Characterization and Measurement of Magnetic Materials (Electromagnetism)

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Measurement and Characterization of Magnetic Materials

A Volume in the Elsevier Series in Electromagnetism

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Elsevier Series in Electromagnetism (Series formerly known as Academic Press Series in Electromagnetism)

Edited by ISAAK MAYERGOYZ

Electromagnetism is a classical area of physics and engineering that still plays a very important role in the development of new technology. Electromagnetism often serves as a link between electrical engineers, material scientists, and applied physicists. This series presents volumes on those aspects of applied and theoretical electromagnetism that are becoming increasingly important in modern and rapidly development technology. Its objective is to meet the needs of researchers, students, and practicing engineers.

Books Published in the Series Giorgio Bertotti, Hysteresis in Magnetism: For Physicists, Material Scientists, and Engineers Scipione Bobbio, Electrodynamics of Materials: Forces, Stresses, and Energies in Solids and

Fluids Alain Bossavit, Computational Electromagnetism: Variational Formulations,

Complementarity, Edge Elements M.V.K. Chari and S.J. Salon, Numerical Methods in Electromagnetism Goran Engdahl, Handbook of Giant Magnetostrictive Materials Edward P. Furlani, Permanent Magnet and Electromechanical Devices Vadim Kuperman, Magnetic Resonance Imaging: Physical Principles and Applications John C. Mallinson, Magneto-Resistive Heads: Fundamentals and Applications Isaak Mayergoyz, Nonlinear Diffusion of Electromagnetic Fields Isaak Mayergoyz, Mathematical Models of Hysteresis and Their Applications Giovanni Miano and Antonio Maffucci, Transmission Lines and Lumped Circuits Shan X. Wang and Alexander M. Taratorin, Magnetic Information Storage Technology

Related Books John C. Mallinson, The Foundations of Magnetic Recording, Second Edition Reinaldo Perez, Handbook of Electromagnetic Compatibility

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Measurement and Characterization of Magnetic Materials

FAUSTO FIORILLO Istituto Elettrotecnico Nazionale Galileo Ferraris Strada delle Cacce 91 Torino, 10135 ITALY

2004

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Foreword

This volume in the Electromagnetism series presents a modern, in-depth, comprehensive and self-contained treatment of the characterization and measurement of magnetic materials. These materials are ubiquitous in numerous industrial applications that range from electric power generation, conversion and distribution to magnetic data storage. Currently, there does not exist any book that covers the physical properties of magnetic materials, their characterization and modern measurement techniques of various parameters of these materials in detail. This book represents the first successful attempt to give a synthetic exposition of all these issues within one volume. The author, Dr Fausto Fiorillo, is a well-known expert in the field. He has an extensive experience in the area of magnetic measurements as well as intimate and firsthand knowledge of the latest technological innovations and has thus managed to put together an extraordinary amount of technical information in one volume. The salient and unique features of the book are its scope and strong emphasis on the metrological aspects of magnetic measurements. The book also reflects the broad expertise and extensive knowledge accumulated over the years by the highly visible and respected research group of the IEN Galileo Ferraris Materials Department based in Turin, Italy. I maintain that this book will be a valuable reference for both experts and beginners in the field. Electrical engineers, material scientists, physicists, experienced researchers and graduate students will find this book to be a valuable source of new facts, novel measurement techniques and penetrating insights. Isaak Mayergoyz, Series Editor

vii


Contents

Foreword

vii

Preface

xiii

Acknowledgments

xvii

Part I.

Properties of Magnetic Materials

1. Basic Phenomenology in Magnetic Materials 1.1 Magnetized Media 1.2 Demagnetizing Fields 1.3 Magnetization Process and Hysteresis

3 3 8 16

2. Soft 2.1 2.2 2.3

25 26 33 38 39 43 49 51 61 65 73

2.4 2.5 2.6 2.7

Magnetic Materials General Properties Pure Iron and Low-Carbon Steels Iron-Silicon Alloys 2.3.1 Non-oriented Fe-Si alloys 2.3.2 Grain-oriented Fe-Si alloys 2.3.3 Fe-(6.5 wt%)Si, Fe-A1 and Fe-Si-A1 alloys Amorphous and Nanocrystalline Alloys Nickel-Iron and Cobalt-Iron Alloys Soft Ferrites Soft Magnetic Thin Films

3. Operation of Permanent Magnets 3.1 Magnetic Circuit and Energy Product 3.2 Dynamic Recoil 3.3 Electrical Analogy and Numerical Modeling

89 90 95 98 ix

x

Contents

Part II.

Generation and Measurement of Magnetic Fields

103

4. Magnetic Field Sources 4.1 Filamentary Coils 4.1.1 Single current loop 4.1.2 Thin solenoids 4.1.3 Helmholtz coils 4.2 Thick Coils 4.3 AC and Pulsed Field Sources 4.4 Permanent Magnet Sources 4.5 Electromagnets

105 105 106 108 113 117 123 132 145

5. Measurement of Magnetic Fields 5.1 Fluxmetric Methods 5.1.1 Magnetic flux detection 5.1.2 Signal treatment and calibration of fluxmeters 5.2 Hall Effect and Magnetoresistance Methods 5.2.1 Physical mechanism of Hall effect and magnetoresistance 5.2.2 Measuring devices 5.3 Ferromagnetic Sensor Methods 5.3.1 Fluxgate magnetometers 5.3.2 Inductive magnetometers 5.3.3 Magnetostriction, magneto-optical, and microtorque magnetometers 5.4 Quantum Methods 5.4.1 Physical principles of NMR 5.4.2 NMR magnetometers 5.4.3 Electron spin resonance and optically pumped magnetometers 5.5 Magnetic Field Standards and Traceability

159 161 161 169 175

Part III.

Characterization of Magnetic Materials

175 185 196 197 202 209 217 218 227 251 262 279

6. Magnetic Circuits and General Measuring Problems 6.1 Closed Magnetic Circuits 6.2 Open Samples

281 282 295

7. Characterization of Soft Magnetic Materials 7.1 Bulk Samples, Laminations, and Ribbons: Test Specimens, Magnetizers, Measuring Standards

307

309

Contents 7.1.1 Bulk samples 7.1.2 Sheet, strip, and ribbon specimens 7.1.3 Anisotropic materials and two-dimensional testing 7.2 Measurement of the DC Magnetization Curves and the Related Parameters 7.2.1 Magnetometric methods 7.2.2 Inductive methods 7.3 AC Measurements 7.3.1 Low and power frequencies: basic measurements 7.3.2 Low and power frequencies: special measurements 7.3.3 Medium-to-high frequency measurements 7.3.4 Measurements at radiofrequencies

xi 309 315 326 336 337 340 362 364 385 409 432

8. Characterization of Hard Magnets 8.1 Closed Magnetic Circuit Measurements 8.2 Open Sample Measurements 8.2.1 Vibrating sample magnetometer 8.2.2 Alternating gradient force magnetometer 8.2.3 Extraction method 8.2.4 Pulsed field method

475 481 499 500 521 531 536

9. Measurement of Intrinsic Magnetic Properties of Ferromagnets 9.1 Spontaneous Magnetization and Curie Temperature 9.2 Magnetic Anisotropy

549 549 564

10. Uncertainty and Confidence in Measurements 10.1 Estimate of a Measurand Value and Measuring Uncertainty 10.2 Combined Uncertainty 10.3 Expanded Uncertainty and Confidence Level. Weighted Uncertainty 10.4 Traceability and Uncertainty in Magnetic Measurements 10.4.1 Calibration of a magnetic flux density standard 10.4.2 Determination of the DC polarization in a ferromagnetic alloy 10.4.3 Measurement of power losses in soft magnetic laminations

581

581 586 589 595 600 603 605

xii Appendix Appendix Appendix Appendix

Contents A: B: C: D:

The SI and the CGS Unit Systems in Magnetism Physical Constants Evaluation of Measuring Uncertainty Specifications of Magnetometers

613 621 623 629

List of Symbols

631

Subject Index

635

Preface

Magnets and measurements are everywhere. Magnetic materials are key pieces of a complex puzzle and are fundamental in satisfying basic demands of our society such as the generation, distribution, and conversion of energy, the storage and retrieval of information, many types of media and telecommunications. With their use in so many critical applications, these materials play a crucial role in our daily life and the present pace of research provides good reason for believing that their importance will continue to increase. With an annual global market valued at approximately EUR s the economic relevance of magnetic materials in industry is clear. Just as magnetic materials are important so accurate measurements are indispensable to science, industry, and commerce and are the prerequisite for any conceivable development in the production and trading of goods. They have relevant costs (about 5% of GNP in industrial countries) and require highly specialized organizations (such as the National Metrological Institutes) to develop and maintain the standards. Taken together these two elements are both scientifically and economically significant. This is the only book that takes that approach. Magnetism has a popular reputation of being a difficult subject. Part of this notion is as a result of the unfortunate duality of unit systems, which has greatly complicated life for students, researchers, and practitioners for many years. Nowadays, the SI system, recommended by the Conf&ence Poids et Mesures under the MKSA label since 1946, is establishing itself as the dominant system, despite resistance from many workers in the field. The SI system is preferentially adopted in most technical journals and recent books on the subject. There are plausible reasons for preferring the CGS system, not least the avoidance of redundant fields in free space, but diffusion of knowledge on magnetism and magnetic materials will certainly benefit from generalized adoption of the SI system. The topic of magnetic measurements is traditionally treated in textbooks as a branch of electrical measurement and the peculiar role of the materials and their physical properties are seldom emphasized. xiii

xiv

Preface

Textbooks on magnetic materials typically devote a chapter to experimental methods, but they obviously follow a concise approach to this matter, which is seen as a corollary to the treatment of physical topics. No modern treatise devoted to magnetic measurements and characterization of magnetic materials is therefore available nowadays. The standard text in the field is the two-volume book by H. Zijlstra Experimental Methods in Magnetism (North-Holland), which was published in 1967. Since the publication of that work there have been many changes such as the discovery of novel compositions and properties and the improved phenomenological understanding of the behavior of the materials. In addition, the digital revolution has brought about widespread changes in the way that measurements are taken both in research laboratories and in industry. Never has there been a greater need for a book that summarizes the principles and the present state of the art in the field of magnetic measurements. This book fills that need whilst bearing in mind materials scientists, the practical impact on everyday test activity, quality control in the laboratory and the education of scientists engaged in the basic characterization of materials. This is a consistent book drawn from the author's own long experience in the lab. It looks at measuring problems from a practical viewpoint and, by placing the treated topics within a clear physical f r a m e w o r k it will be useful both to those approaching the subject for the first time as well as to experienced researchers. It is intended for technicians in the lab and materials scientists in industry, university, and research centers. It aims at answering the basic questions and dilemmas people engaged in this field are faced with, enabling the reader to find straightforward answers without tiresome recourse to scattered literature. The various aspects of standardization of measurements are illustrated and constantly referred to. This goes hand in hand with a discussion on the metrological issues, which include intercomparison, traceability, and measuring uncertainty problems. The book is organized in three parts and 10 chapters. Part I is made of three introductive chapters. Chapter 1 illustrates the general physical concepts and introduces the quantities constantly referred to along the treatise. Chapter 2 consists in a synthetic presentation of soft magnetic materials and includes a description of the preparation methods and a discussion on their physical properties. Chapter 3 is focused on the operation of permanent magnets, the related energetic aspects, and the classical electrical analogy of the magnetic circuits. No attempt is made to delve into the specific physical properties of permanent magnets. Contrary to the case of soft magnets, where scant recent review literature exists, the reader can easily retrieve information

Preface

xv

on the physics of permanent magnets in a good number of comprehensive up-to-date books. Part II is devoted to the discussion on generation and measurement of magnetic fields, a necessary step in any characterization process, but one which also has value in different contexts, including environmental studies and medical applications. Generation techniques are presented in Chapter 4. Distinction is made there between coil-based sources (DC, AC, and pulsed fields) and generation by means of permanent magnets and electromagnets. It is stressed how the field generating capabilities of permanent magnet based sources can be strongly enhanced with the use of extra-hard rare-earth based compositions. Chapter 5 provides a comprehensive review of the physical principles exploited in the measurement of magnetic fields and of the solutions adopted in actual measuring devices. It is stressed that the basic problem of precise absolute measurement and traceability to the base SI units can be solved by use of quantum resonance magnetometers, where the determination of the field strength is reduced to a frequency measurement. The characterization of magnetic materials is discussed in Part III. After a preliminary introduction on general measuring problems and methodologies (Chapter 6), theory and practice in the measurement of the properties of soft and hard magnets are treated. Reference is made, whenever appropriate, to written measuring standards (e.g. IEC, ASTM, JIS standards). The discussion on the characterization of soft magnets is carried out by separately discussing the measurements under DC, low-frequency, medium-to-high frequency and radio frequency excitation (Chapter 7). In hard magnetic materials, distinction is made instead between closed magnetic circuit testing, where electromagnets are used at the same time as field sources and soft return paths for the magnetic flux, and open sample testing (Chapter 8). The latter methods often combine versatility with measuring sensitivity and are nowadays increasingly applied in the characterization of permanent magnets, besides being the natural choice for thin films and weak magnets. After a discussion in Chapter 9 regarding the measurement of intrinsic material parameters (Curie temperature, saturation magnetization, and magnetic anisotropy), Chapter 10 examines the very often neglected topic of measuring uncertainty and its crucial relationship with the metrological issues raised by intercomparisons and traceability to the relevant base and derived SI units. Specific examples regarding magnetic measurements are provided. The SI system of units has been adopted throughout the whole text. When data and graph scales expressed in CGS had to be taken from

xvi

Preface

the literature, the appropriate conversion to SI was made. Because of its persisting use, the measure of the magnetic moment has been provided also in e.m.u., in association with the corresponding SI unit (A m2). Conversion rules for translating SI equations in Gaussian equations and vice versa and a comprehensive conversion table are given in Appendix A. Whilst comprehensive this book is not meant to be exhaustive. A major part of it is devoted to methods for the determination of material properties having relevant interest for applications, i.e., the parameters associated with magnetic hysteresis. In this respect, the measuring written standards are constantly referred to guidelines. Magneto-optical, magnetostrictive, and superconductive effects are among the topics not discussed here. If the reader wishes to explore these further they can be found in many recent textbooks. For example, magnetostriction measurements are described to full extent in the E. du Tr6molet de Lacheisserie book Magnetostriction (CRC Press, 1993). Magneto-optical methods and phenomena are exhaustively discussed in the outstanding work, Magnetic Domains by A. Hubert and R. Sch/ifer (Springer, 1998).

Acknowledgments In preparing this book, I received contributions, suggestions, encouragement, and help from many friends and colleagues. I am indebted to all of them. I am especially grateful to Giorgio Bertotti, who assisted me in many ways, willingly engaging in many clarifying discussions, and allowing me to benefit from his deep knowledge of electromagnetism and magnetic phenomena. The series editor, Isaak Mayergoyz, fully supported my effort, fostering my confidence in the project and generously handling my outrageously delayed delivery of the manuscript. I would like to acknowledge that this project could only be pursued thanks to the special cooperating milieu, the broad expertise on the physics of magnetic materials, and the array of experimental researches developed by fellow scientists at the Materials Department of IEN. I found advice and support in all of them. I am also indebted to elder scientists in my lab who educated me in the early years of my careen The late Andrea Ferro Milone introduced me to materials science and Piero Mazzetti taught me the basic virtues of the experimental physicist. Aldo Stantero helped me in many ways and on innumerable occasions for more than twenty years. His untimely death was an untold loss to me and to the lab at IEN. Giorgio Bertotti, Vittorio Basso, Carlo Appino, and Alessandro Magni read substantial parts of the manuscript and Oriano Bottauscio provided me with crucial help by expressly performing numerous electromagnetic field computations. The field maps presented in Chapters 4 and 8 are due to him. Vittorio Basso, Cinzia Beatrice, Enzo Ferrara, and Eros Patroi kindly supplied me with their own experimental data and Marco Co'isson clarified to me specific aspects of magnetoimpedance measurements. Anna Maria Rietto carried out careful experiments to elucidate a few important details in the magnetic lamination testing with the Epstein test frame and Luciano Rocchino assisted me in assessing the problems related to reference field sources and their traceability to the base units. Sigfrido Leschiutta enhanced my sensitivity to metrological issues and the role of metrology in the physical sciences. I also need to thank all the many colleagues in Europe and elsewhere with whom I shared cooperative research activity and discussions on various scientific x-vii

xviii

Acknowledgments

matters. It is finally a pleasure to acknowledge the help provided by Christopher Greenwell and Sharon Brown at Elsevier, who assisted me in the various stages of the book production, and Lucia Bailo, Francesca Fia, and Emanuela Secinaro at the Publication Department of IEN, for their help in literature retrieval.

PART I

Properties of Magnetic Materials


CHAPTER 1

Basic Phenomenology in Magnetic Materials

1.1 M A G N E T I Z E D

MEDIA

In September 1820 H. C. Oersted demonstrated that electrical currents and magnets displayed equivalent effects. In a matter of weeks, A. M. Amp6re, elaborating on Oersted's discovery and making his own experiments, boldly interpreted the magnetism of materials as electricity in motion, i.e. the result of hidden microscopic currents, circulating around "electrodynamic molecules". The pedestal of electromagnetism was built in those few weeks, to be crowned in less than 50 years by the towering achievement of Maxwell's equations. Nowadays, we know that these currents exist, but they are quantum-mechanical in nature. They naturally slip into the classical Maxwellian scenario through the concept of permanent magnetic moment and the useful intermixing of classical and quantum concepts in the description of their relationship with the electronic angular momenta. A material sample is fundamentally described, from the viewpoint of magnetic properties, as a collection of magnetic moments, resulting from the motion of the electrons. Classically, orbiting electrons generate microscopic currents and are endowed with a magnetic moment m = -(e/2me).L, if e and me are charge and mass of the electron, and L is the angular moment. Quantum mechanics makes the view of electronic magnetic moment physically consistent, besides providing the additional basic concept of magnetic moment associated with spin angular momentum. When writing the classical equations of electromagnetism and assigning a meaning to the value of the physical quantities involved, we look at the material as a continuum. This means that all atomic scale intricacies are lost. In particular, the internal currents of quantummechanical origin are retained as averages over elementary volumes AV, sufficiently small to be defined as local over the typical scale of the problem, but large enough with respect to the atomic scale. These currents

CHAPTER 1 Basic Phenomenology in Magnetic Materials (Amperian currents), resulting from electron trajectories at the atomic scale, do not convey any flow of charge across the body. Let us call jM(r) the associated current density. Because of its solenoidal character, jM(r) can be expressed as the curl of another vector function M(r) jM(r) = V X M(r).

(1.1)

Remarkably, it can be demonstrated that the vector function M(r) represents the magnetic moment per unit sample volume [1.1]. This quantity takes the name "magnetization". If the previous elementary volume contains a certain number of moment carriers, it is M ( r ) = ~imi/~V~where the summation runs over the moments contained in such a volume. Thanks to Eq. (1.1), we are in a position to describe the magnetic effects ensuing from steady external currents when media are involved. If such currents are made to circulate in the absence of media, the induction vector B (often called "B-field") is given by the Biot-Savart law, which is expressed in differential form as V x B =/~0je,

(1.2)

where je is the density of the supplied currents and ~ = 4Ir x 10 -7 N / A 2 is the magnetic constant (sometimes called permeability of vacuum). Analysis of the Biot-Savart law additionally shows that the B vector is solenoidal, i.e. it obeys the equation V.B = 0. It can be shown that this equation and Eq. (1.2) determine B uniquely for given je. We recall here that the operative definition of the vector B is provided by Lorentz's law, which describes the coupling of electrical and magnetic fields with electrical charges. It states that a charge q moving at velocity t is subjected to a force F = q(E + t x B).

(1.3)

In the presence of magnetic media, both the externally supplied currents and the internal microscopic averaged currents will contribute to the B-field. The two base equations will be V.B -- 0,

V x B --/~0(je + jM).

(1.4)

When dealing with experiments on magnetic materials and their applications, we endeavor to drive the magnetic state of the material by means of external currents, i.e. acting on the quantity je- We can single out je in Eq. (1.4), introducing through Eq. (1.1) the magnetization M, a quantity directly accessible to experiments, in place of the awkward internal currents jM. In this way we define the so-called H-field Vx

~- -M

=VXH--je.

(1.5)

1.1 MAGNETIZED MEDIA

5

H is the quantity conventionally defined as the magnetic field as it is the quantity susceptible to direct control by means of the external currents. On the other hand, B appears to be the ffmdamental field vector because it is characterized by the condition V.B = 0 everywhere, in the free space and inside the matter, and Lorentz's equation everywhere applies to it. According to Eq. (1.5), the general relationship connecting the vectors B, H, and M is B = p,oH + / z o M .

(1.6)

In the SI unit system the magnetization M is expressed, like the magnetic field, in A / m , putting in evidence the Amperian origin of the magnetic moment. In the absence of media, M = 0 and B = / ~ H , i.e. magnetic field and induction (i.e. H-field and B-field) are equivalent quantities, as they are related by the proportionality constant/z0. In many kinds of experiments, we exploit the Faraday-Maxwell law V x E = - O B / O t , where E is the electric field, in order to determine.the magnetic behavior of the material. We detect in this case the electromotive force generated in a linked search coil by the time variation of the induction. Normally, we wish to get rid of the term /z0H in Eq. (1.6), because we are only interested in the contribution/z0M deriving from the material. This contribution is called magnetic polarization, J =/z0M, a quantity having the same dimensions as B (tesla, T) and the same properties as M. We then write B =/z0H + J. This general relationship will be specialized to the magnetic properties of the investigated medium by means of some constitutive equation J(H) or B(H). In ferromagnetic materials, these relationships can be very complex and very difficult to predict. In some well-defined instances, it is meaningful to define the relationships B = / z H =/Zr/z0H and M -- xH, where/z is the permeability, /d, r is the relative permeability, and X is the susceptibility. The quantifies/zr and X are related by the equation ~r

m

1 + X.

(1.7)

Note that, in the old Gaussian system, the base Eq. (1.6) is written as B = H + 4rrM,

(1.8)

i.e. field, induction, and magnetization have all the same dimensions (though different names, oersted (Oe) for H and gauss (G) for 4rrM and B). In this book, the SI system will be used throughout and little reference will be made to the increasingly obsolete Gaussian units. A complete set of conversion formulae is nevertheless provided, together with a discussion on their logical foundation, in Appendix A.

CHAPTER 1 Basic Phenomenology in Magnetic Materials Ferromagnetic materials are characterized by hysteresis. A residual magnetization M is always left when the external field, associated with the presence of a current density je, is completely released having once attained a certain peak value. We can say that some internal mechanism, associated with the nature and structure of the material, preserves a non-zero curl of the microscopic average current jM (Eq. (1.1)). With je -- 0, we obtain from the previous Eqs. (1.4) and (1.5), that induction and H-field are described by the equations V.B = 0~

(1.9)

V X B -- /J~)jM

V.H = -V.M,

V x H = 0.

(1.10)

Equation (1.9) simply states that the B-field is now uniquely generated by the internal currents and it preserves its solenoidal character. The two equations (Eq. (1.10)) are instead formally equivalent to the equations for the electrostatic field V.E--p/8o and V x E = 0, with p the electric charge density and ~0 the electric constant. Thus, in the absence of external currents, the field H, whose divergence can by analogy be written as V.H - PM, with PM -- -V.M, can be considered as the gradient of a scalar magnetic potential H = - V ~ M. This potential satisfies the Poisson's equation V2(I)M----PM"The electrostatic analogy then permits us to introduce, in a purely fictitious way, magnetic charges of volume density PM acting as sources of the field H, whenever it occurs that V.M # 0. Although devoid of physical reality, the concept of magnetic charges is constantly applied in the investigation of magnetic materials and in magnetic measurements because of the simplifications it introduces in the description of many phenomena and in the calculations. It permits one, for example, to derive fields from scalar potential functions, which are solutions of Poisson's equation 1 ~ V.M(f) d3 f ~M(r)

=

- - 4---~

Ir -

el

'

(1.11)

thereby applying the conventional methods of electrostatics. Figure 1.1 provides a classical example where the role of magnetic charges can be invoked. It is the case of a cylindrical permanent magnet, where the magnetization M is uniform and axially directed. Since M suffers a discontinuity at the sample ends, the conditions are created for quasisingular behavior of the divergence V.M. It turns out that the potential function can be written as

1 ~ CrM(lJ) d2f, (I)M(r)-- ~ A Ir- rrl

(1.12)

1.1 MAGNETIZED MEDIA

I

(a)

7

lt[/f

(b)

(c)

FIGURE 1.1 Induction B and field H in a cylindrical permanent magnet in the absence of an external applied field (je = 0). It is assumed that the sample remanent magnetization M is uniform. The induction B =/~0H +/z0M is solenoidal (V.B = 0) and the field H satisfies the condition V x H = 0. This means that H can be expressed, in formal analogy with the electrostatic field, as the gradient of a scalar potential. In this respect, it is as if fictitious magnetic charges of equal densities and opposite signs were uniformly distributed over the top and bottom surfaces of the cylinder.

where the integration is performed over the total area A of the top and bottom surfaces. The quantity crM(1~) = n.M(r~), where n is the unit vector normal to these surfaces, plays the role of surface magnetic charge density. The correspondingly calculated magnetostatic field H(r) and the induction B(r) =/z0H(r) + / z 0 M are schematically s h o w n in Fig. 1.1b,c. Notice that within the sample, H(r) is directed in such a w a y as to oppose the magnetization. It is for this reason called a "demagnetizing field". If the magnetization is not uniform or the material is inhomogeneous, internal demagnetizing fields can also arise. In the free space, B(r) and H(r) (which takes the n a m e stray field) coincide (but for the proportionality factor/z0). Note further that the condition V.B = 0 implies that, on traversing the sample surface, the normal c o m p o n e n t B.n is preserved. The condition V x H - 0, however, implies that the same occurs to the tangential c o m p o n e n t of H.

CHAPTER 1 BasicPhenomenology in Magnetic Materials A magnet brought under the permanent condition shown in Fig. 1.1 is endowed with a certain magnetostatic energy content. Part of this energy is contained within the sample and part is associated with the stray field. Under very general terms, we can write the total energy as (1.13)

Et = -~ P,o H2dV,

where we have defined as H d the demagnetizing field and the integration extends all over the space. This is the energy that must be spent for the formation of the magnetic charges and it can be equivalently written as Et - - - ~ - / ~ v

(1.14)

Hd.M dV,

where integration is made over the sample volume.

