1984). For these compounds, all oxygens were assumed to have the same effective charge. In the case of triclinic symmetry the number of crystal field parameters is 27, which makes an interpretation very complex. In general, for rare earth insulators, no cubic or hexagonal compounds are found. The highest found symmetry is tetragonal. In general, the energy levels (eigenvalues) and the eigenfunctions are determined by diagonalization of the above shown Hamiltonians.

3.3 Magnetic Interaction When we include the interaction of the rare earth moments with the molecular field present at these sites, the Hamiltonian H becomes H ¼H

cf


gJ mB HM $J

(16)

In this expression the quantity gJ is the Lande´ g-factor and HM is the molecular field. Alternatively, this equation is often written as H ¼H

cf

where Hex is the exchange field.

þ 2mB ðgJ
1ÞHex $J

(17)

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

157

In this equation the relation between Hex and HM can be written as
gJ Hex ¼ HM (18) 2gJ
1 In most of the models, the magnetic coupling energy between two localized moments is taken proportional to Si$Sj. In the case of rare earths the 4f electrons interact in an indirect way. Usually, results are explained with the RKKY interaction. In this case the magnetic interaction is in an indirect way, that is, the magnetic coupling proceeds via the spin polarization of the rare earth 6s conduction electrons. The essential form of indirect interaction between localized moments was introduced by Rudermann and Kittel (1954) to explain the magnetic interactions between nuclear moments. This model was applied by Kasuya (1956) and Yosida (1957) to the interaction of localized moments. Alternatively, Campbell (1972) proposed a coupling interaction based on an indirect interaction via the 5d electrons, which have a far less localized character than the 4f electrons. Roughly, the variation of the magnetic interaction between the 4f moments scale with the de Gennes factor is as G ¼ ðgJ
1Þ2 JðJ þ 1Þ

(19)

which is proportional to the magnetic ordering temperature. Especially for the heavy rare earth elements, this is true and regarded as a prove for the validity of the RKKY model. The spin polarization of the exchange interaction between the localized spin moments can be described by the Heisenberg Hamiltonian as X H ¼
Ji;j Si $Sj (20) i;j

where Ji,j is the exchange parameter between the localized spins residing on sites i and j. For rare earths, we use the spin operator SR ¼ (gJ
1)JR. Then the Hamitonian between the rare earths becomes X H ¼
JR;R ðgJ
1Þ2 JR $JR (21) R;R

with JRR the exchange parameter between the 4f spins SR. In a mean field approximation, one can replace all spins moments by their mean value hSRi. In this case the exchange field acting on the spin moment is 1 X Hex ¼ (22) JRR hSR i 2mB R;R where the summation runs over the surrounding spins. The molecular field and the corresponding exchange field Eq. (20) can be estimated from the magnetic ordering temperature with the equation Jz HM ¼ 3kB Tc;N (23) gJ mB JðJ þ 1Þ where Jz is the eigenvalue of the ground state level.

158 Handbook of Magnetic Materials

Alternatively, Noakes and Shenoy (1982) have developed a model by including the expectation value of the eigenfunctions due to the crystal fieldda model in which the maximum of the magnetic ordering temperature TM in the rare earth series moves from Gd toward Tb or Dy with equation   TM ¼ 2JRR ðgJ
1Þ2 Jz2 ðTM Þ cf (24) In this equation, hJz2 ðTM Þicf is the expectation value of Jz2 under influence of the Hcf alone, without any exchange term, evaluated at a certain temperature. In uniaxially symmetric materials the magnetocrystalline anisotropy energy can be defined as E ¼ K1 sin2 q þ K2 sin4 q

(25)

where K1 and K2 are the anisotropy constants and q is the angle between the easy magnetization direction and the c axis. The single ion contribution due to the rare earth atoms can be obtained from crystal field theory by means of the following expressions     3     K1 ¼
aJ r 2 A02 O02 þ 5bJ r 4 A04 O04 2

(26)

where hO02 i and hO04 i denote the Boltzman average of the corresponding O02 and O04 terms and K2 ¼


35  4  0  0  b r A4 O 4 8 J

(27)

If the temperature is not too low, one has only to consider the term 3     K1 ¼
aJ r 2 A02 O02 2

(28)

From this equation, it appears that the second-order term of the crystal field is the most important one and in most cases determines the easy axis of magnetization. We then can simplify the Hamiltonian of Eq. (9) to H ¼ aJ hr 2 iA02 Jð2J
1Þ
gJ mB HM $Jz for an easy axis of magnetization parallel to the c axis and to H ¼
ð1=2ÞaJ hr 2 iA02 Jð2J
1Þ
gJ mB HM $Jx for an easy axis of magnetization perpendicular to the c axis. After diagonalizing this Hamiltonian, one can determine the case with a minimal energy. For all the rare earths in the presence of a crystal field, one finds that the magnetic anisotropy is parallel to the c axis if (K1 > 0) A02 < 0 and aJ > 0, while the magnetic anisotropy is perpendicular to the c axis (K1 < 0) if A02 < 0 and aJ < 0. In the case of A02 > 0, the opposite behavior is obtained. Since higher order crystal field terms or the magnetic anisotropy contribution of the 3d sublattice may disturb this picture, it has to be regarded as a simplified one. The magnetic coupling of rare earth spins with 3d spins is antiparallel. For compounds in which the rare earth is a light one (J ¼ L
S), this implies that

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

159

the total rare earth moment (gJJmB) is coupled parallel to the 3d moment. By contrast, for the heavy rare earth (J ¼ L þ S) and the total rare earth, moment is coupled antiparallel to the 3d moment. In the case of light rare earth, one can expect gradually an increase of total magnetic moment from Tc down to 4.2K, whereas in the case of heavy rare earth compensation points (M ¼ 0) can be found due to the larger temperature dependence of the magnetic moment of the heavy rare earths compared to the smaller temperature-dependence of the 3d magnetic moments, to whom they are antiparallely coupled. In general, the 4fe3d coupling strength varies as (gJ
1)J throughout the rare earth series. This means that usually for the Gd compound in the rare earth series, a maximum in Tc has been found. The molecular field magnetic part in Eq. (15), as shown by Gubbens and Buschow (1982), can be written as HM ¼ 2ZRM JRM ðgJ
1ÞhSM ðTÞi=ðgJ mB Þ:

(29)

The mean number of M neighbors (M ¼ 3d metal) of the R atoms in RxMy compounds is represented by ZRM and the ReM coupling constant by JRM. Commonly, a mean field model as shown by Gubbens and Buschow (1982) has been used to describe the variation of Tc in a certain rare earth-3d transition intermetallic compound. In such a model Tc can be written as 3kTc ¼ aMM þ aRR þ ½ðaMM
aRR Þ2 þ 4aRM aMR 1=2

(30)

where axy represents the magnetic interaction between the x and y spins. These energies can be expressed in terms of the corresponding coupling constants JRR, JRM and JMM by means of the relations aRR ¼ ZRR JRR ðgJ
1Þ2 JðJ þ 1Þ

(31)

aMM ¼ ZMM JMM SM ðSM þ 1Þ

(32)

2 aRM aMR ¼ Z1 Z2 SM ðSM þ 1ÞðgJ
1Þ2 JðJ þ 1ÞJRM

(33)

where ZRR and ZMM represent the average number of similar neighbor atoms to an R atom and a 3d atom, respectively. The quantities Z1 and Z2 represent the number of 3d neighbors to an R atom and the number of R neighbors to a 3d atom, respectively. Since the RKKY exchange between the R atoms has a longrange character, it is not sufficient to consider only the nearest neighbors. However, the ReR magnetic interaction is relatively weak. For this reason the aRR term can be neglected in a rare earth-3d intermetallic. In that case, the variation of Tc can be written in a more simplified form

1=2 (34) 3kTc ¼ aMM þ a2MM þ 4aMR aRM

160 Handbook of Magnetic Materials

3.4 Relation to Mo¨ssbauer Parameters As we have shown earlier in Section 2 for rare earth Mo¨ssbauer spectroscopy the two measurable most important parameters are the magnetic hyperfine field (Heff) and the electric quadrupole-splitting (QS). The temperature dependence of the hyperfine field determined by the energy levels and the eigenfunctions of the crystal fields of the rare earth has the expression   4f  (35) Heff ðTÞ ¼ Heff hJz iav  J where hiav indicates a thermal average over the energy levels of the crystal 4f is the value of the hyperfine field for the free ion. In a rare field scheme. Heff earth (with exception of Gd) the orbital contribution is by far the largest contribution to the hyperfine field. The transferred hyperfine field due to the surrounding magnetic moments has only a minor influence, as shown for some 3de4f compounds. This means that one can simply calculate the magnetic moment of a Tm atom by means of the simple relation M ¼ gJ m B J

Heff ðTÞ 4f Heff

(36)

The temperature dependence of the quadrupole-splitting is given by the equation  2  3Jz
JðJ þ 1Þ av 4f DEQ ðTÞ ¼ QS þ QSlatt (37) Jð2J
1Þ where hiav indicates a thermal average over the energy levels of the crystal field scheme. QSlatt is the lattice contribution of the electric quadrupolesplitting and QS4f is the free-ion value. A schematic representation is given in Fig. 2. Since in Gd the 4f electrons fill half of the shell, there is no orbital moment present (L ¼ 0). This means that the 4f-contribution of the electric quadrupolesplitting in 155Gd Mo¨ssbauer spectroscopy is zero and only the lattice contribution has to be considered. In this way the lattice contribution in Eq. (39) can be determined independently from a separate 155Gd Mo¨ssbauer spectroscopic investigation. When the symmetry is lower than cubic the lattice contribution of the quadrupole-splitting is QSlatt ¼ 1=2e2 Vzzlatt Q in the case of axial symmetry as shown by Eq. (3). Then

1 (38) 3cos2 q
1 þ hsin2 qcos 2f 2 where q is the angle between the easy axis of magnetization and the symmetry axis of the electric field gradient, f the angle of the projection of the magnetization in the plane and h ¼ A22 =A02 ¼ B22 =B02 . Alternatively, since the eVzzlatt ¼
4$C$A02 $

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

161

FIGURE 2 Schematic representation of the two crystal field contributions to the electric tensor at the rare earth nucleus: The lattice contribution due to the crystal field of the surrounding magnetic moments and the 4f-contribution due to the asymmetry of the 4f-charge cloud. This figure has been taken from Stewart, G.A., 1994. Mater. Forum 18, 177.

4f term in the quadrupole-splitting is averaging out at very high temperatures, the lattice contribution can be determined by measuring the 169Tm Mo¨ssbauer spectra at such high temperature as shown in Fig. 3 for TmNi2B2C. Traditionally, the constant C was taken to be equal to (1
gN)/(1
s) with gN the antishielding factor of Sternheimer (1966) and s the screening factor. Values of gN and s are calculated by Gupta and Sen (1973). For a long time such an interpretation is still valid for insulators as shown by Stewart (1985) and Stewart and Gubbens (1999). For metallic systems a more modern interpretation has been given on the basis of electronic band structure calculations on the GdM2Si2 compounds performed by Coehoorn et al. (1990). Eqs. (37) and (38) are very useful tools for determining the crystal field potential A02 . Since for rare earth-3d-rich intermetallic compounds the molecular field experienced by the rare earth ion (Eq. 30) is very large, the exchange splitting of the 2J þ 1 ground multiplet is much larger than the crystal field splitting. In that case Jz in Eq. (37) is equal to J and the 4f-contribution of the quadrupolesplitting is the free-ion value. The difference between the free-ion value and the measured quadrupole-splitting gives then the lattice contribution. From Eq. (38) one then can determine the A02 term as shown in Table 5. The constant C can also be determined experimentally. For instance, this value can be determined from a combined inelastic neutron experiment, which determines the eigenfunction of the ground state and a Mo¨ssbauer experiment, which gives

162 Handbook of Magnetic Materials

TmNi2B2C

sample A sample B

QSlatt

FIGURE 3 The quadrupole-splitting (QS ¼ 1/2eVzzQ) as observed by 169Tm Mo¨ssbauer spectroscopy in two samples of TmNi2B2C (A and B). In sample A, only a quadrupole-splitting is observed as shown by the empty dots. In sample B, the filled triangles are deduced from the corresponding subspectrum, which shows a moment of 4.3 mB at 0.3K and the filled dots are deduced from the other subspectrum. Note that the filled and empty dots are identical to each other within the experimental error. The solid curve is obtained from a tentative set of crystal field parameters and can be considered as a guide to the eye. At high temperatures the QSlatt determined by 169Tm and 155Gd Mo¨ssbauer spectroscopy coincide. The Gd result is taken from Mulder et al. (1995). For further explanation see ref. of Mulders et al. (1998).

latt , Used TABLE 5 Lattice Contributions to the Electric Field Gradient, Vzz C-Factors and Corresponding Values of the Crystal Field Potentials A02 for the Rare Earth Mo¨ssbauer Spectroscopy Measured RCo5 þ x Compounds

Compound

latt ð1017 V cm
2 Þ Vzz

C

A02 ðK=a20 Þ

GdCo5

þ8.2

320


206

DyCo5.1


7.0  2.0

285


400  100

ErCo5.9

þ8.0  1.5

270  30


230  50

TmCo6.1

þ5.6  1.0

243  30


185  30

TmCo8.5

þ4.2  1.0

243  30


140  30

(Tm2Co17)

þ1.4  1.0

243  30


46  30

GdCo8.5

þ4.3

320


108

(Gd2Co17) It has to be noted that hexagonal Tm2Co17 has two different Tm sites and rhombohedral Gd2Co17 one-only Gd site. Furthermore, the C-factors were experimentally determined for Er and Tm. The others were estimated.

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

163

the lattice contribution. For TmNi5 (Gubbens et al., 1985c) the C value amounts about 243. Using this value for the data of RCo5 þ x C-factors were determined by Gubbens et al. (1988a, 1988b, 1989). For the GdxCoy compounds the A02 values were determined directly and for the other rare earth from the difference of the free-ion value and the measured quadrupole-splitting in both cases using Eq. (38). They give a good impression about of the sensitivity of the different types of rare earth Mo¨ssbauer isotopes. It is clear that 155Gd is the most sensitive one, since no orbital and hence no 4f-contribution is present. For the other cases the sensitivity decreases in the sequence 169 Tm, 166Er and 161Dy as shown in Table 5. Recently, Bertin et al. (2012) has determined with inelastic neutron scattering measurements the crystal field parameters of Tb2Ti2O7. They found for the A02 parameter a value of 40:5  1:0 meV=a20 , which is equal to 470K=a20 . Results on Ho2Ti2O7 and rescaled Tb2Ti2O7 by Malkin et al. (2004) give approximately the same value. Moreover, results from Yb3þ ions diluted Y2Ti2O7 give also this value as found by Mirebeau et al. (2007) and Rosenkranz et al. (2000). This means that this A20 value is well established. Bonville et al. (2003) and Armon et al. (1973) have found from 155Gd Mo¨ssbauer spectroscopy a lattice contribution of
5.6 mm/s. In this structure the asymmetry parameter h ¼ 0. With Eq. (38) using a Sternheimer value of gN ¼
61 and screening coefficient s2 ¼ 0.67 Bertin et al. (2012) has computed for A02 a value of 95 meV=a20 for Gd2Ti2O7, which gives after scaling 97 meV=a20 for Tb2Ti2O7. This is a factor 2.4 larger than the result from the inelastic neutron scattering on Tb2Ti2O7. Calculation of the Constant C in Eq. (38) gives a value of 385, which is 20% higher than the value used for the intermetallic compound GdCo5 in Table 5. This should mean that metals and insulators are probably not too different. However, this is only one result. Therefore, it is clear that a more extended study both experimental and theoretical is necessary to modernize the use of Mo¨ssbauer spectroscopy on rare earth (especially Gd) insulators for determination of the second order parameter of the crystal field. It might be that covalency effects have an important contribution to the value of C ¼ (1
gN)/(1
s).

3.5 Magnetic Relaxation In studies of rare earth compounds with Mo¨ssbauer spectroscopy, a broadening of the Mo¨ssbauer lines of the hyperfine field due to magnetic relaxation effects are observed. At low temperatures, in the slow relaxation limit (slower than 10
7 s), mostly no line broadening is observed. Usually, with increasing temperature, lines of hyperfine fields show an increasing line broadening, until at a certain temperature one reaches a relaxation time, which is close with the Larmor precession time of a certain Mo¨ssbauer nucleus, where the hyperfine field collapses and only a broadened quadrupole spitting is left. This last broadening will disappear at the fast relaxation limit. For instance, in the case

164 Handbook of Magnetic Materials

of 169Tm Mo¨ssbauer spectroscopy, one observes just below this fast relaxation limit the same asymmetric doublet as is described by Blume (1965) for 57Fe Mo¨ssbauer spectroscopy. Blume and Tjon (1968) have considered a model with stochastic fluctuating magnetic spins parallel and perpendicular to the electric field gradient. A more complex treatment is given by the model of Clauser and Blume (1971). As an example, DyPO4 (TN ¼ 3.39K), which has been studied by Forester and Ferrando (1976a), is shown in Fig. 4. This figure shows that above TN ¼ 3.39K the Mo¨ssbauer spectrum is fully split. Above 16K, increasing relaxation broadening of the spectrum is observed until at 73K the spectrum is completely collapsed and only a broadened single line is observed typically for a 161Dy Mo¨ssbauer spectrum above the magnetic ordering temperature. The crystal field of DyPO4 consists of eight Kramers doublets with an overall splitting of approximately 570K. At low temperature the hyperfine field is close to the free-ion value, which indicates that the ground state doublet is Seff ¼ 1/2. At low temperature, the Eigen functions of the almost purified Kramers doublets j
15/2i and jþ15/2i by “local strong dynamic fields” are moving up and down parallel to the c axis as indicated by 161Dy Mo¨ssbauer spectroscopy. Although at higher temperatures more levels are becoming populated and principally the spin upespin down model is not valid any more,

FIGURE 4 Temperature dependence of the 161Dy Mo¨ssbauer spectra of DyPO4 below and above TN ¼ 3.39K. The term U/G indicates the value of the broadening of the Larmor precession frequency as indicated for the spectra at each temperature. This figure has been earlier published by Forester, D.W., Ferrando, W.A., 1976a. Phys. Rev. B 13, 3991.

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

165

the 161Dy Mo¨ssbauer spectra are becoming more and more populated and the dynamic process is becoming much faster. In the literature, this slow relaxation behavior above the magnetic ordering temperature is called “ferromagnetic relaxation”. In these cases, there is no direct transition possible between these two low-lying energy levels of the Kramers doublet. The relaxation is dominated by an Orbach (1961) spin-lattice process over an intermediate level. In the case of DyPO4, this level is at an excited of about 100K above the ground state doublet as described by Forester and Ferrando (1976a). Further discussion will be given in Section 4.4. In the magnetic ordered region the magnetic relaxation is often mentioned “electronic relaxation”. In the slow magnetic relaxation limit, one observes in a Mo¨ssbauer spectrum the hyperfine fields, which are related to the eigenfunction of each of the populated energy levels. Each level has its own hyperfine field. In this case information over the distance between the crystal field levels at low temperature can be determined from the intensity of the subspectra in relation with the Boltzman distribution. In the fast relaxation limit, one observes only the average of the hyperfine fine fields belonging to these eigenfunctions. However, usually one observes broadened Mo¨ssbauer lines with a strong overlap near the Larmor precession time. If the crystal field diagram is not too complex, simulations of these spectra can be made with relaxation models as described by Blume and Tjon (1968) and Clauser and Blume (1971).

3.6 Analysis Procedure: Examples In this paragraph, two examples will be shown. The first example is a description of the determination of the crystal field with help of the temperature dependence of the 169Tm electric quadrupole-splitting of the superconductor TmBa2Cu3O7
x using Eqs. (37) and (38). Furthermore, a second example will be shown about the temperature dependence of the 179Yb hyperfine field and the deduced behavior of the magnetic relaxation of the compound Yb2Ti2O7.

3.6.1 TmBa2Cu3O7
x Since the quadrupole-splitting in 169Tm Mo¨ssbauer spectroscopy through its crystal field is quite sensitive for small distortions as shown by Gubbens (2012), it was originally interesting to study the temperature dependence of the electric quadrupole-splitting of the high-temperature superconductor TmBa2Cu3O6.9 as initially shown by Gubbens et al. (1988a, 1988b). Fig. 5 shows the 169 Tm Mo¨ssbauer spectra of TmBa2Cu3O6.9 measured between T ¼ 4.2 and 680K as studied by Gubbens et al. (1988b). In Fig. 6 the values of the temperature dependence of the 169Tm quadrupole-splitting of TmBa2Cu3O6.9 and TmBa2Cu3O6.6 are shown. Near the superconducting (SC) transition Tc a broad minimum is observed in the temperature dependence of the quadrupole-splitting. This phenomenon might be attributed to a change in the orthorhombic symmetry of the crystal. On the

166 Handbook of Magnetic Materials

FIGURE 5 Temperature dependence of the 169Tm Mo¨ssbauer spectra of TmBa2Cu3O6.9. The drawn line is the best fit. This figure has been earlier published by Gubbens, P.C.M., van Loef, J.J., van der Kraan, A.M., de Leeuw, D.M., 1988b. J. Magn. Magn. Mater. 76 & 77, 615.

other hand, such a temperature behavior of the quadrupole-splitting can also be explained by temperature dependence of the crystal field. In a first approximation the higher order terms in Eq. (14) were neglected. From the results of the 155Gd Mo¨ssbauer spectroscopy measurements on GdBa2Cu3O6.9 by Smit et al. (1987) the lowest order crystal field terms B02 ¼ 1:9 and B22 ¼ 1:0K terms of Tm in TmBa2Cu3O6.9 were deduced. Then with Eqs. (14), (37) and (38) the temperature dependence of the electric quadrupole-splitting as shown in Fig. 6 was calculated (drawn curve). The 4fcontribution of the electric quadrupole-splitting is opposite in sign with the lattice contribution and negative. Therefore, the data are also negative as plotted in Fig. 6 (open dots). Due to the large discrepancy between the measured and calculated values higher order terms as shown in Eq. (14) are included. To analyze TmBa2Cu3O6.9 a tetragonal symmetry was used ignoring the orthorhombic distortion as shown by Nekvasil (1988). By using an iterative procedure of Eqs. (14), (37), and (38) with respect to the data points, the crystal field values shown in Table 6 on the first line were found. These Bm n values calculated are in good agreement with the values scaled from the inelastic neutron scattering measurements on HoBa2Cu3O6.9 by Fu¨rrer et al. (1988) as shown on the third line in Table 6. This result is given in Fig. 6 as the dashed curve. Hence the temperature dependence can be described with common crystal field parameters.

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

167

FIGURE 6 Temperature dependence of the electric quadrupole-splitting of 169Tm Mo¨ssbauer spectra of TmBa2Cu3O6.9 and TmBa2Cu3O6.6. For the drawn dashed lines, see text. This figure has been earlier published by Gubbens, P.C.M., van Loef, J.J., van der Kraan, A.M., de Leeuw, D.M., 1988b. J. Magn. Magn. Mater. 76 & 77, 615.

In order to discriminate between the two possible explanations for the observed minimum in the quadrupole-splitting, the 169Tm spectra of TmBa2Cu3O6.6 has been measured with 169Tm Mo¨ssbauer spectroscopy on a compound with a lower oxygen content and hence a lower Tc as shown by Cava et al. (1987). From the measurements on TmBa2Cu3O6.6 with a Tc of 58K, it appears that the minimum in the temperature of the quadrupole-splitting is still found at a temperature of about 90K as shown in Fig. 6 (full dots). So it can be concluded that this minimum should be attributed to crystal field effects and not to a distortion at Tc Moreover from Fig. 6, it appears that oxygen removal in these high-Tc superconductors does not alter the crystal field parameters. Bergold et al. (1990) have repeated the measurement on the 169Tm Mo¨ssbauer spectra of TmBa2Cu3O6.9. Their spectra are slightly broadened with respect to spectra shown in Fig. 5. The temperature of the electric quadrupole-splitting shows qualitatively the same behavior as shown in Fig. 6. Results of 155Gd Mo¨ssbauer spectra of GdBa2Cu3O6.9 by Wortmann et al. (1988a, 1988b) were used to determine the crystal field terms B02 and B22 as shown on the second line in Table 6. They used the scaled higher order terms Fu¨rrer et al. (1988). The fit with these crystal terms (second line Table 6) is quite good, indicating that the higher order terms are quite well determined by the inelastic neutron scattering measurements by Fu¨rrer et al. (1988). Fritz and Dixon (1992) have restudied the 169Tm Mo¨ssbauer spectra of TmBa2Cu3O6.9 between T ¼ 0.06 and 92K. Their results could be well explained by the scaled crystal field parameters from Fu¨rrer et al. (1988). Rudowicz

Publication

B02 (K)

B22 (K)

B04 (mK)

Gubbens et al. (1988a, 1988b)

þ1.90

þ1.0


20

Bergold et al. (1990)

þ1.75

þ0.67


44.1

Fu¨rrer et al. (1988)

þ2.41

þ1.05


44.1

B24 (mK)

B44 (mK)

B06 (mK)

B26 (mK)

B46 (mK)

B66 mK

120


250

þ2.91

þ203


183

þ110


5.48

þ77.1

þ2.91

þ203


183

þ110


5.48

þ77.1


5.0

Results were taken from Gubbens et al. (1988a, 1988b), Bergold et al. (1990) and a scaled result from the inelastic neutron scattering measurement of HoBa2Cu3O6.9 by Fu¨rrer et al. (1988).

168 Handbook of Magnetic Materials

TABLE 6 Comparison of the Crystal Field Parameters Bm n for TmBa2Cu3O6.9

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et al. (2009) have reanalyzed the crystal field parameters of TmBa2Cu3O6.9. Their results are roughly in agreement with the results shown in Table 6. From the description of all these results, it is clear that inelastic neutron scattering are additional and crucial to come to a full description of the electric crystal field in these type of compounds.

0.8 0.4 0.0 100

50

Yb2Ti2O 7

0 0.0 0.1 0.2 0.3 0.4

3+

mag. fract. (%)

1.2

moment (µ B / Yb )

3.6.2 Yb2Ti2O7 The pyrochlore structured compound Yb2Ti2O7 has been studied with 170Yb Mo¨ssbauer spectroscopy. Selected 170Yb Mo¨ssbauer absorption spectra are shown in the left panel of Fig. 7. At T ¼ 0.036K, a five-line spectrum is observed, indicating a “static” hyperfine field of 115T. In the present case, “static” means that the fluctuation frequency of the field is slower than the lowest observable relaxation limit of 15 MHz. Knowing that for Yb3þ the hyperfine field is proportional to the 4f shell magnetic moment, it was found that each of the Yb atoms carries a magnetic moment of 1.15 mB. In the absence of a significant quadrupole hyperfine interaction, the local direction of the Yb3þ magnetic moment cannot determined using Eqs. (37) and (38) with the principal axis of the electric field gradient along the [111] direction. Instead, for an anisotropic Kramers doublet, the size of the spontaneous magnetic moment is linked to the zero angle with the local [111] symmetry axis through the relation 
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 MYb ¼ gt mB (39) cos q r 2 þ tan2 q 2

Temperature (K) 170

FIGURE 7 Left panel: Yb Mo¨ssbauer spectra of Yb2T2O7 above, within, and below the firstorder transition occurring near 0.24K. Right panel: thermal variation of the size of the Yb3þ magnetic moment obtained from the hyperfine field (top) and relative weight of the static magnetic fraction (bottom). The lines are eye guides. These figures have been earlier published by Hodges, J.A., Bonville, P., Forget, A., Yaouanc, A., Dalmas de Re´otier, P., Andre´, G., Rams, M., Kro´las, K., Ritter, C., Gubbens, P.C.M., Kaiser, C.T., King, P.J.C., Baines, C., 2002. Phys. Rev. Lett. 88, 077204.

170 Handbook of Magnetic Materials

where r is the anisotropy ratio gt/gz as shown by Bonville et al. (1978). With gt/gz y 2.5 q ¼ 44(5) was found. Thus each moment does not lie perpendicular to its local [111] axis as would be expected if the orientation were governed only by the crystal field anisotropy. With increasing temperature up to 0.24K, an additional single-line subspectrum appears. It is linked with the fraction of the Yb3þ whose moments fluctuate “rapidly” so that the magnetic hyperfine splitting becomes “motionally narrowed”. The two subspectra (see Fig. 7, left panel at 0.24K) are both present up to 0.26K, evidencing the coexistence of temperature regions with “static” and “rapidly fluctuating” moments. The right panel in Fig. 7 shows that, as the temperature increases, there is a progressive decrease in the relative weight of the static hyperfine field subspectrum. This behavior shows a clear evidence of a first-order transition. The single-line subspectrum progressively narrows as the temperature increases. Since magnetic correlations are still present above 0.24K, we attribute this change to the progressive increase in nM, the fluctuation rate of the hyperfine field (Heff) The relation between the dynamic line broadening, DGR, and nM is written DGR ¼ (mIHeff)2/nM, where mI is the 170Yb nuclear moment as given by Dattagupta (1981). As shown in Fig. 8 below, when the temperature is lowered from 1K to just above 0.24K, the rate decreases

Fluctuation rate (106 s−1)

100000

Yb2Ti2O7

10000 1000 ¨ Mossbauer lower limit

100 10

νμ νM

1 0.1

1 Temperature (K)

10

FIGURE 8 Estimate of the Yb3þ fluctuation rates as obtained from 170Yb Mo¨ssbauer (nM) and mSR (nM) measurements. The first-order transition in the fluctuation rates takes place at the L transition. Above w0.24K, the fluctuation rates deduced from mSR and Mo¨ssbauer spectroscopy match for a Yb3þ muon spin coupling of DHT/gm equal to x80 mT. This value agrees reasonably well with the value of 85.6 mT obtained by scaling from Tb2T2O7 as published by Gardner et al. (1999). Below w0.24K, the fluctuation rate is independent of temperature and has dropped below the lowest value, which is measurable with the Mo¨ssbauer method (dashed line). The solid line follows a thermal excitation law. This figure has been earlier published by Yaouanc, A., Dalmas de Re´otier, P., Bonville, P., Hodges, J.A., Gubbens, P.C.M., Kaiser, C.T., Sakarya, S., 2003. Physica B. 326, 456.

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from z15 to z2 GHz. This decrease is linked to the slowing down of the spin fluctuations. Below 0.24K, nM drops to a value, which is less than the lowest observable 170Yb Mo¨ssbauer relaxation value. In conclusion, magnetic fluctuations in materials can be followed with rare earth Mo¨ssbauer spectroscopy. Scientific descriptions can be made from the results in combinations with results of other measuring techniques like mSR, neutron diffraction, and specific heat. In Section 4.7 more results of R2M2O7 will be shown and discussed.

4. OVERVIEW OF RARE EARTH-BASED OXIDES 4.1 Introduction In the following paragraphs the rare earth Mo¨ssbauer effect measurements of the different rare earth-based oxides will be discussed and compared with measurements of other techniques.

4.2 R2O3 Compounds In this paragraph the Mo¨ssbauer results of rare earth oxides are discussed. With exception of Dy2O3 (Forester and Ferrando, 1976b) and PrO2 (Moolenaar et al., 1996) all other rare earth oxides, Gd2O3 (Cashion et al., 1973), Er2O3 (Cohen and Wernick, 1964), Tm2O3 (Barnes et al., 1964), and Yb2O3 (Meyer et al., 1995), studied with rare earth Mo¨ssbauer spectroscopy, are not magnetically ordered. From a physical point of view, Tm2O3 is very interesting for the determination of its crystal field. For Tm2O3 the temperature dependence of the electric quadrupole-splitting of both Tm sites (C3i and C2) have been measured and analyzed up to room temperature by Stewart et al. (1988) and Stewart (2010) as shown in the left side of Fig. 9. Moreover, on the right side, the temperature dependence of the unsplit quadrupole interaction for both sites is shown as measured by Barnes et al. (1964). The quadrupole-splitting at the monoclinic C2 site is about 5% smaller than that obtained with a single doublet analysis. The two sets of data converge above 150K. The quadrupole-splitting measured for the C3i site is consistently larger than that for the trigonal C2 site in the temperature range between T ¼ 0 and 300K. The relative intensity of the doublet of the two C2 and C3i Tm sites is 3:1. The crystal field Hamiltonians for the C3i and C2 are given by Eqs. (12) and (15). The solid curves in Fig. 9 represent theoretical fits to the data for each site using the respective sets of parameters of the crystal field Hamiltonians for the C3i and C2 as given by Eqs. (12) and (15). The used Bm n for the C2 Tm site were taken from Leavitt et al. (1982) and for the C3i Tm site from Gruber et al. (1985). Traditionally, QS4f is proportional to r1 ¼ Q(1
RQ)hr
3i, where Q ¼ 1.5b is the quadrupole moment of the 169Tm nucleus (Olesen and Elbek, 1960), RQ ¼ 0.128 the antishielding factor (Gupta and Sen, 1973) and

172 Handbook of Magnetic Materials

FIGURE 9 Electric quadrupole-splitting DEQ versus temperature of Tm2O3: (A) Resolved data for the two Tm sites (C3i and C2): full symbols (Stewart et al., 1988). (B) Original unresolved data (Barnes et al., 1964). The solid curves (Stewart et al., 1988) representing the parameter fits of theory to the resolved data are included in both plots.
3 ¼ 12:105 au
3 (Dunlap, 1971) for the 169Tm nucleus. The shielding r4f parameters as calculated by Stewart et al. (1988) fit very well for the 4fcontribution for Tm2O3 and thulium ethylsulfate (TmES) as determined by Barnes et al. (1964). Whereas the atomic shielding represented by r1 is consistent with theory, the lattice EFG shielding parameter, r2, is seen to be reduced considerably for both sites in Tm2O3. The found values for C by Stewart et al. (1988) are 94(4) and 45(13) for the C2 and the C3i of Tm2O3, respectively. Stewart et al. (1988) argues that (1
gN) in C ¼ (1
gN)/(1
s) will not vary significantly from one material to another. Therefore, this reduction is most likely due to the influence of the TmeO covalent bonding on the factor (1
s). Comparison with the calculated field gradients from the 155Gd Mo¨ssbauer measurements of Gd2O3 by Cashion et al. (1973) is good for the C2 site and bad for the C3i site. However, Stewart et al. (1988) regards the crystal field results of the C3i site less reliable. It is apparent that the extent of covalent bonding is similar for the two Tm3þ sites despite their differing symmetries. This is supported by the work of Ryzhkov et al. (1985) who performed numerical calculations for TmO9
6 clusters in Tm2O3 and concluded that the electronic structures of the two Tm3þ sites are practically the same. In general, this kind of behavior implies that covalency effects have influence on the value of C ¼ (1
gN)/(1
s) as has been shown earlier in the case of Yb2Ti2O7 in Section 3.4. An 161Dy Mo¨ssbauer spectrum obtained at T ¼ 7K for Dy2O3 is shown at the top of Fig. 10 as measured by Forester and Ferrando (1976b). In this figure

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the predominant spectral features are associated with ions at sites with C2 symmetry. However, a second, weaker and broadened spectrum is noted with an overall splitting greater than the C2 site spectrum, which can be ascribed to the C3i site. The solid line drawn through the data in Fig. 10 is the result of a least-squares computer analysis using an effective spin S ¼ 1=2, hyperfine interaction Hamiltonian for each of the C2 and C3i sites. The broadening of the spectrum of the C3i site was included in the analysis by a spin-upespin-down relaxation. The frequency at T ¼ 7K is 6  10
6 s
1. Whereas the C2 site is reduced to the free-ion value, the C3i site has the same hyperfine field as shown for DyPO4 in Section 3.5, which has an almost pure j15/2i doublet ground state. Then, the 4f-contribution of the quadrupole-splitting has the free-ion value. Therefore, from the lattice contribution of the quadrupole-splitting the crystal field potential A02 ¼ 200K, which is much smaller than the value of 480K, used by Stewart et al. (1988) for his analysis of the crystal field in Tm2O3. Moolenaar et al. (1994, 1996) have studied with 141Pr Mo¨ssbauer spectroscopy the isomer shift of Pr2O3, Pr6O11 and PrO2 as a function of the valency. In Section 4.5, this subject will be further discussed in combination

FIGURE 10 161Dy Mo¨ssbauer spectrum taken at T ¼ 7K. The solid curve through the data points is a computer fit for both C3i and C2 site ions. The two solid curves at the bottom are calculated C3i and C2 spectra that make up this composite fit. This figure has been earlier published by Forester, D.W., Ferrando, W.A., 1976b. Phys. Rev. B 14, 4769.

174 Handbook of Magnetic Materials

of the result of PrBa2Cu3O7. Since at ambient oxygen pressure only Pr6O11 is stable, it is, however, possible that PrO2 is oxygen-deficient (and Pr2O3 oxygen-redundant). Using 141Pr Mo¨ssbauer spectroscopy the magnetic order and the oxygen content of PrO2 could be checked. The spectrum splits below the Ne´el temperature of T ¼ 14K (Kern et al., 1984) into a magnetically split and a broad unsplit contribution as shown in Fig. 11. When the magnetically split part is ascribed to stoichiometric PrO2 and the unsplit part to oxygen deficient PrO2
x, assuming that the recoilless fractions are not too different, it was found that about 6% of PrO2 is oxygen deficient as shown in Fig. 11. The determined hyperfine fields amount 78, 71, and 30T at T ¼ 4, 8, and 12K, respectively. The result of PrO2 is in good agreement with those of Bent et al. (1971), Kapfhammer et al. (1971) and Groves et al. (1973), which were measured with a scattering method of 141Pr Mo¨ssbauer spectroscopy.

FIGURE 11 141Pr Mo¨ssbauer spectra PrO2 measured with a CeF3 source. The curves drawn give the fit to the data points. The spectrum at 4.2K has a second contribution, approximately indicated by the dotted curve. This figure has been earlier published by Moolenaar, A.A., Gubbens, P.C.M., van Loef, J.J., Menken, M.J.V., Menovsky, A.A., 1996. Physica C 267, 279.

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The high-pressure behavior of the cubic C-type Yb2O3 was investigated by 170Yb Mo¨ssbauer spectroscopy and up to a pressure of 20 GP at T ¼ 4.2K by Meyer et al. (1995). They confirm the occurrence of a pressure-induced phase transition from a cubic to a monoclinic structure. The monoclinic phase is retained after releasing the pressure. The Mo¨ssbauer quadrupole splitting values were evaluated in combination with the structural data in a coherent way.

4.3 RMO3 Compounds In this paragraph the magnetic properties of perovskite RMO3 compounds are discussed. These compounds are quite interesting and are well investigated because of their properties, which range from colossal magnetoresistance, charge ordering, and multiferroicity in hole-doped RMnO3. Recently, it was found that RMnO3 compounds with light rare earth, for example, Pr have a first-order transition near room temperature and can be used for magnetic cooling. The compound GdAlO3 has is an orthorhombically distorted perovskite with a c axis lying along one pseudocubic axis. The magnetic ordering direction in GdAlO3 is the orthorhombic a axis as determined by Cook and Cashion (1976, 1980) with a Ne´el temperature of 3.870K. At temperatures well below TN, GdAlO3 can be well described in terms of a molecular-field theory for a two-sublattice uniaxial antiferromagnet. At temperatures close to TN the exchange interaction is small and the magnetic sublattices are aligned close to the principal axis of crystal electric field and will cant to the a axis as the temperature is decreased. 155 Gd Mo¨ssbauer spectroscopy has been performed on single crystals of GdAlO3. The spectra were analyzed using Eqs. (37) and (38) for Gd. For the lowest temperatures, it was found that the values of the hyperfine fields vary linearly with T3. The hyperfine fields just below TN were proportional with d(1
T/TN)b. From a logarithmic plot the calculated critical exponents are d ¼ 1.24  0.05 and b ¼ 0.37  0.02. An estimate of the values of b and d has been made using the theoretical studies of Rushbrooke et al. (1974). The Heisenberg model for a ferromagnet predicts, for S ¼ 7/2, the value of b ¼ 0.368  0.033, which is in agreement with the value found for GdAlO3. For simple-cubic lattice structure the value of d has been extrapolated, using the Ising model, to be 1.32  0.02, which is larger than the value found for GdAlO3. Near the magnetic ordering temperature of 3.87K the electric field gradient makes a canting angle of 43  4 to the a axis. The sublattices rotate toward a axis very quickly with decreasing temperature below TN and remain nearly constant below 3.2K on a value of 7.2  1.5 . This behavior could well be described with an electronic Hamiltonian by Cook and Cashion (1980).

176 Handbook of Magnetic Materials

The compounds DyCrO3 and DyFeO3 were studied with by 161Dy Mo¨ssbauer spectroscopy by Eibschu¨tz and Van Uitert (1969) and Nowik and Williams (1966), respectively. The Mo¨ssbauer spectra of DyCrO3 (orthorhombic symmetry) are below and above TN ¼ 2.16K well split and similar as in the case of DyVO4 (Section 3.5). Above T ¼ 20K the spectra are broadening and show typical paramagnetic relaxation behavior arising from the isolated and almost pure j15/2i Kramers-doublet ground. The magnetization axis is strongly anisotropic as shown from the g-tensor with gz ¼ 19.7 and gx ¼ gy ¼ 0. In DyFeO3 the Fe sublattice orders magnetically at 635K. Down to TN ¼ 4.5K the Dy sublattice in this compound stays paramagnetically until it orders antiferromagnetically. The Mo¨ssbauer spectra above TN show also typical paramagnetic relaxation behavior up to 300K. From the hyperfine splitting and the g-tensor (with only one component along the z axis), it appears also that the Kramers-doublet ground state is almost pure j15/2i. In this respect, DyFeO3 is comparable with DyCrO3 (above) and DyVO4 as described in Section 3.5. Spectra above 50K magnetic hyperfine fields of excited (Stark) levels. Apparently, the relaxation time is long enough to observe them due to the magnetic field of 0.29T caused by the magnetic Fe sublattice. In this respect the observation of the higher lying Kramers doublet levels above TN ¼ 4.5K in DyFeO3 is so far known as a unique observation. The distorted perovskite compound TmAlO3 is studied by Hodges et al. (1984). In the temperature range of 4.2e300K, the observed absorption was in the form of a symmetric quadrupole-splitting without any magnetic interaction. This is compatible with the expected nonmagnetic nature of the Tm3þ ion. The absolute value of temperature dependence of the quadrupole interaction could be used to calculate the crystal field parameters. However, this attempt was not successful. The distorted perovskite compound TmVO3 has a Ne´el temperature of 106K, associated with the ordering of the 3d lattice of Vanadium. This compound was studied with 169Tm Mo¨ssbauer spectroscopy by Hodges et al. (1984). Three temperature zones were defined. In the first one, down to TN ¼ 106K, the Tm3þ ion is nonmagnetic. From TN down to about 15K, the Tm3þ ion is weakly magnetic and then below about 15K, the spectra show more strongly magnetic behavior. At that temperature the hyperfine field increases rapidly with more than a factor five until it reaches a saturated Tm magnetic moment of 2.1 mB, which is still much lower than the free ion value of 7 mB. These three behaviors are represented by three different 169Tm Mo¨ssbauer absorption line shapes. To obtain adequate line shape fits below 7K, it was necessary to allow the widths of the individual absorption lines to be independent of each other; the intensities of the lines being imposed by the fitting procedure. The electric quadrupole-splitting shows changes at

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TN ¼ 106K and at 15K. At T ¼ 4.2K the 169Tm Mo¨ssbauer spectrum is clearly relaxation broadened. No clear explanation could be given. The distorted perovskite compound TmCrO3 has a Ne´el temperature of 124K, associated with the ordering of the 3d lattice of Chromium. This compound was studied with 169Tm Mo¨ssbauer spectroscopy by Hodges et al. (1984). The compound TmCrO3 has two excited Tm3þ singlet levels at 26 and 63K. The 169 Tm Mo¨ssbauer absorption spectra show a line shape of a symmetric quadrupole doublet down to TN and a magnetic splitting at lower temperatures. The electric quadrupole-splitting, the hyperfine field, and q (the angle between the electric field gradient and the magnetization direction) all vary with temperature. As the parameters DEQ and Heff are approximately the same size, this is a favorable case for determining the sign of DEQ and for accurately obtaining q. DEQ is found to be negative and the experimental accuracy of theta is estimated to vary from close to 90 at 1.4K, to close to 45 at 80K. Since the electric field gradient is not changing in direction, this behavior can ascribed to a direction change of the hyperfine field. This result could well be explained by a model of Bertaut et al. (1966). With this structural model and knowing the Tm3þ moment from the Mo¨ssbauer data, the canting angle of the Cr3þ lattice could be calculated. The most convenient temperature for the calculation is the compensation point near 28K, where the resultant ferromagnetic Cr3þ moment is equal and opposite to the Tm3þ moment as shown by Bertaut et al. (1966). At 28K, our value for the Tm3þ moment is 0.33  0.02 mB and the individual Cr3þ moment is 2.56 mB. This latter value is obtained by interpolating the data at 4.2 and 80K as given by Bertaut et al. (1966) using a Brillouin function B3/2. A comparison of the total spontaneous moment and the individual Tm3þ and Cr3þ moments at other temperatures between 4.2 and 80K leads to the same result. From the results, it is unclear whether the Tm3þ moment is magnetic by itself or induced by the Cr3þ moment. The 169Tm Mo¨ssbauer of h-TmMnO3 are studied by Salama and Stewart (2009). Representative spectra are presented in Fig. 12. There is evidence of magnetic splitting over a wide temperature range and all of the spectra were able to be fitted with the superposition of a magnetic sextet and a quadrupole split doublet in the intensity ratio of 2:1. This confirms that the MneTm exchange interaction acts only at the more prevalent Tme4b site. The magnetic moment induced on the Tm3þ 4b site and the associated hyperfine field, Bhf (which is Heff in our notation), acting at its 169Tm nucleus are expected to align with the c axis. This is because the triangular arrangements of Mn spins in the layers above and below combine to produce a molecular field acting along the c axis. Based on symmetry arguments, the c axis is also expected to be the principal of the total electric field-gradient tensor acting at the nucleus. Although the 4b site symmetry of 3 (C3) allows for a nonzero asymmetry parameter, the 2a site symmetry of 3m (C3v) does not. However, the two sites have very similar local environments so that we might expect the asymmetry

178 Handbook of Magnetic Materials

FIGURE 12 Left panel: Representative 169Tm Mo¨ssbauer spectra for h-TmMnO3. The fitted theory curve is the sum of a magnetic sextet [red online] and an asymmetric doublet [green online] corresponding to the Tm-4b and 2a sites, respectively. Right panel: Temperature dependence of the 169 Tm hyperfine field, Bhf, at the 4b site of h-TmMnO3. The fitted theory curve is based on a simple two-singlet crystal field ground-state model (see text) with a ¼ 6, D ¼ 20K and a saturation MneTm molecular field of BM(MnTm) ¼ 1.27T. The temperature dependence of the quadrupole interaction, ð1=2ÞeQVZZ , is shown in the inset for both the 4b site (solid diamonds) and the 2a site (solid circles). These figures have been earlier published by Salama, H.A., Stewart, G.A., 2009. J. Phys. Condens. Matter 21, 386001.

parameter to be negligible for the 4b site. This approach was employed successfully with the analyses of magnetic 170Yb Mo¨ssbauer spectra of hYbMnO3 as published by Salama et al. (2008, 2009), discussed later in this paragraph. The analysis of the 4.2-K spectrum yielded a hyperfine field of 311.8  2T, which is less than half of the free ion value for insulators of 662.5T. This means that magnetic moment on the 4b site is 3.29  mB, which is approximately 47% of the maximum free ion moment of 7 mB and is indicative of significant crystal field quenching as shown in Fig. 12. The fitted quadrupole interaction parameters yielded 1/2DEQ ¼ 41.9  2 mm/s and 22.3  mm/s, respectively for the 4b and 2a sites. These values are significantly smaller than the free ion value of 58.6 mm/s, ignoring the lattice contribution. Again a strong reduction of the quadrupole interaction measured for the 4b site sextet, which is about twice that measured for the 2a site doublet as shown in Fig. 12. With increasing temperature the magnetic splitting of the 4b site collapses and eventually vanishes at 82e83K, which is in good agreement with the values of TNMn z81e84K reported. The fitted Bhf values are shown as a function of temperature in Fig. 12 and the fitted quadrupole interactions are shown in the figure’s inset. The quadrupole interactions appear to be temperature

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

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independent over the full range of temperature that is associated with the Mn magnetization. This behavior indicates two well-isolated singlets in the crystal field scheme. Using a set of recalculated parameters of h-YbMnO3 for the 4b site as assumed earlier by Divis et al. (2008), the crystal field of h-TmMnO3 was calculated by diagonalizing Eq. (12) for a trigonal symmetry. The most important result is a well-isolated two-singlet nonmagnetic ground state. Since the singlet states are nonmagnetic, the inducement of a net Tm3þ magnetic moment on the 4b site requires that the molecular field associated with the weak MneTm exchange interaction brings about a mixing of these levels. Under these circumstances, the induced magnetic moment is given by  0  2g2 m2 a2 BM D m ¼ J B0 tanh (40) D 2kB T where BM is the molecular field acting in the z direction (the crystallographic c axis), a ¼ h0jJzj1i is the Jz coupling parameter between the two singlets, and D0 ¼ ½D2 þ ð2gJ mB aBM Þ2 1=2 is the field enhanced energy separation of the two singlets (D is the energy separation in absence of a molecular field). In order to arrive at a theoretical estimate of the induced moment, it was assumed that the molecular field acting at the 4b site is proportional to the Mn moment. This problem was solved by using a temperature dependence described by an empirical formula by Lonkai et al. (2002). The theory curve (solid line) drawn through the experimental data in Fig. 12 corresponds to the values of BM(T ¼ 0K) ¼ 1.27  1 and the two-singlet state parameters D ¼ 20.2  2K and a ¼ 6.00  2K. Although the authors argue that the solution is not unique due to insufficient information about the crystal field, the form of the theory curve matches closely with the temperature dependence of the experimental data and is a nice example to describe such a two-singlet crystal field system. Salama et al. (2010) has studied with 169Tm Mo¨ssbauer spectroscopy the orthorhombic phase of o-TmMnO3. AC susceptibility and specific heat measurements confirm that the Mn sublattice of o-TmMnO3 orders magnetically at TNMn z41K with a weaker feature at Tc z 32K. From 169Tm Mo¨ssbauer spectra of o-TmMnO3 it appears, that down to 10K, there is no evidence of magnetic splitting and all of the spectra were fitted in terms of a single asymmetric, quadrupole split doublet. However, there is a more subtle effect of line broadening that increases with decreasing temperature. From the spectra the authors argue that the half-width of the absorption lines undergoes a sharp increase in the vicinity of Tc z 32K. Therefore, it is reasonable to assume that the transition of the Mn spin order from incommensurate to collinear antiferromagnetic results in a nonzero exchange field at the Tm site. Similar behavior was reported in an earlier 155Gd Mo¨ssbauer investigation of oGdMnO3 by Zukrowski et al. (2003), where the Mn sublattice orders at 40K, but the 155Gd hyperfine field is not observed until the lower temperature of 21K. As the temperature decreases, the quadrupole-splitting, DEQ, increases

180 Handbook of Magnetic Materials

from 1.1 cm/s at room temperature to 9.2 cm/s at 4.2K with abrupt increases close to both TNMn and Tc. Since the temperature-dependent 4f shell contribution to the electric field gradient involves a thermal average over the crystal field levels, it is unusual to observe such changes in DEQ. This aspect warrants further investigation to possible structural changes associated with the two transitions. Finally, there appears to be some additional structure in the 4.2-K spectrum, indicating the onset of a distinct magnetic hyperfine field. The distorted perovskite compound TmFeO3 has a Ne´el temperature of 630K, associated with the ordering of the 3d lattice of iron. This compound was studied with 169Tm Mo¨ssbauer spectroscopy by Hodges et al. (1984). As in the case of TmCrO3, TmFeO3 has first two excited states at 25 and 56K. Above 95K, the Fe sublattice of TmFeO3 has an antiferromagnetic component parallel to the a axis and a ferromagnetic component is parallel to the c axis. Below 80K, the antiferromagnetic component is parallel to the c axis and the ferromagnetic component is parallel to the a axis. The field inducing Tm3þ moment is parallel to the Fe3þ weak ferromagnetic component, so that the Tm3þ magnetic moment is expected to be parallel to the c axis above 95K and perpendicular to the c axis below 80K. These magnetic moments are a perturbation on the electric quadrupole-splitting. Since the authors argue that DEQ is negative, a correct interpretation can be made with the above found direction of the Tm3þ moment. For the higher temperature range above 95K the moment is increasing with decreasing temperature, until a Tm3þ moment is found of 8T (0.08 mB), and below 80K the moment increases again with decreasing temperature, until at 4.2K Tm3þ moment is found of 8.5T (0.085 mB). A possibility could be that induced hyperfine field on Tm is caused by a transferred field from the Cr sublattice to the Tm sublattice. However, that idea has not been taken into consideration. Mo¨ssbauer spectroscopy measurements have been carried out on 170Yb in some rare earth orthoaliminates (RAlO3) by Bonville et al. (1978). The relaxation behavior Yb3þ ions diluted in TmAlO3 and YAlO3 was studied. The parameters measured in the slow relaxation limit indicate an almost pure j7/2i Kramer’s doublet ground state. In this case, this diluted compound is highly anisotropic. The relaxation time in the spectra was fitted from the line shape using a simple stochastic model with diagonal hyperfine interactions by Gonzalez-Jimenez et al. (1974). The temperature dependence of the relaxation time could be fitted with a mixture of Raman and Orbach contributions between 40 and 70K. At lower temperatures just above 40K, the Orbach contribution is more dominant. In YbA1O3 a long range-ordering temperature has been found with a Ne´el temperature of 0.8K by Bonville et al. (1978). The value of the electric quadrupole-splitting 4.05 mm/s, which is the same value observed in YbAlO3, above TN. This indicates that the ground-state wave functions are consisting mainly of a j7/2i Kramer’s doublet, which is well separated from the j5/2i doublet, agreeing with a gz value of about 6.9. The hyperfine field increases as

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

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the temperature is lowered below TN ¼ 0.8K. Between 0.55 and 0.34K, no further increase in this field is observed, clearly showing that it has reached its maximum value. The data fit gives a maximum field of 263  5T, corresponding with a magnetic moment of 2.49  0.05 mB along a direction that is within a few degrees of the local z axis. In the spinespin-driven relaxation region, two temperature zones were observed, where the relaxation rate has, respectively, a conventional temperature-independent value and anomalous temperature-dependent values. For the third case, the YbxY1
x AlO3 compounds, the dependence of the relaxation rate on the Yb3þ concentration was studied. For YbAlO3 using a formalism based on the Fermi golden rule, the relaxation rates in the paramagnetic region arising from different types of interactions between the spins were calculated. The magnetic exchange between nearest neighbors in the ab plane is dominant for the relaxation mechanism. The origins of the moment lowering and short-range ordering observed in this system were also discussed in terms of the Yb concentration. With increasing Yb, a decreasing percentage of the relaxation of the Yb3þ is in the slow relaxation limit. For x ¼ 0.50, both groups of Yb ions show finite relaxation rates. 170 Yb Mo¨ssbauer spectroscopy measurements have been performed on hexagonal manganite h-YbMnO3 by Salama et al. (2008, 2009). Representative 170Yb Mo¨ssbauer spectra are presented in Fig. 13. The spectrum measured

170 FIGURE 13 Representative Yb Mo¨ssbauer spectra for h-YbMnO3. The fitted theory curve is the sum of relaxation subspectra for Ybe4b site [red] and the Ybe2a site [green], respectively. This figure has been earlier published by Salama, H.A., Voyer, C.J., Ryan, D.H., Stewart, G.A., 2009. J. Appl. Phys. 2010 (5), 07E110.

182 Handbook of Magnetic Materials

at the base temperature of 1.5K was able to be fitted in terms of two static, five-line, and magnetically split subspectra with intensities in the ratio of 2:1 as expected for the relative occurrence of the Yb 4b and 2a sites. Since the electric quadrupole interaction is small, it was not possible to determine the alignment of the magnetic direction with respect to the principal axis of electric field gradient. Therefore, a simple coaxial model was used. For both sites, both the hyperfine field and the electric quadrupole interaction are significantly smaller than the maximum possible Yb3þ free ion values of 412.5T and 55.77 mm/s, respectively, indicating a strong crystal field quenching. For the 4beYb site, this could be determined with the crystal field parameters as determined by Divis et al. (2008) with infra red. With increasing temperature the 2a site subspectrum collapses at 5K and presents a broad, motionally narrowed line above this temperature. However, a residual magnetic structure persists up to 20K for the 4b-site subspectrum. All of the 170Yb Mo¨ssbauer spectra could be analyzed in terms of a superposition of two relaxation broadened, magnetic subspectra with integrated intensities in the ratio of 2:1. The spectra were analyzed with the relaxation model of Wickman et al. (1966). In this model, the Kramers-doublets relaxation was described with effective spin Seff ¼ 1=2. At all temperatures, the linewidth, isomer shift, and quadrupole interaction strength were fixed at values obtained for the 1.5-K spectrum. Only the hyperfine field and the Kramers-doublets splitting were allowed to vary. The relaxation time for both sites is decreasing with increasing temperature. The 4beYb site has a longer relaxation time than the 2a site. This last shows a rapid decrease with increasing temperature below a Ne´el temperature of 5K. In Table 7 an overview has been given about the results of the rare earth Mo¨ssbauer spectroscopy of the perovskite compounds RMO3 with M ¼ Al, V, Cr, Mn and Fe and R ¼ Gd, Dy, Tm, and Yb. Whereas the rare earth sublattice orders magnetically below 5K, the 3d sublattice can have even a much higher transition up to 635K. Moreover, in the case of Tm and Yb compounds, the magnetic moments are strongly reduced with respect to the free ion magnetic moments of 7 (for Tm) and 4 (for Yb) mB above the rare-earth magnetic ordering temperature. This means that the ReM magnetic interaction is most likely the prevailing magnetic exchange in these compounds on the rare earth sites.