1.2 D E M A G N E T I Z I N G

FIELDS

Demagnetizing effects are ubiquitous. Even in accurately closed specimens, (e.g. ring samples), one cannot get rid of them completely. This has fundamental consequences from the point of view of magnetic characterization and it requires measuring strategies aimed at minimizing and/or precisely controlling the demagnetizing fields. We shall discuss and clarify practical methods devised to this purpose, both in soft and hard magnets, in later chapters. In this section, we shall briefly discuss the basic problems connected with the prediction of the demagnetizing fields under different sample geometries. Calculations of demagnetizing fields date back to the 19th century. They were pursued by, among others, Maxwell [1.2], Lord Rayleigh [1.3], and Ewing [1.4]. One chief problem at that time involved a ship's magnetism and the correction to be made on the apparent declination of the magnetic compass to determine a ship's position. It was recognized that only in samples shaped as ellipsoids (or spheres) could the demagnetizing field be homogeneous and susceptible of full mathematical treatment. The general approach consists in determining the volume and surface charge densities pM(r~) = -V.M(r') and O'M(I d) -~ n.M(r') of the uniformly magnetized body and in correspondingly expressing the potential 1 f ~M(r)-- ~

pM(IJ) d3rI + 1 ; V I r - r'l

~

crM(r/) d2r/ A I r - r'l

(1 15) '

"

1.2 DEMAGNETIZING FIELDS

9

from which the demagnetizing field can be derived as the gradient Hd(r) -- --VCI)M(r). With M constant in modulus and direction everywhere inside the spheroidal sample, the volume charge density pM(r') is zero and the surface charge density crM(r~) is easily calculated. If M lies along one of the principal axes (a, b, c), the corresponding homogeneous demagnetizing field is N~ H d ___ _ X d M

= _ --u j,

(1.16)

/z0 where the proportionality factor Nd is called "demagnetizing coefficient". For a generic direction, we have H d -- -[[Ndl[M, where [[Nd[[ is a tensor having only the diagonal elements different from zero. These are the demagnetizing coefficients along the three principal axes Nda,, Ndb, Ndc. They obey the constraint Nda + Ndb + Ndc = 1. In the general ellipsoid, a # b # c and the demagnetizing factor Nda is obtained as the integral N d a _~ abc

2

jo[

(a 2 q-

~)

;(

a 2 q- O ( b 2 if- ~)(c 2 q- ~)

d~'.

(1.17)

which can be numerically calculated, together with Ndb and Ndc = 1 -- Nda -- Ndb. Results are reported in the literature (see, for instance, Ref. [1.5]). Closed expressions are found for ellipsoids of revolution (Fig. 1.2). In the limiting case of a sphere, we have, for reasons of synm~etry, Nda = Ndb = Ndr = 1/3. For a prolate spheroid, where a = b and the rotational symmetry axis c > a,b, the demagnetizing factors are given, for the defined ratio r = c/a -- c/b > 1, by the expressions Ndc

1 [ ~Jy2__r 1 /r+ r2 1/ 1]

r2 - 1

1

(1.18)

Nda -----Ndb = -~(1 -- Ndc ). z

For r >> 1, the approximation Ndc = In 2 r - 1/r 2 holds. If the same axis c < a, b (oblate spheroid, r < 1), we have

1[

r

Ndc-- 1 - r 2 1 -

~arcsin(x/1 x/1-r 2

1

-r2)], (1.19)

Nda -- Ndb ----- ~ ( 1 -- Ndc ). Z

Tables and graphs reporting the value of the coefficient Ndc, calculated for r varying over many decades, have been published over many years and can be found in several textbooks (see, for example, Refs. [1.6, 1.7]). It is

10

CHAPTER 1 Basic Phenomenology in Magnetic Materials

(a)

(b) FIGURE 1.2 Prolate (a = be c > a, b) and oblate (a = b, c < as b) spheroids, c is the rotational symmetry axis. The demagnetizing field lid is always uniform in ellipsoids if the magnetization M is uniform, whatever its direction. The demagnetizing coefficients calculated along the three symmetry axes, which completely define the problem, are given by Eqs. (1.18) (prolate spheroid) and (1.19) (oblate spheroid). They depend only on the ratio r = c/a = c/b. apparent here that the demagnetizing field does not depend on the sample volume, but only on its geometrical properties (the ratio r). Demagnetizing fields can never be ignored in measurements. To recover the intrinsic magnetic properties of the material under test, a correction is required, where the effective field H = Ha - H d ~ obtained as the difference between the applied field H a and the demagnetizing field, is calculated. The problem is apparent with bulk soft magnets having a relatively low aspect ratio in the direction of magnetization, but also there is heavy interference by demagnetizing effects with strips and ribbons. In a typical experiment, a 20 ~m thick, 200 m m long, and 10 m m wide high permeability amorphous ribbon is tested. It is found by hysteresis loop shearing analysis that this sample has what could be deemed a very low demagnetizing factor (Nd "" 1.3 X 10-5), corresponding to a value r---700 in a prolate ellipsoid. For peak polarization value, Jp--0.8 T, the demagnetizing field is, according to Eq. (1.16), Hd -~ 8.5 A / m , which is about eight times larger than the effective field H. In hard magnets, it is fortunately possible in principle to perform accurate measurements with open samples also, thanks to the very high fields intrinsically required for their magnetization and demagnetization. However, permanent magnets generally come as bulk specimens

1.2 DEMAGNETIZINGFIELDS

11

(cylinders, parallelepipeds, spheres), which have high demagnetizing coefficients. It may also happen that some kinds of tapes or thin films preferentially magnetize normal to their plane, thereby approaching the value Nd = 1, and the correction procedure may become very complex. Ellipsoidal test samples are seldom employed. A notable exception is represented by some open sample methods applied in permanent magnet testing (for example, with the vibrating sample magnetometer; see Section 8.2.1), where small spherical specimens (diameter around few mm) are adopted. Use of cylindrical, parallelepipedic, or disk-shaped test specimens is generally the rule, but these geometries engender significant complications in the definition and determination of the demagnetizing coefficient. In fact, even with homogeneous magnetization, the demagnetizing field is not homogeneous. In addition, it is also dependent on the material permeability. In soft magnets, this brings about notable complications, because permeability is high and, even if we disregard hysteresis, corrections become difficult and inaccurate. Large errors are expected to occur when, as is often the case, Ha and Hd have very close values. The problem of predicting demagnetizing fields in non-ellipsoidal bodies can be attacked in principle by computation of the integrals in Eq. (1.15), a relatively complex and time-consuming approach. Threedimensional and two-dimensional finite element calculations have been developed for this purpose. They are indispensable for treating complex micromagnetic problems and the analysis of specific domain patterns [1.8, 1.9]. For regular cylindrical or parallelepipedic shapes and structurally and magnetically homogeneous materials (no domains and constant susceptibility value X everywhere), calculations of the demagnetizing coefficient have been performed to various degrees of approximation. For uniformly magnetized samples and X--0, analytical approaches have been carried out and reasonable approximations can often be given, although, in a strict sense, they apply only to diamagnets, paramagnets, and saturated ferromagnets. For example, a uniformly magnetized cylinder has magnetic charge density r at its top and bottom surfaces. The demagnetizing field at the center of the cylinder is straightforwardly calculated by integrating the field generated by an infinitesimally thick annulus of surface charges and summing up the contributions of the top and bottom ends (details are given in Section 4.4). For a cylinder with height-to-diameter ratio r, such a field turns out to be

H d -= - N d M

-- - ( 1 -

1 )M. ~/1+ 1 / r 2

(1.20)

12


For r = 1, this provides Nd = 0.293, as compared with Nd = 0.333 in a sphere. The demagnetizing field at the center of a prism with square cross-section and height-to-side length ratio r, uniformly magnetized along the axial direction, is similarly obtained as NclM = --_2r arcsin ( l +1r

Hd ._

) M.

(1.21)

We see in Fig. 1.3 how the associated demagnetizing factor compares with that of an inscribed spheroid as a function of the ratio r. In particular, it turns out that, for a cube ( r - 1) Nd -- 1/3, exactly the same as for the sphere. Magnetic materials are normally characterized using regularly shaped samples, where either the measurement of the magnetic flux upon a well-defined cross-section or the determination of the total magnetic m o m e n t of the test specimen is performed and related to the material magnetization. At the same time, the concept of effective field must be given a practical meaning. In connection with these two measuring approaches, one talks of fluxmetric and magnetometric methods. When dealing with non-ellipsoidal open samples, it is therefore expedient to distinguish between fluxmetric (or ballistic) and magnetometric

1.0 0.8 0

.~

0.6

9

0.4

0.2

0.0

.

0

.

.

.

!

1

.

.

.

.

|

2

.

.

.

.

|

3

.

.

.

.

!

4

.

.

.

.

5

r=cla=clb

FIGURE 1.3 Behavior of the demagnetizing factor Nd = H d / M , with Hd the field at the center, in a uniformly magnetized prismatic sample with square crosssection and X = 0 (dashed line, Eq. (1.21)). Comparison is made with the same quantity calculated for an inscribed spheroid.

1.2 DEMAGNETIZING FIELDS

13

demagnetizing factors. Let us consider, as in the earlier example, a cylindrical or prismatic sample. The material is homogeneous, does not show hysteresis, and the susceptibility is isotropic and constant everywhere. A qualitative idea of the non-homogeneity of the demagnetizing field in connection with homogeneous magnetization is provided as a sketch in this case by Fig. 1.1. An example of quantitative derivation of the dependence of the axial component of the magnetization and the demagnetizing field for different susceptibility values in a cylindrical sample is illustrated in Fig. 1.4 [1.10]. We thus define the fluxmetric demagnetizing factor as the ratio of the average demagnetizing field to the average magnetization over the midplane perpendicular to the sample axis (z-direction)

Hd~dA N(df) =

MzdA

1. The opposite occurs if r < 1.4. The ratio 7~ 9~T(m) d ~d increases with increasing r, for given X, and decreases with increasing X, for given r. The subject of analytical determination of N(df) and N~dm) in prisms of rectangular cross-section has been treated by Aharoni, who provided expressions for both of them under the assumption of saturated sample (X=0) [1.12, 1.13]. The results were applied to experiments on 2.35 ~m x 95 I~m x 250 p~m Ni80Fe20 thin films with easy axis directed along the major side. It was verified that a minimum applied field equal to the calculated fluxmetric demagnetizing field H d = -N(df)Ms (M~, saturation magnetization) was required to reverse the magnetization in the central part of the sample.

1.3 M A G N E T I Z A T I O N

PROCESS AND HYSTERESIS

We primarily identify the expression "magnetic characterization" with the experimental process by which we determine the constitutive law of a magnetic material. This is the functional dependence M(H) of the magnetization on the effective field or, according to Eq. (1.6), the equivalent laws J(H) and B(H). This book will be chiefly devoted to reviewing and discussing the measuring methods by which such laws can be experimentally derived. Diamagnetic and paramagnetic materials are described by simple single valued relationships between field and magnetization. This by no means implies that such relationships are easily accessible to experiments because the associated susceptibilities are very low and, consequently, it is difficult to discriminate between the response of the sample under test and that of the background. Ferromagnets (and ferrimagnets) display intrinsically large responses. The landmark property of these materials is that they are endowed with a molecular field of quantum-mechanical origin, which implies magnetic ordering, i.e. possible extraordinarily high values of the magnetic

1.3 MAGNETIZATION PROCESS AND HYSTERESIS

17

moment per unit volume. However, the phenomenology of the magnetization process in these materials has the traits of complexity, as embodied in the manifold manifestations of hysteresis. This appears as the macroscopic outcome of an intricate combination of microscopic processes, centered on the existence of domains (or, in limiting cases, single-domain particles), which lead to collective rearrangements of the magnetic moments under a changing applied magnetic field. The experimental investigation of magnetic hysteresis and the general testing of materials require that some kind of accessible reference state is defined. Magnetic saturation and demagnetized state are two such reference conditions. All trajectories in the H - M (or H-J) plane converge when reaching the saturated state. On recoiling from it, the system follows a unique trajectory. The demagnetized state acquires a similar distinctive character when the condition H - - 0 and M = 0 is attained. To this end, the sample is either brought beyond the Curie temperature and cooled in a zero-field environment or an alternating field of progressively and finely decreasing peak amplitude is applied, starting from saturation and ending at zero value. The demagnetized state should realize, from the microscopic viewpoint, that special arrangement of the internal magnetic structure corresponding to the condition of absolute minimum of the magnetic free energy. Indeed, an infinite number of trajectories can lead to sample demagnetization without leading to the demagnetized state. If, starting from such a state, the applied field amplitude is increased, a curve in the (H-J) or, equivalently, (H-M) plane is described. It is the initial magnetization curve. In current practice, thermal demagnetization is seldom adopted. When this is the case, it is not expected to lead to the same set of microscopic states obtained with the conventional demagnetization under an alternating field. The curve described after thermal demagnetization takes the name virgin curve and it can be slightly different from the initial curve. Figure 1.6 provides examples of initial magnetization curves in soft and hard magnets with both the q - H ) and (B-H) representations shown. These curves have been obtained using a closed magnetic circuit, where the applied field Ha ~-H. If open samples are tested or the flux closure is imperfect, Ha = H + (Nd/t2,o)J and the curves (J-Ha) and (B-Ha) will appear sheared with respect to the intrinsic curves (J-H) and (B-H). We see that in practical soft magnets there is a detectable difference between these two representations only on approaching the saturated state, where the term/z0H can be appreciated (Eq. (1.6)). In order to bring the material along the magnetization curve, we must spend energy. Let us assume that a field, slowly increasing with time, is applied to a sample forming a closed magnetic circuit

18


B(H) ~

/

"....,,,

2.0

/

4"

1.5 r

1.0

0.5

0.0 (a)

_. J i

i ~

100

i

Non-oriented Fe-Si i i iIinl

101

I

i i iiiiii

102

I

I i lllul

103

I

I I llllll

104

I I I II

10 5

H(A/m)

.~ /

s

"s

i

(Hp, Bp)

1

,,,

2

/

c~

"

1-

0

(b)

f

B(H) J(H)

,

/)

500

1000 H(Nm)

1500

FIGURE 1.6 Initial magnetization curves in non-oriented Fe-Si laminations and in a N d - F e - D y - A 1 - B sintered magnet. The soft magnetic laminations have been tested as strips forming a closed magnetic circuit within an Epstein frame. The permanent magnet has been demagnetized and magnetized as a parallelepipedic sample inserted between the pole faces of Ban electromagnet. The shaded area provides the energy per unit volume E = ~0p HdB to be spent for bringing the material from the demagnetized state to the peak induction value Bp.


19

(e.g. a ring specimen) by use of a suitably linked winding supplied with a current i(t). At any instant of time, the supplied voltage is balanced by the resistive voltage drop of the winding Rwi(t) and the induced e.m.f, d ~ / d t UG(t) -- Rwi(t) + d~/dt.

(1.25)

Starting from the demagnetized state, a certain final state with induction value Bp is reached after a time interval to. The correspondingly supplied energy E = ~0~uG(t)i(t)dt is partly dissipated by Joule heating in the conductor and partly delivered to the magnetic system

E=

Rwia(t)dt 4-

X:

NwAi(t)-dTdt

(1.26)

where Nw is the number of turns of the winding and A is the crosssectional area of the sample. We are interested in the second term on the right-hand side of this equation. If, to simplify the matter, we consider a ring sample of average circumferential length lm, we can write i(t)= (lm/Nw)H(t) and we find that the energy delivered by the field-supplying external system in bringing the magnet to the final state is

dB dt = V fSp HdB. U - V f~ H(t) --~

(1.27)

0

If we refer to the graphical representation of the initial magnetization curve in Fig. 1.6, we conclude that the energy per unit volume to be supplied in order to reach the induction value Bp (or, equivalently, the polarization value Jp), is given by the area between the B(H) curve and the ordinate axis. Introducing Eq. (1.6) in Eq. (1.27), we obtain

U= V

ia,oHdH + V

HdJ,

(1.28)

and we can distinguish between the energy stored in the magnetic field and the energy stored in the material. Part of the latter is expected to be lost during the process. Except in somewhat unusual cases, the magnetization process is always associated with measurable energy dissipation. This is mirrored in hysteresis, the phenomenon of output lagging behind input. Figure 1.7 provides two examples of hysteresis loops in the previous soft and hard magnets, as obtained by cycling the field between two symmetric peak values +Hp. If the integration in

20

CHAPTER 1 Basic Phenomenology in Magnetic Materials 1.5

Jr, er 1.0 0.5 0.0 -0.5

-1.5

Non-oriented Fe-Si

m i

-1.0

I

.

.

.

.

-1000

I

.

.

.

-500

.

.

.

.

.

I,

0

(a)

,,,~

,

500

,

I

1000

H(Nm) Nd-Fe-Dy-AI-B

Br, Jr

""4HI

-2 -3 "''

I

. . . . .

-1500

(b)

I

. . . . .

I

. . . . . . . . . .

-1000 -500

~ ~'""

0 500 H(Nm)

'"1

. . . . .

1000

I ' ' "

1500

FIGURE 1.7 Examples of hysteresis loops in soft and hard magnets. In the soft Fe-Si laminations, there is no detectable difference between the B(H) and J(H) curves for magnetizations and fields of technical interest. The difference is instead apparent in permanent magnets. It leads to two different definitions of coercive field: HcB is the field required to bring the induction to zero value starting from the saturated state and Hcj is required to reduce to zero the polarization J (i.e. magnetization M). It is always Hcl > HcB (courtesy of E. Patroi).


21

Eq. (1.28) is carried out over a full cycle, the energy balance is obtained

W= ylzoHdH4- yHdJ= yHdJ= ~HdB.

(1.29)

The quantity W is the energy lost per unit volume in the sample. In fact, the purely reactive term ~ la,oHdH averages out to zero over a cycle. It then turns out that the area of the B(H) loop is equal to the area of the J(H) loop. Notice in the loops represented in Fig. 1.7 the remanence point, where Jr = Br, and the distinction existing between the coercive fields HcB and Hcj, the latter being the field value where the material is demagnetized (indeed, a demagnetized state far removed from the one previously discussed). The phenomenology of magnetic hysteresis is extremely complex but endowed with a certain mathematical regularity, which has attracted relevant modeling efforts starting from the milestone approach of Preisach [1.14]. A full account of the mathematical aspects of hysteresis is found in Ref. [1.15], while the physics of magnetic hysteresis is treated with deep insight in Ref. [1.16]. We only remark here that, if we limit ourselves to material testing at low field strengths, we can fully describe initial curve and symmetric hysteresis loops by means of a defined function, the Rayleigh Law. Its discovery was a triumph of physical intuition and experimental skill in measurements [1.17], which is paralleled by the physical explanation advanced by N6el [1.18], based on the statistical description of the reversible and irreversible displacements of the domain walls. An example of loops determined in the Rayleigh region is reported in Fig. 1.8, where it is observed that the material polarization follows a quadratic dependence on the magnetic field. In particular, any hysteresis loop determined between the peak field values +Hp has ascending and descending branches described by the equations

b (Hp - H2), I(H) = (a + bHp)H -T- -~

(1.30)

where a and b are called reversible and irreversible Rayleigh constants. The tip points of the loops (Hp, Jp) describe the initial magnetization curve (also called normal magnetization curve), and follow the law

Jp = aHp 4- bH2.

(1.31)

In the limit of very low fields, the magnetization curve becomes linear (as is apparent in Fig. 1.8) with the constant a proportional to the initial

22

CHAPTER 1 Basic Phenomenology in Magnetic Materials (Hp, J p ) ~ . .

Pure Ni foil

0.06

/

5/

0.040.02-

aH

o.oo -0.02

J

-0.04 -0.06 " '

I

'

-300

'

' "

I

"

-200

'

'

'

I

.

-100

.

.

.

.

.

.

.

I

. . . .

0 100 H(A/m)

I

. . . .

200

I

r ,

300

FIGURE 1.8 Experimental hysteresis loops in the Rayleigh region. The ascending and descending branches of the loops and, a fortiori, the initial magnetization curve connecting the tip points of the loops, follow the quadratic law (1.30). The energy per unit volume dissipated upon completing a full cycle is given by the equation W = -~bH~, where b is a structure dependent coefficient and Hp is the peak field value (courtesy of C. Beatrice). susceptibility Xi lim Jp Sp--.0 Hpp "-- [d'OXi"

a--

(1.32)

Integration of the loop area provides the hysteresis loss per unit volume

W=~4 blip,

(1.33)

where, as expected, the reversible coefficient a has disappeared. Equivalently, we can express the energy loss for a given peak polarization value Jp in the Rayleigh domain as

1[

W = -~

-a +

2 + 4blp

]3 .

(1.34)

Remarkably, N6el's theory predicts that the quantities aHc and bile~a, where Hc is the coercive field, as obtained with a major loop, are independent of the structure of the material. This conclusion derives from

1.3 MAGNETIZATIONPROCESS AND HYSTERESIS

23

the assumption of scale invariance of the equations describing the interaction of the Bloch walls with the pinning defects. Our discussion has so far highlighted some essential facts concerning the so-called "quasi-static" magnetic behavior of the materials, which is observed when the rate of change of the magnetization is so low that dynamic viscous-type effects do not interfere with the magnetization process. When this condition is no longer fulfilled, we observe ratedependent hysteresis effects. In metallic materials, they are almost exclusively due to eddy currents, whose circulation in the sample takes patterns depending on, besides the magnetization rate, the material resistivity, sample geometry, and domain structure. We shall consider dynamic loss behavior in connection with the properties of soft magnetic materials (Chapter 2) and the related measurements (Chapter 7). We only remark here that the fundamental consequence of the viscous effects associated with long-range eddy currents is the increase of the energy dissipated in any cycle, as manifest in the observed broadening of the hysteresis loop with increasing magnetizing frequency (Fig. 1.9).

'

0.2

Co71Fe4B15Silo '

'

'

I

'

'

'

'

I

.

.

.

.

amorphousribbon

0.1

f

o.o

//~~_,flO0k5~150 kHz

-0.1

300kHz

-0.2 .

-60

1 kHzi

/

.

.

|

I

-40

,

,

,

,

I

-20

.

.

.

.

|

0

,

H(Nm)

|

,

I

20

.

,

.

.

I

40

,

,

,

,

60

FIGURE 1.9 Broadening of the hysteresis loops with increasing magnetizing frequency in a soft magnetic alloy. The measurements refer to an amorphous ribbon (thickness 20 ~m), wound into a many layer ring sample and tested under controlled sinusoidal induction waveform.

24


The whole phenomenology can be assessed in most cases by applying the concept of loss separation [1.16].

aefeyences 1.1. L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media (Oxford: Pergamon Press, 1989). 1.2. J.C. Maxwell, A Treatise on Electricity and Magnetism (London: Clarendon, 1891, 3rd edition. Reprinted by Dover, New York, 1954), Vol. 2, p. 59. 1.3. J.W. Strutt (Lord Rayleigh), "Notes on magnetism. On the energy of magnetized iron," Philos. Mag., 22 (1886), 175-183. 1.4. J.A. Ewing, "Experimental researches in magnetism," Philos. Trans. Roy. Soc. London, 176 (1885), 523-640. 1.5. A. Hubert and R. Sch/ifer, Magnetic Domains (Berlin: Springer, 1998), p. 184. 1.6. B.D. Cullity, Introduction to Magnetic Materials (Reading, MA: AddisonWesley, 1972), p. 58. 1.7. D. Jiles, Introduction to Magnetism and Magnetic Materials (London: Chapman & Hall, 1991), p. 39. 1.8. S.W.Yuan and H.N. Bertram, "Fast adaptive algorithms for micromagnetics," IEEE. Trans. Magn., 28 (1992), 2031-2036. 1.9. D.V. Berkov, K. Ramst6ck, and A. Hubert, "Solving micromagnetic problemsmtowards an optimal numerical method," Phys. Status Solidi, A-137 (1993), 207-225. 1.10. D.X. Chen, J.A. Brug, and R.B. Goldfarb, "Demagnetizing factors for cylinders," IEEE. Trans. Magn., 27 (1991), 3601-3619. 1.11. W.F. Brown, "Single domain particles: new uses of old theorems," Am. I. Phys., 28 (1960), 542-551. 1.12. A. Aharoni, "Demagnetizing factors for rectangular ferromagnetic prisms," J. Appl. Phys., 83 (1998), 3432-3434. 1.13. A. Aharoni, L. Pust, and M. Kief, "Comparing theoretical demagnetizing factors with the observed saturation process in rectangular shields," J. Appl. Phys., 87 (2000), 6564-6566. 1.14. F. Preisach, "Uber die magnetische Nachwirkung," Z. Phys., 94 (1935), 277-302. 1.15. I.D. Mayergoyz, Mathematical Models of Hysteresis (New York: SpringerVerlag, 1991). 1.16. G. Bertotti, Hysteresis in Magnetism, (San Diego: Academic Press, 1998). 1.17. J.W. Strutt (Lord Rayleigh), "On the behaviour of iron and steel under the operation of feeble magnetic forces," Philos. Mag., 23 (1887), 225-245. 1.18. L. N6el, "Th6orie des lois d'aimantation de Lord Rayleigh. I. Les d6placements d'une paroi isol6e," Cahiers de Physique, 12 (1942), 1-20.

CHAPTER 2

Soft Magnetic Materials

A magnetic material is considered "soft" when its coercivity is of the order of, or lower than, the earth's magnetic field. A soft magnetic material (SMM) can be employed as an efficient flux multiplier in a large variety of devices, including transformers, generators, and motors, to be used in the generation and distribution of electrical energy; and in a wide array of apparatus, from household appliances to scientific equipment. With a market around s billion/year, SMMs are today an ever more important industrial product, offering challenging issues in properties understanding, preparation and characterization. SMMs have been at the core of the development of the early industrial applications of electricity. Steel production was sufficiently developed at the turn of the century to satisfy the increasing need of mild steel for the electrical machine cores. In 1900, Hadfield, Barrett, and Brown proved that, by adding around 2% in weight Si to the conventional magnetic steels, one could achieve an increase of permeability and a decrease of energy losses [2.1]. Fe-Si alloys were more expensive and more difficult to produce and gained slow acceptance. In addition, the poor control of the C content was to mask the prospective performance of this product compared with mild steels. It took more than two decades, characterized by a gradual improvement of the metallurgical processes, for Fe-Si to become the material of choice for transformers. An empirical attitude towards research in magnetic materials was prevalent at the time and applications came well before theoretical understanding. This was the case for the Goss process, developed in the early 1930s, by which the first grain-oriented Fe-Si laminations could be produced industrially [2.2]. In the years 1915-1923, Elmen and co-workers at the Bell Telephone Laboratories systematically investigated alloys made of Fe and Ni, discovering the excellent properties of the extra-soft permalloys (78% Ni) [2.3]. Snoek is credited for the successful industrial development of ferrites in the 1940s [2.4], following attempts dating back to the first decade of the century. The discovery in 1967 of the soft magnetic amorphous alloys again occurred nearly by chance [2.5], but it provided a fertile field for technologists and 25

26

CHAPTER 2 SoftMagnetic Materials

theorists. The discovery enriched the landscape of applicative magnetic materials, while straining existing theories on magnetic ordering.

2.1 GENERAL PROPERTIES The rough attribution of magnetic softness, based on the value of coercivity, is completed and made useful by the q,H) hysteresis loop. Figure 2.1 provides a comparison of major DC hysteresis loops in a number of representative soft magnetic alloys. All measurements have been performed under closed-flux conditions. The Ni-Fe (Mumetal type) and the Co-based amorphous alloys reach the highest values of permeability and the lowest coercivities, but their saturation magnetization is somewhat reduced with respect to the Fe-Si and Fe-based amorphous alloys. The ferrites do not display prominent soft DC properties but, being nonconductive, become the best choice at frequencies in the MHz range. The actual and the prospective applications of magnetic materials have thus to be evaluated against a number of parameters, such as initial and peak permeabilities, coercive field, remanence, AC energy losses, squareness ratio, etc., which are the result of both compositional and structural properties. The composition determines the values of the so-called intrinsic magnetic parameters, such as the saturation magnetization, the magnetic anisotropy constants and the magnetostriction constants, which, in turn, affect the magnetization process in a way related to the material structure

1.0 [ 0.5 ~-

er

]

~,

!.0 0.5

0.0

~" 0.0

-0.5

-0.5

-1.0

-1.0

-1.5 -150-100-50 0 50 100 150 H (A/m)

-1.5 -20

............