4.4 RMO4 Compounds The rare earth compounds of the general formula RMO4 oxides with R ¼ rare earth and M ¼ P, V, As, and Cr crystalize in two structural types depending on the size of the rare earth cation. The first member of the vanadates and chromates LaMO4 as well as the phosphate with R ¼ LaeTb and the arsenates with R ¼ LaeNd have the monazite-type structure. All the other rare earth compounds with M ¼ P, V, As, and Cr have the tetragonal zircon

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

183

(ZrSiO4)-type crystal structure. This means that all the rare earth compounds studied with rare earth Mo¨ssbauer spectroscopy, which will be overviewed in this paragraph, have this tetragonal zircon (ZrSiO4) type of crystal structure. The slow relaxation above the magnetic ordering temperature of the compounds DyPO4 (TN ¼ 3.39K) and DyVO4 (TN ¼ 3.0K) were studied with 169 Dy Mo¨ssbauer spectroscopy. In Section 3.5 a study of DyPO4 by Forester and Ferrando (1976a) is shown as example of typical slow relaxation behavior above the magnetic ordering temperature. The slow magnetic relaxation of DyVO4 above TN has been studied by Gorobchenko et al. (1973). In Fig. 14 the inverse temperature dependence of the logarithm of the spin-lattice relaxation is shown. For DyPO4 and DyVO4 the data fall on a straight line between 36 and 16K and between 4 and 11K, respectively. The transition between the two levels of the ground-state Kramers doublet is thus an indirect scattering process, involving an excited electronic energy state at D above the ground state with U ¼ U0e
D/T, known as an Orbach process (Orbach, 1961). From the slope of the straight line, it was obtained D ¼ 110K for the excited state in DyPO4 and D ¼ 40K in DyVO4, respectively.

TABLE 7 Tabulation of Magnetically Ordered Gd, Dy, Tm and Yb Compounds With a Perovskite Compounds RMO3 With M ¼ Al, V, Cr, Mn, and Fe Compound

Heff (T)

MR (mB)

TNRE ðKÞ

TNM ðKÞ

GdAlO3


29.1  0.2

6.1

3.87

e

DyCrO3

565  14

10

2.16

e

DyFeO3

582  14

10

4.5

635

TmVO3

210

2.0

15?

106

TmCrO3

82

0.8

?

124

h-TmMnO3 4b site

311.8  0.2

3.29  0.01

e

z100

o-TmMnO3

Very small

e

?

Tc ¼ 31, TN ¼ 41

TmFeO3

9

0.09

?

630

YbAlO3

263  5

2.49  0.05

0.8

e

h-YbMnO3 4b site

165.9  0.2

1.61

e

89

2a site

109.2  0.1

1.06

5

89

The hyperfine fields, the magnetic moments, and the magnetic ordering temperatures are shown for the rare earth sublattice and the magnetic ordering temperatures for the 3d sublattice.

184 Handbook of Magnetic Materials

FIGURE 14 Inverse temperature dependence of the logarithm of the spin-lattice relaxation frequencies determined from DyPO4 by Forester and Ferrando (1976a) and DyVO4 by Gorobchenko et al. (1973). This figure has been earlier published by Forester, D.W., Ferrando, W.A., 1976a. Phys. Rev. B 13, 3991.

169

Tm Mo¨ssbauer measurements were reported on TmVO4 by Triplett et al. (1974), TmPO4 by Hodges (1983), and TmAsO4 by Hodges et al. (1982). Furthermore, 170Yb Mo¨ssbauer measurements in TmMO4 and in YbMO4 (M ¼ P, V and As) were performed by Hodges (1983) and Hodges et al. (1982). The 170Yb Mo¨ssbauer experiments on the TmMO4 compounds were obtained from a source experiment on neutron-activated TmMO4 using an YbB6 absorber. The ytterbium is present as 170Yb (Ie ¼ 2, Ig ¼ 0, E ¼ 84.4 keV) at very low concentrations so that the bulk structural JahneTeller transition temperatures in TmAsO4 (TD ¼ 6.0K) and TmVO4 (TD ¼ 2.15K) are expected. The measured hyperfine parameters were used to provide information concerning the electronic properties of the rare earth ions. A set of crystalline electric field (CEF) parameters is obtained for Tm3þ in TmPO4 and in TmAsO4 in the tetragonal phase above the JahneTeller transition temperature (TD ¼ 6.0K) as shown in Fig. 15 by Hodges et al. (1982). Previously proposed CEF parameters for Tm3þ in TmVO4 (TD ¼ 2.15K) as determined by Knoll (1971) and later refined by Wortman et al. (1974) were discussed. TmAsO4 and TmVO4 have a doublet ground state with approximately the same ground-state wave functions. On the other hand, TmPO4 has a singlet ground state and consequently no JahneTeller transition. The measured ground-state properties of Yb3þ diluted into the two TmMO4 can be generated by sets of CEF parameters close to those obtained for Tm3þ in the corresponding matrix.

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FIGURE 15 Thermal variation of a ¼ DEQ, the electric quadrupole-splitting for Tm3þ in TmAsO4 from 10 to 295K (A) and from 1.4 to 24K (B). The solid curve is calculated for the tetragonal phase using the crystal field parameters given in Table 8. A clear difference exists between the extrapolation of these calculated values to low temperatures (dashed curve) and the measured values of the quadrupole-splitting in the orthorhombic phase. This figure has been earlier published by Hodges, J.A., Imbert, P., Je´hanno, G., 1982. J. Phys. 43, 1249.

Spinespin relaxation rates were measured in the paramagnetic state of the YbMO4 compounds and the origin of this relaxation rate in YbVO4 was discussed by Hodges et al. (1982). Magnetic ordering is observed in YbVO4 below about 0.15K and also in YbPO4 in a comparable temperature zone. The CEF parameters (in K) relevant to Tm3þ and Yb3þ in TmPO4, TmVO4 and TmAsO4 are given in Table 8. For Tm3þ the parameters concern the tetragonal phase (above the JahneTeller transition temperature, if it exists). For Yb3þ the tetragonal CEF parameters also account for the ground-state magnetic hyperfine parameters in the orthorhombic phase. It can be noticed that in TmAsO4 and TmVO4 Tm3þ has approximately the same ground-state wave function. However, the ground-state wave functions for Yb3þ are different in these two matrices. As mentioned by Hodges et al. (1982) this behavior is compatible with one common set of CEF parameters for both Tm3þ and Yb3þ in TmAsO4 and one common set for both Tm3þ and Yb3þ in TmVO4. Rare earth chromates belong to tetragonal compounds (I41/amd) with the general formula RMO4 with M ¼ V, P, As, and Cr, as shown by Buisson et al. (1964). These RCrO4 oxides allow us to study the effect of the magnetic interaction between the S ¼ 1/2 Cr5þ sublattice and the rare earth sublattice on the overall magnetic properties. The obtained results can be compared with

186 Handbook of Magnetic Materials

3þ n TABLE 8 The Crystal Field Parameters Am and n hr i in K Relevant to Tm 3þ Yb in TmMO4With M ¼ As, P, and V

Tm3þ in TmAsO4 Yb

3þ

A46 hr 6 i


53

22.6

980


60

86

5.9

924


63

15.6

60.7

1253


55.1


50.7

190

in TmPO4

Similar

in TmVO4


125.7

Tm Yb

A06 hr 6 i

in TmPO4

3þ

3þ

A44 hr 4 i

Similar

Tm Yb

A04 hr 4 i

in TmAsO4

3þ

3þ

A02 hr 2 i

in TmVO4

Similar with

A02 hr 2 i

¼
208

3þ

Tm

Ground-State Wave Functions

3þ

in TmAsO4

þ0.88j5i
0.44j1iþ0.19jH3i

3þ

in TmPO4


0.32jþ4iþ0.90j0i
0.32j
4i

3þ

in TmVO4

w0.89j5i
0.42j1iþ0.19jH3i

Tm Tm

Tm

But probably slightly richer in j5i

Yb3þ

Ground-State g Values

Ground-State Wave Functions

Yb3þ in TmAsO4

gz ¼ 0.4 and gt ¼ 3.60

aj5/2i þbjH3/2i

Yb3þ in TmPO4

gz ¼ 1.2 and gt ¼ 3.19

Probably aj5/2iþbjH3/2i

Yb3þ in TmVO4

gz ¼ 6.45 and gt ¼ 0.6

aj7/2i þbjH1/2i

For Tm3þ the parameters concern the tetragonal phase above the JahneTeller transition temperature if it exists. The Tm3þ ground-state wave functions and the Yb3þ experimental ground-state g values and wave functions are shown in TmMO4. This table has been earlier published by Hodges, J.A., 1983. J. Phys. 44 833.

other RXO4 compounds, of which M is not magnetically ordered. We focus our study on the derivates where R ¼ Nd
Lu showing the same tetragonal structural type, so that only a different electronic constitution of the rare earth ion could produce a change in the properties of these oxides. Formerly, with neutron diffraction, it was found that ferromagnetic TbCrO4, as shown by Buisson et al. (1976) undergoes at T ¼ 48K a second-order phase transition from a tetragonal to an orthorhombic symmetry. Since only GdCrO4, TmCrO4, and YbCrO4 have been studied with rare earth Mo¨ssbauer spectroscopy, this paragraph will be restricted to these compounds, respectively published by Jimenez-Melero et al. (2006), Jimenez et al. (2004a, 2004b). The results on TmCrO4 by Jimenez et al. (2004b) are further supplied with later-performed measurements. Study of the crystal structure of GdCrO4 has been performed by the roomtemperature X-ray diffraction data. The derived structural parameters have

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

187

been employed as initial values in the refinement of the neutron diffraction data between 2 and 300K. In this way, accurate oxygen coordinates have been obtained which, in turn, have yielded appropriate bond distances and angles for the coordination polyhedron of both Cr5þ and Gd3þ. Furthermore, bulk magnetic measurements indicated the presence of a ferromagnetic order in this compound below Tc ¼ 22K. The analysis of the neutron diffraction pattern at lower temperatures has allowed us to determine the established magnetic structure. The ordered magnetic moment of the ion is located along the crystallographic c axis, while that associated with the Gd3þ ion forms an angle of z 24 with the mentioned axis. Moreover, specific heat measurements reveal the presence of a second weaker magnetic transition at lower temperatures. Such a transition has been confirmed by subsequent 155Gd Mo¨ssbauer spectroscopy experiments as shown in Fig. 16. The Mo¨ssbauer spectra indicate that only 20% of Gd orders

1240000 1220000 1000000

Intensity (counts)

960000 840000

800000 480000 460000 1260000

1200000 -4

-2

0

2

4

FIGURE 16 155Gd Mo¨ssbauer spectra of GdCrO4 measured between 4.2 and 36K. Two subspectra were observed with an intensity ratio of 80% and 20%. The 80% subspectrum show magnetic order below 8K, whereas the 20% subspectrum show magnetic order below 22K. This figure has been earlier published by Jimenez-Melero, E., Gubbens, P.C.M., Steenvoorden, M.P., Sakarya, S., Goosens, A., Dalmas de Re´otier, P., Yaouanc, A., Rodrı´guez-Carvajal, J., Beuneu, B., Isasi, J., Sa´ez-Puche, R., Zimmermann, U., Martı´nez, J.L., 2006. J. Phys. Condens. Matter 18, 7893.

188 Handbook of Magnetic Materials

magnetically at 22K, while the remaining 80% do not show any magnetic order down to around 10K as shown for the hyperfine fields in Fig. 17. This 80% Gd site may be attributed to a low-temperature orthorhombic phase. The following mSR results indicate that the whole Cr5þ sublattice presents a magnetic order at the mentioned value of the Curie temperature. Besides that, short-range GdeCr magnetic correlations have been clearly observed in the paramagnetic state. Bearing all these facts in mind, we can conclude that the Cr5þ ion presents a ferromagnetic order at the temperature of 22K, and induces the magnetic order in the 20% Gd3þ sublattice via a relatively large GdeCr exchange field. Since the ordered moments of both sublattices are not fully collinear (as expected due to the isotropic nature of the Gd3þ ion), a small anisotropic contribution should be present in the GdeCr magnetic exchange interactions. The magnetic order of the remaining 80% Gd3þ sublattice, resulting from an orthorhombic distortion, takes place at a lower temperature of around 10K. However, further experimental evidence seems to be necessary to shed more light on the nature of this second magnetic transition. The experimental observation of the magnetic order in the rare earth sublattice being induced by the Cr5þ order can help us to better understand the magnetic properties in the remaining RCrO4. Since it was found that the Tm-ordered magnetic moment in TmCrO4 was axial, but uncommonly strongly reduced with respect to the free ion value of 7 mB, it is of interest to study TmCrO4 further with magnetization, specific heat, neutron diffraction, 169Tm Mo¨ssbauer spectroscopy, inelastic neutron 8

GdCrO4

7 6

Heff (T)

5 4 3 2 1 0 -1

0

5

10

15

20

T (K)

25

30

35

40

FIGURE 17 Temperature dependence of the magnetic hyperfine field of GdCrO4. Whereas the 20% subspectrum of the tetragonal phase 1 shows an hyperfine field over the whole temperature range below Tc ¼ 23K, no clear hyperfine field has been found for the 80% subspectrum of the orthorhombic phase 2 between 8 and 22K. This figure has been earlier published by JimenezMelero, E., Gubbens, P.C.M., Steenvoorden, M.P., Sakarya, S., Goosens, A., Dalmas de Re´otier, P., Yaouanc, A., Rodrı´guez-Carvajal, J., Beuneu, B., Isasi, J., Sa´ez-Puche, R., Zimmermann, U., Martı´nez, J.L., 2006. J. Phys. Condens. Matter 18, 7893.

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189

scattering, and muon spin relaxation (mSR) to understand the magnetic behavior of the Tm3þ and Cr5þ sublattices and their mutual magnetic interplay. Temperature dependence of the magnetization of TmCrO4, measured in external fields of 50 and 250 Gauss, shows clearly hysteresis in the magnetization below Tc ¼ 18.75K. The magnetic specific heat of TmCrO4 at zero field shows clearly quite a broad peak with a maximum at 15.3K. In fact the magnetic transition starts at around 20K. From T ¼ 20 to 15.3K, magnetic order starts to build up. The experimental change in the entropy due to the transition is 5.73 J/mol K. The theoretical value for Cr5þ (S ¼ 1/2) is 5.76 J/mol K, which confirms that the Cr5þ is fully ordered. However, there seems to be no contribution coming from the Tm3þ sublattice. It may be due to the fact that Tm3þ is not yet magnetically ordered in the sample around the transition. A study on the magnetic structure of TmCrO4 at T ¼ 2K determined with neutron diffraction shows two ferromagnetic Tm and Cr sublattices both parallel to the c axis with magnetic moments of 0.83  0.04 mB and 3.64  0.06 mB for the Cr5þ and Tm3þ, respectively. Moreover, the neutron diffraction measurements show that above z 40K the structure of the sample is tetragonal, whereas below z 40K, it is mainly orthorhombic (about 80%). This means that the Tm magnetic moment is relatively very small to the free ion value of 7 mB. The total angular momentum might partly quenched due to the competition with the Cr5þ sublattice. On the other hand, high-order crystal field terms might also influence the ground state, leading to such a smaller magnetic moment. In Fig. 18 the temperature dependence of some of the 169Tm Mo¨ssbauer spectra measured from 4.2 up to 300K are shown. The spectrum at T ¼ 4.2K consists of two magnetic sextets with a proportion of four to one and a nonmagnetic doublet. As shown, the spectrum at T ¼ 28K had to be clearly analyzed with two quadrupole doublets. This is also the case for the spectra up to 300K. Thus, the Mo¨ssbauer results do not confirm directly the picture found by neutron diffraction that above 40K the structure of the sample is only tetragonal, whereas below 40K, it is mainly orthorhombic (75%). In Fig. 19 the temperature dependences of the 169Tm hyperfine fields of the two structures are shown together with the percentage of magnetic ordering. This behavior indicates also the first-order character of the magnetic transition of the Tm sublattice. In Fig. 20 at 20K, some inelastic bumps were observed, which are not any more present at 70K. This behavior gives support for the interpretation that these bumps belong to the crystal field of the orthorhombic structure of TmCrO4 below 40K. In Fig. 21 the temperature dependence of the relaxation rate lZ is shown. Three different anomalies are shown. The first one below T ¼ 10K is indicative for the first-order magnetic transition of the Tm sublattice. The second one at Tc ¼ 18.75K represents the second-order magnetic transition of the Cr sublattice. Finally, the drop of lZ around 30e40K might be indicative either for the structure transition from orthorhombic to tetragonal and subsequently a crystal field change. This drop might also

190 Handbook of Magnetic Materials

1.000 0.985 1.000 0.986 1.000 0.986 1.000 0.993

-400

-200

0

200

400

FIGURE 18 Several 169Tm Mo¨ssbauer spectra measured in the magnetically ordered phase below T ¼ 17.5K (Tc ¼ 18.75K). As shown at T ¼ 4.2K the spectra consist of a nonmagnetic and a magnetic part. The last one is decreasing with increasing temperature. At T ¼ 4.2K the magnetic part of the Mo¨ssbauer spectrum has been analyzed with two subspectra in an intensity ratio of 75% and 25%. Since between 10 and 17.5K the two magnetic spectra are not discernable, only one averaged magnetic spectrum has been analyzed. As shown at T ¼ 18K, above 17.5K no magnetic splitting was observed. The nonmagnetic part of the Mo¨ssbauer spectra has been analyzed with two electric quadrupole doublets with intensity ratios of 75% and 25%. This figure was published by Jimenez E., Gubbens, P.C.M., Sakarya, S., Stewart, G.A., Dalmas de Re´otier, P., Yaouanc, A., Isasi, J., Sa´ez-Puche, R., Zimmermann, U., 2004b. J. Magn. Magn. Mater. 272e276, 568.

indicate the energy peak of about 30K, representing a crystal field transition, observed in the inelastic neutron scattering spectrum. One can conclude that from the magnetization, mSR and 169Tm Mo¨ssbauer measurements of TmCrO4 show that the Cr sublattice orders with a secondorder magnetic phase transition at Tc ¼ 18.75K and the Tm sublattice orders gradually at lower temperatures with a first-order magnetic phase transition, which agrees with the hysteresis observed in the bulk magnetization measurements. This means that in the Tm sublattice a first-order magnetic phase transition is observed and the magnetism is induced in the Tm sublattice by the TmeCr magnetic interaction. Therefore, one can expect that the crystal field scheme of Tm has a nonmagnetic singlet ground state. We estimate an eigenfunction aj
4iþbj0i
cjþ4i with c ¼ a as shown in Table 9. Under an

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3 100

500

Hyperfine Field [T]

191

400 300 [%] 200 100 0

0 0

5

10

15

20

Counts [Arbitrary units]

FIGURE 19 Temperature dependence of the 169Tm hyperfine field below Tc ¼ 18.75K. At T ¼ 17.8 and 18.1, no hyperfine field splitting has been observed. Above T ¼ 10K, only an averaged hyperfine field is observed. The lines are drawn to guide the eyes. The crosses indicate the percentage of the intensity ratio of the nonmagnetic part of the Mo¨ssbauer spectrum. This figure is not yet published.

70 K

20 K

−10

−5

0

5

10

15

Energy Transfer [meV] FIGURE 20 Inelastic neutron scattering measurements of TmCrO4 taken at T ¼ 20 and 70K. It had to be noted that the bump at an energy of about 2e3 meV (z30K), observed at T ¼ 20K, is not present at T ¼ 70K. Therefore, these bumps have to be ascribed to the orthorhombic and not to the tetragonal structure. The higher points are an enlargement of the lower ones. This figure is not yet published.

192 Handbook of Magnetic Materials

( s )

TmCrO4

Z

zero-field

FIGURE 21 Temperature dependence of the relaxation rate lZ. At Tc ¼ 18.75K the asymmetry has a loss of 2/3 of its signal and the lZ a sharp maximum, indicating the magnetic ordering temperature of the Cr sublattice in TmCrO4. The anomaly below 10K is indicative for a first-order magnetic transition of the Tm sublattice in TmCrO4. The drop around T around 30e40K can be related with a structure transition from orthorhombic to tetragonal and subsequently a crystal field change. An alternative explanation is the coincidence with an energy of z30K in the inelastic neutron scattering spectrum, as shown in Fig. 20. This figure is not yet published.

influence of magnetic exchange, this eigenfunction will purify in the direction of a pure j
4i, as shown above in Table 9. This provisional approach fits nicely for the orthorhombic structure (site I), but much less for the tetragonal structure (site II). Likely, higher energy inelastic neutron scattering is needed to determine a full crystal field diagram both in the tetragonal and orthorhombic phases than as shown in Fig. 20. In contrast with neutron diffraction

TABLE 9 The Values of the 169Tm Hyperfine Fields of TmCrO4 in Tesla, Their Corresponding Magnetic Moments in mB and the 4f-Contribution of the 169 Tm Electric Quadrupole-Splitting Measured at T ¼ 4.2K Site I (75%)

Site II (25%)

Free Ion Value

Pure j
4iValue

Heff(T)

412

367

662.5

441.7

Magnetic moment (mB)

4.4

3.9

7.0

4.67

1 eQV 4f ðmm=sÞ zz 2

þ12.0


7.1

þ176

þ16

The values of the last parameter are determined by subtracting the lattice contribution (determined from 155Gd Mo¨ssbauer Spectroscopy on GdCrO4) from the total measured electric quadrupole4f is proportional to 3J 2
JðJ þ 1Þ, this value is easy to calculate for a pure splitting. Since ð1=2ÞeQVzz z j
4i state.

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at high temperatures above 40K, TmCrO4 shows still two electric quadrupolesplittings. That would mean that there still two crystal structures are present. No explanation for this behavior has been yet given. In contrast with TmVO4 and TmAsO4, which show JahneTeller transitions due to doublets as ground states, TmPO4 and TmCrO4 do not show this effect due to a singlet ground state. On the other hand, the occurrence of two slightly different crystallographic Tm sites even at T ¼ 28K is indicative of a JahneTeller transition in TmCrO4. Since we found the same type of results in GdCrO4 compound in which Gd has no crystal field, one can argue that this JahneTeller behavior can be ascribed to the Cr5þ sublattice. The 170Yb Mo¨ssbauer spectra on YbCrO4 were measured between 4.2 and 30K by Jimenez et al. (2004a). Some selected spectra are represented in Fig. 22. An extra absorption has been detected near the center of the spectrum coming from an impurity phase, the Yb2O3 oxide, whose known spectrum has been included in Fig. 22 as a dashed line. Its substantially high Debye temperature, when compared with the one corresponding to the YbCrO4 compound, can probably be the cause for its seemingly large Mo¨ssbauer percentage, which amounts to 35%  10%. The YbCrO4 Mo¨ssbauer spectra comprise only a quadrupolar contribution between 25 and 30K, with a quadrupole-splitting of 1.75 mm/s. Below 25K, a mixed magneticequadrupolar hyperfine pattern is clearly present. In local tetragonal symmetry the principal axis of the electric field gradient tensor is along the fourfold symmetry axis. The spectrum at 4.2K has a hyperfine field of 55T, corresponding to a Yb magnetic moment of 0.55 mB, which is significantly reduced with respect to the free ion value of gJ ¼ 4 mB. Moreover, the hyperfine field is perpendicular to the c axis. In Yb compounds, an isolated Kramers is usually the ground state of Yb3þ ion. Such doublet can be described by an effective spin S ¼ 1=2 and a spectroscopic g-tensor, which, in the case of axial symmetry, has two components gz and gt. In local tetragonal symmetry, these crystal field eigenfunctions of the doublets are mixtures. From the measured values of the quadrupolesplitting and gt ¼ 1.1 at 4.2K, the approximate wave function for the ground state is 0.87j7/2iþ0.49j
1/2i. This wave-function results in gz ¼ 5.8 These values are relatively close to those obtained for the Yb3þ ion in YbVO4: gz ¼ 6.46 and gt ¼ 0.77, and in YVO4: gz ¼ 6.08 and gt ¼ 0.85 as shown by Bowden (1998). Surprisingly, in the magnetically ordered state, the Yb moments do not lie along the easy c axis but in the ab plane, which is the hard magnetic plane. Therefore, the YbeCr exchange imposes the direction of the Yb magnetic moments with the Cr5þ ion having an isotropic g-tensor. The saturated moment of the Cr5þ ion is 2S ¼ 1 mB. The magnetic moment of the Yb3þ ion depends on the orientation of the magnetic field. For a polycrystalline sample, the saturated magnetization of an extremely anisotropic doublet (i.e., gt/gz  1) averaged over all orientations is ð1=4Þgz mB . From this result for Yb in YbCrO4 (gt/gz z 0.2), it is 1.45 mB. Then, the remanence

194 Handbook of Magnetic Materials FIGURE 22 170Yb Mo¨ssbauer spectra of the YbCrO4 oxide. The dashed line represents the spectrum of the impurity phase Yb2 O3. This figure is published by Jimenez E., Bonville, P., Hodges, J.A., Gubbens, P.C.M., Isasi, J., Sa´ez-Puche R., 2004a. J. Magn. Magn. Mater. 272e276, 571.

moment at 2K is of 0.55 mB per formula unit. This gives a ferrimagnetic structure with the YbeCr coupling being antiferromagnetic. In contrast with GdCrO4 and TmCrO4, no structure change behavior has been found in YbCrO4 as far as known. In this respect, YbCrO4 resembles ErCrO4.

4.5 RBa2Cu3O7 Compounds In this paragraph the magnetic influence of rare earth on the high-temperature orthorhombic SC compounds RBa2Cu3O7 (Tc ¼ 90K) as studied with rare earth Mo¨ssbauer spectroscopy will be discussed. A comparison will be made with the tetragonal nonsuperconducting (NSC) but magnetic compounds RBa2Cu3O6 (TN z 400K). Since the measuring technique of rare earth Mo¨ssbauer spectroscopy is not sensitive for superconductive parameters, only direct or indirect magnetic parameters of RBa2Cu3O7 compounds can be determined. The compound PrBa2Cu3O7
y has a rather special place in the yttrium and rare earth-substituted RBa2Cu3O7
y compounds, because this compound is not a superconductor as shown by Soderholm et al. (1987), but a semiconductor and the CueO planes order antiferromagnetically near 280K. This is in contrast to all other compounds of this stoichiometry, for which no such ordering is observed. Moreover, it has by far the highest transition temperature for the antiferromagnetic ordering of the rare earth sublattice (17K), much

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

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larger than the corresponding Gd compound. Originally, the suppression of the SC state in PrBa2Cu3O7
y was attributed to the filling of the SC holes due to a tetravalent or mixed valency of Pr as was shown by X-ray absorption spectroscopic studies by Dalichaouch et al. (1988). However, later studies excluded the presence of any Pr4þ in PrBa2Cu3O7
y. The hybridization of the Pr-4f levels with the CueO plane bands was instead postulated as a cause for the suppression of the SC state, but questions remained because some of these studies did show some features that could not be explained on the basis of solely trivalent Pr as argued by Lytle et al. (1990). Since the isomer shift and the hyperfine field in Mo¨ssbauer spectroscopy are the most direct ways of measuring the valency and the ground state magnetic moment of Pr in PrBa2Cu3O7
y, 141Pr Mo¨ssbauer spectra of PrBa2Cu3O7
y were measured at 4.2 and 25K. In addition, the formally tetravalent PrO2, the intermediate valent Pr6O11 and the trivalent Pr2O3 compounds were measured at 25K, as shown by Moolenaar et al. (1994, 1996). These spectra are shown in Fig. 23. In Fig. 24, the isomer shift is plotted against the valency. One can see that the isomer shift of PrBa2Cu3O7
y lies in between that of Pr2O3, Pr6O11, and PrO2. However, because of reasons of entropy and because of the fact that at ambient oxygen pressure only Pr6O11 is stable, it is almost inevitable that PrO2 is to some extent oxygen deficient as shown Section 4.2. Because, due to this oxygen deficiency, the isomer shift is presumably too low, we therefore compare the isomer shift of PrBa2Cu3O7y at 25K with that of Pr2O3 and Pr6O11. Assuming a linear relationship between the valency and the isomer shift, for PrBa2Cu3O7
y, a valency for the isomer shift was found of 3.4  0.1. However, the Mo¨ssbauer peak of PrBa2Cu3O7y is much broadened with respect to that of Pr2O3. Such a broadening can probably not be attributed to an electric field gradient, for its influence is negligible in 141 Pr Mo¨ssbauer spectroscopy. The broadening of the peak must then be ascribed to a distribution in isomer shifts. Furthermore, Fig. 23 reveals that the spectrum at 4.2K is magnetically split. Analysis of this spectrum gives a hyperfine field of 29(3)T. This is only 10% of the free-ion value of 326T and less than that of PrO2, indicating for this sample a small magnetic moment of 0.32  0.03 mB only, which is about half the value found by Li et al. (1989). However, the resonance absorption effect we observe at 4.2K is less than that at 25K, in sharp contrast to PrO2. The analysis furthermore required a rather large peak width. Therefore, we suggest that in Fig. 23 essentially the inner part of a relaxation spectrum is shown. This means that the moment of the Pr ion is fluctuating on the characteristic time scale of 141 Pr Mo¨ssbauer spectroscopy. Another explanation would be that the spectrum is composed of two subspectra, one from a tetravalent and magnetically ordered part, and one from a trivalent part which has only a small induced moment. The tetravalent part must then be hidden in the background. Then it can exclude the possibility of a single hyperfine field, that is, there must be a wide distribution in hyperfine fields. However, such a distribution is in

196 Handbook of Magnetic Materials

FIGURE 23 141Pr Mo¨ssbauer spectra of Pr2O3, Pr6O11 and PrO2 at 25K, and PrBa2Cu3O7
y at 4.2 and 25K. This figure is published by Moolenaar, A.A., Gubbens, P.C.M., van Loef, J.J., Menken, M.J.V., Menovsky, A.A., 1994. Hyperfine Interact. 93, 1717.

contradiction with the results of Li et al. (1989). Concluding, 141Pr Mo¨ssbauer spectroscopy on PrBa2Cu3O7
y shows clearly an intermediate valency and at low temperature ( 0.6 the system is no longer SC. This behavior is probably due to the influence of hybridization between the Pr3þ orbitals and those of the Cu subsystem as shown above. Mo¨ssbauer

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spectra in the fully substituted (NSC) PrBa2Cu3O7 are shown in Fig. 26. They can be interpreted with a single spectrum at each temperature. The thermal variation of the derived parameters are also shown in Fig. 26. At all temperatures, the Yb3þ ion experiences a molecular field of
0.2T coming from the ordered Cu(2) sublattice (TN z 300K), and below 16K, they experience a molecular field coming from the ordered Pr3þ sublattice. This relaxation is driven by the coupling between the Yb3þ and the collective excitations of the Pr3þ sublattice. Below 16K, the Yb3þ relaxation rate drops sharply and it follows a thermal excitation law with a characteristic energy of 54K. This energy scale is linked to that of the separation between the lowest crystal field levels of the Pr3þ ion. At intermediate x levels, inhomogeneous local behavior is observed and is attributed to the influence of statistical variations in the Pr3þ/Y3þ occupancies. The results of this study agree with those shown above from the 141Pr Mo¨ssbauer spectroscopy on PrBa2Cu3O7
y. The temperature dependence of the isomer shift and the recoil-free fraction of the 21.6-keV Mo¨ssbauer resonance of 151Eu is studied in the high-Tc

FIGURE 26 Mo¨ssbauer absorption on 170Yb substituted into PrBa2Cu3O7 (left). The line shapes are governed by the size of the static molecular field acting on the Yb3þ and by the Yb3þ paramagnetic relaxation rate (right top). Below 16K, the relaxation rate follows an excitation law-dependence (right bottom). This figure is earlier published by Hodges, J.A., le Bras, G., Bonville, P., Imbert, P., Je´hanno, G., 1993. Physica C 218, 283.

200 Handbook of Magnetic Materials

superconductor EuBa2Cu3O7
x between 4.2 and 300K by Wortmann et al. (1988a, 1988b), Nagarajan et al. (1988), Eibschu¨tz et al. (1987), and Coey and Donnelly (1987). These four studies show the same results. For Eu, it was found that it has a stable 3þ ion value from its isomer shift. The Mo¨ssbauer Debye temperature is typical for an ionic oxygen surrounding and no anomaly near Tc was found. The electric field gradient tensor Vzz ¼
3.1  5 1017 V/cm2 and the asymmetry parameter h s 0 have been determined by Wortmann et al. (1988a, 1988b). The magnetic ordering of the Gd sublattice in the orthorhombic and tetragonal structure of GdBa2Cu3O7
d have been studied by Mo¨ssbauer spectroscopy using the 86.5-keV gamma resonance of 155Gd by Wortmann et al. (1987), Smit et al. (1987) and Bornemann et al. (1987). Below the Ne´el temperature of TN ¼ 2.4K, the magnetic hyperfine field at the Gd nucleus reflects the increasing local sublattice magnetization extrapolating to a saturation value of Heff(T ¼ 0K) ¼ 31.5T (S ¼ 7/2). The effective magnetic hyperfine field is found to be parallel to the main axis of the electric field gradient tensor [Vzz ¼
(4.6  0.2)  1017 V/cm2] with an asymmetry parameter of h ¼ 0.40  0.05. The observed isomer shift and the value of Heff are typical for trivalent Gd. For both, the orthorhombic and tetragonal structures of GdBa2Cu3O7
d, the same type of values have been found by all three groups. The high-Tc superconductor DyBa2Cu3O7
x as shown Fig. 27 and its oxygen deficient tetragonal NSC counterpart have been studied by 161Dy Mo¨ssbauer spectroscopy down to 0.05K as shown by Hodges et al. (1988). Comparable spectra of DyBa2Cu3O7
x have been published by Wortmann et al. (1989). In Table 10 the results of the fit and the calculations of the parameters are given. The low temperature Dy3þ magnetic properties, which are similar in the two compounds, show the influence of crystal fields and the DyeDy magnetic correlations. The difference between the values of the 4f shell parameters in the SC and NSC samples is much smaller than the widths of the distributions in either sample. The observed Dy3þ sublattice magnetic ordering is chiefly attributed to superexchange with a smaller contribution coming from the dipoleedipole interaction. The ground states show only a modest anisotropy. The Dy3þ paramagnetic fluctuation rates also show the influence of these correlations. Magnetic ordering occurs within the Dy sublattices near 1K. The Dy3þ saturated magnetic moments show fairly large distributions around mean values of 7.0 (SC) and 7.3 mB (NSC). These distributions are attributed to crystal field inhomogeneities linked to the sample microstructure. Both the susceptibility and the Mo¨ssbauer paramagnetic relaxation rate measurements show that DyeDy magnetic correlations are present above the Dy3þ long-range magnetic ordering temperature. However, temperature dependence indicates an Orbach behavior further above the magnetic ordering temperature in agreement with a first excited state of 40K. In the same way as described in Section 3.6.1 for TmBa2Cu3O7
x, Bergold et al. (1990) have determined the crystal field parameters of the

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FIGURE 27 161Dy Mo¨ssbauer absorption in orthorhombic superconducting state DyBa2Cu3O7
x. At 0.05K in the magnetically saturated state, the line fits are obtained in terms of linearly correlated Gaussian distribution in the hyperfine field and in the electric field gradient taking into account the Boltzmann populations of the nuclear levels. At the three other temperatures, which are above the Dy3þ long-range ordering temperature, the line fits are obtained in terms of a paramagnetic relaxation model. This figure is earlier published by Hodges, J.A., Imbert, P., Marimom da Cunha, J.B., Hammann, J., Vincent, E., Sanchez, J.P., 1988. Physica C 156, 143.

tetragonal phase of DyBa2Cu3O7
x. Due to the magnetic character of the CF-split doublets of Dy3þ (Kramers ion with J ¼ 15/2), the spectra exhibit magnetic broadenings, which increase with decreasing temperature. Their 161 Dy Mo¨ssbauer spectra were analyzed in terms of an axial quadrupole interaction and the magnetic relaxation line broadening mechanism described by Wegener (1965). The electric field gradient values were derived as a function of temperature. Although an asymmetric electric field gradient is expected for the orthorhombic phase, inclusion of h as an additional parameter failed to improve the quality of their spectrum fits. Therefore, a crystal field analysis of the quadrupole-splitting was performed only for the data of the tetragonal phase. The results of tetragonal HoBa2Cu3O7
x of Allenspach et al. (1989a) have been used. The determined crystal field

202 Handbook of Magnetic Materials

TABLE 10 161Dy Mo¨ssbauer-Derived Hyperfine Parameters in the Dy3þ Magnetically Saturated Region for Superconducting (SC) and Nonsuperconducting (NSC) DyBa2Cu3O7
x Values Superconducting State

Values Nonsuperconducting State

Heff

415  40

430  25

hjJzjiT¼0

5.3  0.5

5.5  0.3

MT ¼ 0 (mB)

7.0  0.7

7.3  0.4

exp VZZ ð1021 V=m2 Þ

16.2  7.6

18.2  5.1

lat ð1021 V=m2 Þ VZZ


6.0


5.3

4f ð1021 V=m2 Þ Vzz

22.2  7.6

23.5  5.1

h3Jz2


JðJ þ 1ÞiT ¼0

45.6  15.6

48.5  10.5 exp VZZ

4f are obtained from the experimental values The VZZ after correcting for the lattice contribution lat estimated by extrapolating from isomorphous Gd3þ compounds. The Dy3þ 4f shell properties VZZ derived from these hyperfine parameters are also given as shown by Hodges et al. (1988).

parameters are included in Table 11 of where they are seen now to compare more favorably with their counterparts measured via inelastic neutron scattering for the orthorhombic DyBa2Cu3O6.9 phase as determined by Allenspach et al. (1989b). The results of both are roughly in the same range. 166 Er Mo¨ssbauer measurements, down to 0.05K, have been reported for the high-Tc superconductor ErBa2Cu3O7
x (SC) and for its oxygen deficient, tetragonal, NSC counterpart by Hodges et al. (1989). In Fig. 28 several of the 166 Er Mo¨ssbauer spectra in orthorhombic SC state ErBa2Cu3O7
x are shown. A rough estimate for the magnetic transition temperature for the SC and NSC samples is 0.6K. However, magnetic fluctuations are observed around this temperature. From the relaxation broadening (D) of the 166Er Mo¨ssbauer spectra the temperature dependence of the splitting of the ground state doublet around the magnetic ordering has been determined. For the SC compound a decreasing value of D with increasing temperature has been found. On the other hand, for the NSC sample D is roughly constant near the magnetic ordering temperature. No explanation for this behavior has been given. However, magnetic fluctuations are observed around this temperature. In Table 12 the found values for the hyperfine field and the electric quadrupolesplitting are given. Using the relation (Heff/m) y 850T/mB between the hyperfine field and the magnetic moment of the Er3þ ion saturated magnetic moments were found of 4.2  0.1 and 3.7  0.1 mB for the SC and NSC samples, respectively. In both materials the magnetic moments are strongly reduced compared to the free ion value of 9 mB owing to the influence of the

B02 (K)

B22 (K)

B02 (mK)

B24 (mK)

B44 (mK)

B06 (mK)

B26 (mK)

B46 (mK)

B66 (mK)

Tetragonal (NSC)


1.04

0

þ13.1

0


59.2

þ32.1

0

þ1.05

0

Orthorhombic (SC)


1.16


0.36

þ16.3


3.9


77.8

þ22.1


4.85

þ0.604

þ4.85

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

TABLE 11 Comparison of the Determined Crystal Field Parameters Bm n for Nonsuperconducting (NSC) Phase of Tetragonal DyBa2Cu3O7
x by Bergold et al. (1990) With the Parameters of the Superconducting Orthorhombic Phase of DyBa2 Cu3 O7
x as Determined With Inelastic Neutron Scattering by Allenspach et al. (1989b)

203

204 Handbook of Magnetic Materials

FIGURE 28 166Er Mo¨ssbauer spectra in orthorhombic superconducting state ErBa2Cu3O7
x. The line shape in the magnetically saturated Er3þ region at 0.05K was fitted with an effective hyperfine field model. The line shapes at the two other temperatures were fitted with a longitudinal relaxation model (see text). This figure is earlier published by Hodges, J.A., Imbert, P., Marimom da Cunha, Sanchez, J.P., 1989. Physica C 160, 49.

CEF. Inelastic neutron scattering experiments on tetragonal and orthorhombic ErBa2 Cu3O7
x seem to confirm the similarity of the leading CEF parameters in the two compounds as performed by Allenspach et al. (1989a). In addition to the 169Tm Mo¨ssbauer spectroscopy measurements on TmBa2Cu3O7
d as shown in Section 3.6.1, Stewart et al. (1998) performed new Mo¨ssbauer spectroscopy measurements of the temperature-dependent 169 Tm quadrupole-splitting for TmBa2Cu4O8 and an oxygen-depleted TmBa2Cu3O6.64 as a complement to earlier measurements for TmBa2Cu3O7

TABLE 12 Hyperfine Parameters in the Er3þ Magnetically Saturated Region for Superconducting (SC) and Nonsuperconducting (NSC) ErBa2Cu307
x Heff (T)

eVzz (mm/s)

SC

360  5


0.05  10

NSC

315  5

1.35  10

Heff is the Hyperfine field and eVzz the quadrupole interaction.

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205

and TmBa2Cu3O6 by Bergold et al. (1990). The 1-2-3 and 1-2-4 systems are members of a homologous series of compounds. In this series the rare earth layers are essentially unchanged. The measured curves reveal a smooth decrease in maximum DEQ with oxygen depletion of the 1-2-3 type compounds, with the TmBa2Cu4O8 value located approximately midway between the TmBa2Cu3O7 and TmBa2Cu3O6 extremes. In Table 13 the rank-2 parameters A02 and A22 are shown as determined by Stewart et al. (1998). In general A02 and subsequently the overall splitting is decreasing marginally with decreasing d value and is smallest for TmBa2Cu4O8. The 170Yb Mo¨ssbauer results on the YBa2Cu3O7 type of compounds described above can be divided into two groups. The first concerns the intrinsic properties of the Yb3þ ions, while the second concerns the use of the Yb3þ ion as a probe to follow the evolution of the local properties of YBa2Cu3O7 when its SC properties are weakened (or removed). The high critical temperature superconductor YbBa2Cu3O7 has been studied over the range 0.05e95K with 170 Yb Mo¨ssbauer spectroscopy by Hodges et al. (1987). Magnetic ordering is found at TN ¼ 0.35K. The saturated magnetic moments amount 1.7 mB. In YbBa2Cu3O7 above 0.35K, the Yb3þ paramagnetic relaxation rate is driven by the dipoleedipole and super-exchange interactions between the Yb3þ ions. The evolution of the local properties has been followed as the SC state in YBa2Cu3O7 is weakened (and eventually removed) by decreasing the oxygen content as studied by Hodges et al. (1994). When the oxygen level is lowered, the Yb paramagnetic relaxation rate remains low and the transition from the SC state to the state where the Cu(2) are magnetically ordered is accompanied by the appearance of a molecular field on the Yb3þ probe. From a study of this field, it was found that at the x ¼ 6.35 level, there is no well-defined magnetic ordering temperature as shown but rather the fluctuation rate of the correlated Cu(2) magnetic moments decreases progressively as the temperature decreases as shown in Fig. 29. In the SC samples with intermediate oxygen levels for which Tc is 60K and below, it was found that part of the sample continues to show fluctuating correlated magnetic moments.

TABLE 13 A Summary of Rank-2 Crystal Field Parameters of TmBa2Cu3O7
d and TmBa2Cu4O8 Obtained From Theory Fits to Experimental DEQ(169Tm) Data as Determined With the Help of Converted Ho INS Parameters d¼0

d ¼ 0.34

d¼1

TmBa2Cu4O8

A02 ðKÞ

89.6

66.3

27.3

41.8

A22 ðKÞ

60.9

29.3

0.0

96.1

206 Handbook of Magnetic Materials

FIGURE 29 170Yb Mo¨ssbauer absorption substituted into YBa2Cu3O6.35. The evolution is governed by the thermal dependence of the fluctuation rate of the Cu(2)derived molecular field. This figure is earlier published by Hodges, J.A., Bonville, P., Imbert, P., Je´hanno, G., 1994. Hyperfine Interact. 90, 187.

In Table 14 a summery is given of the antiferromagnetic ordering temperatures and magnetic moments of the R3þ ions in SC RBa2Cu3O7 and NSC RBa2Cu3O6 compounds. The PrBa2Cu3O7 compound is NSC due to hybridization of the Pr3þ ion.

4.6 R2BaMO5 Compounds Since the discovery of high-Tc superconductors of the type RBa2Cu3O7
d with R ¼ rare earth, the related R2BaCuO5 have been studied extensively in an effort to gain a better understanding of the magnetic exchange interaction between the Cu and R sublattices. For most R, the Cu sublattice orders antiferromagnetically in the higher temperature range of 15K  TN1  20K (the detection of this transition is often assisted by the induced magnetic behavior of the weakly coupled R sublattice) and the R sublattice orders at a lower temperature, TN2. The orthorhombic structure of R2BaCuO5 (space group Pnma) has two R sites, each with monoclinic point symmetry. A 155Gd Mo¨ssbauer spectroscopy study of Gd2BaCuO5 determines TN2 ¼ 11.75  5K for the Gd sublattice as studied by Strecker et al. (1998). In Fig. 30 and Table 15 the results of the 155Gd Mo¨ssbauer spectroscopy measurements on Gd2BaCuO5 are shown. It is clear that the Gd magnetic moments have a canted direction with angles as given in Table 15. The Beff values are rather close to the saturation value of z 32T at 1.5K. However, they differ

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TABLE 14 A Summary of the Magnetic Ordering Temperatures TN (K) and Magnetic Moments MmB of the Rare Earth in RBa2Cu3O7 and RBa2Cu3O6 Obtained From the Temperature Dependence of the Hyperfine Field Behavior as Determined From Rare Earth Mo¨ssbauer Spectroscopy RBa2Cu3O7

RBa2Cu3O6

R

TN (K)

M (mB)

TN (K)

M (mB)

Pr

16.5

0.32  3

7.7

?

Eu

0.0

0.0

?

?

Gd

2.3

7.0

?

?

Dy

z1.0

7.0

z1.0

7.3

Er

z0.6

4.2

z0.6

3.7

Tm

0.0

0.0

0.0

0.0

Yb

0.35

1.7

?

?

The symbol ? means not measured.

FIGURE 30 Temperature dependence of the magnetic hyperfine field, Beff, at the Gdl and Gd2 sites of Gd2BaCuO5 (insert defines angles a, b and c with respect to principal electric field gradient axes x, y, and z and the crystallographic axes a, b, and c). This figure is earlier published by Strecker, M., Hettkamp, P., Wortmann, G., Stewart, G.A., 1998. J. Magn. Magn. Mater. 177e181, 1095.

208 Handbook of Magnetic Materials

TABLE 15 Hyperfine Interaction Parameters for Gd2BaCuO5 Determined From the Measured 155Gd Mo¨ssbauer Spectra Site Gd1

Site Gd2


3.77  0.08


5.83  0.06

h

0.71  0.07

0.99  0.09

b (deg)

60  2

37  2

a (deg)

45  5

90  5

Beff (at 1.5K) (T)

26.9  0.03

22.3  0.03

21

Vzz (10


2

Vm )

The angles b and a are shown in Fig. 30.

clearly from each other. In contrast with the other R2BaCuO5 compounds no difference in magnetic ordering temperature between the Gd and Cu sublattice has been found. Therefore, Gd2BaCuO5 has been studied with 57Fe Mo¨ssbauer spectroscopy on a 57Fe (1/2%) doped sample and with muon spin relaxation on an undoped sample. CueCu and GdeGd magnetic orders are found to occur simultaneously at TN ¼ 11.8K. The Mo¨ssbauer results suggest a first-order magnetic transition for the Cu sublattice in Gd2BaCuO5 as studied by Stewart et al. (2003). The orthorhombic compounds Tm2BaCoO5 and Tm2BaNiO5 (Harker et al., 2000), Tm2BaCuO5 (Stewart and Gubbens, 1999), and also Tm2Cu2O5 (Stewart and Cadogan, 1993) were studied with 169Tm Mo¨ssbauer spectroscopy to determine the crystal field from the temperature dependence of the quadrupole-splitting. In Fig. 31 the temperature dependence of the measured 169 Tm Mo¨ssbauer spectra of Tm2BaCuO5 (TN ¼ 19K) are shown. Since this type of compounds have the space group Pnma (D2h16), the local symmetry of the two Tm sites is lower and have each a monoclinic point symmetry. This is also the case for Tm2BaCoO5 and Tm2BaNiO5. In Tm2Cu2O5 the point symmetry of the two Tm site stays orthorhombic. No difference has been found for the measured results of the hyperfine field and quadrupole-splitting between the two Tm sites in this compound. Then, the crystal Hamiltonian for the 3H6 ground term of Tm3þ takes the conventional form as shown Eq. (15) in this chapter, where the Om n are Stevens operator equivalents (Stevens, 1952) and the Bm are crystal field parameters. Further details are given in Section 3.2. n Since Gd (L ¼ 0) does not experience any crystal field effects the 155Gd Mo¨ssbauer results for isotructural Gd2BaCuO5 as shown above can be enlisted to describe the three-rank two-crystal field parameters for each site in terms of just one unknown parameter for that site following the same procedure as in Eq. (38) for this Hamiltonian in Eq. (15). For the case of Gd2BaCoO5 and Gd2BaNiO5 the electric field gradient values of the 155Gd Mo¨ssbauer results of

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

209

FIGURE 31 Representative 169 Tm Mo¨ssbauer spectra of Tm2BaCuO5 (TN ¼ 19K) measured between T ¼ 0.3 and 300K. Note that the hyperfine field is a perturbation on the electric quadrupolesplitting. This figure is earlier published by Stewart, G.A., Gubbens, P.C.M., 1999. J. Magn. Magn. Mater. 206, 17.

Strecker et al. (1998) were used to calculate the second-order crystal field parameters in the same way as shown above. By a rotation transformation of the second Tm site over 104.6 about the b axis to place the oxygen atom in the same position as for the first Tm site, both Tm sites can be regarded as equal. This manipulation highlights the similarity of the two local environments and brings the calculated ratios for the two sites into closer agreement with respect to both sign and magnitude. As expected, the rank 6 ratios are more sensitive than the rank 4 ratios to the type of calculation employed. They are observed to decrease in magnitude as the radial dependence is varied over the three models, as mentioned above. Furthermore, experimental data published from optical spectroscopy and inelastic neutron scattering can be used to provide an estimate of the low-lying crystal field levels energy levels. In Fig. 32 the results of the calculations are shown. The insets show both crystal field energy diagrams of the two Tm ground-state multiplets. In Table 16 the results of the used crystal field parameters for the calculation of the temperature dependence of the quadrupolesplitting of the two Tm sites in Tm2BaMO5 (M ¼ Co, Ni and Cu) are shown, which were determined by Harker et al. (2000) and Stewart and Gubbens (1999). The low-lying singlet states of the Tm3þ ions in Tm2BaCuO5 (TN ¼ 19K) are nonmagnetic as shown by Stewart and Gubbens (1999). Hence, the

210 Handbook of Magnetic Materials

FIGURE 32 Quadrupole-splitting, DEQ, as function of the temperature in Tm2BaCuO5. The fitted theoretical curves correspond to the crystal field schemes for sites Tm1 (solid line) and Tm2 (broken line) as shown in the inset. This figure is earlier published by Stewart, G.A., Gubbens, P.C.M., 1999. J. Magn. Magn. Mater. 206, 17.

inducement of net magnetic moments by the magnetically ordered Cu sublattice requires that the molecular field associated with the weak CueTm exchange interaction brings a mixing of these levels. Full crystal field calculations show that at 4.2K only the lowest two levels of the various fitted crystal field schemes are significantly perturbed by a small molecular field. Magnetic moments are induced only along the main z axis of the crystal field (the crystallographic b axis). In other words, the crystal field analysis of the experiment predicts that all the induced Tm-site moments will be directed along the crystallographic b axis. In the absence of a precise knowledge of the crystal field schemes for the two Tm sites, it is useful to apply a two-singletinduced moment model. Then, the averaged moment induced at a Tm site is given by  0  2g2J m2B a2 BM D tanh (41) hmi ¼ D0 2kB T which is practically the same formula as Eq. (40). In this equation, BM is the molecular field acting in the z direction (the crystallographic b axis) proportional to the Cu sublattice magnetization, which follows a mean field Brillouin curve with S ¼ 1=2. Furthermore, a ¼ h0jJZj1i is the JZ coupling parameter between the two singlets and D0 ¼ ½D2 þ ð2gJ mB aBM Þ2 1=2 is the field enhanced energy separation of the two singlets (D is the energy separation in absence of a molecular field). In Fig. 33 curves have been calculated for the parameter combinations D ¼ 19K, a ¼ 5.3 (solid line) and D ¼ 29K, a ¼ 4.5 (dotted line) as derived for the two Tm sites from the CF analysis (Table 16).

TABLE 16 Crystal Field Parameters for the Two Tm Sites of the Pnma Phases of Tm2BaMO5 (M ¼ Co, Ni and Cu) and Orthorhombic Tm2Cu2O5 as Calculated by the Point Charge Model Using the Oxygen-Nearest Neighbors, Together With the Preferred Values for the B0n Parameters Obtained by Grid Searches Tm2BaCoO5

Tm2BaNiO5

Tm2BaCuO5

Tm2Cu2O5

Tm1

Tm2

Tm1

Tm2

Tm

B20 ðKÞ


1.356


0.44


3.846


0.12


1.038


1.869

2.23

r22

1.47

1.31

3.64

0.10

1.14

0.96


5.1

r2
2


0.28

0.59


1.82

1.82


0.70


0.33

e

B40 ðmKÞ


40.8


30.4


45.9


22.4


29.2


52.5

62.8

r42


2.34


3.19


2.28


3.77


2.62


3.95

0.454

r4
2

0.52

1.95

0.52

1.31

0.49

1.06

e

r44


3.82


1.66


4.04


4.07


4.63


5.14


4.54

r4
4

2.94

5.51

3.07

4.15

2.94

4.08

e

B60 ðmKÞ

26.2


59.8


47


49.4

50.9

70.3


266.5

r62


14.36


8.93


15.97


13.28


16.36


12.98


1.205

r6
2

4.22

6.65

4.71

5.38

3.46

4.19

e

r64

23.04

6.94

26.63

24.43

24.50

25.43

8.36

r6
4


21.45


35.55


22.45


31.88


21.9


27.9

e

r66


12.38

3.11


15.28


10.78


16.0


15.49

9.96

r6
6

17.43

19.22

22.85

16.18

16.5

14.1

e

These values were earlier published by Harker, S.J., Stewart, G.A., Gubbens, P.C.M., 2000. J. Alloy. Compd. 307, 70 and Stewart, G.A., Cadogan, J.M., 1993. J. Magn. Magn. Mater. 118, 322.

211

Tm2

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

Tm1

212 Handbook of Magnetic Materials

FIGURE 33 Induced magnetic moments in mB as a function of temperature for the two Tm sites of Tm2BaCuO5 (TN ¼ 19K). The theory curves are for the two-singlet-induced moment models with D ¼ 19K, a ¼ 5.3 (solid line) and D ¼ 29K, a ¼ 4.5 (broken line) and are labeled with the fitted values of the saturation molecular field, BM(sat). This figure is earlier published by Stewart, G.A., Gubbens, P.C.M., 1999. J. Magn. Magn. Mater. 206, 17.

Within the context of this simple model, the only remaining free parameter is BM(sat), the saturation value of BM at T / 0K. For each curve, this parameter was adjusted to give the observed low-temperature value of the local moment. The resultant values of BM(sat) are 0.09 or 0.18T for the site with the smaller moment and 0.19 or 0.40T for the site with the larger moment. These values are included with their corresponding curves in Fig. 33. For Tm2BaCuO5 the magnetic behavior of the two Tm sites are distinguishable with saturation moments of 0.503  0.006 mB and 0.23  0.01 mB along the b axis. This is in contrast with the result in Tm2Cu2O5, where the BM(sat) ¼ 1.65T for both Tm sites and the undistinguishable magnetic moment 3.90  0.08 mB. The antiferromagnetic ordering temperatures for Tm2BaCoO5 and Tm2BaNiO5 are 3.5 and 4.85K, respectively, the order being induced by the transition metal as found by Harker et al. (2000). For Tm2BaNiO5 an additional first-order transition is observed at T  1.4K, which is identified with the independent magnetic order of the Tm sublattice. The indirect coupling between NieOeNi chains of the Immm-phase nickelates for Gd2BaNiO5 and Tm2BaNiO5 results in an interesting magnetic behavior as studied by Harker et al. (1996). In this work, 155Gd Mo¨ssbauer spectroscopy data for Gd2BaNiO5 (TN ¼ 55K) are interpreted in terms of a constant electric field gradient tensor and a temperature-dependent magnetic hyperfine field with a saturation value of 24.5T. The sensitivity of the spectra to the projection of the hyperfine field enables a magnetic feature at T ¼ 24K to be identified with magnetic reorientation from the a axis to the b axis. Mo¨ssbauer spectroscopy measurements are presented for 169Tm nuclei in the

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

213

Immm phase of Tm2BaNiO5 by Stewart et al. (2000). The temperaturedependent hyperfine interactions are consistent with a Tm moment induced by the antiferromagnetically ordered Ni sublattice, and a Ne´el temperature of TN ¼ 14.5K. A crystal field parameter determination of Tm2BaNiO5 with the 155 Gd Mo¨ssbauer results of Gd2BaNiO5 has been performed on the temperature dependence of the quadrupole-splitting of Tm2BaNiO5 using the method ascribed above by Stewart et al. (2000). The most favorable solution of crystal field parameters are B02 ¼ 3:38K, r2 þ 2 ¼
0.4, B04 ¼ 24:2 mK and B06 ¼
127 mK with D10 ¼ 23K and D20 ¼ 32.7K as distances from the ground state to the first- and second excited states, respectively. Moreover, Harker et al. (1996) has studied the temperature dependence below TN ¼ 14.5K as described in Eq. (41) using an S ¼ 1 Brillouin function. The results are D ¼ 18  1K, a ¼ 5.7  3 and BM(T ¼ 0K) ¼ 1.4  0.1K. In Fig. 34 170Yb Mo¨ssbauer spectra of Yb2BaCuO5 (TN ¼ 15K) measured at 2.35K and 0.05K by Hodges and Sanchez (1990) are shown. Since energy levels of the crystal field of Yb3þ consist of doublets, the ground-state doublet will be split by the magnetic molecular field of the Cu2þ sublattice below TN ¼ 15K. As shown in Table 17, two hyperfine fields of the two Yb sites are observed with magnetic moments of 2.2 and 1.1 mB. From the temperature dependence of the hyperfine field in the same way as in Eq. (41), the groundstate splitting D together with the molecular/dipole field of the doublets can be determined as shown in Table 17. 151 Eu Mo¨ssbauer measurements on Eu2BaNiO5, where the Ni2þ form well separated S ¼ 1 antiferromagnetic chains, were measured by Hodges and Bonville (2004). The appearance of magnetic moments in this compound evidences a crossed bootstrap polarization since neither the Eu3þ nor the Ni2þ are carrying the magnetic moments. However, it is confirmed that the Eu3þ ion carries a magnetic moment below 30K implying that Ni2þ (S ¼ 1) ordered moments also appear below this temperature. Both show distributions that can be linked to the role of defects in the nickel chains. The measurements also FIGURE 34 170Yb Mo¨ssbauer absorption in Yb2BaCuO5 at 2.35K (top) and 0.05K (bottom). The line fits are obtained in terms of two equally intense quadrupole plus magnetic hyperfine subspectra. This figure is earlier published by Hodges, J.A., Sanchez, J.P., 1990. J. Magn. Magn. Mater. 92, 201.