-10

0 H (A/m)

10

20

FIGURE 2.1 Representative DC hysteresis loops in different soft magnetic alloys. (a) Grain-oriented Fe-(3 wt%)Si, non-oriented Fe-(3 wt%)Si, MnZn ferrites. (b) Fe78B13Si9 annealed amorphous ribbons (1), Co71Fe4B15Si10as-quenched amorphous ribbons (2), Fe-Ni (Mumetal type) alloys (3).

2.1 GENERAL PROPERTIES

27

(e.g. crystallographic texture, grain size, foreign phases, lattice defects, etc.). By a proper choice of composition and suitable metallurgical and thermal treatments, extra-soft magnets are obtained, where the coercive field and the relative permeability attain values of the order of 0.1 A / m and 106, respectively. However, it should be stressed that a number of additional properties, such as thermal and structural stability, stress sensitivity of the magnetic parameters, mechanical properties and machinability, and thermal conductivity, have to be considered. The final acceptance of a material in applications will result from a cost-benefit evaluation of all these properties. The magnetization process in an SMM occurs by means of two microscopic mechanisms: motion of the domain walls and uniform rotation of the magnetization inside the magnetic domains. Rotations require high field strengths in the conventional Fe-based crystalline alloys because the Zeeman energy EH = -I-I.Js (with H and Js the applied field and the saturation polarization, respectively) must balance a magnetocrystalline anisotropy energy term FK roughly of the order of the anisotropy constant K1. Given that K1 is of the order of few 104 J / m 3, fields in the 10 3 A / m range must be applied to achieve substantial rotations. A soft magnetic behavior can possibly be achieved in these materials only through displacements of the domain walls. Frictional forces, inevitable in real defective materials, resist these displacements. The coercive field measures the typical field strengths at which the domain walls are unpinned from defects and a substantial part of the magnetization is reversed. Energy is lost in this process, and magnetic hysteresis is accordingly observed. The subject of coercivity and hysteresis is classically treated by theorizing the motion of a domain wall, assumed either as a rigid [2.6, 2.7] or a flexible [2.8] object, in a perturbed medium. Basically, it is assumed that the structural perturbation generates a random wall energy profile, whose spatial derivative represents the pressure to be applied by a field in order to achieve wall motion. The hindering effect of the structural defects is chiefly controlled by the value of the anisotropy constant K1, implying that the value of coercivity tends to increase with the strength of the anisotropy effects. However, even with the relatively high values attained by K1, very soft magnetic behavior can be achieved in Fe and Fe-Si when the microstructure is suitably controlled. In practice, this means having the least content of precipitates, voids, dislocations, and point defects, together with large and favorably oriented grains. By having the applied field directed as far as possible alongside one of the (100) easy axes, i.e. in the plane of the main 180~ domain walls, one gets an obvious directional advantage for the wall displacements, as remarkably demonstrated by the grain-oriented Fe-Si

28

CHAPTER 2 Soft Magnetic Materials

laminations. The role of the microstructural defects is clearly observed in Fe. Here, one can reach coercive fields as low as Hc-" 1 A / m upon prolonged purification and annealing treatments [2.9], leading to very low dislocation densities and C and N concentrations (some 10-20 parts in 106 (ppm)). Coercivities of a few hundred A / m can be found instead when these concentrations are in the 100 p p m range [2.7]. C and N are basically insoluble in Fe and tend to form carbides and nitrides, which act as strong pinning centers for the domain walls. With much higher C content (say around I wt%), graphite precipitates and martensitic domains are additionally formed, and Hc can reach values typical of hard magnets (several 104 A/m). Soft and extra-soft magnetic properties are naturally associated with very low values of the magnetic anisotropy (say with K in the range of a few tens J / m 3 and less). This is the case, for example, of F e - N i alloys, with composition around Fe20-Nis0. K1 is positive in ~-Fe (bcc cell) and negative in Ni (fcc cell) and it passes through zero on the high Ni side in the F e - N i alloys. Vanishing anisotropy can equally be obtained in amorphous and nanocrystalline alloys because the structural order in these materials is extended over limited distances, from a few atomic spacings to a few nanometers. The characteristic length L controlling the magnetization process, represented by the domain wall thickness, encompasses a large number of the local ordered structural units, so that the magnetocrystalline anisotropy is effectively averaged out. The residual anisotropy is calculated to be [2.10]

K4 6 K0-"

A3 ,

(2.1)

where A is the exchange stiffness constant and 3 is the average size of the structural units. By taking 3 = 10 -9 m, K1 = 4.8 x 104 J/m 3 (as in Fe) and A--- 10 -11 J/m, we find the negligible value K0 = 5.3 x 10 -3 J/m 3. With the elimination of the magnetocrystalline effects, other sources of magnetic anisotropy can be brought to light in these materials. For instance, in the highly magnetostrictive Fe-based amorphous alloys, a substantial anisotropy can result from the magnetoelastic coupling between frozen-in or applied stresses and the magnetization. In the typical alloy of composition, Fe78B13Si9, the saturation magnetostriction is As "~ 30 x 10 -6 and the long-range internal stresses generated by the rapid solidification process are of the order of 50-100 MPa [2.11]. Induced anisotropies K r 3As~r, in the range of several hundred J / m 3, can therefore arise and annealing treatments are required in order to achieve excellent soft magnetic behavior. Co-based amorphous alloys (like


29

Co71Fe4B15Si10 or Co66Fe5Cr4B15Si10) and F e - N i alloys of the permalloy type have vanishing magnetostriction (in the 10-8-10 -7 range) and, lacking also the magnetocrystalline anisotrop36 attain the lowest coercivities and record values of permeability. In addition, their properties can be adjusted by inducing calibrated uniaxial anisotropies through annealing treatments under saturating fields. In a saturated F e - N i alloy, the magnetization interacts with the Fe-Fe and N i - N i atomic pairs in such a way that, if the temperature is sufficiently high, they tend to distribute preferentially along the field direction, although the alloy preserves its character of random solid solution. In the amorphous alloys, anisotropic atomic rearrangements in the local ordered units, with symmetry influenced by the direction of the magnetization, are expected to occur [2.12]. In all cases, dramatic changes of the magnetization curve can be obtained by means of field annealing, as illustrated for a Co-based amorphous alloy by the example shown in Fig. 2.2. It should be noted that

0.8

Amorphous alloy C071Fe4B15Si10

1

0.4 Field annealed, H• 0.0

-0.4

\

-0.8 -20

J Field annealed, H// I

-10

I

0 H (A/m)

,

I

10

,

20

FIGURE 2.2 DC hysteresis loops in amorphous ribbons of composition Co71Fe4_ Bt5Sil0 after annealing under a saturating magnetic field. A rectangular loop with Hc "--0.5 A / m and peak relative permeability /Zr "~ 106 is obtained after longitudinal field annealing (HII, 1000 s at 340 ~ Transverse field annealing (H,, T - 260 ~ leads to linearization of the loops, as shown here after 600 and 3000 s long treatments. The correspondingly induced transverse anisotropies are Ku "" 5 J/m 3 (2) and Ku "~ 25 J/m 3 (3).

30


the hysteresis loops in these amorphous ribbons can be linearized by means of a transverse induced anisotropy around some 10 J/m 3. In this case the rotation of magnetization becomes an easy process, leading to quite high initial permeabilities (of the order of s o m e 104), a welcome property in many applications. SMMs find most applications in magnetic cores of AC apparatus, from 50 Hz to several MHz. The quantity of basic technical interest is in this case the energy loss. Eddy currents are generated by time-varying magnetic flux, leading both to shielding of the core by the associated counterfields (skin effect) and generation of heat by Joule effect. Excluding ferrites, which are basically insulating materials, all soft magnets have to be used in sheet form in order to minimize skin effect and losses. The theoretical assessment of these effects is by no means a simple one because it is seldom possible to treat the material as a continuum, characterized by a given magnetic permeability, and apply to it the Maxwell equations. The magnetic structure is made of domains and domain walls and the distribution of eddy currents can be extraordinarily complex and nonuniform. Williams, Shockley and Kittel were the first to take such a complexity at face value by investigating the dynamic behavior of a 180~ Bloch wall in single crystals of Fe-Si. They theorized it through a balance equation involving, on the one hand, the applied magnetic field and, on the other hand, the structural pinning field and the eddy current counterfield (WSK model) [2.13]. Pry and Bean generalized the WSK model to a system of 180~ walls, emulating the real domain structure of a grain-oriented Fe-Si lamination [2.14]. The whole problem has in recent times received a complete assessment by Bertotti, who has shown that the complexity of the dynamic magnetization process in real structures can be properly described by means of statistical methods [2.15]. The general phenomenology of losses in magnetic laminations is summarized in Fig. 2.3, illustrating the dependence of the hysteresis loop shape and area (i.e. energy loss) on the magnetizing frequency in a grain-oriented Fe-Si alloy. Bertotti's theory provides a physically based demonstration of the concept of loss separation, as expressed by the equation: W = Wh q- Wcl q- Wexc.

(2.2)

The energy loss per cycle W is taken, at a given magnetizing frequency f and peak polarization ]p, as the sum of three components, having the following meaning. Wh, called hysteresis loss, is the residual energy dissipated in the limit f--, O. Wd is the loss component calculated by applying the Maxwell equations to the material when it is assumed completely homogeneous from the magnetic viewpoint (absence of domains). By summing up Nh and Wd, one falls short of the measured


1.5

.....

.

j----

31 ......

i

9 .~

30

!

-0.5 -1.0

(a)

-60

,i..

. . . . . .

,...,

. . . . . . . .

J,......

. . . .

,. | . ,

,~ 20

0.0

-1.5

j

GO Fe-(3wt%)Si ..~ Jp=1.32T ~ w l

1.0 0.5

. . . .

i i,/

...............

-40

-20

lO

2;2 z

i 0

H (A/m)

~~

E

20 40 60

. . . . . . . . .

(b)

0

,..,

50

. . . . . . . .

, . . . . . . . . . .

1O0

f(Hz)

,..,..,~

150

. . . . .

2(10

FIGURE 2.3 Hysteresis loops (a) and specific energy loss per cycle (b) vs. magnetizing frequency in grain-oriented Fe-Si laminations (thickness 0.29 mm) under sinusoidal time dependence of the polarization (Jp = 1.32 T). The energy loss in (b) is proportional, at a given frequency f, to the area of the corresponding (J,H) hysteresis loop. The loss separation concept formulated in Eq. (2.2) is illustrated in (b). value. The remainder, We• is called excess loss. The three loss components are associated with different eddy current mechanisms and different space-time scales of the magnetization process. The classical loss Wd is a sort of background, always present and independent of any structural feature. For a lamination of thickness, d, conductivity ~r and density 3, one finds, under sinusoidal time dependence of the magnetic polarization, the classical loss per unit mass: Wcl-

~T2

o-d2j2f

6 T '

(2.3)

where 3 is the density of the material and it is assumed that complete flux penetration in the lamination cross-section occurs. Figure 2.3 demonstrates how an elementary approach to losses based on the classical approximation largely underestimates the measured loss in grain-oriented laminations at 50-60 Hz, the frequencies at which the large part of electrical energy is generated and produced. In addition, the dependence of W o n f is non-linear, in contrast with the prediction on Wd (Eq. (2.3)). The components Wh and We• derive from the heterogeneous nature of the magnetic structure of the material and the inherently discrete behavior of the magnetization process. Flux reversal is concentrated at moving domain walls and, even under quasi-static excitation, eddy currents arise because the domain wall displacements, hindered by the pinning centers, occur in

32


jerky fashion (Barkhausen effect). Intense current pulses, with lifetimes around 10 -9 s, are generated, localized around the jumping wall segments. In this way, the hysteresis loss Wh is generated and, since the time constant of the eddy current pulses is always many orders of magnitude smaller than the typical magnetization period T = l / f , it is concluded that Wh is independent of frequency. For a given value Jp, Wh gives a measure of the coercive field. This is totally consistent with the fact that the Barkhausen mechanism is a volume effect, independent, as the coercivity should be, of the material conductivity. The excess loss Wex c is associated with the largescale motion of the domain walls. The theory shows that, in most SMMs, the following expression holds:

kexc~-~J~/2

Wexc =

,

(2.4)

where kexc is a parameter related to the type of existing domain structure and its relationship with the material structure. Very broadly, it can be stated that the larger is kexc the more discrete is the magnetization process. Very large domains are therefore not desirable from this viewpoint and, as discussed below, methods have sometimes to be devised to increase the number of domain walls in the material. In conclusion, if minimization of the AC energy losses is sought, not only the lamination thickness and 10

'

I

'

7

E~

/

I

'

I

'

I

Low-carbon steel

I

=SOHz .. 9

\._

(/) O t..

O

-o~ 9 ~

Grain-oriented Fe-Si

O Q.

'

Jp=IT

"\.\

Fe-Si

'

0.1

/

f

~O--o._,

Amorphous FeBSi "

1880

,

I

1900

,

I

1920

=

I

1940

,

l

1960

,

I

1980

,

2000

Year

FIGURE 2.4 Record loss figures over a century in soft magnetic laminations for transformer cores.

33

2.2 PURE IRON AND LOW-CARBON STEELS

TABLE 2.1 Representative soft magnetic materials and typical values of some basic magnetic parameters H c (A/m) Js (T)

Composition

/d, m a x

Fe NO Fe-Si GO Fe-Si Fe-Si 6.5% Sintered powders Permalloy Permendur Ferrites

Fel00 Fe(>96)-Si( 700 2800 2.5 800 135 x 10 -8 3. The effective magnetocrystalline anisotropy of the material results from the average of the local anisotropies over distances of the order of L, which leads to a very low final value K0. According to Eq. (2.1) we obtain

Ko ~ K(3/L) 6,

(2.5)

where L = x/-A/K. For a structural wavelength 3--- 10-9m and a value A---10 -11 J/m, we can estimate an effective anisotropy K0--" 10 -610 -1 J / m 3, depending on the value of K. This negligible value of the average magnetocrystalline anisotropy is the key to the soft magnetic properties of the amorphous alloys. In fact, under these conditions, coercivity and permeability are due only to residual anisotropies of magnetoelastic origin, or induced anisotropies created by suitable treatments. A stress cr causes a uniaxial anisotropy Kr = (3/2)AsCr in a material characterized by the saturation magnetostricfion constant As. This value provides a sort of "a priori" indicator of the achievable ultimate soft magnetic properties of a given amorphous alloy. Figure 2.15 shows that, in the representative composition Fes0-xCoxB20, As strongly depends on the relative proportions of Fe and Co. It ranges from positive to negative values (from ---30 x 10 -6 to ---- 3 x 10 -6) on passing from the Fe-rich to the Co-rich side and intersecting the value As "" 0 at Fe concentrations around 5-8 at.%. In the highly magnetostrictive Fe-rich alloys, the random distribution of internal stresses introduced during the rapid solidification, typically of the order of 50-100 MPa, is the source of complex anisotropy patterns, with K values in the range of some 102103 J / m 3, and, consequently, of coercivity [2.38]. These stresses can never be completely relieved by annealing, as the treatment temperatures are in any case limited by the necessity to avoid the slightest precipitation of crystalline phases. Even after carefully controlled annealing under a saturating longitudinal field, the Fe-based ribbons reach, at best, coercive fields of 2-3 A/m. The influence of stress anisotropies becomes negligible in the vanishing magnetostriction Co-rich alloys. Accordingly, these materials exhibit the lowest energy losses and the highest permeability at all frequencies. In addition, their properties can be tuned to specific needs by suitable thermal treatments under a saturating magnetic field.

2.4 AMORPHOUS AND NANOCRYSTALLINEALLOYS

55

These can induce a large-scale anisotropy, Ku, as a consequence of localized atomic rearrangements having a definite directional order. Being the only form of anisotropy present in the material, Ku fully governs coercivity, permeability and loop shapes. Figure 2.2 provides an example of the magnetic softness and versatility of the near-zero-magnetostrictive Co-based alloys, which, prepared as very thin ribbons (8-15 ~m), favorably compete with ferrite and Fe-Ni cores up to the MHz region, where they can display initial relative permeabilities approaching 104 [2.39]. With a dominant transverse induced anisotropy Ku, the rotation of the magnetic moments is the chief magnetization mechanism and the associated permeability is

/d'r/d'0-

2Ku'

(2.6)

where/~0 is the vacuum permeability. With the domain wall processes basically suppressed, the loss is minimized and the best high frequency properties are obtained. The extra-soft magnetic properties of the Co-rich compositions are obtained at the expense of a substantial reduction of the saturation polarization with respect to the Fe-based alloys (0.9-0.5 vs. 1.3-1.6 T). This compounds with the obvious cost problems associated with the use of Co so that the related alloys are reserved for specialized applications. Amorphous wires prepared by the in-water-quenching technique exhibit a bistable magnetic behavior, regardless of the sign of magnetostriction. This property derives from the special domain structure that is formed in the wire, typically made of an active longitudinal core, reversing its magnetization with a single Barkhausen jump, and an outer shell, having either radial or circumferential domains. The origin of such a structure is to be found in the anisotropies induced by the large stresses frozen-in during the solidification process, in association with the anisotropy of magnetostatic origin (shape anisotropy). The switching-like behavior of the magnetization reversal in amorphous wires can be exploited in a number of applications, such as jitter-free pulse generators, digitizing tablets, speed and position sensors, and antitheft devices. A further remarkable property of amorphous wires is that their reactance at MHz frequencies can change to a large extent upon application of a DC field (giant magnetoimpedance effect). For instance, variations &X/X-~ 0.1-1 under an applied field of 100 A / m can be found in Co-based amorphous wires [2.40]. This effect originates in the strong variation of skin depth with the variation of the domain

56


structure imposed by the DC field and has potential for many types of magnetic field sensors. Table 2.8 summarizes the behavior of the main physical parameters in a number of common amorphous alloys. Fe-based alloys are used in applications like the distribution transformer cores, where they can often replace the high permeability GO Fe-Si laminations. A total loss reduction by a factor 2-3 can be obtained at 50 Hz on passing from GO to amorphous Fe78B13Si9 laminations (Fig. 2.16). The loss analysis, schematically illustrated in Fig. 2.16b, demonstrates that this is due to a drastic reduction of the excess and the classical loss components brought about, according to Eqs. (2.3) and (2.4), by the combination of low ribbon thickness and high material resistivity. In recent years, increasing emphasis on energy saving has favored the introduction by electrical utilities of distribution transformers made of amorphous alloy cores, especially in the single-phase low power range (10-50 kVA). These devices are characterized by reduced total ownership costs, regarding both purchasing and operating costs through the device lifetime, as well as exhibiting good stability over time [2.41]. Less favorable economic conditions are attached to applications in three-phase power transformers. Significant use of amorphous alloys is made in electronics [2.42]. For instance, Co-based alloys are ideal as cores of inductive components to be employed up to frequencies of the order of I MHz, as found, for instance, in the switched-mode power supplies and in digital telecommunication circuits. Their low Js value is not a disadvantage in these cases, where, in order to limit core heating, the working induction is always kept small. The unique combination of high elastic limit and high magnetostriction in the Fe-based materials is exploited in high-sensitivity sensors and transducers. Further applications include electromagnetic interference filtering, magnetic heads, various types of magnetic shielding and ground fault interrupters. Amorphous alloys tend to crystallize heterogeneously upon annealing, with scattered nucleation and growth of microcrystals taking place at temperatures well below the bulk transition to the crystalline state. This has detrimental consequences on the soft magnetic behavior of the material, besides being associated with drastic mechanical embrittlement. Fe-based alloys actually need stress-relief annealing in order to achieve optimized magnetic properties, but if the treatment temperature is brought to values of the order of 400 ~ a sharp increase of the coercive field is observed due to the heterogeneous formation of microcrystals of 0.1-1 ~m size [2.43]. In Fe-Si-B alloys, however, it is possible to achieve both homogeneous accelerated grain nucleation and restrained grain coarsening by the addition of Cu and Nb [2.44].

o

~o

0

0

0

L'b 0 0

t~

O 0

0

e

"

I.~

o

0

0 O0 O'b ~'--~

t'xl 0 0

~. b~ ~.o o~b~ o"1

~ ~C-,~

eeeee

0

~ ~ d c 5 c 5 ~

~

0

t'xl P'~, I..~ O0 " ~ t'~l O~ O"b 0 r

Cxl txl ~--~

~ ~ . c 5 c 5 c 5

o~,

o ~o

o

~o~

un

~o~ o~

9

~o

o o .n9 ".n

2.4 AMORPHOUS A N D NANOCRYSTALLINE ALLOYS

o

,.-.-i

o ,.~ o

..~ t~

r~

I-i

t~

~~,,,i

v

o

57

58

CHAPTER 2 Soft Magnetic Materials .

"

.

.

!

.

.

.

.

!

.

.

.

.

!

[i; s0.z]

1.5

C3~

.

.

.

.

.

o /

/

Fe-(6.5 wt Yo)Si

1.0

v O~ O)

._o n O

0.5 ,,,

,,, ,, "

............. 0.0 0.75

.

.

.

.

i

.

.

.

AmorphousFe78B13Si9

.

i

1.00

(a) 1.50 r

.

,

,

1.25 Jp (T)

.

I

.

1.50

.

.

.

1.75

f= 50 Hz J_ = 1.25 T

P

1.25 "~ 1.00 ffJ

._o 0.75 O

O

a. 0.50 0.25 0.00 (b)

Ill NO (0.35 mm)

Fe-(6.5wt%)Si (30 mm)

GO

Amorphous

(0.30 mm)

(20 mm)

FIGURE 2.16 Power loss at 50 Hz as a function of peak polarization Jp (a) and its decomposition at Jp = 1.25 T (b) in a number of representative soft magnetic alloys: NO Fe-(3 wt%)Si laminations, 0.35 and 0.50 mm thick; GO Fe-(3 wt%)Si laminations, 0.30 and 0.23 mm thick; Fe-(6.5 wt%)Si rapidly quenched ribbons, 30 ~m thick; Fe78B13Si9 amorphous ribbons, 20 ~m thick. The excess loss component Pexr is the largest one in the GO laminations, whereas the classical component Pd is negligibly small in the high Si alloys and in the amorphous ribbons.

2.4 AMORPHOUS AND NANOCRYSTALLINEALLOYS

59

In particular, by treating amorphous ribbons with composition Fe73.5a homogeneous nanocrystalline structure is obtained, composed of oL-Fe-(---20 at.%)Si grains, having dimensions of the order of 10 nm, embedded in a residual amorphous matrix. The crystallites occupy about 70% of the material volume and are separated by amorphous layers 1-2 nm thick. Quite a similar structure can be obtained in alloys with composition Fe91_84(Zr, NB)7B2_9, the crystalline phase now being made of ~-Fe grains [2.45]. Since the grain size 3 is smaller than the correlation length L = x/A/K and the intervening amorphous phase ensures grain-to-grain exchange coupling, conditions similar to those found in amorphous structures are created, leading to vanishing crystalline anisotropy. In particular, with 3 - 10 nm, K = 104 J/m 3, and L = 50 nm, Eq. (2.5) yields a value of the average magnetocrystalline anisotropy K0-" 0.5 J/m 3. This feature is accompanied by magnetostrictive anisotropies averaging out to vanishing values. In fact, there is a balance in the material between the negative magnetostriction of the crystalline phase and the positive magnetostriction of the amorphous phase. Any applied or residual stress may generate anisotropies at the nanometer scale, having directions dictated by the nature of the stress (tensile/compressive) and the sign of the magnetostriction constant, but, again, the exchange interaction acts to suppress any mesoscopic and macroscopic anisotropy. Equation (2.5) suggests that the coercive field, which can be roughly estimated to be proportional to the average anisotropy constant K0, increases with the sixth power of the grain size 3. Experiments show that this relationship is verified to a good approximation up to 3---100 nm [2.46], a limit beyond which the coercive field starts decreasing with increasing 3. Febased nanocrystalline alloys emulate the properties of the amorphous Co-based alloys, with the advantage that one can deal with inexpensive raw materials while achieving higher saturation magnetization (e.g. 1.24 T in Fe73.5CulNb3B9Si13.5 and 1.63 T in Fe91Zr7B2 vs. 0.61 T in Co67Fe4B14.5Si14.5 and 0.86 T in Co71Fe4B15Si10) and improved thermal stability. The hysteresis loop of nanocrystalline alloys is sensitive to field annealing, although the ordering mechanism, investing the crystalline Fe-Si phase, is less effective than in the Co-based amorphous alloys. In any case, nearly linear, low-remanence hysteresis loops can be achieved by suitable treatments under transverse field, which yield low power losses and high permeabilities up to frequencies of several hundred kHz. An example of loss and permeability behaviors up to the MHz range in nanocrystalline Fe73.5CulNb3B9Si13.5 ribbons is provided in Fig. 2.17. They compare favorably with the properties of other types of soft magnetic alloys prepared for medium-to-high frequency applications.

CulNb3B9Si13.5 at a temperature around 550~

60

CHAPTER 2

103

.

dp= 0.2 T

.

.

.

.

.

i

s"

. . . . .

"

,',.';

G" .."; /

_.

,./" ~k,/'"

z~ S

\

aanocrystalline:

Z

/ ~/.;/'Permalloy ,,. , . ) i " ~F~based

ffJ

L 0

,",, ~" /, "

o ] 01 a.

S

9 , " ~ Co-based , amorphous alloy

t 10 0

.

.

.

.

10 4

105

'

.

.

.

.

o. 104 ! ._= .__.

I

.

.

105 Frequency (Hz)

. . . . . . .

I

~

..Q

alloy

amorphous alloy

f ,'/,

! ~ ~"

E

"./

9 ./.

Mn-Znferrite~

--~ 102

O

Soft Magnetic Materials

.

.

.

.

.

.

.

.

.

.

.

.

I

.

.

'

106 9

9

r---'-- Nanocrystalline

Co!based "":-.~~ / "'.."~

amorphous alloy

I

O

,, ~

Permall~

,_

-12

0

Mn-Zn ferrite

nr"

Fe-based amorphous alloy

103 103

~~~~

.

.

.

.

.

.

.

.

I

" "

I Hp= 0.4 A/m I .

.

.

.

.

.

.

.

n

104 105 Frequency (Hz)

|

FIGURE 2.17 Power losses (peak magnetization Jp = 0.2 T) and relative initial permeability (Hp = 0.4 A/m) in 18 ~m thick nanocrystalline Fe73.sCulNb3BgSi13.5 ribbons (solid lines). The ribbons have been suitably annealed under a saturating transverse magnetic field. Their properties are compared with those of Fe-based and Co-based amorphous ribbons and of Fe-Ni tapes of Permalloy type, all having comparable thickness. Data from a M n - Z n ferrite are also reported (adapted from Ref. [2.44]).

2.5 NICKEL-IRON AND COBALT-IRON ALLOYS

61

2.5 N I C K E L - I R O N A N D C O B A L T - I R O N ALLOYS Nickel-iron alloys display a broad range of magnetic properties and a well-defined structure in the range 35%-< N i - 80%. A stable random fcc solid solution (~-phase) is obtained above 35% Ni by a suitable choice of annealing temperatures, cooling rates, and the possible addition of elements like Mo, Cu, and Cr. In fact, the ~--* ~ phase transition on cooling from high temperatures occurs at T G 500 ~ and, because of the low diffusion rates, it can consequently be easily restrained, together with the formation of the ordered NiBFe phase. Structural stability and homogeneity are conducive to good mechanical properties and ease in cold rolling, down to thicknesses in the 5-10 ~m range. The variety of magnetic behaviors achieved in the final F e - N i laminations are rooted in the remarkable evolution of the intrinsic magnetic parameters with composition and treatment (see Fig. 2.18). It is noticed, in particular, that both the first magnetocrystalline anisotropy constant K1 and the magnetostriction constants ~100 and ~111 pass through the zero value in the Ni-rich side, with a positive K1 value (i.e. a (100) easy axis) co-existing with ~-phase up to ---75 wt% Ni. Compositions with Ni < 30 wt% are illdefined from the structural point of view and bear little interest as magnetic materials. The Fe70-Ni30 alloy is characterized by a singular drop of the Curie temperature Tc, which becomes of the order of room temperature. It increases in a near-linear fashion in the range 30-35 wt% Ni, a property that is sometimes exploited in magnetic shunt devices. At 36 wt% Ni concentration, Tc has already reached the value of 230 ~ the thermal expansion coefficient is extremely low (---1 x 10 -6 K -1, the socalled Invar behavior) and the resistivity is quite high (75 x 10-8 f~ m). This latter feature is conducive to low losses at high frequencies and makes the Fe64-Ni36 tapes interesting for applications like radar pulse transformers. The Fe50-Ni50 alloys are characterized by a high saturation polarization of 1.6 T. They can be prepared as strongly (100)[001] textured sheets by means of severe cold rolling to the final thickness (>95%) and primary re-crystallization annealing around 1000 ~ The favorable directional feature provided by the texture can be reinforced by the anisotropy Ku induced by means of a longitudinal magnetic annealing at a temperature T---450 ~ (To "" 500 ~ and followed by slow cooling. In this way, a squared hysteresis loop is achieved, to be exploited, for instance, in magnetic amplifiers and saturable reactor cores. By increasing the Ni content towards 55-60 wt%, the induced anisotropy Ku, of the order of 300-400 J / m 3, is not far from the value of the magnetocrystalline anisotropy K1. By annealing under a transverse

62

CHAPTER 2 Soft Magnetic Materials 2.0

,..

.,.

,,.

600

.,..

s s

,.,.

500

1.5

400 A

I-v~.aco 1.0

300

bY

200

0.5

/ I

100

i

0.0 30

~111

2O o, o

10

o

-10 -20

,

|

,

|

|

2 K 1 disordered A O3

E

,

0

..

,.,.

.

.

.

.

.

.

.

.

, . .

,

,,,

o

2

K 1 orde

~F-2

', I

t t

-3

e t

! t

-4

3o

Ni (wt %)

e

8o

FIGURE 2.18 Dependence of the magnetic intrinsic parameters on the Ni concentration in Ni-Fe alloys. Js, saturation polarization; To, Curie temperature; K1, magnetocrystalline anisotropy constant; A100 and Am, magnetostriction constants. Tc approaches the room temperature for Ni--- 30 wt% and interesting magnetic properties are observed only at higher Ni concentrations. The anisotropy constant depends, besides the composition, on the degree of structural ordering, associated with the formation of the Ni3Fe phase. It is thereby related to the annealing temperature and the cooling rate. A uniaxial anisotropy Ku is induced by cooling below Tc under a saturating magnetic field.

2.5 NICKEL-IRON AND COBALT-IRON ALLOYS

63

saturating field, one can therefore achieve a sheared low remanence hysteresis loop characterized by a large unipolar swing (0.9-1.2 T). This feature is welcome in devices like unipolar pulse transformers or ground fault interrupters. The highest permeabilities and lowest coercivities are obtained around 75-80 wt% Ni concentration because it is possible to approach vanishing values for both magnetostriction and anisotropy. Figure 2.18 shows that it is not actually possible to simultaneously achieve zero values for K1, &00, and Am. It is therefore expedient to make calibrated additions of elements like Mo, Cu, and Cr by which one can achieve, at the same time, an isotropic magnetostriction constant As ~" 0 and a good control of the FegNi phase ordering through annealing and cooling. Since the anisotropy constant K1 depends on ordering, it is possible to devise a thermal treatment leading to K1 "-" 0 [2.47]. A further advantage introduced by the additives is a substantial increase of resistivity (e.g. from 20 x 10 -8 to 62 x 10 -8 12 m by introducing 5 wt% Mo in 78 wt% Ni alloys) at the cost of a certain reduction of Is. As reported in Table 2.9, a coercive field lower than 1 A / m and relative initial permeabilities higher than 105 can be obtained in these alloys, generally known under the trade name of permalloys. A typical DC hysteresis loop in a permalloy tape is given in Fig. 2.19a, illustrating, through comparison with the loop of a GO Fe-Si lamination, a somewhat extreme example of magnetic softness. By field annealing at temperatures ranging between 250 and 380 ~ a substantial manipulation of the hysteresis loop can be obtained because the magnetocrystalline anisotropy can be overcome by the anisotropy Ku induced by magnetic ordering (less than 1 0 0 J / m 3, see Fig. 2.18). On then passing from longitudinal to transverse field annealing, the hysteresis loop shape may undergo a change as shown in Fig. 2.19b. This is associated with a change of the mechanism of the magnetization process, which becomes dominated by the rotation of the magnetic moments. It should be TABLE 2.9 Properties of some basic Fe-Ni and Fe-Co alloys

Fe64-Ni36 Fe50-Ni50 Fe15-Ni80-Mo5 Fe14-Ni77-Mo4-Cu5 Fe49-Co49-V2

Js (T)

Tc (~

p (10-812 m)

Hc (A/m)

/~i (103)

1.30 1.60 0.80 0.78 2.35

230 490 400 400 930

75 45 60 60 27

40 7 0.4 1.5 100

2 15 150 40 2

Js, saturation polarization; Tc, Curie temperature; p, electrical resistivity; Hc, coercive field;/~, relative initial permeability. Compositions are given in wt%.

64

CHAPTER 2 Soft Magnetic Materials 1.5 / (T) ......... ; "7"'~ s-

GOFe-SI

/'f

""" "- "--

.

t

;' 1.0~-

,

s

;'

9

"

~, / . /

0. !

I[1

~ Fe15-Ni80-Mo5

:

"

/i

.,,."/

--.1-: . . . . . . . . . . . .

H (A/m)

: _t.., -'.i --i ."5

0.75 Fe15-Ni80-Mo5

I

/

J (T)

~..

"Jll

.i

i'1/

Long. f i e l d ~ "i anneal ~ ' ~ ~I / / -10 -5 [,, i; !/

----

Transverse field anneal _ 5 10 H (Nm)

FIGURE 2.19 (a) DC hysteresis loops in GO Fe-(3 wt%)Si laminations and in permalloy (Fe15-Ni80-Mo5) tapes. (b) Loop shearing in permalloy by means of annealing under a transverse saturating magnetic field. remarked that K1 increases with decrease in temperature. Consequently, the treatments should be calibrated for the temperature at which the magnetic core is eventually employed. For instance, in permalloys for cryogenic applications, the annealing temperature and the cooling rate are conveniently reduced in order to achieve K1 "" 0 at such temperatures (i.e. K1 < 0 at room temperature) [2.48].

2.6 SOFT FERRITES

65

Cobalt-iron alloys do not display outstanding soft magnetic properties, but represent a unique solution in terms of Curie temperature and saturation polarization, both remarkably higher than in pure Fe. In the classical Fe50-Co50 alloy, we have, for instance, Tc = 980 ~ and Js = 2.40 T. This is useful for a number of applications where volume reduction and high working temperature may be required, as in the case of onboard high-speed generators for aircraft and spacecraft, without concern for the cost of Co. The Fe50-Co50 alloy transforms from fcc (~/) to bcc (o0 solid solution at 1000 ~ on cooling and undergoes rapid long-range ordering below 730~ The ~-~/ transformation can to some extent limit the re-crystallization process and solid state refining, while ordering detrimentally affects the mechanical properties, leading to a brittle material. Ordering can, however, be retarded by the addition of 2 wt% V and rapid quenching, so that Fe49-Co49-V2 alloys can eventually be prepared as thin sheets by cold rolling, with the further benefit of a large increase of resistivity with respect to Fe50-Co50 (27 x 10 -8 vs. 7 x 10-811 m). By adjusting the cooling rate, one can also dramatically affect the anisotropy constant, which can be made to approach the zero value, but the magnetostriction always remains extremely high, with A100 "" 150 x 10 -6 and '~111 "" 30 X 10 -6, which hinders the achievement of a really soft magnetic behavior. The value of the coercive field in regular Fe49Co49-V2 alloys (Permendur) is around 100A/m, with a relative permeability ---2 x 103. A substantial property improvement can be obtained by very careful control of the material purity and magnetic field annealing. The high purity alloy, called Supermendur, can exhibit Hc --" 10 A / m a n d / z --- 8 x 104.

2.6 S O F T FERRITES Soft spinel ferrites are largely applied at frequencies above the audio range, up to a few hundred MHz, because of their non-metallic character. They have the general composition MO.Fe203, where M is a divalent metal ion such as Fe 2+ , Mn2 + , Ni 2+ , Zn 2 + , Mg 2 + . Typical applications include pulse and wide-band transformers for television and telecommunications, inductor cores in switched-mode power supplies, antenna rods, cores for electromagnetic interference suppression, and magnetic heads. For frequencies in the range 500 MHz-500 GHz, the so-called microwave ferrites are employed. Some types of spinel ferrites, hexagonal ferrites (like BaFe12019), and garnets (like Y3Fe5012) belong to this class of materials. They are used in a variety of devices, such as waveguide

66


isolators, gyrators, and modulators, to control the transmission or absorption of electromagnetic waves. The magnetic properties of ferrites are due to the magnetic moments of the metal ions. The i o n - i o n interaction is antiferromagnetic in nature and leads to the distinctive temperature dependence of the inverse of susceptibility shown in Fig. 2.20 [2.49]. The oxygen ions in spinel ferrites are arranged in a close-packed face-centered cubic structure and the small metal ions slip into interstitial positions, at either tetrahedral (A) or octahedral (B) sites, which are surrounded by four and six oxygen ions, respectively (Fig. 2.21). In a unit cell, which contains eight formula units (i.e. 32 0 2- ions, 16 Fe 3+ ions, and 8 M 2+ ions), 8 of the available A sites and 16 of the available B sites are occupied by the metal ions. When the M 2+ ions and the Fe 3+ ions are in the A and B sites, respectively, we have the so-called normal spinel structure. The inverse spinel structure is obtained when the 16 Fe 3+ ions are equally subdivided between the A

1/X

,so'* ,Is,,,~

(a) 0.75

(b)

T(K)

...................

0.50

0.00

T.

C~

0

25O

~l=e O

500

75O 1000 T(K) -

-

FIGURE 2.20 (a) Predicted non-linear temperature dependence of the inverse of susceptibility in a ferrimagnetic material [2.49]. The paramagnetic transition occurs at the N6el temperature TN. (b) Saturation polarization vs. temperature in a number of cubic ferrites [2.50].

2.6 SOFT FERRITES

67

.... @'%, B %

....-"

9 9%

~

%o

s

so

osoS~ oS oS / e s -,r.,..--

/

FIGURE 2.21 Portion (one-eighth) of a unit cell of a cubic spinel. The 0 2 - ions (dark) are arranged in an fcc structure. The metal ions (white) are interstitially arranged in tetrahedral (A) and octahedral (B) sites. and B sites, the latter being shared with the M 2+ ions. However, intermediate cases are very frequent. The spontaneous magnetization of ferrites and its temperature dependence were explained by N6el [2.49], by assuming that the spin moments of the metal ions in the A and B sublattices are antiferromagnetically coupled through indirect exchange interaction. Actually, since the cations are separated by the oxygen anions, direct exchange interaction between their 3d electron spins is negligible and an indirect coupling mechanism, the superexchange [2.50], is expected to take place. This involves the spins of the two extra 2p electrons in the 0 2- ion, which interact by direct exchange with the 3d spins of two neighboring metal cations. The mediating effect of the oxygen spins, which are oppositely directed, is such that, if the two cations have five or more 3d electrons (half-full or more than half-full 3d shell), their total magnetic moments are bound, according to Hund's rule, to antiparallel directions. This is the case of the common ferrite ions Mn 2 + , Fe 2 + , Co 2 + , Ni 2 + . The strength of the superexchange interaction is the greatest when a straight line connects the cations through the 0 2+ ion. The A - B coupling, which is associated to an A - O - B angle around 125~ is then much stronger than the A - A and B-B couplings, where the angles are 90 and 80 ~ respectively. At the end, one is left with a system made of two coupled arrays of antiparallel magnetic moments of unequal magnitude, which results in a net magnetic moment. This uncompensated antiferromagnetic behavior is called ferrimagnetism and the resulting magnetic moment per unit cell can be calculated through N6el's hypothesis. These calculations are in good agreement with the experimental values of

68


the saturation magnetization and are s u p p o r t e d b y neutron diffraction experiments. A l t h o u g h the magnetic m o m e n t per formula unit m a y be very large in terms of n u m b e r of Bohr magnetons, the saturation polarization Js of spinel ferrites is low (typically a r o u n d or below 0.5 T at room temperature), because of the low density of the u n c o m p e n s a t e d magnetic ions. In addition, the temperature d e p e n d e n c e of Js, which results from the composition of the temperature variations of the magnetization of the individual sublattices, m a y give rise, according to N6el's theory; to a variety of behaviors [2.51]. Most magnetic spinel ferrites, like FeFe204, NiFe204, and CoFe204, are of the inverse type. In this case, the magnetic m o m e n t per formula unit equals that of the M 2+ ion because the Fe B+ ions are evenly distributed a m o n g the A and B sublattices. The ZnFe204 ferrite is of the n o r m a l type, but, since the Z n 2+ ion has a closed 3d shell and zero magnetic moment, it is paramagnetic at r o o m temperature. MnFe204 is an example of partly n o r m a l and partly inverse spinel structure, where the M n 2+ and Fe 3+ share in certain proportions the A and B sites. Table 2.10 provides a few examples of ion and m o m e n t distribution a m o n g sites in a few types of spinel ferrites. General properties are s h o w n in Table 2.11. The differences observed TABLE 2.10 Cation occupancy and magnetic moments in different types of spinel ferrites. (1) Inverse ferrite NiFe204 (Fe3+ cations in the tetrahedral A sites). (2) Mostly normal ferrite 1V[nFe204. (3) Normal ferrite ZnFe204 (no net magnetic moment). (4) Mixed ferrite ZnxMn(1-x)Fe204.The addition of the non-magnetic Zn 2+ ion increases the magnetic moment per formula unit. (5) The same occurs with the mixed ferrite ZnxNi(1-x)Fe204 Ferrite

Tetrahedral sites A

Octahedral sites B

Bohr magnetons per formula unit

NiFe204

Fe3+ ~ 5/~B

Ni 2+ 1"2/~B Fe3+ T5~B

2~B

MnFe204

Mn 2+ 10.8 x 5/~B Fe3+ ~ 0.2 X 5/~B

Mn 2+ T0.2 x 5/~B Fe3+ T0.8 x 5/~B Fe 3+ 1"5/~B

5/~B

ZnFe204

Zn 2+ ~ 0/U,B

Fe3+ T5/~B Fe3+ ~ 5/~B

0~B

ZnxMno_x)Fe204 Zn 2+ ~ 0/J,B Fe3+ T5/J,B Mn 2+ ~ (1 - x) x 5/~B Fe3+ T5/~B ZnxNi(1-x)Fe204

Zn 2+ ~ 0/d,B Fe3+ 1 (1 - x) • 5/.sB

(1 + x) x 5/~B

Ni 2+ T (1 - x) x 2/J,B (1 q- 4x) X 2p,B Fe3+ T5/J,B Fe3+ Tx x 5/~B

2.6 SOFT FERRITES

69

TABLE 2.11 Properties of some basic spinel ferrites

r/B,th r/B,exp Js (T) FeFe204 NiFe204

4 2 CoFe204 3 MgFe204 1 MnFe204 5

4.1 2.3 3.7 1.0 4.6

0.603 0.340 0.534 0.151 0.503

Zc (~

(103kg/m 3) (f~m)

/~

p

K1

As

585 585 520 440 330

5.24 5.38 5.29 4.52 5.00

10-5 102 105 105 102

- 12 - 7 200 -4 -- 4

40 - 26 - 110 -6 -- 5

(103j/m 3) (10 -6)

FeFe204, NiFe204, and CoFe204 are inverse spinel ferrites, MgFe204 is mostly inverse (90% of A sites occupied by Fe3+, 10% by Mg2+), MnFe204 is mostly normal (80% of A sites occupied by Mn 2+, 20% by FEB+). r/B,th and r/B,exp are the calculated and experimental magnetic moments at 0 K per formula unit (Bohr magnetons). Js, saturation polarization at room temperature; Tc, Curie temperature; 3, density; p, electrical resistivity; K1, anisotropy constant; As, saturation magnetostricfion. between predicted and measured magnetic moments are ascribed to a number of factors, such as imperfect quenching of the orbital magnetic moments, changes in the ion valence, and fluctuations in the cation distribution between the A and B sites. Such differences are small, in general, except for CoFe204, where the orbital contribution is important. Ferrites, being ionic compounds, are insulators in principle and display in practice a wide range of resistivity values, always orders of magnitude higher than in typical Fe-Si or amorphous alloys. The most important conduction mechanism is the transfer of electrons between Fe 2+ and Fe B+ ions in the octahedral sites. Magnetite (FeFe204) therefore exhibits a nearly metallic behavior, with resistivity p--- 10-5 f~ m. Most technical spinel ferrites are of the mixed type, where the presence of two or more metal ions M 2+, often introduced in non-stoichiometric proportions, can provide great versatility in the magnetic properties. M n - Z n and N i - Z n ferrites are the two basic families of mixed soft ferrites, where, by tuning the relative concentrations of the metal ions and making suitable additions and thermal treatments, material tailoring to specific applications can be achieved. Although, as previously stressed, normal ZnFe204 has zero magnetic moment, its addition to the inverse MnFe204 or NiFe204 ferrites leads to an increase of the global saturation magnetization at 0 K. This can be understood, in terms of N6el's theory, as due to the parallel alignment of the magnetic moments of the Fe 3+ ions in the B sites of ZnFe204, which is enforced by antiferromagnetic coupling with the ion moments in the A sites (see Table 2.10). The price one has to pay for mixing is a progressive decrease of the Curie temperature with

70


increase of the ZnFe204 proportion. This is due to the weakening of the A - B coupling, as summarized for the Mnl-xZnxFe204 and Nil-xZnx Fe204 ferrites in Fig. 2.22 [2.52, 2.53]. The simple spinel ferrites, having cubic symmetry, generally display a negative value of the anisotropy constant K1 ((111) easy axis). This negative anisotropy derives, according to the single ion model, from the sum of the opposite contributions of the Fe 3+ ion moments occupying

,.,% 9

1.0

9

9

i

-

-

-

,

-

-

9

i

9

-

9

|

9

",'7

'

~l

0.8 ~,,,

0.6

~~ 1 ~ - ~

" 9,~

II I

0.4 0.2 0.0

0

200

(a)

400

600

800

600

800

T(K)

1.0 0.8

0.6 0.4 0.2 0.0 (b)

0

200

400 T (K)

FIGURE 2.22 Effect of Zn substitution on the saturation polarization and its temperature dependence in Mnl-xZnxFe204 (a) and Nil-xZnxFe204 (b) ferrites (from Ref. [2.53]).

2.6 SOFT FERRITES

71

the A and B sites, respectively, where the negative KIB term eventually prevails over the positive term K1A. This occurs because the orbital angular moment in the octahedral sites is not fully quenched by the crystal field. In CoFe204, however, K1 is large and positive ((100) easy axis), because the large spin-orbit coupling of the Co 2+ ions predominates (see Table 2.11). By acting on both the starting composition and the processing method, mixed M n - Z n and N i - Z n ferrites can be prepared having very low anisotropy values in a range of temperatures suitable for applications (20-100 ~ It has been shown that Zn substitution in Mn and Ni ferrites leads to weakening of the exchange field acting on the octahedral (B) Fe ions and, consequently, to weakening of the negative KIB constant on approaching the room temperature [2.54]. If, in addition, calibrated replacement of divalent cations with Fe 2+ or Co 2+ ions is made, one can combine the related positive anisotropy with the negative K1 value of the host in such a way that the resultant anisotropy constant K1 will cross the zero value around a convenient temperature (e.g. room temperature). Full anisotropy compensation is, for example, obtained at 300 K with the composition Mn0.s3Zn0.40Feo.07Fe2 2+ 3+04. As remarked in previous sections, a small anisotropy value straightforwardly leads to soft magnetic behavior. It can be stated, in fact, that the coercivity and the initial susceptibility approximately follow the relationships: Hc oc

K1/2 Js.<S> '

~i ~

j2

9 oc -K '

dw

oc

J2"<s> K1/2

(2.7)

'

where ~i ~ and Xdw are the contributions to the initial susceptibility deriving from coherent rotation of the magnetization and domain wall displacements, respectively, and <s> is the average grain size [2.55]. Besides the crystalline structure, applied and residual stresses and magnetic ordering induced by field annealing may contribute to the magnetic anisotropy. The effect of these various terms is summarized by the constant K in Eq. (2.7). According to this equation, the temperature stability of permeability, which is important in many applications, is determined by the dependence of Js and K on temperature. This can be controlled by acting on the addition of Fe 2+ (mainly in the M n - Z n ferrites) or Co 2+ (mainly in the N i - Z n ferrite). The highest permeabilities are reached in the M n - Z n ferrites, whereas somewhat lower values are obtained in the N i - Z n ferrites. However, the latter display much higher resistivities (some 107 vs. 10 -2 to ---10 f~ m, depending on the amount of doping with Fe 2+ in M n - Z n ferrites). The near-insulating character of ferrites is conducive to a nearly constant value of the initial susceptibility over many frequency decades, typically up to the MHz region in M n - Z n

72

CHAPTER 2 Soft Magnetic Materials 104

........

,

........

,

........

,

........

MnZnFe204

.jz" 1oa "

-d.

NiZnFe204

102

10= 0.1

.................... 1

, .............. 10

100

1000

f(MHz)

FIGURE 2.23 Dependence of the real (/~) and imaginary (/z') components of the initial permeability in selected commercial Mn-Zn and Ni-Zn ferrite samples (from Ref. [2.56]). ferrites and the 100 MHz region in N i - Z n ferrites, respectively. This is illustrated in Fig. 2.23, where the behavior vs. frequency of the real and imaginary parts of the relative initial permeability / z - / z ~ -i/z" are presented for selected industrial products [2.56]. In the low induction regimes and at sufficiently high frequencies, the dissipation of energy can be related to the phase shift 3 between J(t) and H(t) according to the equations: tan 3 (2.8) /~,, tan 3 - /z" W = "rrJpHp ~/1 + tan 23 , where W is the energy loss per unit volume and the field and the induction have peak amplitude Hp and Bp, respectively. Besides possible damping effects by eddy currents, which in anisotropy-compensated M n - Z n ferrites critically depend on the addition of Fe 2+ ions, resonant absorption of energy is generally invoked for the high frequency losses. In any case, whatever the predominant magnetization mechanism, either coherent magnetization rotation or domain wall displacements, it appears that high susceptibility and high limiting frequency of operation are conflicting requirements (see Fig. 2.23). By denoting with f0 the relaxation frequency, it is predicted, in particular, that for spin rotations f0x[ ~ oc J~ and for domain wall processes f0x~iWoc J2/(s)[2.57]. This suggests that

2.7 SOFT MAGNETIC THIN FILMS

73

the dispersion of the susceptibility is shifted towards higher frequencies in small-grained ferrites. The accurate control of the intrinsic and structural properties of spinel ferrites (e.g. ion valence, stoichiometry, grain size, and porosity) during material preparation is accomplished through the well-established routes of powder metallurgy. The conventional production process starts with the preparation of the base oxides, typically by calcination of suitable iron salts, and their mixing by prolonged wet grinding. This leads to a homogeneously fine powder, where the dimension of the single granules is around or less than I ~m. The resulting mixture is then dried and prefired in air at 900-1200 ~ During this stage, the spinel ferrite is formed by solid state reaction of Fe203 with the other metal oxides present (MO or M203). The so-prepared powders are then compacted, either by die-punching or hydrostatic pressing, and pieces of the desired shape are obtained. The filling factor of the so-obtained assembly of particles is around 50-60%. In the final main step, the pieces are brought to a temperature of 1200-1400 ~ in an oxidizing atmosphere, with or without application of external pressure. The desired final magnetic and structural properties of the material are thus achieved through: (i) particle bonding by interdiffusion and grain growth; (ii) densification, by elimination of the interparticle voids, up to -~95-98% filling factor; and (iii) chemical homogenization, by completion of unfinished reactions. The resulting product is hard and brittle and, if required, it is eventually machined with precision abrasive tools in order to meet the final tolerances.

2.7 S O F T M A G N E T I C

THIN FILMS

Trends towards miniaturization of components and devices and the need for soft magnets for high frequency applications are placing increasing emphasis on the preparation and the properties of thin soft magnetic films. Three applicative areas, in particular, benefit from the use of soft film cores: (1) magnetic recording heads; (2) sensors and actuators; and (3) high-frequency inductors. While with the thin-film geometry one cannot attain the extra-soft behavior of bulk materials, novel structures can be achieved and novel properties can be demonstrated. Thin magnetic films are prepared by a number of chemical or physical methods. Electroplating and CVD are the most frequently used chemical methods. Electroplating applies to conducting materials and requires metallic substrates. It can provide very high growth rates (up to --- 1 ~m/s), depending on the current density. The CVD method is based on the transport of the constituents of the film, usually as vapors of a halide

74


compound, over the surface of the substrate. Ar is normally employed as the carrier of the compound. The plate to be coated, which is kept at a conveniently high temperature, is placed in a continually renewed gaseous environment, with which it reacts. Garnet single-crystal films are often prepared by this technique. The thin-film preparation method which is by far the most prevalent in the laboratory and in the industrial milieu today is the sputtering deposition. It is a flexible technique, which can be applied to a wide range of materials, both metals and insulators. It is characterized by good control and ease of change of the deposition parameters. Sputtering is based on a process of ejection of particles (normally atoms) from the surface of a target bombarded by ions and their deposition on a substrate. Figure 2.24 offers a schematic representation of a DC sputtering setup [2.58]. An inert gas (usually Ar), kept at a partial pressure of 0.1-1 Pa in a vacuum chamber (10-4-10 -6 Pa), is ionized in a strong electrical field and brought to a regime of self-sustained glow discharge. The positively charged ions are attracted by the target, made of the material to be deposited, where they extract by m o m e n t u m transfer either single atoms or atom clusters. Electrons generated by a heated filament are useful in assisting the glow discharge and a magnetic field can additionally confine the plasma (magnetic bottle), increasing the efficiency of the available

I coil I

VT

Vs

Target Ar

stmte

Vacuum pump

Th. emitter

FIGURE 2.24 Schematic representation of a DC bias sputtering setup for deposition of thin magnetic films. The target voltage VT is high and negative (around I kV or higher). The substrate voltage Vs is slightly negative with respect to the anode (---50-100 V). The glow discharge column, thermionically and magnetically assisted, is sketched at the center of the vacuum chamber (from Ref. [2.58]).


75