DEQ (mm/s)

h

Heff (T)

q (degrees)

m (mB)

D (K)

HCu
Yb (T)

Site 1

1.95  0.06

0.28  0.04

216  6

30  5

2.2  0.1

3.6  0.1

1.2  0.1

Site 2


1.33  0.04

0.78  0.10

113  4

75  5

1.1  0.1

3.2  0.1

The sizes of the derived 4f shell-saturated magnetic moments (m), the ground doublet splitting (D), and the molecular field acting on the Yb given.

3þ

2.2  0.1 Cu
Yb

4f shell (H

) are also

214 Handbook of Magnetic Materials

TABLE 17 For Each of the Two Sites in Yb2BaCuO5: 170Yb3þ Mo¨ssbauer Values for the Electric Quadrupole Interaction Above 15K, the Values of the Low-Temperature Saturated Effective Hyperfine Field (Heff) and Its Polar Direction (q)

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

215

provide the thermal dependence of the relative size of the Ni2þ moments. From the measurements, it appears the Eu magnetic moments are of the order of a fraction of a mB. The peak in the distribution of the exchange field acting on the Eu3þ occurs at 7.4T and the magnetic moment at 0.17 mB. In Table 18 the antiferromagnetic ordering temperatures and the magnetic moments are given for R2BaMO5 with R ¼ Gd, Tm and Yb and M ¼ Co, Ni and Cu for Pnma and Immm structures, and Tm2Cu2O5. Most of the compounds are magnetic induced systems from the Co, Ni, and Cu sublattice. Only in the Immm phase of Gd2BaNiO5 likely the Gd sublattice is magnetically ordered by itself. Moreover, for the Pnma phase of Tm2BaNiO5, it was found that the Tm sublattice orders magnetically below 1.4K.

4.7 R2M2O7 Compounds The pyrochlore lattice compounds R2M2O7 (space group Fd3m), where R is a rare earth and M is a transition or sp metal, are very interesting compounds. The main motivation is that the R ions form corner-sharing tetrahedra such that the interionic interactions are prone to geometrical frustration. A number of different situations have been found depending on the form, sign, size, and

TABLE 18 Magnetic Ordering Temperatures and Magnetic Moments of R2BaMO5 With R ¼ Gd, Tm, and Yb and M ¼ Co, Ni, and Cu for Pnma and Immm Structures, and Tm2Cu2O5 TN (K)

Magnetic Site 1

Moment mB Site 2

Origin of Rare Earth Magnetic Moment

Gd2BaCuO5

11.75

5.9

4.9

Induced?

Tm2BaCoO5

3.5

2.0

0.4

Induced

Tm2BaNiO5

4.85

1.18

0.28

Induced

1.4

3.8

0.8

Tm order

Tm2BaCuO5

19

0.50

0.23

Induced

Yb2BaCuO5

15

2.2

1.1

Induced

Gd2BaNiO5

55

5.4

Gd order

Tm2BaNiO5

14.5

3.6

Induced

Tm2Cu2O5

17.5

3.9

Induced

Pnma

Immm

Free ion

7.0

216 Handbook of Magnetic Materials

anisotropy of the various possible interionic interactions as reported by Gardner et al. (2010). The search for the spin liquid state, a ground state where the spins are strongly correlated, show no long-range order and are dynamical, has motivated important theoretical and experimental efforts. For instance, Ho2Ti2O7, which has a large axial anisotropy and a net ferromagnetic interaction, gives evidence for a magnetic frustration, which shows analogies with the positional fluctuations of the protons in ice. Such systems have been mentioned “spin ice”. Their behavior is essentially driven by dipolar interaction between the spins. Recently, the interest has focused on the quantum spin liquid model where exchange interactions determine the low-temperature behavior and the possibility has been raised that the pyrochlore Yb2Ti2O7 is a physical realization of this model as shown by Savary and Balents (2012). Other more newer models are now in progress of development and discussion. In this paragraph the description of the R2M2O7 will be mainly limited to the results of the rare earth Mo¨ssbauer spectroscopy. A comparison with muon spin relaxation (mSR) and rare earth Mo¨ssbauer results of R2M2O7 will be made. In Section 3.6.2, these combined results for Yb2Ti2O7 were already shown. First, some 155Gd Mo¨ssbauer data of Gd2M2O7 compounds with M ¼ Ti, Sn and Mo will be discussed. Second, the 161Dy Mo¨ssbauer results of the compound Dy2Ti2O7 will be shown. Furthermore, 170Yb Mo¨ssbauer results and mSR will be compared for the compounds Yb2M2O7 with M ¼ Ti, Sn, mixed GaSb and Mo. Bonville et al. (2003) have combined specific heat, 155Gd Mo¨ssbauer, magnetic susceptibility and magnetization measurements to examine Gd2Sn2O7 and Gd2Ti2O7. The specific heat measurements evidence a single, strongly first-order magnetic transition near 1.0K in Gd2Sn2O7 and confirm the existence of two transitions, near 1.0K and 0.75K, in Gd2Ti2O7 as earlier shown by Ramirez et al. (2002). In Gd2Sn2O7, it was shown previously by Bertin et al. (2002). that spin dynamics with frequencies below this value persists down to the very low temperature of 0.03K. In Fig. 35 155Gd Mo¨ssbauer spectra of Gd2Sn2O7 and Gd2Ti2O7 are shown in panels (A) and (B), respectively. For Gd2Sn2O7 down to 1.1K, only a quadrupole interaction is observed. The quadrupolar interaction (splitting) is
4.0 mm/s, with the sign obtained from the data below 1.0K. An additional magnetic hyperfine interaction appears between 1.10 and 1.05K. Below 1.05K the data analysis gives the size of the hyperfine field and its direction relative to the principal axis of the electric field gradient tensor, which is the local [111] axis. At each temperature, the Mo¨ssbauer line shape is well described by a unique hyperfine field that is oriented 90  5 to the principal axis of the electric field gradient. Since the Gd3þ spontaneous magnetic moment m(T) is proportional to the hyperfine field Heff(T), this shows that at each temperature the four moments of a tetrahedron have a common size and each is perpendicular to the local [111] axis. The magnetic moment is saturated at low temperature and the Gd3þ moment of 7 mB has the free ion value. The thermal variation of m(T) is given

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

217

FIGURE 35 155Gd Mo¨ssbauer spectra of Gd2Sn2O7 [left-hand panel, (A)] and Gd2Ti2O7 [middle panel, (B)] as a function of temperature. Thermal variation of the Gd3þ magnetic moment in Gd2Sn2O7 (right-hand panel, top) and Gd2Ti2O7 (right-hand panel, bottom) obtained from the 155 Gd Mo¨ssbauer measurements. The dashed line marks the position of the maxima of the specific heat peaks (z1.05K) in the two compounds. The chain line marks the position of the second transition in Gd2Ti2O7 (z0.75K). Above this temperature, two different Gd3þ moments are evidenced (full and open triangles) (see text). This figure is earlier published by Bonville, P., Hodges, J.A., Ocio, M., Sanchez, J.P., Vulliet, P., Sosin, S., Braithwaite, D., 2003. J. Phys. Condens. Matter 15, 7777.

in the top part of the right-hand panel of Fig. 35. The moment shows a smooth decrease as the temperature is increasing from 0.03 to 1.065K. At 1.065K, it amounts to 75% of the saturated value and then drops abruptly to zero at 1.1K, indicating a possible strong first-order character in Gd2Sn22O7. The observed hyperfine fields (and the associated correlated magnetic moments) are “static” with a Larmor frequency scale of 120  10
6 s
1 with magnetic moments directions that are perpendicular to each local [111] axis. 155 Gd Mo¨ssbauer spectra of Gd2Ti2O7 are shown in the middle panel (B) of Fig. 35. At 4.2K, an electric quadrupole-splitting is visible, with a value of
5.6 mm/s. As the temperature is decreased from 4.2 to 1.1K, which is just above that of the upper specific heat peak, each of the two absorption lines start to broaden considerably. This broadening is due to the slowing down of the fluctuations of the short range-correlated Gd3þ magnetic moments. No such comparable line broadening was found in Gd2Sn2O7. This line broadening masks the magnetic transition near 1.0K. Well below this temperature, the line shape evidences the characteristic structure associated with combined quadrupole and magnetic hyperfine interactions. Up to 0.6K, the spectra of Gd2Ti2O7 are very satisfactorily fitted with a single hyperfine field perpendicular to the principal axis of the electric field gradient, like in the case of Gd2Sn2O7. This means that the four Gd3þ ions of

218 Handbook of Magnetic Materials

the tetrahedron carry equal magnetic moments and each is perpendicular to the local [111] axis. The moments were obtained from the hyperfine field values using the scaling law (36), with a measured saturated hyperfine field Heff(0) ¼ 28.3T, which is slightly smaller than in Gd2Sn2O7. Their thermal variation is shown in the bottom part of the right-hand panel of Fig. 35. At low temperatures, the Gd3þ moment is essentially temperature-independent, whereas at 0.8K and above, it decreases with increasing temperature. At 0.8K and above, the Mo¨ssbauer analysis indicates that the four moments of a tetrahedron are no longer identical and no reliable values for the moments could be obtained above 0.9K due to the influence of the dynamical broadening of the Mo¨ssbauer lines. Attempts to fit the spectrum of Gd2Ti2O7 measured at T ¼ 0.8K with different models were not successful. It is clear that the transition of the two compounds near 1.0K has a very complex character. Its main feature is that it is strongly first order in Gd2Sn2O7 and there is some evidence of weak first-order behavior in Gd2Ti2O7. Although the pyrochlore lattice is capable of displaying a first-order magnetic transition, which does not involve long-range magnetic ordering, the transition at 1K in both compounds is associated with the onset of long-range order. The magnetic structure, however, is probably different in the two cases. The rare earth molybdates R2Mo2O7 are formed with the rare earths from Nd3þ to Lu3þ. For R from Nd to Gd, these compounds show metallic behavior, whereas for R from Dy to Lu (and with Y), they show semiconducting behavior with an activation energy of the order of 15 meV, as shown by Greedan et al. (1987). The change in behavior (metalesemiconductor crossover) has been related to variations in the MoeO bond lengths associated with the changing lattice parameter (lanthanide contraction) as explained by Katsufuji et al. (2000) and in the MoeOeMo bond angles as studied by Moritomo et al. (2001). The metalesemiconductor crossover leads to profound changes in the magnetic properties. In the R2Mo2O7, the dominant exchange interaction is that within the d-ion sublattice. The next most important interaction is the intersublattice exchange. The interaction within the f-ion sublattice is the weakest of the three. For the dion sublattice, the dominant exchange mechanism is different on either side of the metalesemiconductor crossover. In the semiconducting compounds, the antiferromagnetic superexchange prevails, whereas in the metallic compounds, the direct ferromagnetic coupling involving the Mo 4d-spin density at the Fermi level dominates, as shown by Kang et al. (2002). The 155Gd Mo¨ssbauer absorption measurements were performed over the temperature range between T ¼ 0.027 and 80K by Hodges et al. (2003a). At 80K, in the paramagnetic phase, the absorption takes the form of a nearly symmetric doublet. It corresponds to an electric quadrupole-splitting with a value of
5.1 mm/s, not much different from the other R2M2O7 compounds with M ¼ Sn and Ti as shown above. A hyperfine field, proportional to the Gd3þ magnetic moment, appears when the temperature is lower than 75K,

Mo¨ssbauer Spectroscopy on rare Earth-Based Oxides Chapter j 3

219

which is the magnetic transition. In Gd2Mo2O7, the magnetic hyperfine interaction is much smaller than the quadrupole interaction, and acts only as a broadening on each of the two absorption lines. Therefore, the asymmetry h and subsequently the angle q are hardly to determine as discussed by Hodges et al. (2003a, 2003b). At 0.027K, the saturated hyperfine field is 19.8T, which is markedly smaller than that usually found in insulating Gd compounds (z30T). This reduced value in this metallic compound is probably linked to the exchange polarization of s-type conduction electrons, which contributes a hyperfine field opposite to that arising from the polarization of the core selectrons by the 4f shell moment. Fig. 36 shows the thermal variation of the Gd3þ magnetic moment, obtained from the hyperfine field value by scaling, using the relation that the measured saturated hyperfine field of 19.8T corresponds to a saturated Gd3þ moment of 7 mB. Then, the temperature dependence of the Gd3þ magnetic moment is calculated with a modified molecular field model including a Brillouin function for S ¼ 7/2 and a derived exchange field for Mo (Tc ¼ 75K) as described by Hodges et al. (2003a). The MoeGd exchange field amounts 5.5T. This is about 30% lower than found from specific heat analysis by Raju et al. (1992). Furthermore, in Gd2Mo2O7, it was observed that the steady-state Gd hyperfine populations at 0.027K are out of thermal equilibrium, indicating that Gd and Mo spin fluctuations persist at very low temperatures. Frustration is thus operative in this essentially isotropic pyrochlore where the dominant Mo intrasublattice interaction is ferromagnetic. A study on quadrupole interactions and relaxation phenomena of Dy2Ti2O7 with increasing temperature up to 750K have been performed using the 161Dy Mo¨ssbauer effect by Almog et al. (1973). 161Tb in 160Gd2Ti2O7 at 150K was used as source in order to emit a very narrow line. Quadrupole interaction

FIGURE 36 Temperature dependence of the Gd3þ 4f shell magnetic moment in Gd2Mo2O7 obtained from the 155Gd Mo¨ssbauer measurements. The solid line is a fit with the molecular field model. This figure is earlier published by Hodges, J.A., Bonville, P., Forget, A., Sanchez, J.P., Vulliet, P., Rams, M., Kro´las, 2003a, Eur. Phys. J. B 33, 173.

220 Handbook of Magnetic Materials

parameters and relaxation times, as function of temperature, were deduced from the measurements. It was found that at T ¼ 4K from the hyperfine field that Dy3þ has values for the hyperfine field and the electric quadrupolesplitting, indicating a JZ ¼ 15/2 Kramers doublet. At higher temperatures, magnetic relaxation broadening was observed. In this respect, Dy2Ti2O7 resembles the same relaxation behavior as DyPO4 and DyVO4 as described in Section 4.4. In Section 3.6.2, Yb2Ti2O7 was already discussed. Recently, the Higgs ferromagnetic phase has recently been proposed for this compound at low temperature by Chang et al. (2012), since a longstanding controversy exists as to the intrinsic presence of magnetic Bragg reflections in Yb2Ti2O7 as discussed by several authors (e.g., Hodges et al., 2002; Ross et al., 2009). Beside the R2Ti2O7 series, the pyrochlore rare-earth stannates R2Sn2O7 have also attracted strong interest since both series share similar R anisotropies and a nonmagnetic M sublattice. Comparing the two series, it seems that the titanates are more influenced by exchange interactions beyond the nearest neighbors than the stannates. Therefore, Yb2Sn2O7 is a better test than Yb2Ti2O7 for the current theoretical works dedicated to quantum spin liquids. For Yb2Sn2O7, bulk macroscopic properties were studied first. Then microscopic results were obtained from 170Yb Mo¨ssbauer spectroscopy, powder neutron diffraction and muon spin relaxation (mSR) as studied by Yaouanc et al. (2013). The heat capacity, measured in the range 0.08e4K, is shown in Fig. 37A. Its most important result is a fairly symmetric narrow peak at the transition at Tt z 0.15K, and a broad hump centered at about 2K. This hump is ascribed to the exchange splitting of the ground-state Kramers doublet associated with the onset of magnetic correlations, which is a commonly observed feature in frustrated magnets. The 170Yb Mo¨ssbauer spectroscopy measurements on Yb2Sn2O7 as shown in Fig. 37B were measured down to T ¼ 0.045K. At 4.2K a pure quadrupole hyperfine spectrum is observed with a temperature-independent size and symmetry. With decreasing temperature, the line shapes broaden progressively as shown in Fig. 37B at 0.2K due to the increase of the short-range correlations among the Yb3þ moments as also evidenced by the specific-heat measurements. These correlations lead to a decrease of the Yb3þ spin fluctuation rate vcM so that it enters the 170Yb Mo¨ssbauer frequency window as displayed in Fig. 38. As the temperature is further lowered through Tt, an additional magnetic hyperfine interaction initially appears on some of the Yb3þ spins. The relative weight of this fraction increases as the temperature is decreased as shown in Fig. 37D. At T ¼ 0.09K the pure quadrupole subspectrum has disappeared and only the mixed quadrupole with magnetic hyperfine spectrum was observed. This behavior evidences that paramagnetic and magnetically ordered moments coexist in a small temperature range around the transition, which is an indication of first-order transition. Below the transition, the fluctuation rate is below the sensitivity limit of this technique (108 s
1).

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FIGURE 37 (A) Low-temperature heat capacity of Yb2Sn2O7. (B) 170Yb absorption Mo¨ssbauer spectra at selected temperatures. The decomposition of the spectra at 0.12 and 0.10K in terms of two subspectra, pure quadrupole and quadrupole with magnetic hyperfine interactions, is indicated. (C) Thermal evolution of the magnitude of the Yb3þ magnetic moments detected by Mo¨ssbauer spectroscopy and (D) the percentage fraction of the Yb3þ ions carrying a magnetic moment. Here the lines are guides to the eye. The behaviors shown in panels (C) and (D) are the signatures of first-order transitions. This figure is earlier published by Yaouanc, A., Dalmas de Re´otier, P., Bonville, P., Hodges, J.A., Glazkov, V., Keller, L., Sikolenko V., Bartkowiak, M., Amato, A., Baines, C., King, P.J.C., Gubbens, P.C.M., Forget, A., 2013, Phys. Rev. Lett. 110, 127207.

The hyperfine field of the Yb3þ ion, proportional to the spontaneous magnetic moment, amounts 110  2T, which corresponds to 1.1 mB. These moments are parallel to the hyperfine field and lie at an angle of 65 relative to their local [111] direction. The spectroscopic g-factors of the anisotropic Yb3þ groundstate Kramers doublet are determined. For Yb2Sn2O7 gz y 1.1 and gt y 4.2, which gives a stronger XY anisotropy than in Yb2Ti2O7. The neutron diffraction measurements show magnetic reflections observed at the position of the nuclear Bragg peaks. Therefore, a long-range magnetic order is found characterized by a k ¼ 0 propagation wave vector. A Rietveld refinement was performed according to a linear combination of two basis vectors of the sixth-order irreducible representation, of which only one of the four possibilities exist for a k ¼ 0 magnetic structure in the Fd3m space group

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FIGURE 38 Temperature dependence of the fluctuation rates of the correlated Yb3þ moments in Yb2Sn2O7 obtained from 170Yb Mo¨ssbauer and mSR spectroscopies, and comparison with the Yb2Ti2O7 data studied by Yaouanc et al. (2003). The full lines above the transition temperatures result from fits to activation laws. The dashed lines are guides to the eye. At the temperature of the respective specific heat peaks they are vertical, indicating a sharp change in the spin dynamics at those temperatures. The few points below Tt for Yb2Sn2O7 correspond to paramagnetic moments that coexist with ordered moments as shown in Fig. 37C. This figure is earlier published by Yaouanc, A., Dalmas de Re´otier, P., Bonville, P., Hodges, J.A., Glazkov, V., Keller, L., Sikolenko V., Bartkowiak, M., Amato, A., Baines, C., King, P.J.C., Gubbens, P.C.M., Forget, A., 2013, Phys. Rev. Lett. 110, 127207.

allowing for an angle between the ordered moment and the local [111] axis different from 0 and 90 . Setting this angle to 65 a very good fit of the data was obtained with a magnetic moment of 1.05  2 mB, a value close to the 170 Yb Mo¨ssbauer spectroscopy result shown above. Further information was derived from mSR measurements, which were performed in the temperature range from 0.014 to 2K. These measurements give access to the so-called asymmetry a0 Pexp Z ðtÞ, where a0 is an experimental parameter and Pexp the muon polarization function, which reflects the physics Z of the compound under study. All the spectra were fitted to a0 Pexp Z ðtÞ ¼ as PZ ðtÞ þ abg , where the second time-independent component accounts for the muons implanted in the sample surroundings. Above Tt PZ(t) ¼ exp (
lZt), where lZ is the muon spin-lattice relaxation rate. This means the system is in the fast fluctuation limit, that is, the fluctuation rate vc,m of the Yb3þ dipolar field at the muon site verifies the relation vc,m [ gmDpara, where Dpara is the root-mean-square of the field distribution at the muon site and gm is the muon gyromagnetic ratio. It was found that lZ increases on cooling. This is a usual behavior reflecting the slowing down of the fluctuations of exchange-coupled Yb3þ moments in Yb2Sn2O7. In this temperature range, Dpara can be taken to be temperature-independent and it was found that Dpara y 73 mT, which is close to the value of 80 mT for Yb2Ti2O7 from

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Yaouanc et al. (2003). Fig. 38 displays vc,m deduced from the above analysis together with vc,M. An activation law is rather well obeyed, with an activation energy of 0.20K in temperature units. Below Tt y 0.15K, that is, in the presence of long-range magnetic order, it might be expected that the spectral shape would show rapid depolarization or pronounced oscillations due to muon spin precession in the spontaneous field. Instead the spectral shape shows little change on crossing Tt. A similar unusual behavior was already found, for example, Tb2Sn2O7 (Dalmas de Re´otier et al., 2006) and explained by a persistent dynamics in the ordered state, with a fluctuation rate larger than the muon precession frequency. Since there is no evidence of spontaneous precession below Tt, the spectral shape has changed and it no longer follows a simple exponential form. It means that the fast fluctuation limit no longer applies and therefore the Yb3þ spins have undergone a drastic slowing down at Tt. This is confirmed by weak longitudinal field measurements. The weak minimum seen below 1 ms indeed reveals the presence of a field distribution at the muon site with dynamics in the microsecond time scale. The shape of the spectra recorded below Tt is reminiscent of the dynamical Kubo-Toyabe function, however, with a slight modification as shown by Hodges et al. (2002). The main result of the mSR study is the abrupt decrease of vc,m to an essentially temperature-independent value in the megahertz range below Tt y 0.15K. It is consistent with the upper bound of 108 s
1 derived from Mo¨ssbauer spectroscopy measurements. In this respect, Mo¨ssbauer spectroscopy and mSR are partly overlapping and additive. This unusual transition in the dynamics in Yb2Sn2O7 is similar to that in Yb2Ti2O7 as shown in Fig. 38. The 170Yb Mo¨ssbauer measurements on Yb2GaSbO7 were performed over the temperature range from 0.03 to 80K. The results of these measurements were compared with mSR measurements on Yb2GaSbO7 as studied by Hodges et al. (2011). At 1.6K and above, the Mo¨ssbauer spectra are similar to the one measured at T ¼ 0.9K. They are all well described using only a quadrupole hyperfine interaction. The asymmetry parameter (h) as defined in Eq. (3) is not equal to zero as for the other crystallographically ordered pyrochlores. Another feature of these quadrupole spectra is that the measured line width is broader than that in crystallographically ordered Yb2Ti2O7 and Yb2Sn2O7. This broadening had to be attributed to the presence of a small distribution in the quadrupole interaction, which can be ascribed from a small distribution in the local crystal field parameters. These two effects have a common origin linked to the disorder in the Ga3þ/Sb5þ site occupancy. As the temperature is lowered below 0.9K, the line shape progressively broadens and then splits under the influence of a magnetic hyperfine interaction. The spectra from 0.03 up to 0.3K are characteristic of hyperfine field spectra, with large and inhomogeneous line broadenings due to the presence of a distribution in the hyperfine field, which again is related to the distribution of crystal field parameters produced by the Ga3þ/Sb5þ disorder.

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The data-fits of the 170Yb spectra were made with the quadrupolar interaction fixed at the value obtained above 0.9K. The saturated hyperfine field at 0.03K is Heff y 110T, which corresponds to a spontaneous moment of 1.1 mB per Yb3þ ion, identical to that in Yb2Ti2O7. The angle q of the hyperfine field with respect to the principal axis of the electric field gradient (the local [111] axis) is close to 55 . This measurement of the direction of the short-range correlated Yb3þ magnetic moments relative to the direction of the local principal anisotropy axis directly evidences that each moment is essentially oriented at the same angle relative to its local anisotropy axis. Because of the anticipated planar anisotropy of the ground-state g-tensor, the molecular field that gives rise to this moment is not aligned along the direction of the moment but is rather oriented much closer to the local [111] axis. The fact that the Yb3þ magnetic moment does not lie along the direction of the molecular field, which gives rise to the moment directly, shows that the exchange in Yb2GaSbO7 is anisotropic. The spectrum at 0.4K is only partially resolved. As the temperature is further increased, the line shape becomes narrower and tends to the quadrupolar hyperfine pattern observed above 0.9K. This line-shape change can be regarded as a fluctuation induced spectral narrowing due to the dynamic nature of the magnetic correlations. The spectra between 0.4 and 0.9K were fitted using a relaxational line shape involving stochastic fluctuations of the hyperfine field as shown by Hodges et al. (2001). On the same way as ascribed for Yb2Sn2O7 above, the magnetic dynamic time behavior sm can been determined from the measured value of lZ and the derived value of Dpara with the formula lZ ¼ 2g2m D2para sm . In Fig. 39 the composed results for the fluctuation rate of Yb2GaSbO7 and Yb2Ti2O7 are compared. The absence of any sharp anomaly in the temperature dependence of the fluctuation rate in Yb2GaSbO7 [there is also no sharp anomaly in the specific heat as shown by Blo¨te et al. (1969)] suggests that under the influence of the disorder, the thermal behavior is closer to that predicted for a classical collective paramagnet. However, the low-temperature behavior does not fully correspond to this state because the Yb3þ magnetic moments fluctuate only between a subset of the total possible directions since the moments retain a fixed angle of 55 relative to the local [111] axis. The relatively high fluctuation rate of 7  107 s
1 observed at 0.02K indicates that the Ga3þ/Sb5þ disorder does not suppress the frustration-induced spin dynamics. The fact that frustration is operative both in ferromagnetic Yb2Ti2O7 (and Yb2Sn2O7) on the one hand and in antiferromagnetic Yb2GaSbO7 on the other hand may be linked to the fact that in both compounds the magnetic moments have components along a local [111] axis and in a local basal plane, so allowing different microscopic mechanisms to be operative in both cases. Hodges et al. (2003a) have examined the semiconducting pyrochlore compound Yb2Mo2O7, where the Mo sublattice is antiferromagnetic, with 170 Yb Mo¨ssbauer measurements down to z0.03K. The microscopic

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FIGURE 39 Temperature dependence of the fluctuation rates of the correlated Yb3þ moments in Yb2GaSbO7 and in Yb2Ti2O7 as studied by Yaouanc et al. (2003) obtained from the mSR measurements (vcm ¼ 1/sm, over a wide temperature span) and 170Yb Mo¨ssbauer measurements (vcM ¼ 1/shf, over a limited temperature span). The full line is a fit to an activation law as shown by Yaouanc et al. (2003) and the dashed line a guide to the eye. The variation in Yb2GaSbO7 does not result in evidence for a first-order transition, which is present in Yb2Ti2O7. The limiting low temperature fluctuation rate in Yb2GaSbO7 is much higher than the one found in Yb2Ti2O7. This figure is earlier published by Hodges, J.A., Dalmas de Re´otier, Yaouanc, A., Gubbens, P.C.M., King, P.J.C., Baines, C., 2011. J. Phys. Condens. Matter 23, 164217.

measurements evidence lattice disorder, which is important in Yb2Mo2O7. This behavior introduces distributions in the Yb3þ ground states and subsequently broadening in the measured 0.036K spectrum. Magnetic irreversibilities occur at 17K in Yb2Mo2O7. Below TN ¼ 17K, the Yb3þ ions carry magnetic moments that are induced through couplings with the Mo sublattice. The temperature dependence of the hyperfine is analyzed with a reduced exchange field in a self-consistent way with a Brillouin function for S ¼ 1 and s ¼ T/TN.

4.8 R3M5O12 Compounds The rare earth sublattice in garnets with the formula R3M5O12 with R is rare earth and M a metal, where the rare earth ions lie on two interpenetrating corner sharing triangular networks can lead to frustration. Therefore, magnetic frustration will be discussed in Gd3Ga5O12 and Yb3Ga5O12, which were studied respectively with 155Gd and 170Yb Mo¨ssbauer spectroscopy. The spin liquid properties of Gd3Ga5O12 have been examined using 155Gd Mo¨ssbauer spectroscopy down to 0.027K by Bonville et al. (2004). Information has been obtained concerning both the directional properties of the shortrange correlated moments and the thermal dependence of their spin fluctuation rates. The spectrum at the lowest temperature, 0.027K, is a partially resolved

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mixed quadrupolar and magnetic hyperfine field pattern. For Gd3þ, the hyperfine field in insulating compounds is about 30T. The presence of a static hyperfine field, which is directly proportional to the Gd magnetic moment, shows that the Gd3þ spins are strongly correlated at 0.027K. As a long-range magnetic order is not observed down to 0.025K, the existence of dynamic short-range order has to be supposed, with a spin fluctuation rate that is below the lower limit of the 155Gd Mo¨ssbauer window of 3  107 s
1. From the analysis of the spectrum at 0.027K, it was found that the short-range correlated Gd3þ spins fluctuate (with rates lower than 3  107 s
1) in the YOZ plane of the orthorhombic structure such that they present equal probabilities along the electric field axes OY and OZ. For the three sites making up an elementary triangle of the rare earth sublattice, the three local YOZ planes are not parallel, and thus the fluctuations of the spins of a triangle are not coplanar. As the temperature is increased above 0.027K, the individual lines first broaden, and then above 0.2K a two-line spectrum is observed, with linewidths that decrease progressively. At 4.2K, a pure quadrupolar hyperfine spectrum is recovered. All these features are characteristic of the presence of electronic fluctuations with a rate that increases as the temperature increases. The relaxation line shape involves hyperfine field fluctuations along local axes, that is, the principal axes of the electric field gradient tensor. This line shape uses the random phase approximation that assumes that the jumps of the hyperfine field of the short-range correlated moments between the chosen directions are uncorrelated. The results of the relaxation model as described Bonville et al. (2004) are shown in Fig. 40. The only free parameters are (1/s) and the directions between which the hyperfine field fluctuates. The temperature dependence of the correlations could well be fitted with a T2.2 law as shown in Fig. 40. This law as well as a T linear law predicted at much lower temperatures is due to thermal excitations within correlated sin structures. At low temperatures, the Mo¨ssbauer derived relaxation rate lies far below those derived from the mSR measurements of both Dunsiger et al. (2000) (chained line) and Marshall et al. (2002) (dashed line). A clear explanation of this difference in behavior could not be given. In the garnet-structure compound Yb3Ga5O12, the Yb3þ ions with a ground-state effective spin S ¼ 1/2 are situated on two interpenetrating cornersharing triangular sublattices such that frustrated magnetic interactions are possible. Specific-heat measurements have evidenced the development of short-range magnetic correlations below w0.5K and a sharp transition at 54 mK as measured first by Filippi et al. (1980). 170Yb Mo¨ssbauer spectroscopy measurements down to 36 mK were performed by Hodges et al. (2003b). No static magnetic order at temperatures was found below the sharp specific transition. The 170Yb Mo¨ssbauer spectra were analyzed in a way comparable as performed for Gd3Ga5O12 as shown above. At 0.036K the line shape of spectrum of Yb3Ga5O12 is well broadened, and this makes it possible to obtain both the magnitude of the hyperfine field and its fluctuation rate. It was found

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FIGURE 40 Thermal variation of the fluctuation rate of the correlated Gd3þ moments in Gd3Ga5O12, obtained from the 155Gd Mo¨ssbauer data (black squares). Inset: Low-temperature behavior (same units as the main figure), where the horizontal dashed line defines the lower limit of the 155Gd Mo¨ssbauer frequency window of 3  107 s
1. The solid line (main figure and inset) is the law: (1/s) ¼ aT2.2, with a ¼ 17  109 s
1 K
2.2. The chain and dashed lines schematically represent the data from the mSR measurements by Dunsiger et al. (2000) and Marshall et al. (2002), respectively. This figure is earlier published by Bonville, P., Hodges, J.A., Sanchez, J.P., Vulliet, P., 2004. Phys. Rev. Lett. 92, 167202.

that Heff x 140  10T, which corresponds to a Yb3þ moment of x1.4 mB and (1/s)hf ¼ 3  109 s
1. The value for the Yb3þ moment is not far from the mean value expected both from the average g-tensor and from the saturated magnetization measured at 0.09K in of 1.7 mB as found by Filippi et al. (1980). In the fits for the spectra up to 0.2K, an intrinsic half-width of 1.35 mm/s was used, and assumed that the fluctuating hyperfine field has a size that remains constant at the value 140T derived at 0.036K. The thermal variation of (1/s)hf is shown in Fig. 41. As the temperature is lowered over the range 0.2 to 0.1K, the frequency decreases approximately linearly according to a law Zð1=sÞhf ¼ 0:3 kB T. Then, below about 0.1K, it tends to saturate toward the value 3  109 s
1 as shown in Fig. 41. This is quite a high value and, in fact, Yb3Ga5O12 is the only known compound where the T / 0 spin fluctuation rate is rapid enough to fall within the 170Yb Mo¨ssbauer spectroscopy frequency window. There is essentially no difference between the rates either side of the sharp specific-heat transition at 54 mK as measured by Filippi et al. (1980) and shown in Fig. 42. Above w0.5K, the Yb3þ magnetic moments undergo paramagnetic fluctuations. Up to w150K, the fluctuation rate has a temperature-independent value of x3.8  1010 s
1. Above 150K, additional temperature-dependent relaxation occurs through coupling to phonons, according to a two-phonon

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FIGURE 41 Thermal variations, in Yb3Ga5O12, of the Yb3þ hyperfine field fluctuation frequency extracted from the 170Yb Mo¨ssbauer spectra. The dashed line is the law Zð1=sÞhf ¼ 0:3 kB T. This figure is earlier published by Hodges, J.A., Bonville, P., Rams, M., Kro´las, K., 2003b. J. Phys. Condens. Matter 15, 4631.

Orbach process involving the excited crystal field states near 850K, which is quite a high value. From the 170Yb Mo¨ssbauer measurements on Yb3Ga5O12, it is clear that at TL ¼ 54 mK (the peak in the specific heat as shown in Fig. 42) no conventional magnetic phase transition was found. From the temperature dependence of LZ

FIGURE 42 Zero-field muon spin-lattice relaxation rate, lZ, versus temperature measured for Yb3Ga5O12. The solid line is the result of a fit to a model explained in the main text. The two straight dashed lines for T  0.4K down to 21 mK are guides to the eye. The specific heat (from Filippi et al., 1980) of Yb3Ga5O12 is also reproduced. A marked change of slope in lZ occurs at Tl ¼ 54 mK. This figure is earlier published by Dalmas de Re´otier, P., Yaouanc, A., Gubbens, P.C.M., Kaiser, C.T., Baines, C., King, P.J.C., 2003. Phys. Rev. Lett. 91 167201.

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measured with mSR, no onset of magnetic was found at Tl ¼ 54 mK by Dalmas de Re´otier et al. (2003). Furthermore, the observation of an exponential relaxation function below Tl is another important point. It implies that we are in the fast fluctuation limit and therefore the Yb3þ moments continue to fluctuate rapidly down to the lowest temperature investigated, which is in contrast with Yb2Ti2O7 and Yb2Sn2O7 as shown earlier, where the moments abruptly slow down below Tl and are here quasistatic. For a Heisenberg magnet at high temperature, if the thermal energy is larger than the exchange energy as explained by Dalmas de Re´otier et al. (2003) can simply be given by the Curie law. Then it can be deduced that lZ should be temperatureindependent as was effectively observed for 0.4  T  80K. However, above 80K, lZ(T) decreases steadily as the sample is heated. This behavior of lZ is due to the relaxation of the Yb3þ magnetic moments resulting from an Orbach process, which is a two-phonon real process with an excited crystal-field level as intermediate at an energy level at 850K as found by Hodges et al. (2003b). Combining the value of lZ in the range 0.4  T  80K and the value sc w 38 ps obtained from Mo¨ssbauer data in the same temperature interval, DZF x 0.16T which is of the expected magnitude. For comparison DZF x 0.08T was found for Yb2Ti2O7 above Tl as found by Yaouanc et al. (2003). Below T < Tl being in an interstitial site, the muon spin can be strongly influenced by magnetic pair correlations. In contrast, since a 170Yb Mo¨ssbauer nucleus is embedded in a Yb atom that is magnetic, it is mainly sensitive to self-correlations. Taking into account that the dynamics measured by Mo¨ssbauer spectroscopy is not very different above and below Tl it was inferred that the sharp increase of lZ occurring right below Tl is the signature of the building up of magnetic pair correlations as argued by Dalmas de Re´otier et al. (2003). In Sections 4.7 and 4.8 examples were shown, where pyrochlore and garnet lattice compounds show magnetic dynamic behavior down to very low temperatures. In Table 19 an overview of these oxide-based Gd and Yb compounds is given.

5. CONCLUSIONS, JUSTIFICATION, AND ACKNOWLEDGMENT In this overview, we saw quite interesting features. Rare earth Mo¨ssbauer spectroscopy gives clear information concerning crystal field effects and magnetic interaction. In this paragraph the author of this chapter will resume the most interesting results. First, a reanalysis of Tm2O3 Mo¨ssbauer results by Stewart et al. (1988) on the old measurements of Barnes et al. (1964) gives new information about the crystal field. Further, the results on Dy2O3 by Forester and Ferrando (1976b) and on PrO2 by Moolenaar et al. (1996) were discussed. For RMO3 the results of Hodges et al. (1984) and e.g., Salama and Stewart (2009) explain the

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TABLE 19 Transition Temperatures in Kelvin, Types of Transition, and Magnetic Moments in Bohr Magnetons in Gd and Yb Pyrochlore and Garnet Compounds Compound

Tl K

Type of Magnetic Transition

Magnetic Moment mB

Gd2Sn2O7

1.1

First order

7

Gd2Ti2O7

1.0

First order

7

Yb2Sn2O7

0.15

First order

1.1

Yb2Ti2O7

0.24

First order

1.1

Yb2GaSbO7

None

None

1.1

Gd3Ga5O12

None

None

7

Yb3Ga5O12

0.054

Second order

1.4

The values given in the table are approximate.

magnetic behavior based on the crystal field results. In the case of RMO4 the crystal field determinations on TmMO4 and YbMO4 were shown in relation to the existence of a JahneTeller effect, for example, by Hodges (1983). Furthermore, the magnetic interplay between rare earth and chromium as mainly studied by e.g., Jimenez-Melero et al. (2006) leads to some interesting features on magnetic and crystallographic behavior. In the SC RBa2Cu3O7 compounds, studies on magnetic and crystal field behavior by a diversity of authors lead to quite interesting results. The determination of the intermediary valency of 3.4 by Moolenaar et al. (1996) with help of the isomer shift interpolation from the 141Pr Mo¨ssbauer measurements on the PrxOy compounds is an exceptional, however, undervalued result. Crystal field-effect determination has been performed by cooperation of the Mo¨ssbauer groups in Delft (Netherlands) and Canberra (Australia) on the R2BaCuO5 compounds to explain its magnetic behavior. At last the Gd and Yb Mo¨ssbauer results, mainly performed by Joe Hodges and coworkers in Saclay on magnetic dynamic behavior in pyrochlore and garnet compounds, were discussed in combination with mSR results of cooperation between the groups in Grenoble, Delft, and Saclay. The author of this chapter thanks Glen Stewart, Steve Harker, Enrique Jimenez, Regino Sa´ez-Puche, Alain Yaouanc, Pierre Dalmas de Re´otier, Joe Hodges, Pierre Bonville, and others for their cooperation during many years on the field of Mo¨ssbauer spectroscopy on rare earth-based oxides. The author of this chapter thanks Ignatz de Schepper for critical reading the manuscript, Jouke Heringa for solving computer problems, and Ekkes Bru¨ck for pushing me forward to finish this chapter.

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Author Index ‘Note: Page numbers followed by “f” indicate figures and “t” indicate tables.’

A Abe, K., 46e47 Adams, P.W., 51e52 Aharoni, A., 72 Ahmadian, F., 61 Ahmed, W.K., 61 Aktas, B., 29 Alam, A., 3e4, 9e10, 36e42, 38te39t, 44e48, 50e59, 50f, 53fe54f, 56f, 58f Albanesi, E.A., 13 Ale´onard, R., 177 Alfonsov, A., 24e27, 26fe28f, 35e36 Algarabel, P.A., 110 Ali, N., 218 Alijani, V., 4, 36e37, 37f, 38te39t, 43e46, 49, 55, 59e60 Allenspach, P., 200e204, 203t Almog, A., 219e220 Alves, M.C.M., 40 Amato, A., 220, 221fe222f, 223 Ambrose, T., 3, 10e11, 54e55 Amrouni, C., 111e112 Ando, Y., 3e4, 32e33 Andre´, G., 169f, 220, 223e224 Andreev, A.V., 127e128 Anitas, E.M., 110 Appavou, M.S., 111e112 Arkhipov, V.E., 50e51 Aroyo, M.I., 94e95, 99 Asano, S., 13e17, 19e20, 51e52 Assaf, B.A., 9e10 Avdeev, M.V., 110 Awschalom, D.D., 110

B Babcock, E., 111e112 Babdock, K., 110 Bacon, G.E., 68, 76e79 Badurek, G., 110 Baev, A.S., 172 Baines, C., 162f, 169f, 220, 221fe222f, 223, 225f, 228e229

Bainsla, L., 1e62, 2e4, 9e10, 13e19, 15fe18f, 36e42, 38te39t, 41f, 44e47, 48f, 50e59, 50f, 53fe54f, 56f, 58f, 61 Balbashov, A.M., 89 Balents, L., 215e216 Balk, A.L., 119 Balke, B., 9f, 12e14, 20e29, 22fe23f, 25fe30f, 32, 33f, 38te39t, 40e41, 45e48, 59 Ballauff, M., 111e112 Ballou, R., 96e97 Barnes, R.G., 170e172, 172f, 229e230 Barry, A., 3, 10e11, 54e55 Barth, J., 12 Bartkowiak, M., 100, 220, 221fe222f Basit, L., 38te39t, 41, 47e48, 59 Batlogg, B., 167 Bauminger, E.R., 219e220 Becker, P., 108 Beckurts, K.H., 99 Behr, G., 12 Belmeguenai, M., 35 Beno, M.A., 194e195 Bent, M.F., 174e175 Bergold, M., 167, 168t, 200e202, 203t, 204e205 Bernardi, F., 40 Berri, S., 61 Bertaut, E.F., 177 Bertaut, E.H., 94e97 Bertaut, F., 185e186 Bertin, A., 163 Bertin, E., 216e217 Beuneu, B., 186, 187fe188f, 229e230 Bhattacharyya, D., 36e37, 44e45 Bia1as, F., 130 Bindi, L., 98e99 Bird, M., 120 Blackburn, E., 120 Blamire, M., 10e11, 18e19 Blasco, J., 110 Bleif, H.-J., 100

237

238 Author Index Bloch, F., 68 Blomberg, M., 127 Blo¨te, H.W.J., 224 Blu¨gel, S., 8, 22, 44e49, 60e61 Blum, C.G.F., 12, 24e27, 26fe28f, 51, 53e54 Blume, M., 163e165 Blumenro¨der, S., 167, 199e200 Blundell, S., 72 Blundell, S.J., 225e226, 227f Bo´dogh, M., 196e197 Boesecke, P., 117 Bombor, D., 12, 51, 53e54 Bonazzi, P., 98e99 Bo¨ni, P., 110 Bonnaud, M., 111e112 Bonville, P., 163, 169e170, 169f, 171f, 180e181, 186, 193, 194f, 198e199, 199f, 205, 206f, 213e220, 219f, 221fe222f, 222e229, 225f, 228f Boolchand, P., 184 Booth, J.G., 13e16 Borchers, J.A., 119 Bornemann, H.J., 200 Bosu, S., 3e4, 33, 34f Bouchaud, J.P., 216e217 Boultif, A., 104e105 Boure´e, F., 96e97 Bourges, P., 99 Bowden, G.J., 193e194 Brabers, J.H.V.J., 127, 162f Bradbury, M., 156 Braga, G., 196e197 Braithwaite, D., 163, 216e217 Brammer, L., 68 Branford, M.R., 10e11, 18e19 Breczewski, T., 99 Broussard, P.R., 3, 10e11, 54e55 Bru¨ck, E., 12, 68, 120e121, 127e128, 133 Bruckel, T., 111e112 Bru¨esch, P., 167e169, 168t, 200e204 Bruˆlet, A., 111e112, 116 Bucher, E., 184 Bu¨chner, B., 12, 24e27, 26fe28f, 51, 53e54 Budnick, J.I., 26e27 Bugoslavsky, Y.V., 10e11, 18e19 Buisson, G., 185e186 Burch, T.J., 26e27 Burlet, P., 91e92, 127e128 Buschow, H.J., 2e3, 13 Buschow, K.H.J., 3, 120e121, 127e128, 146, 159, 161e163, 162f, 165, 168t

Buu¨chner, B., 35e36 Byers, J.M., 3, 10e11, 54e55, 54f

C Cable, J.W., 72e73 Cadogan, J.M., 208e209, 211t Cameron, A.S., 110, 116 Campbell, I.A., 157 Campbell, R.A., 116 Cashion, J.D., 170e172, 175e176 Castellano, C., 110 Catalan, G., 68 Cava, R.J., 163, 167, 216e217 Cervellino, A., 223 Chaboussant, G., 111e112, 116 Chadov, S., 35, 38te39t, 41, 47e48, 59 Chakhalian, J.A., 225e226, 227f Chang, L.-J., 220 Chang, N.C., 171e172 Chapuis, G., 98e99 Chapuis, Y., 163 Chen, C.H., 167 Chen, H., 57 Chen, X., 57 Cheng, S.F., 3, 10e11, 54e55 Cheong, S.-W., 218 Chetverikov, Y.O., 110 Chieda, Y., 46e47 Chien, C.L., 54f Chioncel, L., 12 Chmist, J., 179e180 Choi, E.-M., 110 Cimberle, M.R., 110 Clauser, M.J., 163e165 Clemens, D., 100, 111e112 Clinton, T.W., 195e198 Clowes, S.K., 10e11, 18e19 Coehoorn, R., 161e163, 162f Coelho, A.A., 3e4, 9e10, 37e40, 38te39t, 42, 46e47, 52e55, 53f, 57e58, 58f Coey, J.M.D., 3, 10e11, 54e55, 72, 199e200 Cohen, L.F., 10e11, 18e19 Cohen, R.L., 170e171 Coldea, A.I., 225e226, 227f Colis, S., 35 Collet, E., 99 Continenza, A., 57, 59 Convert, P., 106 Cook, D.C., 175e176 Cook, D.D., 174e175 Coppens, P., 108 Cornei, N., 110

Author Index Cornelius, A.L., 225e226, 227f Cottrell, S.P., 206e208 Crauss, M.-L., 110 Cubitt, R., 111e112, 116 Cui, Y.T., 59 Cullen, J., 110 Czjzek, G., 200

Donnelly, K., 199e200 Du, Y., 3e4, 9e10, 55e57 Dunlap, B.D., 171e172 Dunlop, J.B., 171e174, 172f, 229e230 Dunsiger, S.R., 225e226, 227f Dunsinger, S.R., 171f Dusek, M., 98e99, 103e104

D

E

Dai, X., 4 Dai, X.F., 59 Dalichaouch, D., 194e195 Dalmas de Re´otier, P., 163, 169f, 171f, 186, 187fe188f, 190f, 193, 206e208, 220, 221fe222f, 222e223, 225f, 228e230 Daniel, C., 116 Daniels, P., 120 Das, I., 12 Datars, W.R., 218 Date, M., 120e121 Dattagupta, D., 170 Day, R.K., 171e174, 172f, 229e230 de Boer, F.R., 12, 120e121, 127e128, 133, 162f De Boissieu, M., 99 de Groot, R.A., 2e3, 12e13 de Jongh, L.J., 167, 200 de Leeuw, D.M., 165, 166f, 168t De Oliveira, A.C., 196e197 De Teresa, J.M., 110 de Vries, J., 177 Debrunner, P.G., 174e175 Debska, U., 50e51 Dederichs, P.H., 6e8, 7fe8f, 13e14, 20, 22, 44e49, 60e61 Deka, B., 40, 47 Deng, J., 61 DePasquali, G., 174e175 Deriglazov, V., 116 Desclaux, J.P., 154e155, 155t Descleaux, J.P., 82e83, 95 De´sert, S., 111e112, 116 Devakul, T., 9e10 Dewhurst, C., 110 Dewhurst, C.D., 111e112 Dhar, S.K., 199e200 Di, Z.Y., 111e112 Diao, Z., 35 Dirken, M.W., 161e163, 167, 200 Divis, M., 128, 178e179, 181e182 Dixon, N.S., 167e169, 184 Dolnik, B., 110

Early, E.A., 194e195 Ebbinghaus, S.G., 12 Ebke, D., 32, 33f, 35 Eckerlebe, H., 110 Ecolivet, C., 99 Edge, A.V., 177e178, 181e182 Edge, A.V.J., 212e213 Ehmler, H., 120 Eibschu¨tz, M., 176, 199e200 Elahmar, M.H., 61 Elbek, B., 171e172 Ellis, T., 12 Elmers, H.J., 20e24, 23f, 25f, 45e46 Elouneg-Jamroz, M., 177e178, 181e182 Emmerling, F., 20e24, 23f, 25f Enamullah, 38te39t, 41e42, 48, 58e59 Endo, K., 46e47 Ene´, B.H., 110 Engels, R., 111e112 Engler, E.M., 195e198 Ewert, D., 200

239

F Faber Jr., J., 174e175 Faiz, M.M., 13 Faucher, M., 156 Fecher, G.H., 4, 9e10, 9f, 12e14, 16e17, 20e24, 22fe23f, 25f, 26e29, 28fe30f, 32, 36e37, 37f, 38te39t, 40e41, 43e55, 59e60 Feeman, A.J., 82e83, 95 Feigin, L.A., 111 Felea, V., 110, 116 Felner, I., 197e198, 198f Felser, C., 2e7, 6f, 7t, 9e10, 9f, 12e14, 16e17, 20e24, 22fe23f, 25f, 26e29, 28fe30f, 32, 33f, 35e37, 37f, 38te39t, 40e55, 57, 59e60 Feng, Y., 57 Feoktystov, A., 110 Feoktystov, A.V., 111e112 Fermon, C., 116

240 Author Index Ferrando, W.A., 164e165, 164f, 170e171, 173e174, 173f, 183, 184f, 229e230 Ferretti, M., 110 Fidler, J., 110 Filipov, N.I., 183, 184f Filippi, J., 226e227, 228f Fischer, H.E., 106 Fischer, P., 119 Fitzsimmons, M.R., 110 Flatau, A.B., 110 Forester, D.W., 164e165, 164f, 170e171, 173e174, 173f, 183, 184f, 229e230 Forgan, E.M., 120 Forget, A., 169f, 218e220, 219f, 221fe222f, 223e225 Fournet, G., 111 Fragneto, G., 116 Frauenfelder, H., 145e146 Freeman, A.J., 25e26, 57, 59, 154e155, 155t Freimuth, A., 167, 199e200 Frielinghaus, H., 111e112 Friemel, G., 12 Fritsche, C., 120 Fritz, L.S., 167e169 Fromme, M., 100, 111e112 Fujii, H., 127, 133 Fujji, S., 13e16, 19e20 Fu¨rrer, A., 68, 99, 167e169, 168t, 200e204, 203t Furthmuller, J., 56e57 Furubayashi, T., 3e4, 13, 51e52, 55 Furukawa, N., 89 Furutani, Y., 46e47

G Gabor, M.S., 35 Gahler, R., 111e112 Galanakis, I., 6e10, 7fe8f, 13e14, 19e20, 22, 29, 31e32, 44e49, 56e57, 59e61 Ganguli, A.K., 38te39t, 43, 50 Gao, G.Y., 9e10, 57 Gao, Q., 61 Garcia, D., 156 Garcia, P., 99 Garcia-Munoz, J.L., 89 Gardner, J.S., 171f, 215e216, 225e226, 227f Garg, V.K., 196e197 Gasser, U., 162f Gaulin, B.D., 110, 171f, 225e226, 227f Gavilano, J., 110, 116 Georgii, R., 110

Ghara, S., 38te39t, 43, 50 Gilbert, D.A., 119 Gilbert, E.P., 111e112 Gilder, S., 110 Gingras, M.J.P., 171f Gingras, M.P.J., 215e216 Glazkov, V., 220, 221fe222f Gmelin, E., 219 Gnutek, P., 167e169 Go¨koglu, G., 61 Goldanskii, V.I., 145e146 Go¨lzha¨user, A., 32, 33f Gonzalez-Jimenez, F., 180 Goosens, A., 186, 187fe188f, 229e230 Goripati, H.S., 51e52 Gorobchenko, V.D., 183, 184f Goto, R., 2e3, 13, 54e55 Govorkova, T.E., 50e51 Graf, H.A., 108e109 Graf, T., 2e7, 6f, 7t, 13, 42, 54e55, 57, 59 Grant, P.M., 195e198 Greedan, J.E., 171f, 215e216, 218 Greegor, R.B., 194e195 Grigoriev, S.V., 110 Grillo, I., 111e112 Grodkiewicz, W.H., 199e200 Gro¨ssinger, R., 110 Grover, A.K., 199e200 Groves, J.L., 174e175 Gruber, J.B., 171e172 Gubanov, V.A., 172 Gubbens, P.C.M., 145e230, 146, 149e153, 159, 161e163, 162f, 165, 166f, 168t, 169f, 170e171, 171f, 174e175, 174f, 186, 187fe188f, 190f, 193, 194f, 195e197, 196fe197f, 206e212, 209fe210f, 211t, 212f, 220, 221fe222f, 222e223, 225f, 228e230 Guinier, A., 111 Gupta, L.C., 38te39t, 43, 50 Gupta, R.P., 161e163, 171e172 Gupta, S., 16e17, 38te39t, 41e42, 48, 58e59 Guthrie, M., 119e120

H Haas, C., 3 Haese-Seiller, M., 116 Halder, M., 38te39t, 42e43, 48e49, 59e60 Hall-Wilton, R., 100

Author Index Hamann, A., 110 Hamaya, K., 13 Hammann, J., 200, 201f, 202t Hanna, S.S., 184 Hansen, T.C., 106 Hanslik, R., 111e112 Haque, Z., 38te39t, 43, 50 Harker, S.J., 204e205, 208e209, 211t, 212e213 Harris, M.J., 73 Hartmann-Boutron, F., 169e170, 180e181 Hartwig, S., 98 Havela, L., 12, 120e122, 126e128, 133, 135 Hayasaki, M., 16e17, 19, 51e52 Hayashi, A., 163, 216e217 Hayashi, M., 3e4, 54e55 Hayes, W., 225e226, 227f Hebral, B., 226e227, 228f Heil, W., 111e112 Heiman, D., 9e10 Henry, P.F., 106 Herber, R.H., 145e146 Hess, C., 12, 51, 53e54 Hettkamp, P., 206e208, 207f Heusler, F., 2e3 Hexemer, A., 116 Hickey, B.J., 10e11, 18e19 Hill, H.H., 120e121 Hines, W.A., 26e27 Hinks, D.G., 194e195 Hiroi, M., 13e16 Hodges, J.A., 163, 169e170, 169f, 171f, 176e177, 180e181, 184e186, 185f, 186t, 193, 194f, 198e200, 199f, 201f, 202e205, 202t, 204f, 206f, 213e220, 213f, 219f, 221fe222f, 222e230, 225f, 228f Hoell, A., 117e118 Hoffmann, R.-D., 98 Hohlwein, D., 178e179 Hollingsworth, M.D., 99 Holmes, A.T., 120 Ho¨lsa¨, J., 178e179, 181e182 Homonnay, Z., 196e197 Honda, Y., 35 Honecker, D., 111e112 Hono, K., 3e4, 9e10, 13e19, 15fe18f, 30e32, 36e40, 38te39t, 42, 44e47, 50e58, 50f, 53fe54f, 56f Hrebik, J., 127 Hu, L., 9e10, 57 Huang, Y.-K., 98