electrons and the rate of ionic bombardment of the target. In a DC sputtering setup, the target voltage is high and negative (typically around I kV). The substrate voltage is generally kept slightly negative with respect to the plasma (typically - 5 0 to - 1 0 0 V) in order to favor light ionic bombardment of the deposit and the ensuing removal of impurities. The DC method cannot be used with insulating materials because the Ar ions would rapidly charge the target to a positive potential, repelling further incoming Ar ions. It is expedient in this case to resort to the RF sputtering technique. This is based on the application of a radio frequency voltage ( f = 13.56 MHz) to the target, which is then bombarded for a fraction of a period by the positive ions and for the remaining fraction by the electrons. The latter neutralize the positive charge left by the ions and the extraction process from the target can continue. Vacuum evaporation is widely employed as a thin-film deposition method. The material to be deposited is heated in a crucible and the vapor, resulting from sublimation or melt evaporation, is condensed on a substrate at a rate generally varying between I and 100 n m / s . Heating is realized in several ways, including the passage of a current in a refractory metal crucible (Ta, W, Mo) and the bombardment by an electron gun of the material, held in a cooled crucible. For a practical growth rate, the partial pressure of the obtained vapor must be in the range of some 10 Pa. For Fe this means, for example, a temperature around 1650 ~ A vacuum is required for two reasons. First, the mean free path of evaporated atoms must be high enough to permit their flight to the substrate without collisions. Second, contamination of the source and of the newly formed deposit must be avoided. A special variant of vacuum evaporation is the molecular beam epitaxy (MBE) method. By this term is generally meant a system with a base pressure of 10-9-10 -7 Pa and an "in situ" monitoring, layer by layer, of film growth. The reflection high energy electron diffraction (RHEED) technique, based on the real time analysis of the diffraction of an electron beam at a grazing angle trajectory, is used for this purpose because it does not interfere with the deposition process. MBE is especially suited for the preparation of epitaxial ultrathin films and multilayers. The term "epitaxy" defines a growth process where the crystallographic properties of the deposit and the substrate are tightly related (for instance, growth of a monocrystalline film on a monocrystalline substrate). Artificial lattices, a few atomic layers thick, with unique magnetic properties, rapidly varying with the number of atomic layers, can be created by epitaxial growth (e.g. Fe on Au and W, Ni on Cu and W, Co on Au and Pd) [2.59]. Using either MBE (for single crystals) or sputtering (for polycrystalline samples), complex and interesting structures, made of a sequence of

76


few lattice spacing thick ferromagnetic layers separated by metallic non-magnetic or antiferromagnetic layers, can be prepared. The same exchange or superexchange interaction effects occurring between magnetic moments in bulk materials give rise to coupling between layers. Coupling can be conveniently modulated by changing the nature, the thickness, and the number of layers, subject to the requirement of excellent crystallographic quality of the deposited layers and their interfaces. It has been observed that the exchange interaction between two ferromagnetic transition metal ultrathin films separated by a spacer made of a non-magnetic transition metal (e.g. Mo, Ru, Pd) or a noble metal (e.g. Au, Ag, Cu) oscillates between antiferromagnetic and ferromagnetic coupling as a function of the spacer thickness. The example reported in Fig. 2.25a, regarding the behavior of the exchange coupling energy in Nis0Co20/Ru multilayers vs. the Ru spacer thickness, shows that the associated wavelength is in the nanometer range [2.60]. The coupling mechanism is assumed to be strictly similar to the indirect exchange interaction of magnetic impurities in a metallic host (RKKY coupling), which oscillates with distance between ferromagnetic and antiferromagnetic coupling. The spin polarized conduction electrons in the metallic non-magnetic spacer act as carriers of the interaction between the magnetic moments of the ferromagnetic layers. The antiferromagnetically coupled layers can have their magnetization forced into parallelism by a suitably high applied field. When this occurs, the resistance R of the multilayered structure is reduced to a considerable extent with respect to the antiparallel magnetization state, as first shown by Baibich et al. in [Fe (3 nm)/Cr (0.9 nm)]60 structures [2.61]. This is the so-called giant magnetoresistance (GMR) effect, where relative resistance changes AR/R around 100% and more can be achieved upon the field induced antiparallel-parallel transition of the magnetization in the layers. This is a much stronger variation than the classical anisotropic magnetoresistance (AMR) can provide, for example, by permalloy, whose resistivity is 2-3% lower along a direction normal to the magnetization direction than along the parallel direction. The current interpretation of GMR is based on the appraisal of the different role of the spin T and spin I conduction electrons in ferromagnets [2.62]. According to Mott's model, two conductivity channels are associated with these electrons and the total conductivity is cr = cr T +or 1. The conduction electrons can be scattered into localized d states; the higher the density of such states the larger the scattering probability. In Fe, Co, and Ni, the density of spin T states at the Fermi level is definitely lower than the density of the spin I states. The spin T channel is thus associated with low or negligible s - d scattering and consequently cr T >>or1. In a multilayer structure subjected


77

0.15

,, !

NisoCo2o/Ru

i

!

0.10.

io

E E

I

|

!

i

i

I

!

c-

0.05,

o) cO O O O~ c0~

multilayer

i

!

antiferromagnetic

9 |

!

|

! I |

!

! I

c= 0.00

0 X I.U

oi

a, i i

_~

_

-,,O_.o.-~

-~o q,

-.

ferromagnetic

% . . . I"

i

. ,~"D",Q,

O'~,,

u

-0.05

|

0

(a)

.

.

.

.

1

u

u

2

3

.

.

.

- tRu (nm) 30

Ni81Fe19/Cu multilayer T=4.2K

.i I. II.

20

i i i i i

v

10

1), an oscillatory damped, a critically damped, or an overdamped i(t) behavior is predicted. Under the condition that at time t = 0 the capacitor is charged at a voltage V0 and the current i = 0, the oscillatory damped solution is --L-- exp - ~-~t

sin wt

,

(4.38)

with

~o= ~I /LC - R 2/4L 2

(4.39)

and time constant z = 2L/R. An example of oscillating decay of i(t) is provided in Fig. 4.10b, where the circuit parameters provide R2C/4L = 2 x 10 -3. The overdamped solution is analogously found to be

i(t)= Vo exp(

-E-

R ) sinh kt k

(4.40) ,

with

k - ~R2/4L 2 - 1/LC.

(4.41)

The oscillatory condition is normally adopted for efficient conversion of the electrostatic energy into magnetic energy. However, it eventually brings the magnetic sample in the demagnetized state and, in order to maintain the sample in the remanent state, the switch $2 is closed at t -- 0. Thus, after the current has reached its maximum value im at time t -- t m and the capacitor is completely discharged, the diode, acting as a short circuit, prevents C from charging with opposite polarity. The current decays then exponentially, with the time constant of the LR circuit Zl = L/R. From Eq. (4.36) and its time derivative we obtain that at the time 1

t m - --.atan(wz) o9

(4.42)

128

CHAPTER 4 Magnetic Field Sources

the current achieves its maximum value

(atan(o~z))

~-~ im = V0 .exp -

oJ~

"

(4.43)

The conversion of the electrostatic energy E c - 89 into magnetic energy EL -- 1 Li 2 peaks at time t = tm. Part of Ec is dissipated as heat in the leads and the internal resistance of the coil. The efficiency of the conversion is obtained from Eq. (4.43), (1/2)Li2m (atan(oJ~') ) 7 q - (1/2)CV2 = exp - 2 - - ~ 0 r "

(4.44)

The efficiency increases with increasing the quantity

roz =

~4L ~T~ - 1.

(4.45)

In a similar way, one finds for the overdamped case Li 2 ( atanh(k~') ) 7q .- C V 2 - exp - 2 kr '

(4.46)

with k'r =

4L R2 C

(4.47)

The efficiency is obviously decreasing with increasing damping, as illustrated in Fig. 4.11 by the dependence, calculated through Eqs. (4.44) and (4.46), of ~/ on the damping coefficient R2C/4L. The advantage of using the circuit of Fig. 4.10a ($2 closed) for the generation of a non-oscillating current transient instead of adopting an overdamped configuration is clear. To be stressed that, for a given stored energy Ec = 89 2, better efficiency is achieved using high voltage and low capacitance. The pulsed field setups employed in permanent magnet testing are generally required to produce peak fields of the order of 5 x 106 A / m or higher with rise time tm > I ms. They should be reasonably uniform over volumes of several cubic centimeters, thereby sufficient to accommodate samples of technical size [4.18]. The magnetizing coil is normally

4.3 AC AND PULSED FIELD SOURCES

1.0-

129

Underdamped

',

0.8-

~

0.6-

0.4-

0.2-

0.0 1 E-4

. . . . . . . .

I

1E-3

0.01

0.1

I

1

10

R2C/4L FIGURE 4.11 Efficiency of the conversion of the capacitively stored energy into magnetic energy in a pulsed field setup as a function of the damping factor

R2C/4L.

a multilayered solenoid, though Bitter type coils can also be used, and the flow of current in the copper conductors is normally not affected by the skin effect. Limited increase of the coil temperature by Joule heating and good mechanical strength are two important factors to be considered in designing the magnetizer [4.19]. Because of the short duration of the current pulse, the coil is expected to behave adiabatically during the transient and nearly all of the energy delivered by the capacitor is retained in the copper winding. If the mass of the copper is m, an upper limit for the temperature increase AT of the coil upon a field pulse is therefore obtained as AT=

(1/2)CV~ Cpm

(4.48)

if Cp is the specific heat of copper. For energies up to 30-50 kJ, no special provisions for cooling are required, unless the system is subjected to high repetition rates, as it often occurs in industrial environments. Stresses in the coil arise because the conductors are subjected in part to the very same field they produce. As schematically illustrated for the case of a thick solenoid in Fig. 4.12, the longitudinal field

130


'r

__.z__....

......... L................................. ~i""

,J

T

~

H,

......

Z

'i

FIGURE 4.12 Stress distribution in a solenoid. The axial field component Hx is the source of a radial stress err which is equilibrated by a tangential tension cre in the conductor. The radial field component Fir generates in turn an axially directed stress crx pointing towards the equatorial plane zz.

component Hx generates by Lorentz force an outwardly directed radial stress err which tends to expand the solenoid diameter, err is equilibrated in each point of the conductor by a tangential stress or0. The radial field component Hr is the source of an axially directed stress Or r pointing towards the equatorial plane, which tends to shorten the solenoid. A Bitter type solenoid has good mechanical strength, while reinforcement of the structure of the solenoid is required for wire-wound and tapew o u n d coils. These can be made rigid by impregnation with suitable resins and by fitting them into circumferential strengthening rings made of some hard material. It is imperative, in any case, that the yield strength of copper is not exceeded anywhere in the coil. We can make an estimate of the m a x i m u m stress to be endured by the coil by equating the field pressure, which is compensated by the stresses in the conductor, with the energy density of the magnetic field E = (1/2)/z0 H2 [4.1]. A field of 5.106 A / m corresponds, for example, to a stress of 16 MPa, safely below the yield stress of hard d r a w n Cu (~ry = 350 MPa). When the field to be produced is so high that the ensuing stresses overcome the yield strength of the conductor in the coil, one can still carry out the experiment, provided the pulse duration is shortened

4.3 AC AND PULSED FIELD SOURCES

131

to the point that it is extinguished before a substantial a m o u n t of kinetic energy can be transferred to the material. The example reported in Fig. 4.10 refers to the case where a pulsed field is p r o d u c e d by a solenoid having the following dimensions (see Fig. 4.6a): L = 102 mm, R 1 = 25 mm, R 2 -- 36 mm. The solenoid is m a d e of 10 layers of h a r d - d r a w n copper wire of rectangular cross-section 5 m m x I m m and its inductance is L = 0.98 mH. The generated axial field can be calculated t h r o u g h Eq. (4.27). The solenoid constant takes at the center the value Hx(O)/i = 1667 m -1. Table 4.1 provides an overview of the main features of this pulse generator, where the storage of energy is accomplished by charging a capacitor bank C - - 8 0 0 ~F at a voltage V0 = 6000 V. A second typical setup, as reported in the literature [4.21], is also considered. Peak field amplitude suitable for saturating high coercivity p e r m a n e n t magnets is obtained with good efficiency in both cases, while keeping the rise of coil temperature and stresses within comfortably low limits.

TABLE 4.1 Significant quantities in pulsed field setups. System 1 is described in Fig. 4.10. System 2 is described in Ref. [4.21] Quantity Inside diameter of the solenoid (R1) Outside diameter of the solenoid (R2) Length of the solenoid (L) Cross-sectional area of the wire (Sw) Resistance (R) Inductance (L) Capacity (C) Voltage (V0) Stored energy (Ec) Damping coefficient (R 2C/4L) Time constant (~-) Rise time (tm) Maximum current value (/m) Maximum field value (Hm) Efficiency (7/) Temperature increase (AT) Maximum stress (o-)

System I

System 2

25 mm

13.5 mm

36 mm

32.2 mm

102 mm 5 mm 2

96.2 mm

0.13 f~ 0.98 mH 800 ~F 6000 V 14.4 kJ 3.3 x 10 -3 15 ms 1.24 ms 4960 A 8.27 X 10 6 A / m 0.84 21 ~ 40 MPa

10 m m 2

0.036 f~ 0.24 mH 8000 ~F 2000 V 16 kJ 1.08 x 10 -2 13.3 ms 2.05 ms 9900 A 12.3 X 10 6 A / m 0.74 40 ~ 170 MPa

132


4.4 PERMANENT MAGNET SOURCES The very first magnetic phenomena observed by man had to do with fields generated by minerals like magnetite and other oxides having the character of permanent magnets. The objective historical importance of permanent magnet field sources is actually associated for the most part with the useful mechanical effects they inexhaustibly display once they are magnetized. They have generally played a minor role in the magnetic characterization of materials, where the flexibility of the current based sources, with or without soft iron cores, has revealed indispensable, for example, in the experimental approach to all the phenomena related to magnetization processes and hysteresis. Advances in the properties of materials, namely the development of the high coercivity, high remanence rare-earth based hard magnets, have changed somewhat this state of affairs and enriched the landscape of field sources useful in measurements. A classical permanent magnet source can be schematically represented by the gapped ring discussed in Section 3.1 and shown in Fig. 3.1. It was shown there that, under the approximation of uniform induction in the whole magnetic circuit, the useful field in the gap Hg is related to the value of the polarization Jm in the material by the equation Hg --

1 Im/Ig Jm, ~o 1 + lm/lg

(4.49)

with lm and lg the lengths of the magnet and the gap, respectively. Equation (4.49) provides in a narrow slit (lm/lg >> 1) the maximum field value Hg ----Jm/p,o. An upper limit is then predicted for the induction in the gap, equal to the saturation polarization Js of the material. As we see in Fig. 4.13b, the idea of complete flux channeling, assumed in the circuit of Fig. 3.1 and in Eq. (4.49), is much too often a rough approximation. On the other hand, the magnetization M of an ideal magnet is rigidly fixed along the axial direction and is constant in modulus. This implies that no volume free charges are present in the material and manageable determination of the gap field by analytical means can be envisaged. Let us therefore consider in some detail this calculation in the model pole pieces of circular cross-section shown in Fig. 4.13a. These cylindrical pole pieces are assumed to be a portion either of a closed circuit or of very long rods. A practical circuit would actually be made of relatively short rods connected by a soft iron return path. In any case, the gap field is attributed to the free charges, of

4.4 PERMANENT MAGNET SOURCES

t

"

_1 ..... ~

?

r

""-,,

M

[

'

i \,

--..

i

Ig

~'

i

/

\,, "'.

--

i

\

..

M

/

'

'-~

..

9

ii X l,ql--, 1/

+1' 0, according to Fig. 5.9), respectively, we obtain that the current density is jx -- jxe + jxh = - - n e e V x - + nheVx+

(5.32)

and the conductivity Crx

Ex

ere + crh -- neela'e + nhela'h"

(5.33)

Notice that, in defining the conductivity of semiconductors, the effective mass for electrons and holes must be considered. The application of Bz makes the holes and the electrons, drifting in opposite directions, deflect towards the same side of the plate. The Lorentz forces, however, are not the same, because hole and electron mobilities, and hence velocities, are different. The equations of motion (5.25) must now be written for both types of carriers, imposing the steady-state condition of net zero transverse current jy "~ jye + jyh -" - n e e v y _

+ nhevy+ = O.

(5.34)

In compact notation, the process can be described by expressing the electric field as [5.26] E -- Ex~ q- EH~ = ---lje O'e + ~eeejeX Bz-- ----1 Orh j h -

~ j h X Bz

(5.35)

with ~ and ~ the unit vectors in the reference system of Fig. 5.9. The hole mobility ~tl,h is defined as in Eq. (5.24). Equation (5.35) can be solved for je and Jh in terms of E and Bz and the total current density j = je + jh, directed along the x-axis, is obtained. The relationships between fields

180

CHAPTER 5 Measurement of Magnetic Fields

and currents with the two-carrier conduction mechanism are summarized in Fig. 5.10. While the zero charge transfer constraint for a metal implies zero transverse drift velocity of the free electrons, in semiconductors it allows for transverse motion of the carriers because the Hall field cannot compensate for both hole and electron lateral drift at the same time. By introducing the condition (5.34) in the equations of motion for the holes and the electrons and by assuming the weak field approximation (which is largely satisfied with the fields ordinarily applied to materials), we obtain the following expression for the Hall coefficient (see for instance Ref. [5.26]) RH -- 1 nh/~ -- he/J,2 e (nh/~h -ff ne~e) 2"

(5.36)

It is apparent that in semiconductors both mobility and concentration of the charge carriers play a role in determining the strength of the Hall coefficient. Minority carriers can actually exert a remarkable effect when they have large mobilities. In all cases, R H is orders of magnitude larger in semiconductors than in metals, which is explainable on account of similarly large differences in the carrier concentration. It is clear that R H

'
v

E

Gr /IR

50" 0 -50 T = 100 o c / / / -100

~ ~~ 9

i

.

.

-8000

.

.

T = 24 oC i

.

.

-4000

.

.

.

.

0

.

.

H (A/m)

i

'-

4000

-

9

9

i

'-

8000

FIGURE 5.20 Response of a NiFe/Ag multilayer GMR bridge sensor (Ref. [5.42]) as a function of the applied field compared with the response of a NislFe19 AMR bridge (Ref. [5.45]). The curves are normalized to the same input voltage. multilayer decrease or increase according to its polarity and the polarity of the bias, and the output voltage will be given, as for the AMR bridge of Fig. 5.18, by Eq. (5.47). Contrary to the field-shunting device, the field bias bridge can thus reveal the polarity of the sensed field. Linearity, large dynamic range, and lack of hysteresis are distinctive features of the GMR magnetic sensors, as illustrated by the output voltage curve of the biased NiFe/Ag multilayer bridge, shown in Fig. 5.20 in comparison with the response of a high-performance Ni81Fe19 AMR bridge [5.45].

5.3 F E R R O M A G N E T I C S E N S O R M E T H O D S A natural way of increasing the sensitivity of the fluxmetric field detection method consists in filling the sensing coil area with a soft magnetic core in order to exploit the flux multiplying properties of the material. There are no special difficulties in the direct use of a soft ferromagnetic material as a simple flux multiplier. However, phenomena like magnetic saturation, eddy currents, non-linearity and hysteresis are detrimental to the measuring accuracy and more indirect methods, not necessarily based on the behavior of the J-H curve, have been developed. Altogether, they belong to the wide domain of magnetic field sensing technology and their applications, ranging from geophysical surveys to space exploration, are in many cases only loosely related to the problem of magnetic materials

5.3 FERROMAGNETIC SENSOR METHODS

197

characterization. Consequently, they will be discussed here to a somewhat limited extent, a more detailed analysis being available in recent literature reviews [5.46, 5.47].

5.3.1 Fluxgate magnetometers A way to get rid of the intricacies of hysteresis loops, while exploiting the symmetry properties of the magnetization curve, consists in imposing an alternating exciting field, strong enough to cyclically drive the soft sensing core to deep saturation, and in analyzing the modifications occurring in the secondary voltage when the core is exposed to the field under measurement. This is the working principle of the fluxgate magnetometers, summarily sketched in Fig. 5.21. We see in this figure how an external DC field Hs, combining with the AC exciting field Hexc(t), brings about hysteresis loop asymmetry and leads to modified periodic behavior of the secondary voltage. The lost symmetry property leads to the presence of even harmonics in this voltage, which suggests that the measurement of the external field can be accomplished through the determination of the amplitude of the second harmonic. This can be appreciated by means of an approximate argument, where the role of e

A ou

11

Time

Hexc+H s

Hexc

~--,

Yout'

9

v

Time

Time

FIGURE 5.21 Generation of the secondary voltage in a fluxgate core, described by means of a simplified parallelogram-shaped hysteresis loop, under a triangular driving field He• If the core is exposed to a DC external field Hs the symmetry of the described B(H) curve is lost and the secondary voltage V1 contains even harmonics.

198


the time-varying permeability of the core on the secondary voltage is considered [5.48, 5.49]. Let us thus assume that the core, having crosssectional area A, is subjected to a field H which is the sum of a largeamplitude alternating driving field Hexc(t)~ cyclically saturating the material, and a DC external field Hs. The latter is sufficiently small to Its effect is taken into account be considered as a perturbation on He• by writing the voltage u ( t ) = - N A d B / d t induced on the N-turn secondary winding as a series development truncated to the first order dB dH

u(t) = - N A d---H dt

~ -NA

dHexc dt

((/d'd)ne•

-}- (di~d/dH)HexcHs)'

(5.48)

where the apparent differential permeability /d,d can be related to the effective differential permeability ~e(t) via the sample demagnetizing coefficient Nd /~e (~e)"

~d=

(5.49)

lff-N d ~ - - 1 Equation (5.48) contains, besides the usual odd symmetry term in the output voltage, the even symmetry term dHexc d/d,d dt dH

d/~d dt ~

having the periodicity of the permeability. By revealing the second harmonic through filtering, one can obtain, according to this equation, a measure of Hs. For a magnetic core excited as in Fig. 5.21, this means, for example, locking to the second harmonic of the output voltage Uout(t) using phase sensitive detection. By adopting the two-core sensing configuration shown in Fig. 5.22 (known as Vacquier-type sensor), one can automatically filter out the odd-harmonic components. Here, the exciting field, whose frequencyftypically ranges between I and 10 kHz, is applied to the two identical cores along opposite directions. In the absence of any other field, the voltage Uo~t induced in the secondary winding is zero. With an external field Hs~ the working point on the hysteresis loop is displaced differently in the two cores and a resulting instantaneous flux ~ s ( t ) ~ 2NAi~d(t)Hs at the frequency 2f becomes linked with the secondary winding (small Hs values). The sensor reveals in principle only the component of the external field parallel to the probe axis and it therefore works as a vector magnetometer. Its sensitivity, proportional to d/~d/dt, and its dynamic range both increase with increasing the peak amplitude of Hexc(t). It also increases with a decrease of the demagnetizing coefficient, somewhat at the expense of the linearity of


199 B

lexc

-Hexc +Hs .,.~ |

-.&

._..~-.'-

.._1 ~

_.~--

_.,i ~

I z'~/',#''

/__t'

~

/3

+G

"4 FIGURE 5.22 Two-core fluxgate sensor (Vacquier type). The two identical cores, which can be obtained by using either soft ferrite rods, permalloy strips or amorphous ribbons, are subjected to the AC exciting field Hexc (frequency in the kHz range) along opposite directions and are exposed to the same DC or lowfrequency external field I-I~.The corresponding working points on the hysteresis loop are asymmetrically displaced by I-I~. This implies that the voltage Vout detected by a secondary winding contains only even order harmonics and it is, according to Eq. (5.48), proportional to Hs. the response. On the other hand, all spurious contributions due to imperfectly balanced cores and residual second harmonic in the input are enhanced with stronger exciting fields, and the Barkhausen effect, which is the main source of noise in fluxgate devices, is known to increase with the apparent differential permeability /.4,d [5.50]. In order to increase linearity and measuring range, the closed-loop feedback technique is often adopted. In this case, the magnetometer is used as a null-sensing device, where a supplementary coil is added and a field exactly balancing the external field Hs is provided. As core materials, Mumetal or permalloy wires or strips are used in most cases because of their obvious extra-soft magnetic behavior, indispensable for low-field sensitivity, vanishing magnetostriction and very low Barkhausen noise. Since the 1980s, Co-based amorphous ribbons, exhibiting near-zero magnetostriction (see Section 2.4), have been increasingly employed [5.51]. Ribbon annealing under transverse field or stress annealing, both leading to uniaxial transverse anisotropy (i.e. favoring homogeneous moment rotations vs. domain wall displacements during the magnetization process) are exploited in order to minimize the noise. Besides the sensor presented in Fig. 5.22, several realizations of the fluxgate principle are reported in the literature and employed in practical setups. Figure 5.23 shows two examples. The ring-core device, which can be considered as derived from the Vacquier-type sensor by bending and

200


Vout

I.

(a)

(b)

FIGURE 5.23 (a) Ring-core fluxgate sensor. It can be considered as a derivation of the Vacquier-type sensor, where the two straight cores are bent and connected at their ends. The output voltage Vout is different from zero only in the presence of the external field Hs, which creates a flux unbalance between the two half-cores. (b) Miniature mixed parallel-orthogonal fluxgate. A circular AC exciting field is generated by the current iexcflowing along the narrow amorphous ribbon, which i * . . . . . s characterized by 45o directed helical anlsotropy (indicated by Ku at the surface). The field H s breaks the symmetry of the magnetization process in the two ribbons and a second harmonic output is generated (from Ref. [5.54]).

connecting the limbs at their ends, is very popular and extensively applied because it offers some intrinsic advantages. It can be fabricated and wound with a high degree of uniformity, and the magnetization in the material is, in the absence of the external field, the same in all crosssections. In addition, it can be rotated within its sensing solenoid in order to minimize offset and other spurious effects due to any residual geometrical imperfection. On the other hand, it is more sensitive to crossfields than the highly anisotropic Vacquier-type sensor, although always faring much better in this respect than magnetoresistance magnetometers. It is typically obtained by wrapping a few turns of a permalloy or amorphous ribbon on a non-magnetic former, but, in recent developments of miniaturized sensors, single foil rings have been prepared by photolithographic techniques and chemical etching and incorporated in suitably prepared printed circuit boards [5.52]. There is no basic difference in the working principle of the Vacquier-type and ring-core fluxgate sensors, both belonging to the category of the so-called "parallelgating" fluxgates, the parallelism referred to being the exciting and the sensing fields. Orthogonal and mixed orthogonal-parallel gated devices have also been developed and patented [5.53]. Figure 5.23b schematically illustrates an example of an orthogonal-parallel fluxgate, the miniature hairpin sensor developed by Nielsen et al. [5.54]. It makes use of two identical near-zero magnetostriction ribbon pieces (composition


201