241

Huiskamp, W.J., 224 Hulliger, F., 200e202, 203t Hun, X., 61 Huse, D.A., 216e217 Husmann, A., 225e226, 227f Hutchison, W.D., 179e180 Hu¨tten, A., 32, 33f Hwang, H.Y., 218

I Ibarra, M.R., 110 Ibrir, M., 61 Ihringer, J., 178e179 Ikeda, N., 2e3, 13, 54e55 Ikenaga, E., 27e29, 28f Ikhtiar, 3e4 Imbert, P., 169e170, 176e177, 180e181, 184e185, 185f, 198e200, 199f, 201f, 202e205, 202t, 204f, 206f, 229e230 Inomata, K., 2e3, 13, 26e27, 54e55 Inosov, D.S., 110, 116 Ioffe, A., 111e112 Irkhin, V.Y., 12 Isasi, J., 186, 187fe188f, 190f, 193, 194f, 229e230 Ishida, S., 13e16, 19e20 Islamov, A.K., 110 Itie´, J.P., 170e171, 175 Ito, M., 13e16 Itoh, A., 3e4 Ivankov, O.I., 110 Ivanov, V.Yu, 89 Izyumov, Y.A., 68, 76, 94e97

J Jacobs, P.J., 12 Jacques, M., 111e112 Jaime, M., 225e226, 227f Jaju, N.P., 171f Jakob, G., 51 Jaksch, S., 111e112 Jamer, M.E., 9e10 Janssen, S., 111e112 Janssen, T., 99 Jedryka, E., 2e3, 13, 26e27, 54e55 Jee, C.S., 195e198 Je´hanno, G., 184e185, 185f, 198e199, 199f, 205, 206f Jenkins, C.A., 12 Jensen, J., 72e73 Jha, S.N., 36e37, 44e45

242 Author Index Ji, Y., 54f Jiang, C., 37e40, 38te39t, 46e47 Jimenez, E., 186, 190f, 193, 194f Jimenez-Melero, E., 186, 187fe188f, 229e230 Jirman, L., 127e128, 133 Johannson, O., 10e11, 18e19 Jorgensen, J.D., 194e195 Joubert, D., 56e57 JulliIre, M., 13 Jung, V., 13e14, 26e27

K Kaindl, G., 167, 199e200 Kainuma, R., 46e47 Kaiser, C., 35 Kaiser, C.T., 169f, 171f, 206e208, 220, 222e223, 222f, 225f, 228e229 Kakurai, K., 220 Kallmayer, M., 20e24, 23f, 25f Kammel, M., 117e118 Kampmann, R., 116 Kandpal, H.C., 12, 20e22, 22fe23f, 25f, 29, 29fe30f, 44, 47e49 Kang, J.-S., 218 Kankeleit, E., 170e172, 172f, 229e230 Kanomata, T., 46e47 Kao, Y.-J., 220 Kapfhammer, W.H., 174e175 Karimian, N., 61 Karthik, S.V., 30e32 Kasai, S., 3e4, 13 Kashiwagi, S., 20 Kashyap, M.K., 61 Kasuya, T., 157 Kato, M., 16e17, 19, 51e52 Katsnelson, M.I., 12 Katsufuji, T., 218 Kebede, A., 195e198 Keiderling, U., 111e112 Keller, L., 220, 221fe222f, 223 Kemmerling, G., 111e112 Kern, S., 174e175 Khenata, R., 61 Khodami, M., 61 Kiefl, R.F., 171f, 225e226, 227f Kienle, P., 174e175 Kikuchi, M., 26e27 Kim, J., 12, 20e22, 22f, 29, 29fe30f Kim, J.eJ., 27e29, 28f King, P.J.C., 169f, 220, 221fe222f, 223, 225f, 228e229

Kirby, B.J., 119 Kittel, C., 72, 157 Klaasse, J.C.P., 120e121 Klaer, P., 45e46 Klein, M.P., 181e182 Kleines, H., 111e112 Klencsa´r, Z., 196e197 Klenke, J., 111e112 Klokkenburg, M., 110 Knigavko, A., 110 Knoll, K.D., 184 Kobayashi, K., 12, 20e22, 22f, 27e29, 28fe30f, 46e47 Koehler, W.C., 72e73, 89e90 Ko¨hler, A., 35 Kohlhepp, J.T., 12, 24e27, 26fe28f Kokado, S., 32e33, 34f Kolodziejczyk, A., 167, 199e200 Kondo, Y., 13e16 Konno, T.J., 3e4, 33, 34f Koopmans, B., 24e27, 26fe28f Kopcansky, P., 110 Korolev, A.V., 50e51 Kossut, J., 50e51 Kostorz, G., 111 Kota, Y., 32e33, 34f Kotsis, I., 196e197 Kozina, X., 4, 36e37, 37f, 38te39t, 44e45, 55 Kozlowski, A., 179e180 Krajewski, J.J., 216e217 Kremer, R.K., 219 Kresse, G., 56e57 Kro´las, K., 169f, 218e220, 219f, 223e229, 228f Krop, K., 179e180 Ksenofontov, V., 13e14, 21e22, 24e27, 26fe28f Ku¨bler, J., 9e10, 16e17, 49e55 Kubota, T., 3e4 Kudryashov, V., 116 Kumar, A., 110 Kurimsky, J., 110 Kuzel, R., 127 Kuzmann, E., 196e197 Kwon, S.K., 218

L Lakshmi, N., 37e40, 46 Lamago, D., 110 Lander, J.H., 174e175 Larson, A.C., 103e104

Author Index Larson, E.M., 194e195 Lasjaunias, J.C., 226e227, 228f Lastusaari, M., 178e179, 181e182 Laver, M., 110 Lavie, P., 111e112, 116 Lawson, A.C., 128 le Bras, G., 198e199, 199f Leavitt, R.P., 171e172, 184 Lee, B.W., 194e195 Lee, P.A., 50e52 Lee, S.-I., 110 Lees, M.R., 220 Lehlooh, A.F., 13e14 Lei, G., 61 Leng, O., 35 Levy, A., 219e220 Lewandowska, M., 167e169 Li, G.J., 56e57 Li, G.T., 60e61 Li, L., 61, 98 Li, S., 3e4, 55 Li, W.H., 195e198 Li, Y., 4, 37e40, 38te39t, 46e47 Lichtenstein, A.I., 12 Lieutenant, K., 100, 111e112 Lin, H.eJ., 21e22 Lin, T.T., 59 Lindner, P., 111e112 Litvinchuk, A.P., 178e179, 181e182 Liu, E.K., 9e10, 56e57 Liu, G., 4 Liu, G.D., 56e57, 59e61 Liu, H., 4, 35, 37e40, 38te39t, 46e47 Liu, K., 119 Liu, N., 9e10, 57 Liu, X.F., 59 Liu, Z.H., 60e61 Lkenaga, E., 12, 20e22, 22f, 29, 29fe30f Lo¨hneysen, H.V., 110 Loidl, A., 110, 116 Lonkai, Th., 178e179 Loong, C.K., 174e175 Louer, D., 104e105 Lovesey, S.W., 68, 76e79 Luca, D., 110 Lukashevich, I.I., 183, 184f Lumsden, M.D., 171f Luo, B., 9e10, 57 Luo, H., 37e40, 38te39t, 46e47 Lynn, J.W., 195e198 Lytle, F.W., 194e195

243

M Ma, X.Q., 60e61 MacFarlane, W.A., 171f, 225e226, 227f Machida, A., 218 Mackintosh, A.R., 72e73 Mahmood, S.H., 13e14 Maier-Leibnitz Zentrum, Heinz, 108e109 Maji, B., 17e18 Makinistian, L., 13 Maletta, H., 120e121, 127e128 Maleyev, S.V., 110 Mallick, A.I., 3e4, 9e10, 36e40, 38te39t, 42, 44e47, 50e58, 50f, 53fe54f, 56f, 58f Mallik, R., 12 Manaka, H., 13e16 Manning, G., 99 Manzin, G., 111e112 Maouche, D., 61 Maple, M.B., 194e195 Maranville, B.B., 119 Mareschal, J., 177, 185e186 Marimom da Cunha, J.B., 200, 201f, 202e204, 202t, 204f Marquina, C., 110 Marrows, C.H., 10e11, 18e19 Marshall, I.M., 225e226, 227f Marshall, W., 68, 76e79 Marsolais, R., 200e204 Martinelli, A., 110 Martı´nez, J.L., 186, 187fe188f, 229e230 Matsuura, M., 3e4 Matthias, T., 110 Maurer, W., 174e175 Mavropoulos, P., 31e32 Mavropoulos, Ph., 6e7, 7fe8f, 13e14, 22, 44e49, 60e61 May, L., 145e146 Mazin, I.I., 10 McIntyre, G.J., 99 McPherson, I.M., 206e208 Meissner, M., 120, 127 Mel’nikopv, E.V., 183, 184f Menken, M.J.V., 170e171, 174e175, 174f, 195, 196fe197f, 229e230 Menovsky, A.A., 170e171, 174e175, 174f, 195, 196fe197f, 229e230 Mesot, J., 68, 99 Metlov, K.L., 110 Meyer, C., 170e171, 175, 200

244 Author Index Meyer, M., 98 Mezei, F., 100, 111e112 Michels, A., 110 Miller, R.I., 225e226, 227f Min, B.I., 218 Mitani, S., 3e4, 13 Miura, Y., 3e4, 20, 31, 33, 34f, 35, 46e47 Miyao, M., 13 Miyazaki, A., 26e27 Miyazaki, T., 3e4 Miyoshi, Y., 10e11, 18e19 Mizukami, S., 3e4 Mizutani, U., 16e17, 19, 51e52 Mochzuki, M., 89 Moges, K., 35 Mondelli, C., 110 Moodera, J.S., 3, 10e11, 52, 54e55 Moolenaar, A.A., 170e171, 174e175, 174f, 195, 196fe197f, 229e230 Mootoo, D.M., 52 Morais, J., 21e22, 40 Moritomo, Y., 218 Morrison, C.A., 171e172, 184 Mortensen, K., 111e112 Mo¨ssbauer, R.L., 170e172, 172f, 229e230 Movshovich, R., 225e226, 227f Mudivathi, C., 110 Mukadam, M.D., 38te39t, 42e43, 48e49, 59e60 Mukhin, A.A., 89 Mulder, F.M., 162f Mulders, A.M., 146, 162f Mu¨ller, F.M., 2e3, 13 Mu¨ller-Buschbaum, P., 116 Murakami, T., 13 Murphy, D.W., 199e200 Murtaza, G., 61 Mydosh, J.A., 98, 127

N Nadgorny, B., 3, 10e11, 13, 54e55 Naganuma, H., 3e4, 32e33 Nagao, K., 20, 31 Nagarajan, R., 199e200 Nagarajan, V., 12, 199e200 Nagasako, M., 46e47 Nagatomo, D., 13e16, 19 Naghavi, S.S., 4, 36e37, 37f, 38te39t, 43e45, 49, 55, 59e60 Naish, V.E., 94e97 Nakatani, T.M., 13

Nakotte, H., 12, 120e121, 127e128 Nath, A., 196e197 Nehra, J., 37e40, 46 Nekvasil, V., 156, 167, 178e179, 181e182 Newman, D.J., 156 Niculescu, V.A., 26e27 Niesen, L., 148 Nieuwenhuys, G.J., 127 Nigam, A.K., 3e4, 9e10, 13e19, 16fe18f, 36e42, 38te39t, 41f, 44e47, 48f, 50e58, 50f, 53fe54f, 56f, 58f Nishibori, E., 218 Nishihara, H., 46e47 Nishimura, K., 179e180 Nishino, Y., 16e17, 19, 51e52 Noakes, D.R., 158 Noakes, T.J., 111e112 Nowak, J., 3, 10e11, 54e55, 176, 219e220 Nozar, P., 120e121

O O’Neill, H., 179e180 Ocio, M., 163, 216e217 Odin, C., 99 Ofer, S., 219e220 Ohkubo, T., 3e4, 30e32 Ohloff, K.-D., 120 Okada, H., 46e47 Okamura, S., 26e27 Okorokov, A.I., 110 Okulov, V.I., 50e51 Okulova, K.A., 50e51 Olesen, M.C., 171e172 Olson, C.G., 218 Omran, S.B., 61 Ono, M., 120e121 Onoda, S., 220 Oogane, M., 3e4, 32e33 Orbach, R., 164e165, 183 Orlowski, M., 167e169 Osofsky, M.S., 3, 10e11, 54e55 Ott, F., 110, 116, 118 Otto, M.J., 3 Ouardi, S., 4, 9e10, 16e17, 27e29, 28f, 36e37, 37f, 38te39t, 44e45, 50e55 Ouladdiaf, B., 96e98 ¨ zdogan, K., 8, 22, 44e49, 60e61 O ¨ zdogan, K., 9e10, 29, 56e57, 59, 61 O Ozerov, R.P., 68, 76, 94e97

Author Index

245

P

R

Pal, L., 16e17 Palatinus, L., 103e104 Palstra, T.T.M., 127 Panguluri, R.P., 13 Pannetier, M., 116 Papaefthymiou, G.C., 26e27 Papanikolaou, N., 6e8, 20, 22, 44e49, 60e61 Park, J.H., 218 Parkin, S.S.P., 2e7, 6f, 7t, 13, 54e55, 57, 59 Pauling, L., 9 Paulose, P.L., 12, 199e200 Pauthenet, R., 177 Perez-Mato, J.M., 94e95, 99 Perillo-Marcone, A., 111e112 Perkins, J., 26e27 Persson, B.I., 174e175 Peters, B., 35e36 Peters, J., 100 Petrenko, V.I., 110 Petr´ıcek, V., 94e95, 98e99, 103e104 Petrisor Jr., T., 35 Petukhov, A.G., 13 Petukhov, A.V., 110 Pfleiderer, C., 110 Philipse, A.P., 110 Picozzi, S., 57, 59 Pierce, D.T., 119 Pipich, V., 111e112 Poindexter, J.M., 170e172, 172f, 229e230 Pooke, D.M., 204e205 Pool, F.S., 50e51 Portnichenko, P.Y., 110, 116 Po¨ttgen, R., 98 Prandl, W., 178e179 Pranzas, K., 110 Pratt, F.L., 225e226, 227f Prestigiacomo, J.C., 51e52 Price, D.C., 171e174, 172f, 229e230 Prokes, K., 12, 98, 120e121, 127e128 Prokhnenko, O., 100 Prowse, D.B., 170e172 Przewoznik, J., 179e180 Pytlik, L., 130

Rabe, C., 111e112 Rabiller, Ph, 99 Rached, D., 61 Rached, H., 61 Radulescu, A., 111e112 Raja, M.M., 3e4, 9e10, 13e19, 15fe18f, 36e37, 38te39t, 40e41, 41f, 44e47, 48f, 50e58, 50f, 54f, 56f, 58f Rajanikanth, A., 3e4, 13, 30e32, 54e55 Rajnak, M., 110 Raju, N.P., 219 Ramakrishnan, T.V., 50e52 Ramazanaglu, M., 110 Ramirez, A.P., 163, 216e217 Rams, M., 169f, 218e220, 219f, 223e229, 228f Rao, T.V.C., 19e20 Rao, V.V., 20 Rawat, R., 12 Rebouillat, J.P., 177 Reifenberger, R., 50e51 Reiss, G., 32, 33f Reller, A., 12 Renker, B., 200 Reshak, A.H., 61 Ressouche, E., 89, 98 Reznik, D., 110 Ribeiro, J.L., 94e95, 99 Richter, D., 111e112 Riegel, D., 167, 199e200 Riegg, S., 12 Rietman, E.A., 167 Rietveld, H.M., 103e104 Ritter, C., 110, 169f, 220, 223 Robinson, R.A., 120e121, 127e128 Rodan, S., 12, 51, 53e54 Rodriguez-Carvajal, J., 96e97, 103e105, 130, 186, 187fe188f, 229e230 Roessli, B., 223 Roisnel, T., 103e105, 130 Rosenkranz, S., 163 Rossat-Mignod, J., 91e92, 105e106, 226e227, 228f Rudermann, M.A., 157 Rudowicz, C., 167e169 Rupp, A., 111e112 Russina, M., 111e112 Ryan, D.H., 177e178, 181e182, 181f Ryzhkov, M.V., 172

Q Quardi, S., 9e10, 16e17, 50e55 Quezel, S., 91e92

246 Author Index

S Sachmann, E., 116 Sa´ez-Puche, R., 186, 187fe188f, 190f, 193, 194f, 229e230 Saha, R., 19e20 Saini, H.S., 61 Saito, K., 3e4, 33, 34f Saito, T., 3e4 Sakarya, S., 171f, 186, 187fe188f, 190f, 193, 222e223, 222f, 225f, 228e230 Sakata, M., 218 Sakuma, A., 32e33, 34f Sakuraba, Y., 3e4, 32e33, 34f Salama, H.A., 177e182, 178f, 181f, 229e230 Saleh, A.S., 13e14 Sampathkumaran, E.V., 12, 199e200 Samson, Y., 116 Sanchez, J.P., 163, 170e171, 175, 200, 201f, 202e204, 202t, 204f, 213, 213f, 216e219, 219f, 224e226 S¸asıo glu, E., 8e10, 22, 29, 44e49, 56e57, 59e61 Sassik, H., 110 Sato, J., 32e33 Sato, K., 3e4, 33, 34f Sato, M., 218 Savary, L., 215e216 Savey-Bennett, M., 120 Sawicki, M., 179e180 Sayetat, F., 185e186 Scheunemann, K., 185e186 Schmalhorst, J., 32, 33f Schneider, H., 51 Schoenleber, A., 98 Scho¨nhense, G., 27e29, 28f, 44, 49 Schreyer, A., 116 Schuhl, A., 176e177, 180, 229e230 Schuller, I.K., 110, 194e195 Schulz, J.C., 111e112 Scott, D.R., 179e180 Scott, J.F., 68 Seaman, C.L., 194e195 Sechovsky, V., 12, 120e122, 126e128, 133, 135 Segre, C.U., 194e195 Sen, S.K., 161e163, 171e172 Shastry, B.S., 163, 216e217 Shenoy, G.K., 158 Shi, J., 110 Shigeta, I., 13e16 Shirai, M., 3e4, 20, 31, 33, 34f, 35, 46e47

Shirane, G., 106 Shirley, D.A., 181e182 Shull, C.G., 68, 72e73 Siddharthan, R., 163 Siebenbu¨rger, M., 111e112 Sikolenko, V., 220, 221fe222f Sikora, W., 130 Simmons, C.T., 167, 199e200 Singh, M., 61 Singh, P., 38te39t, 41e42, 48, 58e59 Skaftouros, S., 9e10 Skanthakumar, S., 195e198 Skumryev, V., 89 Slater, J.C., 9 Smart, J.S., 68 Smeibidl, P., 120 Smit, H.H.A., 167, 200 Soda, K., 16e17, 19, 51e52 Soderholm, L., 194e195 Sonier, J.E., 171f, 225e226, 227f Sonntag, R., 120e121 Sosin, S., 163, 216e217 Soulen, R.J., 3, 10e11, 54e55 Squires, G.L., 68, 76e77 Srinivas, V., 19e20 Srinivasan, A., 3e4, 40, 47, 54e55 Staderrmann, G., 200 Stadler, S., 51e52 Stankevich, V.G., 183, 184f Steenvoorden, M.P., 186, 187fe188f, 229e230 Steer, C.A., 225e226, 227f Stein, W.-D., 100 Steiner, W., 98, 110 Sternheimer, R.M., 161e163 Stevens, K.W.H., 153e154, 208e209 Stewart, G.A., 146, 152f, 161e163, 161f, 167, 168t, 171e174, 172f, 177e182, 178f, 181f, 186, 190f, 193, 200e202, 203t, 204e213, 207f, 209fe210f, 211t, 212f, 229e230 Stra¨ssle, T., 68, 99 Strauser, W.A., 72e73 Strecker, M., 206e208, 207f, 212e213 Strijkers, G.J., 54f Strunz, P., 111e112 Stryganyuk, G., 4, 36e37, 37f, 38te39t, 44e45, 51, 55 Su, Y., 220 Sudheesh, V.D., 37e40, 46 Sugimoto, S., 2e4, 13, 54e55

Author Index Sugiura, E., 120e121 Sukegawa, H., 13 Sundaresan, A., 38te39t, 43, 50 Sunshine, S., 199e200 Suresh, K.G., 1e62, 2e4, 9e10, 13e19, 15fe18f, 36e61, 38te39t, 41f, 48f, 50f, 53fe54f, 56f, 58f Sutton, I., 116 Suzuki, T., 127, 133 Svergun, D.I., 111 Svoboda, P., 120e121, 127e128 Swagten, H.J.M., 12, 24e27, 26fe28f Swainson, I., 171f

T Takabatake, T., 127, 133 Takahashi, H., 46e47 Takahashi, Y.K., 3e4, 9e10, 13e19, 15fe18f, 30e33, 34f, 36e40, 38te39t, 42, 44e47, 50e58, 50f, 53fe54f, 56f Takata, M., 218 Tanaka, C.T., 3, 10e11, 54e55 Tarnawski, Z., 179e180 Tche´ou, F., 185e186, 226e227, 228f Tennant, A., 120 Terada, N., 13e16, 73 Teterin, Y.A., 172 Tezuka, N., 2e4, 13, 26e27, 54e55 Thakur, G.S., 38te39t, 43, 50 Thakur, J., 61 Theroine, C., 100 Thiel, R.C., 161e163, 162f, 167, 200 Thomas, A., 32, 33f Thomasson, J., 170e171, 175 Timko, M., 110 Tiusan, C., 35 Tjon, J.A., 163e165 Tokura, Y., 68 Toperverg, B., 116 Torikachvili, M.S., 194e195 Torregrossa, J., 106 Toudic, B., 99 Tran, V.H., 120e121 Triplett, B.B., 184 Tristl, M., 116 Triyono, D., 110 Troc, R., 120e121 Tsidilkovski, I.M., 50e51 Tsuei, K.-D., 220 Tsunegi, S., 3e4 Tsurkan, V., 110, 116 Tun, Z., 171f

247

Turchanin, A., 32, 33f Turtelli, R.S., 110 Tymoshenko, Y.V., 110, 116

U Ueda, M., 3e4, 33, 34f Ueda, S., 12, 20e22, 22f, 27e29, 28fe30f Uemura, T., 35 Ulhaq-Bouillet, C., 35 Umetsu, R.Y., 46e47 Unguris, J., 119 Unterna¨hrer, P., 167e169, 168t, 200e204 Urcelay-Olabarria, I., 89

V Vaju, C., 223 van Bruggen, C.F., 3 van der Berg, J., 127 van der Kraan, A.M., 146, 161e163, 165, 166f, 168t, 196e197 van der Laan, G., 194e195 van der Valk, P.J., 3 van Engen, P.G., 2e3, 13 van Loef, J.J., 165, 166f, 168t, 170e171, 174e175, 174f, 195, 196fe197f, 229e230 van Smaalen, S., 98 Van Uitert, L.G., 176, 199e200 van Woerden, R.A.M., 3 Varaprasad, B., 13 Varaprasad, B.S.D.Ch. S., 3e4, 9e10, 13e19, 15fe18f, 36e40, 38te39t, 42, 44e47, 50e58, 50f, 53fe54f, 56f Varma, M.R., 38te39t, 41e42, 48, 58e59 Vas, A., 170e172 Vasundhara, M., 20 Venkateswara, Y., 36e37, 38te39t, 41e42, 44e45, 48, 58e59 Venugopalan, K., 37e40, 46 Ve´rtes, A., 196e197 Vidal, E.V., 51 Vincent, E., 200, 201f, 202t Violet, C.E., 194e195 Vlastuin, R.F.M., 127 Vogtt, K., 111e112 Volkonskiy, O., 12, 51, 53e54 Von Dreele, R.B., 103e104 Voyer, C.J., 177e178, 181e182, 181f Vulliet, P., 163, 216e219, 219f, 224e226

248 Author Index

W Wacklin, H.P., 116 Wagner, F.E., 13e14, 174e175 Walsh, G.R., 120 Wang, L.Y., 59 Wang, W.H., 9e10, 56e57 Wang, X.L., 9e10, 54e55 Wang, X.T., 59 Ward, R.C.C., 225e226, 227f Wasniowska, M., 179e180 Watson, R.E., 25e26 Webster, P.J., 21 Weddemann, A., 32, 33f Wegener, H., H., 145e146, 200e202 Werder, D., 167 Wernick, J.H., 170e171 Wertheim, G.K., 145e146 Wickman, H.H., 181e182 Wiedenmann, A., 110e112, 117e118 Wielinga, R.F., 224 Wiesinger, G., 110 Wijngaard, J., 3 Wilkinson, M.K., 72e73 Williams, H.J., 176 Wilpert, T., 100 Winterlik, J., 4, 36e37, 37f, 38te39t, 43e46, 49, 55, 59e60 Wirtz, K., 99 Wit, H.P., 148 Wojcik, M., 2e3, 13, 24e27, 26fe28f, 54e55 Wolf, T., 110 Wollan, E.O., 72e73 Wollmann, L., 35 Wong, J., 194e195 Wortman, D.E., 184 Wortman, G., 167, 168t, 200e202, 203t, 204e205 Wortmann, G., 167, 197e200, 198f, 206e208, 207f, 212e213 Wosnitza, J., 110, 116 Wu, G., 37e40, 38te39t, 46e47 Wu, G.H., 9e10, 56e57 Wurmehl, S., 12, 20e27, 23f, 25fe28f, 35e36, 40, 44, 47e49, 51, 53e54 Wuttig, M., 110

X Xu, G.Z., 9e10, 56e57

Xu, H., 37e40, 38te39t, 46e47 Xu, Sh., 218

Y Yadav, A.K., 36e37, 44e45 Yagmurcu, A., 110, 120e121 Yamada, S., 13 Yamamoto, M., 35 Yang, F.J., 32e33, 34f Yang, F.Y., 35e36, 54f Yang, K.N., 194e195 Yano, I., 13e16 Yao, K.L., 9e10, 57 Yaouanc, A., 163, 169f, 171f, 186, 187fe188f, 190f, 193, 206e208, 220, 221fe222f, 222e223, 225f, 228e230 Yasui, Y., 220 Ye, W., 127, 133 Yoon, S., 13e16 Yosida, K., 157 Young, D.P., 51e52 Yu, X., 37e40, 38te39t, 46e47 Yuan, H., 57 Yusuf, S.M., 38te39t, 42e43, 48e49, 59e60, 110

Z Zaccai, G., 68 Zachariasen, W.H., 108 Zahurak, S.M., 167, 199e200 Zarubicka, V., 177 Zeleny, M., 127 Zerarga, F., 61 Zhang, H.G., 9e10 Zhang, K., 194e195 Zhang, X.M., 9e10 Zhang, Y.J., 60e61 Zheng, Y., 35 Zherlitsyn, S., 110, 116 Zhou, H., 194e195 Zhou, Y., 57 Zhu, W., 37e40, 38te39t, 46e47 Zhu, X., 37e40, 38te39t, 46e47 Zimmermann, U., 186, 187fe188f, 190f, 193, 229e230 Zinkin, M., 73 Zounova, F., 127 Zukrowski, J., 179e180

Subject Index ‘Note: Page numbers followed by “f” indicate figures and “t” indicate tables.’

A Andreev reflection, 10e11, 11f Anisotropic magnetoresistance effect (AMR effect), 32e33 Australian Nuclear Science and Technology Organization (ANSTO), 102 Axial vectors, 74e75

B Bohr magneton, 149 Bound-atom neutron scattering, 79f Bragg’s law, 70

C CEF. See Crystalline electric field (CEF) Co atoms, 8 Co-based alloys, 3e4 Co2-based alloys, 20e36. See also Fe2-based alloys Co2Cr1exVxAl, Co2V1xFexAl, Co2Cr1xFexAl alloys, 30e32 Co2FeAl1exSix alloys, 32 Co2Mn1exFexSi alloys, 20e29, 27f, 30f, 34f thin films and devices, 32e36 Co2(Mn, Fe)Si, 35 Co2Cr1xFexAl alloys, 30e32 magnetic properties, 31 spin polarization, 31e32 structural properties, 30e31 Co2Cr1exVxAl alloys magnetic properties, 31 spin polarization, 31e32 structural properties, 30e31 Co2CrAl alloys, 20, 31e32 Co2FeAl1exSix alloys, 32 Co2Mn1exFexSi alloys, 27fe28f, 30f, 34f electronic structure calculations, 29 magnetic properties, 22e24 NMR, 24e27 structural properties, 20e21, 22f

XPS, 27e29, 28f Co2V1xFexAl alloys, 30e32 magnetic properties, 31 spin polarization, 31e32 structural properties, 30e31 Co2Val alloy, 31 Co2YAl alloys, 20 CoFeCrAl alloy, 55 CoFeCrGa alloy, 40, 58, 58f CoFeCrGe alloys, 41e42, 48 CoFeCrZ alloys, 37e40, 46e47, 52e53, 57e59 zero field electrical resistivity variation with temperature, 53f CoFeMnSi alloys, 36e37, 50e51, 56e57, 56f CoFeMnZ alloys, 36e37, 37f, 44e46, 50e52, 54e57 normalized differential conductance curves, 54f temperature dependence of electrical conductivity, 50f CoFeTiAl alloys, 41, 47e48 CoFeTiZ alloys, 60e61 Cold source, 99e100 Commensurate magnetic structures, 85e87, 86t CoMnCrAl alloys, 41e42, 48 CoMnVAl alloys, 41, 47e48 CoRuFeZ alloys, 40e41, 41f, 47, 48f, 53e54 Cr-rich phase, 31 Crystal fields, 153e156 selected ionic properties of trivalent rare earth, 154t total angular momentum parameters of trivalent rare earth, 155t Crystal structure of Heusler alloys, 4e5, 5fe6f site occupancy, general formula, and structure type, 7t Crystalline electric field (CEF), 184 Crystallographic unit cells, 81e82

249

250 Subject Index Cu2MnAl alloy, 2e3 CuCoMnGa alloys, 49e50, 60 Curie temperature, 92e93

D Data collection methods, 100e120 Laue technique, 102 neutron reflectivity, 118e119 neutrons and nano-objects, 110e111, 113f powder diffraction, 102e106 sample environment, 119e120 SANS, 111e118, 111f, 115f single crystal experiments, 106e109 use of polarized neutrons, 109e110 Debye-Waller factor, 81e82 Density of states (DOS), 5e6, 27e29, 29f Diamagnetism, 71e72 Differential thermal analysis (DTA), 36e37 Diffraction theory for nanoparticles, 112 DOS. See Density of states (DOS) DTA. See Differential thermal analysis (DTA) DyPO4, 164e165

E EDS. See Energy-dispersive X-ray spectroscopy (EDS) Elastic neutron diffraction classification of magnetic structures, 84e92 formulas, 76e78 group representation theory, 96e99 identification of magnetic signal in practice, 92e96 instrumentation, 99e120 magnetic diffraction, 82e84 magnetic materials, 68 and magnetic structure, 70e75 magnetic structure determinations, 120e136 nuclear diffraction, 78e82 properties of neutron, 68e70, 69t Elastic scattering, 77e78 Electric quadrupole interaction, 148 Electrical resistivity, 11e12 Electronic relaxation, 165 Electronic structure, 5e6 calculation, 55e61 Co2Mn1-xFexSi alloys, 29 CoFeCrZ alloys, 57e59 CoFeMnZ alloys, 55e57 CoFeTiAl alloys, 59

CoFeTiZ alloys, 60e61 CoMnCrAl alloys, 59 CoMnVAl alloys, 59 CuCoMnGa alloys, 60 Fe2-xCoxMnSi alloys, 19 NiCoMnZ alloys, 59e60 NiFeMnGa alloys, 60 Electrostatic repulsion, 6e7 Energy-dispersive X-ray spectroscopy (EDS), 43 EQHAs. See Equiatomic quaternary Heusler alloys (EQHAs) Equiatomic Heusler alloys. See Equiatomic quaternary Heusler alloys (EQHAs) Equiatomic quaternary Heusler alloys (EQHAs), 2e3, 36e61, 38te39t. See also Quaternary Heusler alloys electronic structure calculation, 55e61 magnetic properties, 44e50 magneto transport properties, 50e54 results, 36e61 spin polarization using PCAR, 54e55 structural aspects, 36e43 European Spallation Source (ESS), 100 External magnetic field, 12

F Face-centered cubic (FCC), 4 FCHMF. See Fully compensated half-metallic ferrimagnetism (FCHMF) Fe2-based alloys, 19e20. See also Co2-based alloys Fe2-xCoxMnSi alloys, 13e19, 15f Fe2-xCoxMnSi alloys, 16f, 18f electronic structure calculations, 19 magnetic properties, 14e16 magnetotransport properties, 16e18 spin polarization, 18e19 structural properties, 13e14 Fermi golden rule, 181 Fermi pseudopotential, 78e79 Fermi’s golden rule, 77 Ferromagnetic alloys (FM alloys), 2e3 Ferromagnetic relaxation, 164e165 FHAs. See Full Heusler alloys (FHAs) First-order magnetostructural transition, 3 Flipping ratio method, 109e110 FM alloys. See Ferromagnetic alloys (FM alloys) Four-cycle geometry, 106e108 Full Heusler alloys (FHAs), 2e3

Subject Index

251

Full-potential linearized augmented planewave method (FPLAPW method), 57 Fully compensated half-metallic ferrimagnetism (FCHMF), 59

Inverse Heusler alloy (IHA), 4 IS. See Isomer Shift (IS) Isomer Shift (IS), 147 Isothermal magnetization measurements, 16

G

K

Garnet compounds, 229e230 GdAlO3compound, 175 Giant magnetoresistance (GMR), 3 GISANS. See Grazing incident small-angle neutron scattering (GISANS) GMR. See Giant magnetoresistance (GMR) Graphical method, 104e105, 105f Grazing angle GISANS, 116 Grazing incident small-angle neutron scattering (GISANS), 110, 116 Group representation theory, 96e99

K magnetic domains, 91e92

H Half Heusler alloys (HHAs), 2e8 Half-metallic band gap, 27e29 Half-metallic ferromagnetic materials (HMF materials), 3 HMF materials, potential applications of, 13 Slater-Pauling rule for, 9 Half-metallic gap, origin of, 6e9, 7fe8f Hamiltonian function, 153 Helmholtz-Zentrum Berlin (HZB), 100 Heuasler alloys, 13e14 Heusler alloys, crystal structure of, 4e5 HHAs. See Half Heusler alloys (HHAs) High-Tc superconductors, 206 HMF materials. See Half-metallic ferromagnetic materials (HMF materials) Hund’s rules, 71, 153 Hyperfine field, 25e26, 26f HZB. See Helmholtz-Zentrum Berlin (HZB)

I IHA. See Inverse Heusler alloy (IHA) ILL. See Institut Laue-Langevin (ILL) Imma space group, 120e121 Incommensurate magnetic structures, 87e89 Institut Laue-Langevin (ILL), 102, 107f Instrumentation data collection methods, 100e120 neutron sources, 99e100 Intermediary valency, 229e230

L Laboratoire Le´on Brillouin (LLB), 111e112 Laue technique, 102 Line positions and intensities, 149

M Magnetic diffraction, 82e84 field, 16e17 form factor, 82e83 hyperfine interaction, 148e149 interaction, 156e159 vector, 84 materials, 68, 70e75 magnetic structure, 72e75 origin of magnetic moments, 70e72 moments origin, 70e72 ordering, 72e73, 92 in UNiGa, 127 relaxation, 163e165 scattering amplitude, 112 signal identification in practice, 92e96 determination of magnetic structures, 92e95 limitations of neutron technique, 95e96 thin films, 110e111 Magnetic properties Co2Cr1-xVxAl, Co2V1-xFexAl, Co2Cr1-x FexAl alloys, 31 Co2Mn1exFexSi alloys, 22e24 CoFeCrGe alloys, 48 CoFeCrZ alloys, 46e47 CoFeMnZ alloys, 44e46 CoFeTiAl alloys, 47e48 CoMnCrAl alloys, 48 CoMnVAl alloys, 47e48 CoRuFeZ alloys, 47, 48f CuCoMnGa alloys, 49e50 EQHAs, 44e50 Fe2-xCoxMnSi alloys, 14e16 MnNiCuSb alloy, 50 NiCoMnGa alloys, 49e50

252 Subject Index Magnetic properties (Continued ) NiCoMnZ alloys, 48e49 NiFeMnGa alloys, 49e50 Magnetic structure, 70e75 classification, 84e92, 88f commensurate magnetic structures, 85e87, 86t incommensurate magnetic structures, 87e89 K magnetic domains, 91e92, 93f multi-k structures, 89e91 S magnetic domains, 91e92, 93f determination, 92e95 determinations powder sample of UPdSi, 120e126, 124t, 126f single crystal of UNiGa, 126e136, 128fe129f, 131te132t, 135fe137f origin of magnetic moments, 70e72 Magnetism in Heusler alloys, 9e10 Magneto transport properties, 50e54 CoFeCrZ alloys, 52e53 CoFeMnZ alloys, 50e52 CoRuFeZ alloys, 53e54 Magnetoresistance (MR), 12, 17f, 34f ratios, 3e4 Magnetotransport properties, 16e18 Majority spin band henceforth, 5e6 Metamagnetic-like transition (MT), 127 Mn2CoAl alloy, 9e10, 16e17 MnNiCuSb alloy, 50 Mo¨ssbauer effect, 145e147 parameters of rare earth nuclei with, 150te151t Mo¨ssbauer spectroscopy (MS), 13e14, 15f analysis, 165e170 TmBa2Cu3O7-x, 165e169 Yb2Ti2O7, 169e170 crystal fields, 153e156 Hund’s rules, 153 magnetic interaction, 156e159 magnetic relaxation, 163e165 methodology, 149e153 nuclear energy levels, 147e149 pure electric quadrupole interaction, 152f recoilless fraction, 147 on rare earth-based oxides, 146 relationship to Mo¨ssbauer parameters, 160e163 crystal field contributions, 161f lattice contribution, 162t QS, 162f

MR. See Magnetoresistance (MR) MS. See Mo¨ssbauer spectroscopy (MS); Saturation magnetization (MS) MT. See Metamagnetic-like transition (MT) Multi-k structures, 89e91 Muon spin relaxation (mSR), 188e189, 216, 220

N Nano-objects, 110e111, 113f Neutron(s), 68, 110e111 mass, 69 properties, 68e70, 69t reflectivity, 118e119 reflectometry, 118e119 sources, 99e100 technique limitations, 95e96 NiCoMnAl alloys, 43 NiCoMnGa alloys, 49e50 NiCoMnGe alloys, 43 NiCoMnZ alloys, 42e43, 48e49, 59e60 NiFeMnGa alloys, 49e50, 60 NiMnSb alloys, 3, 13 NMR. See Nuclear magnetic resonance (NMR) Nonsuperconducting (NSC), 194 Normal-beam geometry, 106e108 Nuclear diffraction, 78e82 Nuclear energy levels, 147e149 electric quadrupole interaction, 148 IS, 147 line positions and intensities, 149 magnetic hyperfine interaction, 148e149 Nuclear magnetic resonance (NMR), 24e27

O Off-specular scattering. See Scattering in incidence plane Orthorhombic compounds, 208e209

P Paramagnetic state, 72e73 Pauli Exclusion Principle, 70e71 PCAR spectroscopy. See Point contact Andreev reflection spectroscopy (PCAR spectroscopy) Peak overlapping, 103e104 Pnma space group, 120e121, 121f PNR. See Polarized neutron reflectivity (PNR)

Subject Index Point contact Andreev reflection spectroscopy (PCAR spectroscopy), 10, 11f spin polarization using, 54e55 Polar vector, 74e75, 76f Polarized neutron reflectivity (PNR), 110e111 Polarized neutron reflectometry, 119 Polarized neutrons use, 109e110 Position-sensitive detectors (PSDs), 103 Potential applications of HMF materials, 13 Powder diffraction, 93e94, 102e106 Powder neutron diffraction pattern, 125f Propagation vectors, 105e106 PSDs. See Position-sensitive detectors (PSDs) Pseudoternary Heusler alloys, 2e3

Q QS. See Quadrupole-splitting (QS) Quadratic-field dependence, 12 Quadrupole-splitting (QS), 160 Quasi-half-metallic ferrimagnet, 60e61 Quaternary Heusler alloys. See also Equiatomic quaternary Heusler alloys (EQHAs) cartoon showing schematic density of states, 3f crystal structure of Heusler alloys, 4e5 electrical resistivity, 11e12 electronic structure, 5e6 magnetism in Heusler alloys, 9e10 origin of half-metallic gap, 6e9, 7fe8f potential applications of HMF materials, 13 results on substituted, 13e36 Co2-based alloys, 20e36 Fe2-based alloys, 13e20 spin polarization, 10e11

R R2BaMO5 compounds, 206e215 crystal field parameters for Tm Sites, 211t hyperfine interaction parameters for Gd2BaCuO5, 208t induced magnetic moments, 212f magnetic ordering temperatures and magnetic moments of, 215t quadrupole-splitting, 210f representative 169Tm Mo¨ssbauer spectra of Tm2BaCuO5, 209f sites in Yb2BaCuO5, 214t

253

temperature dependence of magnetic hyperfine field, 207f 170 Yb Mo¨ssbauer absorption in Yb2BaCuO5, 213f R2M2O7 compounds, 215e225 155 Gd Mo¨ssbauer spectra of Gd2Sn2O7, 217f low-temperature heat capacity of Yb2Sn2O7, 221f temperature dependence of Gd3+ 4f shell magnetic moment, 219f temperature dependence of Yb3+ moments fluctuation rates, 222f, 225f R2O3 compounds, 170e175 161 Dy Mo¨ssbauer spectrum, 173f electric quadrupole-splitting vs. temperature of Tm2O3, 172f 141 Pr Mo¨ssbauer spectra, 174f R3M5O12 compounds, 225e229 temperature dependence of correlated Gd3+ moments fluctuation rate, 227f thermal variations in Yb3Ga5O12, 228f transition temperatures, 230t zero-field mSR rate vs. temperature, 228f Rare earth Mo¨ssbauer spectroscopy and methodology, 147e153 nuclear energy levels, 147e149 parameters of rare earth nuclei with Mo¨ssbauer effect, 150te151t pure electric quadrupole interaction, 152f recoilless fraction, 147 Rare earth-based oxides, 170e229 Mo¨ssbauer spectroscopy on, 146 R2BaMO5 compounds, 206e215 R2M2O7 compounds, 215e225 R2O3 compounds, 170e175 R3M5O12 compounds, 225e229 RBa2Cu3O7 compounds, 194e206 RMO3 compounds, 175e182 RMO4 compounds, 182e194 RBa2Cu3O7 compounds, 194e206 comparison of determined crystal field parameters for NSC, 203t 161 Dy Mo¨ssbauer absorption, 201f 161 Dy Mo¨ssbauer-derived hyperfine parameters, 202t 166 Er Mo¨ssbauer spectra, 204f hyperfine parameters, 204t IS values, 197f Mo¨ssbauer absorption, 199f 141 Pr Mo¨ssbauer spectra, 196f

254 Subject Index RBa2Cu3O7 compounds (Continued ) rank-2 crystal field parameters, 205t temperature dependence of 155Gd resonance line width, 198f 170 Yb Mo¨ssbauer absorption, 206f Recoil energy, 147 Recoilless fraction, 147 RMO3 compounds, 175e182 RMO4 compounds, 182e194 crystal field parameters, 186t GdCrO4 155 Gd Mo¨ssbauer spectra of, 187f temperature dependence of magnetic hyperfine field, 188f inelastic neutron scattering measurements of TmCrO4, 191f inverse temperature dependence of spinlattice relaxation, 184f temperature dependence of 169Tm hyperfine field, 191f temperature dependence of relaxation rate, 192f thermal variation, 185f 169 Tm Mo¨ssbauer spectra measurement, 190f 170 Yb Mo¨ssbauer spectra of YbCrO4 oxide, 194f Room temperature (RT), 3e4

S S magnetic domains, 91e92 SANS. See Small-angle neutron scattering (SANS) Saturation magnetization (MS), 16 SC. See Superconductor contact (SC) SC transition. See Superconducting transition (SC transition) Scattering in incidence plane, 116 Scattering process, 69 Scattering vector, 76e77 SGS. See Spin gapless semiconductor (SGS) Shubnikov groups, 94e95 Sine waveemodulated collinear magnetic structures, 89 Single crystal experiments, 106e109 Single crystal neutron diffraction, 128, 135. See also Elastic neutron diffraction reciprocal space of UNiGa mapped using, 128fe129f single crystal of UNiGa, 126e136 Single-k structures, 89e90 magnetic structures, 85, 91f

Slater-Pauling curve, 9 Slater-Pauling rule, 22, 44e45 for HMFs, 9 Small angle scattering, 111 Small-angle neutron scattering (SANS), 110e118, 111f, 115f Spin gapless semiconductor (SGS), 3 “Spin ice” behavior, 215e216 Spin polarization (Pc), 3, 10e11 Co2Cr1-xVxAl, Co2V1-xFexAl, Co2Cr1-x FexAl alloys, 31e32 Fe2-xCoxMnSi alloys, 18e19 using PCAR, 54e55 CoFeCrAl alloy, 55 CoFeMnZ alloys, 54e55 Spin-lattice process, 164e165 Spineorbit coupling, 70e71 Spinup band. See Majority spin band henceforth mSR. See Muon spin relaxation (mSR) Structural aspects CoFeCrGe alloys, 41e42 CoFeCrZ alloys, 37e40 CoFeMnZ alloys, 36e37, 37f CoFeTiAl alloys, 41 CoMnCrAl alloys, 41e42 CoMnVAl alloys, 41 CoRuFeZ alloys, 40e41, 41f CuCoMnGa alloys, 43 MnNiCuSb alloy, 43 NiCoMnGa alloys, 43 NiCoMnZ alloys, 42e43 NiFeMnGa alloys, 43 Structural properties Co2Cr1-xVxAl, Co2V1-xFexAl, Co2Cr1-x FexAl alloys, 30e31 Co2Mn1exFexSi alloys, 20e21, 22f Fe2-xCoxMnSi alloys, 13e14 Superconducting band gap, 18e19 Superconducting transition (SC transition), 165e167 Superconductor contact (SC), 10 Superspace symmetry formalism, 99

T Tetragonal zircon (ZrSiO4), 182e183 Theory of neutron scattering, 76, 76f Thermal source, 99e100 Thin films and devices, 32e36 Three-dimensional materials, 110 Time-of-flight method (TOF), 100. See also Data collection methods

Subject Index diffractometers and spectrometers, 100 instrument layout of, 101f TmBa2Cu3O7-x, 165e169 comparison of crystal field parameters, 162t temperature dependence of 169Tm Mo¨ssbauer spectra, 166f of electric quadrupole-splitting, 166f TMR. See Tunneling magnetoresistance (TMR) TOF. See Time-of-flight method (TOF) Top Co2Mn1xFexSi layer thickness (tCFMS), 33 Total magnetic moment (Mt), 6e7 Transport spin polarization, 10 Tunneling magnetoresistance (TMR), 13, 33f Two-dimensional systems, 110

U UB-matrix, 106e108 Ultrahigh-vacuum (UHV), 32e33

255

UNiGa, single crystal of, 126e136, 128fe129f, 131te132t, 135fe136f UPdSi, powder sample of, 120e126, 124t, 126f

V Vienna ab initio simulation package (VASP), 56e57

X X atoms, 9 X-ray absorption spectra (XAS), 22e23, 23f, 25f X-ray diffraction (XRD), 13e14 X-ray magnetic circular dichroism (XMCD), 22e23, 23f X-ray photoemission spectroscopy (XPS), 27e29, 28f

Y Yb2Ti2O7, 169e170

Material Index ‘Note: Page numbers followed by “f” indicate figures and “t” indicate tables.’

A Aluminum, 79e81

B Ba2TiSi2O8, 98e99 BiF3, 7t Boron, 79e81

C Ca2MgSiO7, 98e99 Cadmium, 79e81 CeCu2 type structure, 120e122 CeRuSn, 98 Co/Pd thin film, 119 Co2-based alloys, 20e36 Co2(Fe0.4Mn0.6)Si, 3e4 Co2(Mn,Fe)Si thin films, 35 Co2(Mn,Fe)Si/MgO/Co2(Mn,Fe)Si tunnel junctions, 35 Co2Cr0.5V0.5Al alloys, 31e32 Co2Cr0.6Fe0.4Al alloys, 30e31 Co2Cr1-xFexAl alloys, 30e32 Co2Cr1-xVxAl alloys, 30e32 Co2CrAl alloys, 20, 31e32 Co2Fe(Ga0.5Ge0.5), 3e4, 51e52, 55 Co2Fe1-xVxAl alloys, 30e32 Co2FeAl alloy, 26e27 Co2FeAl0.5Si0.5 films, 35e36 Co2FeAl1-xSix, 32 Co2FeSi, 21e22, 29, 29f Co2FeZ, 40 Co2Mn0.5Fe0.5Si, 29 Co2Mn0.6Fe0.4Si, 22e23 Co2Mn0.6Fe0.4Si/Ag/Co2Mn0.6Fe0.4Si structure, 32e33 Co2Mn1-xFexSi alloys, 20e29, 27f Co2MnAl alloy, 51 Co2MnGe alloy, 8, 23e24 Co2MnSi alloy, 21e24, 29, 29f, 35 Co2MnZ alloy, 20 Co2V1-xFexAl alloys, 30e32 Co2Val alloy, 31e32

Co2VGa alloy, 46e47 Co2YAl alloys, 20 Co2YZ alloy, 20 CoCrTiAs alloy, 59 CoCrVSi alloy, 59 CoFeCrAl alloy, 38te39t, 40e41, 46, 52, 55, 57e58 CoFeCrGa alloy, 36e37, 38te39t, 40, 46e47, 52e53, 57e58, 58f CoFeCrGe alloys, 38te39t, 41e42, 48, 57e59 CoFeCrSi alloys, 57 CoFeCrZ alloys, 37e40, 46e47, 52e53, 57e59 CoFeMnAl alloy, 36e37, 38te39t, 45, 56 CoFeMnAl(Ga) alloy, 45e46 CoFeMnGa alloy, 38te39t, 44e45 CoFeMnGe alloy, 38te39t, 44e45, 51e52, 55 CoFeMnSi alloy, 36e37, 38te39t, 44e45, 50e57, 56f CoFeMnSi(Ge) alloy, 45e46 CoFeMnZ alloys, 36e37, 37f, 44e46, 50e52, 54e57, 61 CoFeScZ alloys, 61 CoFeTiAl alloys, 38te39t, 41, 47e48, 59 CoFeTiGe alloys, 60e61 CoFeTiSb alloys, 61 CoFeTiSi alloys, 60e61 CoFeTiSn alloys, 60e61 CoFeTiZ alloys, 60e61 CoMnCrAl alloys, 38te39t, 41e42, 48, 59 CoMnTiSi alloys, 59 CoMnTiZ alloys, 61 CoMnVAl alloys, 38te39t, 41, 47e48, 59 CoRuFeGe alloys, 38te39t, 40e41, 47, 53e54 CoRuFeSi alloy, 38te39t, 40e41, 47e48, 53e54 CoRuFeZ alloys, 40e41, 41f, 47, 48f, 53e54 Cr2P2O7, 98 CsCl, 7t Cu2MnAl alloy, 2e3, 7t

257

258 Material Index Cubic C-type Yb2O3, 175 CuCoMnGa alloys, 38te39t, 43, 49e50, 60

D Distorted perovskite compound Dy, 89, 154e155 Dy2O3, 170e171 Dy2Ti2O7, 219e220 DyBa2Cu3O7ex, 200 DyCrO3, 176 DyFeO3, 176 DyPO4, 164e165

E Equiatomic quaternary Heusler alloys (EQHAs), 2e5, 38te39t, 40, 61 Er, 90, 154e155 Er2O3, 170e171 Eu1-x PrxBa22Cu3O7-d, 196e197 EuBa2ex$xPrxCu3O7-d, 196e197

F Fe-Nb-B alloys, 117 [Fe1-xCox]2MnAl, 19 Fe1.4Co0.6MnSi, 13e14 Fe1.8Co0.2MnSi, 15f Fe1.95Co0.05MnSi, 15f Fe2-based alloys, 13e20 Fe2-xCoxMnSi alloys, 13e19, 15fe16f, 18f Fe2MnSi alloy, 13e16 Fe2O3, 72e73 Fe2VAl1-xBx, 20 Fe2VAl1-xGex alloys, 19 Fe3O4, 72e73, 117 FeCrMnSb alloys, 61 FeCrVAs alloys, 59 FeMnCrAl alloys, 59 FeMnTiAs alloys, 59 FeMnVSi alloys, 59 Ferrofluids, 110 Full Heusler alloys (FHAs), 2e5, 9e10

G Gadolinium, 79e81 Gd, 146, 229 Gd2BaNiO5, 212e213 Gd2M2O7 compounds, 216 Gd2Mo2O7, 218e219 Gd2O3, 170e171

Gd2Sn2O7, 216e217 Gd2Ti2O7, 163, 217e218 Gd3Ga5O12, 225e226 GdAlO3 compound, 175 GdBa2Cu3O6.9, 167 GdBa2Cu3O7-d, 200 GdCrO4, 186e187 GdM2Si2, 161e163 Ge, 120e121

H h-TmCrO3, 177e178 h-YbMnO3, 181e182 Half Heusler alloys (HHAs), 2e5, 9 Half-metallic Co2MnSi films, 32e33 Heusler alloys, 3e5, 9e10, 13, 44, 51e52 Hg, 100 HgCdTe, 50e51 Ho, 89e90, 154e155 Ho2Ti2O7, 163, 215e216 HoBa2Cu33O6.9, 167 Hole-doped RMnO3, 175

I Iron, 180 Isotropic, scattering, 78e79, 117

L La, 153 LiMgPdSn, 4, 7t Lu, 153

M Magnetic semiconductors, 110e111 Magnetic thin films, 110e111 Metal thin films, 110e111 (Mn + Fe)-deficient Co2(Mn,Fe)Si electrodes, 35 (Mn + Fe)-rich Co2(Mn,Fe)Si electrodes, 35 Mn0.97Ni-Cu0.95Sb, 43 Mn2CoAl alloy, 9e10, 50e51 MnF2, 72e73 MnNiCuSb alloy, 38te39t, 43, 50 MnO, 68, 72e73 MnWO4-related material, 89

N Na2CO3, 98 Nd, 154e155

Material Index NiCoCrGa, 61 NiCoMnAl alloys, 38te39t, 42e43, 59e60 NiCoMnGa alloys, 38te39t, 43, 49e50, 59e60 NiCoMnGe alloys, 38te39t, 42e43, 48e49, 59e60 NiCoMnSn alloy, 38te39t, 42e43, 48e49, 59e60 NiCoMnZ alloys, 42e43, 48e49, 59e60 NiFeMnGa alloys, 38te39t, 43, 49e50, 60 NiFeTiZ alloys, 61 NiMnSb alloy, 3, 6e7, 13

O Orthorhombic compounds, 208e209

P Pb, 100 Pd-doped FeRh thin films, 119 Perovskite RMO3 compounds, 175 Polycrystalline compound EuBa1.3Pr0.7Cu3O7-d, 196e197 Pr, 72e73, 154e155 Pr0.7Ca0.3MnO3, 115e116 Pr0.925Gd0.075Ba2Cu3O7ed, 197e198 Pr1exYxBa2Cu3O7, 198e199 Pr2O3, 174e175 Pr6 O11, 174e175 PrBa2Cu3O7-y, 194e195 PrBa2Cu3O7, 174e175, 198e199 PrO2, 170e171, 174e175 Pyrochlore lattice compounds, 215e216

Q Quasi-half-metallic ferrimagnet, 60e61 Quaternary Heusler alloys, 13e36

R R2BaMO5 compounds, 206e215 R2M2O7 compounds, 170, 215e225 R2Mo2O7, 218 R2O3 compounds, 170e175 R3M5O12 compounds, 225e229 RAlO3, 180

RBa2Cu3O7 compounds, 194e206 RMO3 compounds, 175e182 RMO4 compounds, 182e194

S Sm, 154e155 Soft magnetic Fe-Si-B-(Nb,Cu), 117 Spin gapless semiconductor (SGS), 3

T T-late transition metal, 120e121 Tb, 89, 154e155 Tb2Sn2O7, 223 Tb2Ti2O7, 163 Thulium ethylsulfate (TmES), 171e172 TiNiSi-type of structure, 120e121 Tm, 72e73, 146, 154e155 Tm2BaCoO5, 208e209 Tm2BaNiO5, 208e209, 212e213 Tm2Cu2O5, 208e209 Tm2O3, 170e172 TmAlO3, 176 TmAsO4, 193 TmBa2-Cu33O6.6, 167 TmBa2-Cu3O6.9, 165e167 TmBa2Cu3O6.6, 165e167 TmBa2Cu3O7-x, 165e169, 200e202 TmCrO3, 177 TmFeO3, 180 TmNi5, 161e163 TmVO3, 176e177 TmVO4, 193

U UNiGa, 126e136 UPdSi, 120e126 Uranium atoms, 120e122

V Vanadium, 79e81, 176e177

Y Yb, 146, 154e155, 193e194, 229 Yb2BaCuO5, 213 Yb2GaSbO7, 223 Yb2Mo2O7, 224e225

259

260 Material Index Yb2O3, 170e171 Yb2Sn2O7, 220 Yb2Ti2O7, 165, 169e170, 220 Yb3Ga5O12, 225e229 Yb3Rh4Sn13, 116 YbA1O3, 180e181 YBa2Cu3O7, 205

YbCrO4, 193 Yttrium, 194e195

Z Zircon (ZrSiO4), 182e183 ZnCr2Se4, 116