Fe3.sCo66.sSi12B18) I m m wide and 23 ~m thick, with minimum length 8 mm, characterized by helical anisotropy Ku ~ 50 J/m 3 directed at 45 ~ to the ribbon length. Anisotropy is induced by subjecting the ribbon to torsional stress-annealing at temperatures between 300 and 400 ~ The two ribbon pieces are series-connected as shown in the figure and a 15 kHz exciting current is made to circulate in them. This generates a circular field, which gives rise to a net longitudinal flux in each piece, but no net flux in the sense coil wrapped around the two pieces. The longitudinal external field Hs breaks the symmetric condition so obtained and even harmonics are induced in the sense coil. Detection methods other than revealing the second harmonic of the output voltage are frequently adopted in fluxgates. For example, with the classical "peaking strip" technique one measures the deviation that the external field Hs imposes on the symmetry of the position of the voltage pulses generated each half-period by the saturating exciting field when traversing the hysteresis loop (see Fig. 5.21). A variant of this method consists in the determination of the zero passage of the hysteresis loop, which is displaced by the quantity Hs [5.55]. In the self-oscillating type magnetometer of Takeuchii and Harada [5.56], making use of a single-core amorphous sensor, the generated square wave has a duty ratio depending on Hs, which is revealed as a DC output voltage equal to the mean value of the waveform. In the multivibrator-type magnetometer of Mohri et al. [5.57], two identical sensing cores made of short amorphous ribbon pieces (length 5-50 mm) are used as the arms of a bridge, which generates a DC voltage upon unbalance of the core voltages created by the external field. Conventional fluxgate sensors are bulky objects with mass and volume typically in the 0.1 kg and 100cm ~ range, respectively, and power consumption might represent a problem. Several attempts have been made in recent times to overcome the size problem, either by trying to integrate the magnetic core preparation via thin-film deposition or electroplating with the Si technology [5.58] or by combining millimetersized cores, etched from amorphous ribbons, with planar coils and printed circuit boards [5.59]. At the present time, the miniature fluxgates do not reach the performance of the conventional bulk devices because thin films lack the soft magnetic quality available in sheets and ribbons, the noise increases and it is difficult to drive the core into saturation [5.60]. Fluxgate magnetometers are largely employed for DC and low frequency (typically up to few hundred Hz) field measurements because of their high sensitivity, linearity, directional properties, wide range of operating temperatures, stability, reliability, and ruggedness. They have

202


105 10 4

AMR ""'------~_....~....._

N

-r

03

. FG3

v

a

m < 102 O '""

-

o 101

z

S ~

100 ....

6'1

FG2

.......

i

. . . . . . .

~0

Frequency (Hz) FIGURE 5.24 Amplitude spectral density (ASD) of the noise in different types of magnetic field sensors. FGI: Race-track fluxgate; FG2: ring-core fluxgate. FG3: miniature fluxgate; AMR: permalloy magnetoresistor (from Ref. [5.61]). ME: magnetostrictive magnetometer (from Ref. [5.62]). FO: fiber-optic coupled magnetostrictive magnetometer (from Ref. [5.63]). been widely applied over the past several decades in areas like geomagnetic studies, airborne detection, and space magnetometry and in other instances, like medical investigations, where extremely low field strengths, in the nT range, need to be reliably determined. Commercial setups covering the range I nT-1 mT are offered. The limit to fluxgate sensitivity is chiefly due to the inevitable presence of hysteresis and magnetization noise, while an upper field limit is found when non-linear behavior sets in or, in any case, the field Hs definitely brings the magnetic probe into saturation. Fluxgates are nowadays second only to SQUIDs for sensitivity, with incomparably lower complexity and running costs. One can appreciate their performance when comparing their noise behavior with that of other sensitive magnetometers, as in the example reported in Fig. 5.24 [5.61-63].

5.3.2 Inductive magnetometers There is no clear line between fluxgates and generally inductive magnetometers. The latter will be considered here as those devices not necessarily working under a deeply saturating AC exciting field. In many cases, these cores are made of deposited or electroplated permalloy-type


203

thin films, having a defined uniaxial anisotropy Ku. This is obtained by applying a saturating field in the desired direction during the film preparation. It permits one to identify an EA and, orthogonal to it, a hard axis (HA) in the plane of the film. The magnetization process can consequently be made to occur in such a way that an external magnetic field can be revealed and measured with little or no interference from hysteresis and its memory effects. A thin film is evidently characterized by very low in-plane and very high out-of-plane demagnetizing coefficients. This ensures that there is little difference between in-plane applied and effective fields and that there is no out-of-plane magnetization component. Since the obtained magnetization variations and the cross-sectional areas of the thin film cores are generally small, the exciting field frequency is high, from a few hundred kHz to several MHz, in order to obtain usable voltages across the inductor. The domain configuration in a field-annealed permalloy film generally consists, in the demagnetized state, in a simple balanced array of 180 ~ slab-like antiparallel domains directed along the EA. Such a structure remains unchanged under an AC exciting field Hexc of frequencyfdirected along the HA, because, as shown in Fig. 5.25, the only effect of Hexc is to instigate the homogeneous rotation

v

Hm

....

~ i~'~-~

i

,

Vout

(a)

(b)

FIGURE 5.25 Examples of thin-film inductive magnetic field sensors. The domain structure and the oscillating magnetic moments are schematically depicted. Preparing the permalloy films under a saturating magnetic field creates an EA with anisotropy Ku. An exciting field I~xc at radiofrequencyfis applied along the hard axis (HA) and the DC/low-frequency field FI~under measurement is directed along EA. In the sensor (a) a modulating field Hm of frequency fro > 1, the absorbed power becomes independent of/-/1 and saturates, a condition generally avoided in such a measurement. In the theoretical treatment of NMR, we are basically interested in finding out the evolution of the nuclear magnetization M with time. In quantum mechanical terms, this amounts to obtaining the equation for the expectation value of M. This does not require solving the appropriate time-dependent Schr6dinger equation because the expectation value follows in its time dependence the classical equations of motion. We can therefore write, for the time dependence of M in the absence of relaxation effects, the classical gyroscopic equation dM dt - 7M x/z0H0.

(5.68)

5.4 QUANTUM METHODS

221

This equation describes the free precession of the magnetization M around the longitudinally directed field H0 (see Fig. 5.32b). Equation (5.68), written for the three components as dM~9 dt - )'MY/z~176

dMy _ - ~/Mx/z0H0, dt

d]VIz - 0 dt

(5.69)

provides Mx = m cos(a~ot) ,

My = - m sin(oJot) ,

Mz = Mo,

(5.70)

as expected for a vector M precessing at the Larmor frequency ~o0 = ~'/z0H0. Free precession of M is not self-sustaining in real systems because the moments are subjected, besides the field H0, to internal random fields, which introduce randomizing effects in the precession frequency of each magnetic moment. Such fields are assumed to derive from the interaction by neighboring nuclear spins, but also the spin-lattice interaction, associated with the longitudinal decay, is expected to contribute to moment de-phasing. Whatever the mechanism, it results in a transverse relaxation time T2, with Mx and My eventually decaying to zero. In nonviscous liquids it is generally found T1 ~ 1.5T2 [5.85]. As discussed at the start, steady-state resonance is predicted to occur through the absorption of the energy provided by an alternating field H1, matching the frequency condition o)0 = AE/h. Restated in classical terms, this amounts to saying that, under these conditions, a persistent component of the magnetization vector M (or, equivalently, of its expectation value) is achieved, which rotates at the frequency ~o0. This comes out from the previous gyroscopic Eq. (5.69) by rewriting it through the introduction of the total field H -H0 + H1 and of additional terms phenomenologically accounting for the rate of variation of Mx and My due to transverse relaxation and that of Mz due to longitudinal relaxation. We obtain the "Bloch equations" [5.87] dMx dt = ~//z0(MyHz - MzHy)

Mx T2

dMy = ~/Izo(MzHx - MxHz) dt

My T2

(5.71)

dMz (Mo - Mz) dt - ~/tzo(Mxt-Iy - MyH,.) + T1 ' where the magnetization rates associated with the transverse and longitudinal relaxations are described by the terms (Mx/T2,My/T2) and (Mo - Mz)/T1, respectively. It is expected that, being the rate of exchange of energy between an AC field I-Ii(t) and the precessing magnetization

222


M(t) given by the product izoHl(t).dM(t)/dt , this field should be mostly effective when applied in a plane normal to H0. This is what is done in experiments, where the field H1 (t) = 2H1 cos(cot) can actually be thought of as the superposition of two circularly polarized fields HiL(t) and H1R(t) of same peak amplitude H1, counter-rotating in the same plane. This is a natural way to demonstrate the normal modes of the process. Only the field HIL(t), rotating in the same sense as the precession (see Fig. 5.32b), with components

Hx(t) = H1 cos(~ot),

Hy(t) = - H 1 sin(oJt),

(5.72)

is effective and, at least around the resonance frequency, the other rotating component can be disregarded. In Eq. (5.72), o~ has been conventionally taken as positive for H1L(t). A relatively simple way to find solutions to the Bloch equations is to move to a novel frame of reference (x~,y', z'), with z ' = z, rotating about the z-axis at the frequency a~ In this frame, the circular field of amplitude H1 is fixed. If we make the further assumption of weak circular r.f. field (H1 ~ H0), low enough to avoid saturation (so that we can pose Mz ~" Mo, Mx,My K<Mo and, in particular, (T/z0H1) 2 ~ (1/T1)(1/T2)), the following solutions of the Bloch equations in the rotating frame are obtained [5.88] (~o0 - co)T2 M,~ = X0co0T2 1 q- (a~0 - o~)2T22H1, 1 M~ = Xoo~oT21 + (oJ0 - oj)2T22H1

(5.73)

M~, = M0, with, as previously stressed, X0 - Mo/Ho and ~o0 the resonance frequency. These solutions show that M~ and My are fixed in the rotating frame because H1 is fixed and so is the transverse magnetization component m, given by their composition. In the laboratory frame, m, having constant modulus m = ~M2~ + M~ = ~/1

X~176176 H1, + (~0 o))2T2

(5.74)

rotates at the frequency o4 which suggests that it can be observed, as shown later, by detecting the electromotive force induced in a surrounding coil. We see in Eq. (5.74) that at resonance, where ~o = ~o0, m attains the maximum value m--Xo~oT2H1. Such a value is proportional to the transverse relaxation time T2, which is natural because this quantity provides a measure of the degree of precession coherence of the individual moments. The vector m lags behind the rotating field H1L(t) by the angle ~, which is found by

5.4 QUANTUM METHODS

223

relating the two components of m Mx = m cos(cot - ~),

My = - m sin(cot - ~o)

(5.75)

to the rotating-frame counterparts M.,~ and My, in Eq. (5.73). We find [5.89] cos ~ =

(coo- co)T2 ~1 + (coo -

,

sin ~0=

1

.

(5.76)

~1 4- (coo - co)2T22

co) 2 T 2

If we take the in-phase and out-of-phase components (m cos ~p and m sin ~p, respectively), we can correspondingly define the real and imaginary susceptibilities ~ and &Y~ = m cos ~ = Xo cooT2 (coo - co)T2 1 4- (co0 - 09) 2 T~' 2H1 2 (5.77) &/, = m sin r = X0 co0T2 1 2H1 2 1 4- (co0 - co)2T2" )~ is an odd function of (coo - co) while ~ achieves a m a x i m u m value ~ -x0 2 cooT2 for co = coo as illustrated in Fig. 5.33. ~ and ~ represent the dispersion and absorption parts of the susceptibility. They are often associated with the term "Lorentzian lines". At resonance, only the out-of-phase susceptibility survives and m lags behind the rotating field by the angle ~ = 7r/2. We can write ,f'(~oo) = xo ~o 2 Aco'

(5.78)

where Aco = 1/T2 is the half-width of the absorption line and co0/Aco = Q,, is the related quality factor. From Eq. (5.74), we obtain, recalling that coo-7/z0H0 = 7/z0M0/X0 coo

m(coo) = XoH1 ~

= 2AJ'(co0)H1 = ~/MzT2tzoH1,

(5.79)

having posed Mz ~ M0. ~ is directly related to the power P absorbed per unit volume by the system of nuclear spins subjected to the r.f. field Hi(t) = 2H1 cos(cot) in the presence of the static field H0. In fact, the instantaneous power P, equal to the average power because of the stationary conditions, can be expressed as P =/zoH1L.dm/dt. By the use of the previous Eqs. (5.72)(5.77), we obtain P(co) = 2~~

= P'~176176 1 + (coo1- co)2T2 coH~"

(5.80)

The resonance condition can then be recognized either by detecting the m a x i m u m of the voltage induced in a detecting coil by the rotating transverse

224

CHAPTER 5

Measurement of Magnetic Fields

1.0

0.5

%

0.0 Aco = I l T 2 "--0.5

"

-1.0 '

'~'I

-4

'

'

'

.

I

-2

'

.

.

.

.

.

0

(r162

'

9

I

2

9

,

9

,

I

4

'

'

T2

FIGURE 5.33 Normalized absorption ~! and dispersion ~ susceptibility components, obtained by dividing Eqs. (5.77) by the quantity (Xo/2)~oT2, as a function of (oJ0 -co)T2.~0 is the resonance frequency, X0 is the nuclear paramagnetic susceptibility, and T2 is the transverse relaxation time. The absorbed power is, according to Eq. (5.80), proportional to ~/I.

magnetization m or the maximum value of the absorbed energy per unit volume E(~o) = ;ff P(~o). We have E(oJ0) = 2 ,rlzoXo oo T2H2 .

(5.81)

Once the resonance frequency to0 = h/AE is measured, an operation that can be done with great accuracy, the field strength B0 =/z0H0 is obtained, according to Eq. (5.64), as B0 = Wo. Y

(5.82)

The measurement of the magnetic field is then reduced to the measurement of the resonant frequency. NMR thus provides a means for the absolute determination of a magnetic field with the highest accuracy. If we use pure water probes, the value of the proton gyromagnetic ratio appearing in Eq. (5.82) is slightly different from the value given above for the bare proton because there is a small diamagnetic shielding effect by the enveloping electron cloud and the effective field acting on the nuclei is slightly smaller than the external field. One talks of a "shielded proton gyromagnetic ratio", whose recommended value is (for a spherical container)

5.4 QUANTUM METHODS

225

3/p = 2.67515341 x 108 T -1 s -1 (relative uncertainty 4.2 x 10 -8) [5.90]. The eventually achieved field measuring uncertainty is around 2 - 5 x 10 -6 [5.3]. With a field B0 = 1 T, we have a Larmor frequency f0 = to0/2vr = 42.57639 MHz. For fields higher than a few Tesla, nuclei with a lower value of ~/may be preferred in order to decrease the working frequencies. With deuterons we get, for instance, 3i= 4.10605 x 107 T -1 s -1 and for B0 = 1 T the resonance frequency becomes f0 = 6.53498 MHz. The NMR magnetometers often operate under a weak r.f. field H1, that is, far from saturation. On increasing H1, the condition can in fact be approached where, as remarked in Eq. (5.67), the transition rate from the lower to the higher spin energy level is large enough to overcome the rate at which the energy is relaxed to the lattice (time T1), so that the nuclear populations in the two states tend to equalize. Consequently, the magnetization Mz, associated with the population unbalance created by the field H0, decreases and eventually becomes zero at saturation. This is the result provided by the Bloch equations (5.71) when the condition (T/z0H1)2 ~ (1/T1)(1/T2) is not satisfied (it is understood here and in the following that the notation ~/stands for the effective gyromagnetic ratio). It turns out in this case that, at resonance,

Mz(tOo) =

Mo 1 + (T/zoH1)2T1T2, (5.83)

XooJoT2 1 ~' ( o~o) = 2 1 + (~,/z0H1)2T1 T2 and m(oJ0)= 2X'(w0)H1 is consequently affected. While Mz(oJ0) is a monotonically decreasing function of H1, m(o~0) attains a maximum value [m(w0)]max -- Mo~/T2/T 1

(5.84)

for Hi = 1/')'lZo~/TiT2, to eventually vanish at high H 1 values, together with Mz(oJ0) (Fig. 5.34). On the other hand, the absorbed energy E(oo) = 4~'/Zo~'(wo)H~ = 2rrlZoXoooH~

T2 1 + (-y/zoH1)2T1T2

(5.85)

saturates for H1 >> 1/'yp, o~/TIT2. The resonance conditions can be searched for in experiments either by keeping the field H0 fixed and sweeping the frequency till passing through r 0 = ~//z0H0 = 3'B0 or by changing the field at a fixed frequency. If we associate any field value H with a frequency o~ - ~B = wz0H, we can

226

CHAPTER 5 Measurement of Magnetic Fields 1.0 0.8 0.6 0.4 0.2 0.0

!

,

,

,

o

~

.

.

.

.

.

i.

2 7~toH1 ( T1 T2) 1/2

FIGURE 5.34 The longitudinal magnetization Mz/Mo, with M0 = XoHo the thermal equilibrium magnetization, decreases when increasing the strength of the r.f. field H1. At the same time, the transverse component m(roo) passes through a maximum value for H1 = 1/'ytzox/-T~lT2. express, based on Eq. (5.80), the specific energy absorption line as

E(B) = 4~q~oAY'(w)H2 = 2r

T2

1 + y2(B o - B)2T2

H2

"

(5.86)

Figure 5.35 shows the behavior of the reduced energy absorption line E(B)/E(Bo), together with its derivative E'(B/Bo). It appears here that, in order to make precise determination of the resonance field B0 for a given frequency co0, it is convenient to rely on E'(B/Bo), looking for its zero passage. This can be accomplished in practice, as shown in the figure, by sweeping the field B amplitude through the line and superposing to it a small modulating field Bm (smaller than the line width) at some suitably low frequency, by which the slope of the line E'(B/Bo) can be retrieved. Since B/Bo = offro0 and E(B)/E(Bo)= E(ra)/E(roo), we can, in a perfectly analogous way, apply the modulation to the fixed field B0 and find the zero passage of E'(w/ro o) by scanning the frequency of the r.f. field H1. An alternative technique consists in applying a sufficiently large field Bm and sweeping it back and forth through the resonance condition, which is then revealed by a signal at twice the modulation frequency.

5.4 QUANTUM METHODS

227

_

-x

o

LU

Il

I

/

I

!

I

t

E'

I I v,

0.8

'

1.'0

I

'

B/B o

1.2

FIGURE 5.35 Solid curve: reduced NMR energy absorption line E(B)/E(Bo), with B = oJ/~/. If the field B is made to sweep through the line while amplitude modulated by a low-frequency field smaller than the line half-width value, a signal proportional to the derivative E' is obtained (dashed line). The resonance condition is thus determined with precision in correspondence of the passage of E' through the zero value. An equivalent result is obtained by sweeping the energy absorption line E(~)/E(~o).

5.4.2 N M R magnetometers 5.4.2.1 Continuous-wave magnetometers. The early successful NMR field measurements by Bloch et al. hinged on the determination of the signal generated at resonance by the rotating transverse magnetization m (Eq. (5.74) in a water-filled spherical sample (volume 1.46 cm3)) [5.91]. Figure 5.36 schematically illustrates an NMR field measuring setup based on Bloch's method. The DC field under measurement B0 is amplitude modulated by a co-linear low-frequency field (e.g. f m - - 6 0 Hz) of peak amplitude Bin, provided by a pair of supplementary coils. Normal to Bm and B0, the alternating field Hi is applied using two transmitter coils. The measurement of B0 is performed by sweeping the frequency of H1, thereby scanning the absorption line and detecting the signal Ur induced by the rotating transverse magnetization vector m in a coil wound around the sample, perpendicular to both H1 and B0 (the y-axis in Fig. 5.36). Since the transmitting and receiving windings are placed exactly at right angles, only a very small fraction of the signal generated by the Hi windings can leak directly to the Ur winding. At resonance, the induced r.f. signal Ur is

228


sweep generat~ I H~

]

Bo

' y

Bo ~

x

2

I FIGURE 5.36 Schematic view of an NMR magnetometer based on the nuclear induction method [5.91]. The water-filled sample (e.g. small cylinder or sphere) is immersed in the uniform DC field B0 under measurement, which is modulated at the low-frequencyfm by the co-linear field Bm, having sufficient peak amplitude to sweep back and forth through the resonance condition. The r.f. field H1 is applied in a direction normal to B0 and a signal ur, induced by the precessing magnetization, is detected along the third direction by means of a coil wound around the sample. The r.f. signal Ur, modulated at the frequency 2fm, is r.f. amplified, demodulated and audio-amplified. The resonance frequency ro0 = ~,B0 can then be obtained by search of optimum signal on the scope and direct measure of the generated r.f. frequency by means of a frequency counter.

modulated in amplitude at a frequency 2J:m because of the periodic passage of the total field B0 + Bm through the center of the resonance line. Usually, a shunt condenser is used to resonate the coil and to increase the signal-to-noise ratio by the related Q factor. Notice that the small residual leakage coupling of transmitter and receiver coils provides a reference phase for the signal. After r.f. amplification, demodulation and lowfrequency amplification, the resulting signal of frequency 2fm can be sent to the y-axis of a scope, while the signal of frequency fm from the sweep generator is fed into the x-axis. Very fine frequency tuning is achieved by looking for a precisely centered and symmetric pattern on the screen graticule. The B0 field value is then obtained, according to Eq. (5.82), by determining the frequency of the corresponding field H1 with a frequency counter, a measurement which can be done with great accuracy. One major source of error, with this and other NMR methods, resides in the inhomogeneity of the field B0 over the NMR probe region, which leads to line broadening and flattening. Being the typical NMR probe size

5.4 QUANTUM METHODS

229

of 5-10 mm, the experimentally tolerated field gradient is around some 10 -4 cm -1. The water sample is usually contained in a glass, alumina, or plexiglas holder and special precautions must be taken to ensure the greatest rigidity of the assembly in order to avoid fluctuating signal leakage from transmitter to receiver coils and related microphonic effects. Notice that, given the intrinsically isotropic nature of NMR, a certain tolerance exists regarding the exact orientation of the measuring head with respect to the direction of the field B0. Most NMR magnetometers are nowadays based on the determination of the energy absorption at resonance, a feat that can be accomplished with great sensitivity by incorporating the resonator, made of a single coil inductive head, and the tuning shunt capacitor, into a marginal oscillator circuit. This is an amplifier with positive feedback loop kept at a gain around unity, that is, on the verge of oscillation. Under these conditions, the oscillation level becomes a very sensitive function of the Q factor of the resonator and amplifies the relatively small dip of it ensuing from the absorption of the energy E(Bo)at resonance (Eq. (5.86)). In practice, a large change of the oscillator voltage is obtained with a small change of Q. By frequency scanning the resonance line E(~/Wo)with the r.f. field H1 (or the E(B/Bo) line with the field B) and by superposing to the measuring field B0, a low-frequency modulating field (fm < I kHz) of amplitude Bm smaller than the half-width of the absorption line, we obtain that the oscillator voltage is modulated in amplitude at the frequency fro, with depth following the derivative E~(~/~o).Figure 5.37 provides an example of a setup implementing the power absorption method in the NMR measurement [5.85, 5.92]. The resonator is attuned to the precession frequency w0 by means of a couple of varactors (voltage-driven variable capacitances), controlled by the signal provided by the frequency control circuit. In order to cover an adequate field (i.e. frequency) interval (---50 m T - 1 T in typical commercial setups), in general several coils with different numbers of turns are used. The marginal oscillator, which is endowed with JFET input stage and low-noise wide-band r.f. amplification, feeds back its output current via the conductance Gf. The voltage leaving the oscillator is demodulated, amplified, and fed into the phasesensitive detector. Thus, if the previous condition of small Bm amplitude is satisfied, phase-sensitive detection using the reference signal provided by the 1.f. generator (frequencyfm) permits one to run through the derivative of the absorption line E~(w/wo) while scanning it with the oscillator frequency. The zero passage of E~(w/~o) is detected by means of the resonance discriminator, whose output can be used to lock the oscillator frequency onto the resonance condition. Frequency readout by means of a counter provides, via Eq. (5.82), the value of the field B0.

230


.. J f.back J _ . frequency FLI" ] circuit ~ counter

eo+ ~

r~

~~]dem~176

-- ~~,

~ ,,,~,, J frequencyJ ~ / t) CvI,-4-1 control I-q-ldiscri~natorl---~l~l 1/ ~ I circuit I L.. J i i

~~ 'I nIf J IphaseJ J ge eratorfml = I shifter I '

~"

PSD

i_.-~

FIGURE 5.37 Scheme of an NMR magnetometer based on the detection of the power absorption at resonance and the use of a marginal oscillator circuit (adapted from Refs. [5.85, 5.92]). The resonating circuit is tuned over the appropriate frequency range (-2-50 MHz in advanced commercial devices) using different detector coils and varactor diodes as tuning capacitors. The input stage of the r.f. amplifier in the marginal oscillator circuit employs a couple of JFETs. The derivative of the absorption line is obtained by field modulation at the frequencyfm < I kHz, demodulation of the oscillator voltage and phase-sensitive detection (PSD). NMR field measurements outside the 50 mT-1 T standard interval offer a number of problems that are dealt with by resorting to specific solutions. Measuring increasingly high fields implies correspondingly increasing frequencies and when these approach the 50-100 MHz range the transmission of the signal through conventional coaxial lines may be hampered because the transmission line becomes part of the tuning circuit. Classically, either deuteron, 7Li (f0--16.547 MHz/T), or 27A1 (f0---11.1119 MHz/T) samples in place of proton assemblies are employed in order to decrease the value of ~0- There are cases, however, where the combination of high measuring fields (B0 > 1 T) and large distances between the sample coil and the oscillator tank circuit cannot be avoided. This may happen, for example, in the large superconducting dipole magnets used in particle accelerators, where distances around a few meters may compound with fields larger than 10 T. In such cases, resonator tuning must be carried out by taking into account the length of the connecting cable and the characteristic impedance of the coaxial line [5.93]. Let the input admittance of the ensemble made of the NMR coil and the coaxial cable be, according

5.4 QUANTUM METHODS

231

to the theory of lossless transmission lines, Yi -

1 Z0Ys cos(/3g) 4- j sin(13g) Z0 cos(13g) + jZoYs sin(/3g) '

(5.87)

where Z0 is the characteristic impedance of the coaxial line, Ys the admittance of the NMR coil, g the cable length, and 13 the propagation constant. The NMR coil behaves in practice as a pure reactance and it is usually assumed Ys = 1/j0)Ls. Having assumed that the leakage conductance and the resistance per unit length of the line are negligible, it is 13 = wx/LcCc, with Lc and Cc the series inductance and the leakage capacitance per unit length of the line, respectively. For the same reason, Yi is an imaginary quantity. The resonance frequency is thus obtained by imposing

jYi = 0)0Cv

(5.88)

with Cv the variable tuning capacitance of the oscillator tank circuit. This condition has periodic solutions (oscillation modes). From Eq. (5.87), we obtain the input admittance Yi as a periodic function of 0) 1

Wi m_ .--~-cot(13e + 0), ]L0

(5.89)

where 0 = tan -1

1

jZoYs

- tan -1 ~0)Ls .

Zo

(5.90)

A line of length g terminated in an inductance Ls acts, as far as the input admittance Yi is concerned, like a short-circuited line of length g 4-d, with d -- 0//3. If s is equal to a half-wavelength A/2 (A = 2~rv~/0), with v,p = 1/x/LcCc. the velocity of the electromagnetic wave along the cable), the input admittance Yi is equal to the admittance of the NMR coil. In this case/3g = zr and from Eqs. (5.89) and (5.90) we obtain Yi -- Ys -- 1/jwLs. Everything goes as if no connecting line were present. For a resonant frequency f0 = 50 MHz and v, -- 2 x 10 s m / s we have A/2 = 2 m. In practice, real lines suffer signal attenuation due to losses. Consequently, they are used as short as permitted by the structure of the measuring environment, which requires tuning of the capacitor Cv with the admittance Yi ~ Ys. The tuning condition (5.88) has solutions as long as Yi - 0. For a given length of the coaxial line and sample inductance, the fundamental mode can then be established only up to a maximum frequency O ) o l , provided by the condition Yi -- 0 (i.e. cot(/?~ + 0) - 0) in Eq. (5.89). This amounts to pose ~oolL~L~cC~g,+ 0 = zr/2 or, equivalently, g + d = A/4. To make a numerical example, we calculate the limiting

232


frequency value fol when a conventional NMR probe of inductance Ls = 0.2 ~H is connected to the tuning tank with a 1.5 m long coaxial cable of characteristic impedance Z0 = 50 f~. Since 1/x/LcCc -- v~, we find through Eq. (5.90)fol ~ 23 MHz. With a proton probe, this corresponds, using the fundamental mode, to a field of the order of 0.5 T, rising to about 2 T if a 27A1 probe is used. It has been suggested to overcome this limitation by suppressing the fundamental mode and exciting the upper modes, exploiting the function of a suppressing network inserted between the tuning capacitor and the connecting line [5.93]. The NMR measurements at field levels below some 100 mT are naturally associated with sensitivity problems because of the interference by the background fluctuating fields and the decrease of the frequency resolution. The r.f. field H1 has correspondingly reduced strengths. We can see how the signal decreases with H1, if the weak r.f. field condition H1 > T1 so that an equilibrium magnetization M = XoHp (Hp -- Bp//.~0) is obtained. Then Bp is abruptly switched off. If the switching time is sufficiently short, M remains unchanged and a state is constructed where it starts precessing around B0, the only remaining macroscopic field, at the frequency f0 = ~Bo/2~'. M is much larger than M0, the equilibrium magnetization pertaining to the field B0, and the expected NMR signal strength is accordingly larger than the one achievable with the continuous wave resonance method, with the additional benefit of increased measuring sensitivity because there is no disturbance due to the exciting signal. The condition on the switching time can be defined more precisely with an example. If B0 is the earth magnetic field, that is B0 "" 50 ~T, an orthogonal polarizing field Bp -10 mT is appropriate. The resultant field is basically coincident with Bp and it also remains so after the sudden decrease of Bp down to a value B!p~ a few times larger than B0 (say around 250 ~T). To preserve the magnetization value M, it is then required that the time interval At1 needed in order to pass from Bp to Bp be much lower than the spin-lattice ! relaxation time T1. The further decrease from Bp to zero must instead occur in a time At2 KK1~fo, if f0 is the precession frequency, so that M has no time meanwhile to re-orient in a field different from B0. For the specific case here considered, f0 ~ 2130 Hz (pure water sample), so that At2 KK50 ~s. The sensor generally has a considerable inductance and the energy stored in it is equally significant. It is notable that most of it can be dissipated along a reasonably long time At1. A schematic view of a low-field measuring setup exploiting nuclear free induction decay is provided in Fig. 5.39. In this circuit, the same winding is used both to apply the polarizing field and detect, after switch-off, the nuclear induction signal eN. The sensor can have cylindrical, spherical, or toroidal shape. The latter has the advantage of being omni-directional because, whatever the direction of B0, there is always a portion of the winding perpendicular to the measuring field [5.97]. With the other probes, we instead have a dead cone of orientations of B0 around the coil axis, where the perpendicular component of B0 is too small to be detected and the signal drowns into the noise. The electronic switching circuit provides the necessarily fast transition between the polarizing and the measuring configurations. If an N-turn solenoid of length s is used on a sample of volume V, the peak value reached by the induced

235

5.4 QUANTUM METHODS

~ ~

_~

frequency counter

switching electronics

Bo DCcurrent source

v

~

~

PC

(a)

00 r

r'~

I C~ co0

rv Z (b)

0

1

2 Time (s)

3

FIGURE 5.39 (a) Measurement of magnetic field B0 by free induction decay. The water-filled sample is first subjected for a time t >> T1 to the polarizing field Bp, perpendicular to and much larger than Bo. Bp is then switched-off in a very short time and the nuclear magnetization M startsto precess around the direction of B0 at the frequency to0 = 3'B0. The signal correspondingly induced in the winding (normally the same winding used to polarize the sample, switched between the DC source and the tuned amplifying circuit) after switching decays with time constant determined by transverse relaxation and radiation damping. (b) Example of time decay of the NMR induced signal for Bp = 5 mT and B0 = 20 ~T. The oscillation period results from beating of the induced e.m.f, with a reference signal (from Ref. [5.94]).

signal immediately after switching is calculated from the F a r a d a y Maxwell law as

N2ip eNp = ~0X0c~ s

V~lk,

(5.91)

236


where ip is the peak value of the polarizing current, ~/< 1 a factor taking into account the non-homogeneity of the field over the sample volume due to the finite length of the solenoid, and k the volume fraction of the solenoid occupied by the water [5.98]. Some other proton-rich fluid, like kerosene or ethanol, can be used in place of water. In this case, a small correction to the value of 3' for protons should be applied in order to account for the so-called "chemical shift" (equivalent in these cases to about I nT in the earth's magnetic field). In fact, the magnetic resonance of the proton may occur at different frequencies in different compounds because the specific chemical environment can affect the diamagnetic shielding. Incidentally, it is noticed that chemical shift, making it possible to distinguish between different molecular environments, is a precious tool of biochemical research. The value of ep is very small and it can reach, in accurately designed probes, the microvolt range, with a signal-to-noise ratio around a few hundred (for B0 "" 50 ~T). On the other hand, there are obvious limitations to the values of N, ip~ and V. The number of turns and current are limited by the available power, a critical point in portable instruments, and a large probe volume may be associated with field inhomogeneity, which causes line broadening and decrease of the s p i n - s p i n relaxation time T2. In a typical setup one can find V = 50-300 cm3~ N = 1000-3000, ip = 1 - 2 A. A sufficiently long decay time ~'F is actually needed for the accurate measurement of the precession frequency. Under transient phenomena, rF is related to the intrinsic spin relaxation mechanisms and the damping of magnetic resonance brought about by the tuned electrical circuit. The latter effect, called radiation damping, is associated with the dissipation by Joule effect of the energy provided to the circuit by the precessing spins, the sole source of energy during the transient. According to Bloombergen and Pound [5.99], the time constant for radiation damping is given by ~R = 2/%~oBpQ~,where Q is the quality factor of the coil when connected to the amplifier input. It turns out that, under certain circumstances, ~'R is comparable with T2 and the resultant decay time ~'F =

~RT2 ~R+T2

(5.92)

can be reduced with respect to T2. For instance, with Bp = 10 mT, Q = 50, and 77= 0.7, we obtain ~R = 5.2 S. With water samples, where T2 = 2.4 s, this leads to ~'F -- 1.64 s. Free precession proton magnetometers find relevant applications in geophysical and environmental surveying thanks to their combination

5.4 QUANTUM METHODS

237

of accuracy, sensitivity to low fields, and the absolute character of their measurement. Commercial portable setups are available, whose typical specifications provide a measuring range 20-120 ~T and a resolution of InT. With laboratory instruments, developed and used in a tight metrological environment, a range 10 ~T-2 mT is covered with relative measuring uncertainty varying between 10 -4 and 10 -6 [5.94]. Note that, in detecting local perturbations of the earth magnetic field, a gradiometer configuration is often adopted, where two identical sensors are placed a distance apart and the difference of the local fields is read as a difference in the frequency readings. A transient resonant state useful for the purpose of field measurement can also be obtained by applying the r.f. field as a single pulse of convenient duration. Strict metrological applications are a somewhat minor subject in the vast and fertile area of pulsed NMR, which has led to outstanding progress in materials science, chemistry, biology and medicine, the latter field having benefited enormously, for example, from the development of the magnetic resonance imaging techniques. The method of field measurement by pulsed NMR consists, in principle, in applying to the sample probe, immersed in the field B0, a r.f. pulse of amplitude 2H1 and convenient duration t I along a direction perpendicular to B0 instead of the steady-state r.f. field applied in the conventional continuous-wave method. If, starting from a condition of equilibrium where the system is endowed with the magnetization M0--XoHo = xoBo/~o directed along the z-axis, the r.f. pulse of frequency equal to the resonance frequency f0 is applied for a time t 1 -- (~/2)(1/~/p, oH1), the vector M 0 is tipped down into the x - y plane, describing a 90 ~ angle (Fig. 5.40). This can be understood if, as previously discussed for the solutions (5.73) of the Bloch equations, we move to the frame (x/,y~,z ~) rotating with the angular velocity w0. It can be shown that, in this frame, the field B0 -- 0 and only the field HIL ~fixed in the direction of the x~-axis, remains [5.88]. It then turns out that at the time t -- 0 the magnetization M0 finds, in this frame, the field HIL only and it starts precessing around it at the angular velocity wl -- ~//~0H1. After the time interval tl has elapsed, M0 lies along the yCaxis and there it remains if the r.f. field is switched-off at that instant of time. Going back to the laboratory frame, we eventually find the vector M0 precessing around B0 at the frequency f0, after having suffered transient nutation along the time tl. The signal correspondingly induced in the x- (or y-) directed receiver coil, which can be the same winding used for launching the r.f. pulse, is initially proportional to M0, that is, far higher than the signal induced with the conventional continuous-wave weak r.f. field method, where it is proportional to the transverse magnetization m. A time decay will be observed with

238


I Z'_----Z I JL..

I ~,

Ho

y'

~o

..........

....

',",";'; H1 L ~"....t~,~

..-'"" (a)

Y.~,

x'

""

(b)

d

Mo - ~ tlN--(c)

FIGURE 5.40 Pulsed NMR. (a) In the frame (x~,b/, z'), rotating at resonance with angular velocity to0 -- ~//~0H0,only the rotating r.f. field of amplitude H1 is left. If H1 is applied as an r.f. pulse of duration tl = (fr/2)(1/'),~H1), the magnetization M0 is tipped down in the plane x'-y j (90~ pulse). (b) In the laboratory frame (x, y, z) M0 is observed precessing around H0. The rotating spins fan out because of transverse relaxation and the signal induced in a receiver coil placed along the x- or y-axis decays with time. If longitudinal relaxation is appreciably involved (time T1), the length of the vectors precessing in the plane x - y is progressively shortened. (c) r.f. field pulse of time duration tl and decay of the nuclear induction signal. the progressive fanning out of the rotating spins (transverse relaxation) and recovery of the longitudinal equilibrium magnetization (time T1). Pulsed NMR magnetometers have been developed, which are based on the determination of the frequency of the time-decaying free induction signal. Their notable advantage with respect to the continuous-wave magnetometers is that they do not require field modulation and the related supplementary windings, which are sometimes incompatible with specific measuring configurations. This is the case, for instance, with precise field mapping in the superconducting magnets employed in particle accelerators, where simultaneous NMR decay frequency readings are made on a large number of probes suitably located at different points of the beam pipe [5.100]. These probes satisfy the demanding requirements on spatial resolution and physical restrictions on the probe volume. Since the circuit for pulse generation and signal analysis is normally connected to the probe by a long coaxial cable, the optimum conditions for signal propagation should be satisfied [5.101]. This means, in particular, providing for a cable length s equal to a multiple of the half-wavelength ~./2 = v~/2fo. For a field B0 = 2 T, we have that a length s = 2.34 m is equal to the full wavelength A. The basic drawback of the NMR pulse methods is that a complex coupling scheme between transmitter, probe, and receiver must be realized. In fact, during the time interval where

5.4 QUANTUM METHODS

239

the strong r.f. pulse is applied to the probe, the receiver must be protected from overload (ringing), while it is required that, in a short time after the end of the pulse, the energy conveyed by the transmitted pulse is dissipated and the receiver starts amplifying the small time-decaying nuclear resonance signal. To this end, several coupling schemes have been developed in the literature, which aim at damping the probe for a convenient time interval. This basically implies a decrease of the Q factor of the tuning circuit, which must possibly be obtained with little deterioration of the signal-to-noise ratio [5.102].

5.4.2.3 Flowing-water magnetometers. The measurement of low fields by means of the free-induction magnetometer has a basic limitation in its non-continuous nature, which prevents its application in the active control of magnetic fields besides requiring an often inconveniently big sensing head. It is, however, possible to retain the basic principle of the free-precession method, which is one of forcing a large out-of-equilibrium magnetization in the sensing sample and make it resonate in the low measuring field, while maintaining a continuouswave approach. This is accomplished with the flowing-water NMR method, where the operation of polarization, r.f. excitation, and signal detection are performed over spatially separated regions while maintaining sufficiently short time intervals between subsequent measuring steps to avoid important signal loss due to the relaxation mechanisms. Originally developed by Sherman [5.103], following an idea of E. M. Purcell, in order to measure the magnetic field with high precision over an extended region in space, and further assessed by Pendlebury et al. [5.104], the flowing-water technique can, in principle, span a very large measuring field range, from a few ~T to several T. This is extremely appealing, both from the viewpoint of establishing flux density standards in the laboratory and of achieving a general-purpose, easy-to-use device of superior accuracy and stability. The working principle of the flowing-water NMR magnetometer can be understood by making reference to the setup developed by Kim et al., schematically shown in Fig. 5.41. With this circuit, excellent signal-to-noise ratio down to about a measuring field strength B0 ~ 100 p~T has been achieved [5.105]. The water is pumped at a rate of a few ten cm3/s through a baffled polarizing chamber, where it is subjected to a large field Hp, of the order of some hundred roT, and it spends a time ~-p generally larger than the longitudinal relaxation time T1 ('-"3.5 s in pure water). It thereby acquires a magnetization Mpo -- M0(1 - e -TP/T1 ) close to the equilibrium nuclear magnetization M0 = XoHp. The pipe brings it to the region subjected to the measuring field H0, to which the magnetization becomes

240


|

,~rator

~mpl.

y,,

~

--z~

S PC

~x

Jfllllj L:~

C

pump

FIGURE 5.41 Flowing-water NMR magnetometer. The water is pumped at a rate of a few ten cm3/s through a polarizing chamber, inserted between the pole faces of a permanent magnet, where a field Hp is applied, and spends there time enough for the magnetization Mp to approach the equilibrium value. It then enters the region subjected to the measuring field Ho, where it receives first a transversely directed r.f. field pulse of amplitude 2H1 and duration tl, to eventually pass in the detecting region. Here the signal induced by the precessing magnetization vector in a sensing coil is collected. Such a signal is proportional to product HpHo (adapted from Ref. [5.105]). aligned, and, after a traveling time 71, the water is made to traverse a short region, where it is irradiated by a transversely directed oscillatory field H 1 (of peak magnitude 2/-/1). This corresponds to receiving a r.f. pulse, whose duration corresponds to the time of passage beneath the irradiating coils. The detecting region immediately follows, where a multiturn sensing coil with axis perpendicular to both H 1 and H0 is used to sense the precessing magnetization. Note that the directional change of the field from Hp to H 0 occurs over a sufficiently long time, much longer than the resonating period (of the order of 1/~//~0H0). The conditions are thus respected for the so-called "adiabatic variation" and the magnetization always sticks to the external field when it changes its orientation [5.106]. Because of longitudinal relaxation, the magnetization intensity, which lacks any transverse component, decays along the travel of the water from the polarizer to the sensing region. If 71 is the traveling time, we have that the magnetization arriving at the entrance of the exciting region is Mp = M0(1 - e -~'p/T1)e - r l / T ~ . Here, the effect of the r.f.

5.4 QUANTUM METHODS

241

pulse is one of producing non-adiabatic re-orientation of the magnetization vector Mp~ eventually leaving it precessing a r o u n d H0 with a canting angle 0 determined by the m a g n i t u d e of H 1 and the pulse duration t 1. There is a close analogy between this process and the classical molecular beam experiment, where nuclear and rotational magnetic m o m e n t s are m e a s u r e d by subjecting the traveling atoms to a perturbing localized transverse oscillating field. Beam defocusing is obtained in this case because of the ensuing non-adiabatic re-orientation of the magnetic m o m e n t s [5.107]. As previously remarked, the analysis of transient situations is simplified by m o v i n g to a frame (x~,y~, z ~) solidly rotating with the r.f. field H I L of m o d u l u s H I L = H1 (Fig. 5.42). If co is the angular velocity of H I L } o n e finds that in this new frame the field H0 is substituted by a field H 8 = H0 - ~o//~0~/[5.86], directed along z ~ = z, and, at the entrance of the r.f. tract, Mp starts precessing with angular velocity f~ around the effective field Her -- H 0 - oo//.~03/if- HIL. W e have evidently

AZ'-=Z I I

!i

A~ Z

TM "'~\

Ho [ ..............

,,'"

~'~

. [[ ~/'i~,'/"

a~~,,O?,~,,,,He,,r[[f]7"'-.,k I1__;"-. II M

"'n lL

(a)

Ho " ......................

:, . . . . -~,""

).,,'

. . . . _,~. X'

../,~"

Io

/'

H

1L

%"

~[~r~(~p'}

.. x'

, ' (b)

"

HIL

X

(c)

FIGURE 5.42 Precession of the magnetization Mp in flowing water, as occurring in the region irradiated by the r.f. coils. (a) In the frame (x~,yJ,z~), rotating with angular velocity to, precession occurs around the effective field H e r - H 0 oo/H,0~/q-H1L. (b) At resonance (to = r , the precession angle is 0* = Ir/2, the precessional angular velocity is fl = ~/H,0H1L and the angle cKtl) covered by Mp from t --- 0 (entrance) to t = tl (exit) is cKtl) -- W-l,0H1ctl. (C) If we come back to the laboratory frame (x, y, z), we find that the rotating transverse component Mpt -Mp sin cr either trails or leads the rotating field H1L by 90~according to whether cr < "a"or or(t1) > 7. A signal e(t) = b~ooM P sin a(tl) sin(oJ0t), with b a constant depending on coil geometry, is subsequently induced in a sensing coil perpendicular to both Ho and H1.

242


-- ~//~0Her. Let us assume that at time t = 0 Mp is aligned with H0 (i.e. the z-axis) and that the fluid leaves the r.f. irradiated region after a time interval tl. With elementary geometrical considerations as in Fig. 5.42a we find that the angle a(t~) described by Mp from t = 0 to t = tl satisfies the equation cos

a(tl)= 1 - 2 sin20 * sin 2 f~tl 2 '

(5.93)

if 0 * is the precession angle around Her. It is evident from Fig. 5.42a that 0* = t a n - 1

H1 H0 - to//.t03"

(5.94)

Notably, the correspondence between classical variables and quantummechanical expectation values permits one to write Eq. (5.93) also as cos a ( t l ) = 1 - 2 p ( r o , t l ) , where p(to,tl) is the transition probability P1/2,-1/2 between the two spin q u a n t u m states with mz = + 1 / 2 [5.108]. At resonance, the angular velocity of H1L is ro0 = T/z0H0 and the field H~ = 0. The effective field reduces then to Her -- H1L and the situation in the rotating frame becomes the one shown in Fig. 5.42b, where at time t - 0 the magnetization vector Mp, directed along the z-axis, starts its precessional motion in the plane (z',b/) around HIL (0" = ~r/2, see Eq. (5.94)). Since the angular velocity of Mp in the rotating frame is f~ = 7/~0H1, the angle covered from entrance to exit of the r.f. region is

a(tl) = f~tl = 7/~0H1tl.

(5.95)

Thus, as an example, the time tl required to tip Mp by 90 ~ in an r.f. field of peak value 2H1 = 2 A / m is tl = 4 . 6 7 x 1 0 -3 s. Returning to the laboratory flame (Fig. 5.42c), we eventually find at the time t = tl the magnetization vector Mp precessing with angular velocity ~o0 around H 0. The longitudinal magnetization is now Mpz---Mp cos a(t 1) and the transverse component is Mpt = Mp sin c~(tl). If the transit time t I is regulated in such a way that cr < r Mpt trails the rotating field HIL by 90 ~ If 7r < a(tl) < 2~r, it leads HIL by 90 ~ On leaving the r.f. coils at t - tl, Mp retains its direction in the rotating frame because of the rapid non-adiabatic removal of H1. This means that it continues its precessional motion around the z-axis in the laboratory frame, subjected to longitudinal decay (T1) and dephasing (T2). It immediately enters the detection region (a 3 cm diameter spherical sample in the apparatus of Kim et al. [5.105]), where a signal is induced in a multiturn coil directed perpendicular to both H0 and H1. The signal is e(t)=-d~/dt, where

5.4 QUANTUM METHODS

243

the flux linked with the coil at resonance is ~(t) = bMpt cos(co0t) (as one can easily induce from Fig. 5.42c) and b is a constant accounting for the geometrical parameters of the coil. We get

e(t) = bcooMp sin a(tl) sin(co0t) bTt~,oXoHoHp sin(7/~0Hlt~) sin(~0t),

(5.96)

which is m a x i m u m for ~ ( t l ) = Ir/2, that is, for a 90 ~ r.f. pulse. We see here that e(t) is proportional to the product HoHp. With the conventional NMR continuous-wave method in still water, we would have obtained (using, for example, the expression (5.79) for the transverse magnetization re(co0)) e(t)oc H 2. This implies a sensitivity advantage of the flowing-water method of the order of Hp/Ho, justifying the special interest attached to it in low-field measurement. Tuning and phasesensitive detection of e(t) can eventually provide the power absorption line, examples of which are shown in Fig. 5.43a [5.105]. One can see in this figure that the absorption peak can pass from positive to negative on increasing the magnitude of the r.f. field Hi. This occurs because the angle ~(tl) becomes greater than 180 ~ and the transverse magnetization component Mpt leads HIL. The energy balance is preserved because it involves both the r.f. circuit and the water pump. The technique originally developed by Pendlebury et al. [5.104] differs from the one schematically shown in Fig. 5.41 because in it the detection of the signal is carried out, after excitation by the r.f. pulse under the measuring field I-I0, in a conventional NMR setup. This is adapted for use with the water duct in place of the measuring head and is tuned to the Larmor frequency cood = '~/d'0Hdet. Hdet is a suitably high field provided, for example, by a permanent magnet or an electromagnet. Let us thus assume that the r.f. field Hi is applied. If ~'1 and ~'2 are the times taken by the water, pre-polarized to the magnetization level Mp, to flow between the polarizer and the r.f. coil and from the r.f. coil to the NMR detector, respectively, and t I is the time spent beneath the r.f. coil, we have that the longitudinal magnetization of the water at the entrance of the detector is Mpz = M0(1 - e-~p/T1)e-~l+~2)/rlcos a(tl).

(5.97)

By entering the resonating detector, Mpz suffers a further magnitude change, the mechanism for it being the same as the one that occurred before under the previous r.f. coil. Such a change ~ p z OCCURSunder the DC field Hdet and requires that energy be supplied to the detector coil, which, for a volume flow rate 17of the water through the coil, is in unit time of the order of ~r~/pzHde t. In the limiting case where ~ p z -- 2Mpz [5.104],

244

0

LO

0

(b)~ soo

E

o

0

CHAPTER 5

~0

L E

~u..

O ~

~_o

(s~,!un "qJe)leuB!s uo!;dJosqv

~i cu~

~~

I-i

~

N

~~-

x

~

xA

~§ E

~

Measurement of Magnetic Fields

cS

.-I

(o-

U_

~

5.4 QUANTUM METHODS

245

we thus get, through Eq. (5.97), the expression for the absorbed power P = 2W~0HdetM0(1 -

e-%/T1)e-(Zl+z2)/TIcosor(t1).

(5.98)

By sweeping the frequency of the r.f. field, we find, according to Eq. (5.93), a m i n i m u m of cos a(tl) at 00 = 000 (see Fig. 5.43b) and the magnitude of the field H0 is correspondingly determined as H0 = Wo/~'i~o.Remarkably, Eq. (5.98) shows that P depends on the product of the fields in the detector and the polarizer field. The signal strength being thus quite independent of H0, this method can be used to cover a very wide range of measuring field strengths. Accurate measurements down to around 2 ~T have been demonstrated, for example, by Woo et al. [5.109]. Commercial flowingwater magnetometers are available today by which the range of measuring fields 1.4 ~T-23 T can be covered [5.110].

5.4.2.40verhauser magnetometers. The notable low-field measuring capability exhibited by the free-induction and the flowing-water magnetometer is based on the creation of a magnetization value Mp much stronger than the equilibrium value M0 = XoHo in order to achieve a greatly increased measuring signal. This feature is obtained by applying a conveniently strong polarizing field far from the measuring region (flowing water) or at a different time (free-induction). There is, however, a subtler way to increase Mp beyond equilibrium that does not require any polarization field. It is based on a powerful physical idea by Overhauser, who boldly predicted that the saturation of the electron spin resonance in a metal, brought about by a r.f. field, could produce an enormous increase in the nuclear polarization [5.111]. He stated, in particular, that the steady-state nuclear polarization would be augmented by the amount expected for an increase of the nuclear gyromagnetic ratio to the value of the electron gyromagnetic ratio (~/e -- 21/zel/h -- 1.76085979 x 1011 T -1 s -1 for the bare electron). In short, it would be as if the nuclei were partially to take up, under such conditions, the equilibrium magnetization of the electrons, which had disappeared because of saturation and restoration of equally populated levels. Carver and Slichter, working on Li 7, provided an experimental verification of Overhauser's proposal [5.112]. They additionally showed, by working with a solution of Na in ammonia, that this effect did not require a metal, but, basically, the presence of unpaired electrons. Working with pure hydrofluoric acid, Solomon demonstrated that polarization transfer could also occur between different nuclei [5.113]. The physical mechanism lying behind the Overhauser effect (also called "dynamic nuclear polarization" (DNP)) is the coupling between

246

CHAPTER 5

Measurement

of M a g n e t i c F i e l d s

the nuclear and electronic spins, occurring either by hyperfine interaction (in metals) or dipolar interaction, the latter process also being responsible for the nucleus-nucleus transfer of polarization. This coupling is a route by which the nuclear spin-lattice relaxation processes, which tend to restore the conditions of thermodynamical equilibrium, can take place. It requires that each nuclear spin flip be associated with a simultaneous electron spin flip. Let us therefore consider a system endowed with dominant nuclear spin relaxation via coupling to electrons, where, under the applied field H0, electron spin resonance at the frequency f0 = 7e/~0H0/2vr (f0-28.02495GHz for /~0H0 = 1 T) is sustained by means of a transverse r.f. field H1. For a system of this kind, Overhauser's analysis provides, per unit volume, a rate equation for the difference D = N e - N + between down and up electron spin populations in relation to the nuclear spin population difference A = N + - Nn. For nuclear spin, dD/dt-

(D O - D)/T~i)e + (A0 - A)/T~in) ,

(5.99)

where Do and A0 are the population differences at equilibrium under the field H0. In Eq. (5.99), as in the following, to simplify matters we assume, I = 1/2. T(i) ~le and T(i) ~ln are the longitudinal electronic and nuclear relaxation times arising from the hyperfine or dipolar interaction only. We do not consider, for the time being, other relaxation processes. Since the interaction keeps the total spin momentum constant, we have d D / d t - d A / d t . This implies that, once the steady-state conditions are attained and the electronic resonance is saturated, d A / d t - - 0 . We consequently obtain from Eq. (5.99) that the nuclear spin population difference is

T(i)

A = A o + ~ in7s D 0 ,

(5.100)

where we have introduced the saturation factor s = 1 - D/Do. s = 1 when the two electronic spin populations are equal. With hyperfine interaction we have that, at temperature T,

TIn( i ) i

T(i) le

2 TF 3 T'

(5.101)

where TF is the Fermi temperature [5.112]. The equilibrium spin population differences are related to the susceptibilities and the applied field H0 Do

= XPauliH0/ld, e,

A 0 -" ,]f'0H0/ld, n,

(5.102)

5.4 QUANTUM METHODS

247

where XPauli is the Pauli paramagnetic susceptibility of the conduction electrons, X0 is the nuclear susceptibility, and/d, e and/z~ are the electronic and nuclear magnetic moments, respectively. Using the known expressions for the susceptibilities 3 XPauli - - ~/-i'0

Ne/Ze 2 kTF

'

Xo -

/z0 Nn/z2n I + 1 3 kT I '

(5.103)

we obtain from Eq. (5.102), posing Ne = Nn, Do

=

A0 /~e 23 T--~" T

(5.104)

By introducing Eqs. (5.101) and (5.104) in Eq. (5.100) and recalling that /Ze//Zn = ~/e/~'n (same spin quantum number), we eventually obtain A = A0(1 + s ~/e ). Tn

(5.105)

It then turns out that the saturation of the electron resonance ( s - 1) brings about an enhancement by a factor ye/~/n of the nuclear magnetization with respect to the equilibrium magnetization M 0 /zeA0. With proton nuclei, this factor is around 660 and it is reduced when other interactions, besides electron-nucleus spin coupling, can provide nuclear relaxation. For example, Carver and Slichter find that the theoretical enhancement factor ~'e/~'n is more than 80% reduced in their experiments on Li 7 [5.112]. An example of the evolution of the nuclear resonance signal with the strength of the r.f. field H 1 is shown, for this specific case, in Fig. 5.44. Practical Overhauser magnetometers are generally based on the use of liquid samples, where a free radical, playing the role of electron donor, is diluted in a proton-rich solvent. We deal in this case with dipolar coupling, as presented for the first time by Beljers et al. in the free radical diphenyl-picrylhydrazyl (DPPH) [5.114]. This is an organic salt with one free electron (g factor, g = - 2 . 0 0 3 6 + 0.0002) and a very sharp resonant lineshape, a property deriving from an effect called "exchange narrowing" [5.115]. Nitroxide free radicals are currently applied in DNP magnetometers [5.116]. The nitroxide has a free electron associated with the nitrogen atom, dwelling in the relatively large magnetic field, of the order of 2 mT, provided by the nitrogen nucleus [5.117]. This is extremely interesting for low-field (e.g. earth field) measurements because the ensuing hyperfine splitting of the energy levels (zero field splitting) makes available to the electrons a low energy state, which is then crowded by a spin population much larger than the one expected under

248


100

50

.

0

.

.

.

.

.

.

.

1()0

!

200

.

.

.

.

300

H 1 (A/m)

FIGURE 5.44 Overhauser effect in Li7. Electron spin resonance (ESR) is obtained by applying a r.f. field H1 at a frequency around 100 MHz and nuclear resonance at 50 kHz is simultaneously produced and observed. By increasing the amplitude of Hi and approaching the ESR saturation, the NMR absorption signal is largely increased. The obtained enhancement factor A is larger than 100 (from Ref. [5.112]).

the low-strength measuring field only. By flooding the sample with the saturating r.f. field and, consequently, restoring the electronic spin population balance, a correspondingly larger polarization is transferred to the proton nuclei and the DNP gain can attain, for earth field measurements, the order of a few thousand. Figure 5.45 illustrates this case, where the hyperfine interval factor a corresponds to a shift of the resonance frequency to f0 = 2 x 10 -3 Te/2~r of about 60 MHz. The upper level suffers further Zeeman splitting upon application of the external field. Of the two allowed transitions, the lower one (1), enriches the population of the lower nuclear level, bringing about positive DNP gain on saturation. Transition (2) has the opposite effect, leading to negative DNP amplification. The magnetometer setup developed by Kernevez and G16nat [5.117] is schematically shown in Fig. 5.45. It employs two DNP probes in a bridge configuration. The probes are placed in a resonator, producing the saturation of electronic resonance at the radiofrequencyf0, and are, at the same time, excited at the low NMR frequency. A special design of the coils permits one to always find a part of the sensor where H 1 and H0 are perpendicular one to another and consequently eliminate

5.4 QUANTUM METHODS

,'

i

,

',

"

,I

a/4+e.

;

a/4

DNP probe

a/4-e r.f.

a

,, I ; "

,~,

249

generator 1

2

I

-3a/4

Ho=O Ho>O

-" 1

o i% i I

O!

1

!

DNP probe ____-c___,i !

If1 1

! readout I

FIGURE 5.45 A practical Overhauser magnetometer is often based on the use of a nitroxide free radical as a source of unpaired electrons and a proton-rich solvent. The electrons reside in a relatively strong nuclear field and suffer hyperfine splitting (zero field splitting). The upper level is further split by the Zeeman interaction with the measuring field H0. The two possible electronic transitions lead to DNP gains of opposite sign. Thanks to the accurate choice of the solvents, they are separately obtained at the same r.f. frequency in the two probes employed in the bridge circuit. It is therefore possible to reject the external interference signal, which is instead symmetrically detected. The background noise spectral density turns then out to be lower than 10 pT Hz -1/2 (adapted from Ref. [5.117]). the signal extinction zones. Two different solvents are used with the nitroxide free radical (methanol and dimethoxyethane), which produce different chemical shifts. They are calibrated in such a way that, at the same frequency, f0, the DNP gain is positive in the first probe (proton polarization parallel to the external field H0) and negative in the second (proton polarization antiparallel to H0). It then turns out that, using the bridge circuit, the NMR resonant signals are amplified and the symmetrically detected disturbances are eliminated. The sensitivity claimed for this type of continuous-wave magnetometer is better than 10 pT H z -1/2. It is also possible to design an Overhauser magnetometer operating under transient conditions, like a free-precession magnetometer. In this case, the r.f. field is applied for a time interval sufficient to establish electron saturation and the ensuing proton polarization. After ringing has subsided, a short DC current pulse in the pickup coil aligns the proton moments perpendicularly to the measuring field, around which they are left to freely precess and generate the time-decaying signal in the sensing coil. Commercial Overhauser magnetometers display a typical measuring range of 20-120 ~T and a sensitivity around 0.1 nT. They are prevalently employed in the measurement of the environmental fields, especially the terrestrial magnetic field and its variations due to geophysical

250


phenomena and various man-generated disturbances. To this end, they are generally made portable and respond excellently to the ensuing requirement of low power consumption, requiring typically 1-2 W. In this respect, they favorably compete with the free-induction magnetometers. Power is chiefly required for saturating the electronic resonance and, in order to minimize it, a narrow absorption line would be required. The typical resonance linewidth of nitroxides is, however, fairly broad, being around 100 ~T in the earth magnetic field, but it can be reduced to about 20 ~T by substituting hydrogen atoms with deuterium atoms in the compound [5.118]. An alternative free radical with 2.5 ~T linewidth has been proposed, which can reach saturation with far less power than the perdeuterated nitroxide [5.119]. This compound does not display zero field splitting and, in order to have high DNP gain, separate polarization in a homogeneous high field is normally provided. The NMR magnetometers are assumed to provide absolute measurements. Equation (5.82) shows that the value of the field expressed in T is obtained from knowledge of the fundamental constant 3/ and the determination of the resonance frequency. For the conversion of T in A / m , division is made by the magnetic constant/~0 = 47r x 10 -7 N / A 2 exact by definition. The measurement is therefore traceable to the national standards of time. These are maintained today with an uncertainty of the order of 10 -13 and the generally available frequency counters, calibrated against these standards, have a time base stability better than 10 -6. With pure water probes, the shielded proton gyromagnetic ratio ~/~ should be adopted. There is a long history of ~/~ determinations, which have been carried out in the last 50 years in different national metrology laboratories. The motivation for such experiments is, on the one hand, the obvious desire for more precise magnetic field measurements and, on the other hand, the control of the practical unit of current in the laboratories. Basically, ~/~ is obtained by applying Eq. (5.82) in reverse, where the field is measured with the highest possible accuracy with a force method (high fields) or produced by means of an accurately calculated and realized single-layer solenoid (low fields). Mohr and Taylor provide a detailed critical account of the latest experiments in their comprehensive report on the CODATA recommended values of the fundamental constants [5.90]. The 1998 adjustment provides "}/p = 2.67515341 x 10s T -1 s -1 with relative uncertainty 4.2 x 10 -s. If, instead of pure water, proton-rich substances are used, the related chemical shifts must be taken into account. This correction can be the source of substantial uncertainty, typically some 10 -6 . Further uncertainty contributions can arise. For example, traces of magnetic impurities might remain in the elements of the sensing head and some detrimental effect could be associated with AC

5.4 QUANTUM METHODS

2~1

environmental fields. The precise measurement of the precession frequency in the decaying free-induction signal at low fields might be a problem because of the low frequency, the noise, and the limited time available for the measurement. The adoption of digital methods and specific algorithms, including spectral analysis by fast Fourier transform, can be instrumental in achieving the best measuring accuracy.

5.4.3 Electron spin resonance and optically pumped magnetometers In the Overhauser magnetometers, the resonant absorption of a r.f. signal by the electrons is exploited indirectly, the field measurement solidly relying on the proton resonance. It is, however, possible to make direct use of electron spin resonance, the basic difference with respect to NMR being that the involved frequencies are multiplied by a factor 3'e/~'~ ~ 660. This means that with fields higher than about 10 mTwe fall into the microwave region. ESR resonance magnetometers have been principally developed for low-field measurements, where the high value of ~/e provides an advantage with respect to NMR in terms of signal-to-noise ratio (S/N). However, the electronic relaxation times are generally much smaller than the nuclear ones and the resonance linewidth A~o0 = 2/T2, in particular, is much enlarged with respect to the proton linewidth. Interactions between electron spins are much stronger than between nuclear spins and the coherence of the spin precession is in most cases rapidly destroyed. Since the sensitivity of a resonance magnetometer, that is, the smallest detectable field change 8H0, can be written as [5.120] 8Ho - &tOo(S/N)_I/2,

(5.106)

it is concluded that optimal tradeoff between linewidth and signalto-noise ratio is required for ESR to compare advantageously with NMR. In practice, ESR magnetometers only employ sensing materials (e.g. free radicals) where the unpaired electrons are well separated and the spinspin interactions are minimized. The organic salt DPPH is a classical ESR compound, endowed with a nearly free electron ( g - - - 2 . 0 0 3 6 vs. g - -2.0023193 of the bare electron) and an exchange-narrowed linewidth A~o0/~/e ~ 0.27 mT [5.115]. Other narrow linewidth materials, typically radical cations and anions, have been developed and applied. An example is the fluoranthene radical cation salt (FA)2PF6, where, thanks to an important motional narrowing effect, Aco0/Ye ~ 1.5 ~T is obtained [5.121]. A highly accurate magnetometer based on its use in a small-sized (around 100 m m 3 or less) probe head has been developed by Gebhardt

252


and Dormann for measurements between 50 ~T and 10 mT [5.122]. There are no basic differences in the electronic design of NMR and low-field ESR magnetometers, but for the increase of the resonance frequencies in the latter. ESR obviously displays much larger (S/N) value and it might be preferred for continuous measurements of the earth magnetic field, as required, for example, in land-surveying and defense activities. The associated measuring resolution can be of the order of 10 nT over an angle of + 85 ~ around the measuring axis [5.121]. They do in general not provide absolute measurements because the electron gyromagnetic ratio, subjected to a variety of electronic interactions, can change over different materials. A calibration in a known field by comparison with a NMR device is therefore recommended. The ESR magnetometers, like the NMR ones, are intrinsically scalar devices, providing a measure of the modulus of the field. They can, however, be adapted, at the cost of a somewhat reduced resolution, to vector field measurements. Duret et al. [5.121] combine a known static internally generated field Hi with the measuring field H0. If 0 is the angle made by H i with H 0 and the condition Hi >> H0 is satisfied, the modulus of the resulting field H m -- H i q- H0, which is the quantity measured by the magnetometer, is given to good accuracy by Hm = Hi + H0 cos 0. The auxiliary field Hi, being so much higher than H0, must obviously be very stable in order to achieve the desired measuring resolution. If such stability cannot be achieved, it is expedient to make a double measurement, with the sign of H i reversed. As shown in Ref. [5.121], the two measured dispersion resonant signals can be combined to provide H0 cos 0, independent of Hi and its fluctuations. The ESR magnetometers exploit the resonant behavior of unpaired electrons in some specific compounds, where they act almost like free spins in thermal equilibrium. The magnetization level associated with a given measuring field is the one we expect from Boltzmann statistics. It might be asked whether we can overcome, as already obtained with nuclear magnetism, the thermal equilibrium limitation and correspondingly increase the sensitivity of the field-measuring device via increased non-equilibrium magnetization. The method of optical pumping provides an affirmative answer to this question. This method, while having an importance going far beyond the relatively narrow subject of low-field magnetometry (being the basis, for example, of atomic clocks), provides a practical and widely applied route to measurements in the ~T and nT range. The physical basis of the optically p u m p e d magnetometers can be understood by making reference to the atomic energy level diagrams of two commonly employed sensing elements: He 4 and Rb s7. The first is shown in Fig. 5.46a, the second in Fig. 5.47a. The He 4 atom has no nuclear moment and the two electrons in the ground state 1~S0 have antiparallel

5.4 QUANTUM METHODS

253 r.f.

m j= 0 i

23P~ ; 23 P1

. . . . . . .

,

:

generator I He lamp ( )

.......

3,= 1083 pm 23S1~ ,,~

,, !-I

interference filter

lens

i i__

m j= 1

J

circular polarizer ~

m j= 0

1 He cell

/

/,"

mj= -1

rf

IHo (a)

(b)

i r. detector I

~

output

m

FIGURE 5.46 Optical pumping in He 4. (a) Atoms in a gas cell are raised from the ground state I 1 So to the first excited state 23 $1 of the triplet system by means of a r.f. discharge. Circularly polarized light (Do line at 1.083 ~m) is selectively absorbed by one of the m} = + 1 Zeeman sublevels and the corresponding atoms reach the excited state 23P0. They radiatively decay in a very short time and with the same probability into the three 23S1 sublevels. The absorbing level, whose depopulation is signaled by cell transparency, can then be repopulated by the action of an AC field I-I1 at the Larmor frequency ~0, applied orthogonal to the external field H0. (b) Schematic diagram of the servomode magnetometer. The r.f. field I-I1is slightly modulated in frequency. When the center frequency is equal to ~0, the output signal is made of even harmonics only of the sweep frequency. Phase detection and feedback are used to lock the frequency of H1 to oJ0.

spins Sz = + 1/2 [5.123]. He 4 atoms can be excited from the g r o u n d state to high energy levels or even ionized by r.f. generated electron collisions in a discharge tube, where the gas is held at a pressure of some 102 Pa. A n u m b e r of the excited atoms can decay back to the first excited parallel spin state 2381 (orbital m o m e n t u m L = 0, total angular m o m e n t u m J = total spin m o m e n t u m S = 1), whose radiative decay to the g r o u n d state has a forbidden character, since it w o u l d involve the reversal of a spin. Consequently, this state is sufficiently long-lived (10-20 ms) to be considered metastable. U n d e r steady-state conditions, the singlet 11S0 g r o u n d state population can then be considered as a buffer gas (parahelium) for the triplet 23S1 atoms (orthohelium). The next higherlevel triplet state (L = 1) is 23p, which is s u b d i v i d e d in the three substates 23po, 23p1, 23p2, corresponding to the three values of the total m o m e n t u m

254


I R o7

2 ......

/

A

N

(,

, I

x

i~

I ! , I

\

'

,'~ m~=-2 i! i !

~ i

'

! l !

!

: I I l

I i I

9

I]

i I

I

1

~I

i

'

,, I ,I

i

I I

i I

15,

I

i

,,

_

"-

, ,'i" . :~G,' =

i

/r"-,.?

'

)?

_______mF=.____~ 2

lo,

mF= ' 1

!.~.//-1

\

mF=--4~ oo

A =hf o

2 = 2a

F=4

5.

i I ~.~1r

i

5 2 S1/2

0

a.

I'1.4 ! ! 1 I I I

,'

Z = 0.7948/zm

"

N.N

, I

,

mF=4

I l

; ~ 1

__

'~AmF-+I

~

J

6S1/2

"~.o 20. o"~

AmF=O,• '

C s 133

25.

0.2

0.4 0.6 0.8 Light intensity (a.u.)

1.0

(b)

m F- -2

J= 1/2 ~

/=3/2

F=I

(a) FIGURE 5.47 (a) Relevant energy levels and optical pumping in Rb 87. Atoms in the hyperfine levels F -- I and F -- 2 of the ground state 52S1/2 can be raised to the first excited state 52p1/2by circularly polarized light of wavelength A - 0.7948 I~m. Transitions from the mF -- +2 Zeeman sublevel are forbidden by the selection rule AmF -- -t-1. Decay from the excited state is equally likely to occur in all ground state Zeeman sublevels. The light-absorbing levels consequently depopulate and the cell becomes transparent. Equal populations, restoring light absorption, can again be obtained by applying, at right angle to the external field H0, a resonating r.f. field H1, whose frequency provides a measure of H0. (b) Example of evolution with the intensity of the pumping light of Zeeman sublevel pp~ulations. They belong to the hyperfine level F - 4 of the ground state 6S1/2 of Cs 1~~(adapted from Ref. [5.130]).

J = 0, 1, 2. These substates are slightly separated in energy, their separation being m u c h smaller than their distance from the first excited state 23S1 9In the presence of an external field Ho, the state 23S1 splits into three Z e e m a n sublevels, corresponding to the q u a n t u m n u m b e r s

5.4 QUANTUM METHODS

255

mI -- - 1 , 0, + 1. The Zeeman energy separation is AE -- hco0 = g/xB/x0H0,

(5.107)

where/XB is the Bohr magneton and the g factor is that of the free electron. This is a good approximation because the optical electron is basically decoupled from the atom and behaves as in a vacuum (g = -2.0023193). The field-frequency conversion factor of about 28 kHz/~xT obviously provides a specific advantage in the measurement of very low fields. Let us now suppose that the gas cell containing the mixture of parahelium and orthohelium is invested by a circularly polarized light beam, with optical axis z coincident with the direction of H0. Such a beam can be generated either by a He lamp or a laser. In all cases, the light is tuned to the infrared wavelength A = 1.083 Ixm, corresponding to the Do spectral line, which connects the states 23S1 and 23p0. The atoms, initially assumed to be equally distributed among the available sublevels, selectively absorb the polarized photons because only the Zeeman sublevels endowed with the appropriate mj value can make the transition to the excited state 23p0. Thus, if the absorbed photons transfer, depending on the sense of the circular polarization, the axial angular momentum mj = +1, only the atoms with quantum numbers mj = -T-1 can be excited, respectively, to the mj = 0 state 23p0. Unequal distribution between the mj = 1 and m j - - 1 sublevels amounts to a net magnetization Mz, directed either with or against the field H0. The optically excited atoms are short-lived, having a lifetime around 10 -4 s, during which mixing of the P states induced by the relatively high pressure buffer gas is likely to occur. It is therefore assumed that the optically excited atoms tend to relax with equal probabilities into the different 23S1 sublevels, regardless of the level from which they originated. Under the counteracting effects of optical pumping and relaxation, with their characteristic times ~'p and ~'R,a higher than thermodynamic equilibrium value of the magnetization Mz is reached in a time ~-given by 1 1 1 - -t , "r"

"rp

"rR

(5.108)

of the order of some 10 -4 s [5.124]. With the absorbing state mj - - 1 (or, equivalently, the state mj = +1) depopulated, the polarized light can propagate through the gas cell, which becomes transparent. In order to recover a homogenous distribution of atomic states, a transition between the Zeeman sublevels can be induced by means of an AC field H: applied at right angle (or at least with a component at right angle) to the external field, having a frequency equal to the Larmor frequency to0 = AE/h. If

256


the AC field frequency is swept through the resonance frequency COo,the cell opacity is recovered when CO= COo, with a corresponding dip in the transmitted light intensity. At steady state, the behavior of the magnetization M, precessing around the direction of H0, is the one predicted by Bloch's equations. A typical scheme employed in optically p u m p e d He 4 magnetometers is shown in Fig. 5.46b. The He lamp generates the Do, D1, and D2 infrared spectral lines. D1 and D2 connect the state 23S1 with 23p1 and 23p2, respectively, and can be suppressed by filtering. Alternatively, a laser tuned to the Do line can be used, which has the advantage of a largely increased pumping efficiency [5.125]. With this scheme a servoed magnetometer is realized, where the r.f. field HI is modulated with a small swing +ACOaround a center frequency. If the sweep frequency is f/ and the center frequency coincides with COo,the output signal is made of even harmonics of 1~ only. Any deviation of the center frequency from COo results in the appearance of a fundamental component, which is phasedetected and used in a feedback loop to lock the frequency of H1 at COo, which then provides the value of the field H0 (Eq. (5.107)). The He 4 magnetometer is characterized by a resonance linewidth ACO/2~r of the order of few kHz, corresponding to ACO0/~/~ 0.1 ~T. Field tracking speed is therefore limited by the modulation frequency f~, which is of the order of a few hundred Hz. The sensitivity can be better than 100 pT and is predictably improved, even by two orders of magnitude, by the use of increased cell size and laser pumping in place of the He lamp [5.125]. Like all the Larmor resonance devices, the He 4 magnetometer provides a measurement of the modulus of the external field, whatever the angle 0 made by H0 with the optical axis. The signal amplitude, however, depends on such an angle as cos20 [5.124] and the progressive degradation of the signal on approaching 0 = 90 ~ eventually results in a dead zone, where the noise can be too high for the magnetometer to comply with defined specifications. Methods for automatic orientation of the optical axis with the external field and of the r.f. field perpendicular to it have been devised [5.126]. Alkali elements have a single electron in their outer shell, which is available for magnetic resonance. Na 23, K 39, Rb 87, Rb 8s, and CS 133 have all been used to realize optically p u m p e d magnetometers. The optimal alkali vapor pressure, of the order of 10 -4 Pa, is maintained in a glass cell with a buffer inert gas at a pressure around some 103 Pa. The temperature required to obtain the appropriate vapor pressure ranges between 126 ~ in Na 23 and 23 ~ in Cs 133. The physical mechanism of optical pumping in alkali elements is analogous to the one already7 described in He 4 and is schematically illustrated for the case of Rb 8 , an element frequently

5.4 QUANTUM METHODS

257

employed in devices [5.127], in Fig. 5.47a. Transitions in Rb 87 can be induced from the ground state 5281/2 to the first excited state 52p1/2, lying 3.7725 x 1014 Hz above it by absorption of circularly polarized light of wavelength A = 0.7948 ~m. The nucleus of Rb 87 and the optical electron have total momentum number I = 3/2 and J = 1/2, respectively, and their interaction gives rise to hyperfine splitting of the 5281/2 a n d 52p1/2 states. I and J couple in a manner which is analogous to the previously sketched L-S coupling in the He 4 atom and produce a total angular momentum with quantum number F. This runs from I + J through i I - Ji, thereby attaining, in this specific case, the values F = 1 and F = 2 for b o t h 5281/2 and 52P1/2 states. The interaction energy between nuclear and electron magnetic dipole moments can be written as [5.128] a WF = ~(F(F + 1) - I(I 4- 1) - lq + 1)), (5.109) where the constant a is the so-called interval factor of hyperfine structure. We have WF = 3 a and WF = -- 88 a for F -- 2 and F -- 1, respectively, so that the hyperfine energy splitting is &W12 = 2a. This amounts to the frequency /12 -'- AW]2/h -- 6.8347 x 109 Hz for the split ground state 5281/2 and f12 = 8.18 x 108 Hz for the state 52p1/2 (vs. the much higher dipole radiation frequency f = 3.7725 x 1014 Hz connecting these two states). In presence of the external field I-I0, Zeeman splitting occurs, corresponding to the possible orientations of the vector F with respect to H0. If m F is the associated magnetic quantum number, five (m F -- 2, 2, 1, 0, 1, 2) and three (m F = 1, 0, 1) sublevels within the hyperfine levels F = 2 and F -- 1, respectively, are generated. At high fields, when the Zeeman splitting becomes larger than the zero-field hyperfine splitting AWl2 , I and J become decoupled and quantize independently in the direction of H 0 (Paschen-Back effect), so that no pumping will take place. The energy behavior in the Zeeman and intermediate field amplitude domains can be quantitatively predicted, for the present case of electronic angular momentum J = 1/2, by means of the Breit-Rabi formula. It provides for the energy in the field H0 when the momenta are mF, J = 1/2, I

WI+l/2'mF

a

4 [

aWl2

2

4mFgJi~'BIJ'oHo (gJ[d'B[Ld'oHo)2]1/2

X 1 4- 21 +'------~ •W12

4-

AWl2

,

(5.110)

where gj is the Land6 atomic g factor. The dependence of WI+l/2,mFo n H0 at low and intermediate field values is shown in sketch form in Fig. 5.47a. It is noticed that this formula can predict the transition to the Paschen-Back

258


regime at high fields, where the two mj levels mj = + 1/2 are each split in the four sublevels with quantum numbers m1 = 3/2, 1/2, - 1/2, - 3/2. We are interested in the transitions between sublevels in t h e 281/2 state, for which gl is equal to the g factor and AW12 = 2a. For very low values of H0 (typically H0 < 10 p~T in the alkali atoms), the energy difference between :tifferent adjacent sublevels (AmF -- +__1) is the same and it is provided by the Breit-Rabi formula in the limit gj/~B/~0H0 ~ AW12. This limit amounts to the condition that the energy of the whole atom in the external magnetic field is negligible with respect to the hyperfine structure separation (the distance between the two F levels). Under these conditions, Eq. (5.110) reduces to the linear law

mF

H0),

WI+l/2,mF = _ a-'4 + -- a 1 + -~a gl/~,B~0

(5.111)

where it has been posed I = 3/2. The separation between the sublevels turns out to be 1

bE = ~gjj/,BIU,0H0 = h30/zoHo,

(5.112)

with ~ the effective gyromagnetic ratio. ~ is then reduced by a factor four with respect to the gyromagnetic ratio of the free electron and the corresponding frequency-field conversion factor of Rb s7 is then 7.006 kHz/~T. The same value is obtained for N a 23 and K 39, while it is 4.668 k H z / ~ T in Rb 85 and 3.50 k H z / p X in CS 133. The curvature of the level lines on increasing H0, predicted by Eq. (5.110), is apparent in Fig. 5.47a. It leads to different AE values between adjacent sublevels and thereby to a spectrum of resonance lines. As for the previously discussed case of H e 4, the pumping in Rb 87 from the 5 2$1/2 to the 52p1/2 state with circularly polarized light and the successive radiative decay will lead to unequally populated sublevels in the ground state [5.129]. For one thing, the transitions from the ground state to the excited state occur with different probabilities for different Zeeman sublevels. It is easily understood, for example, that the atoms in the ground state sublevel mF - - + 2 cannot absorb photons if the sense of rotation of the polarization imposes the selection rule A m F = +1. If the sense of the circular polarization is the opposite, it is the transition from the sublevel mF -- - 2 that is inhibited. Because of the relatively high pressure buffer gas contained in the cell, fast mixing of the Zeeman sublevels occurs in the excited state and it can be safely assumed that all sublevels of the ground state have an equal probability of being repopulated in the relaxing atoms. The result is depletion of the absorbing levels, which implies larger than equilibrium magnetization and cell transparency. Figure 5.47b provides an example of

5.4 QUANTUM METHODS

259

evolution of the Zeeman level populations in the ground state of C s 133 upon optical pumping [5.130]. C s 133 has nuclear and electron angular moments I = 7/2 and J - - 1/2, respectively, and is characterized by two hyperfine energy levels in the ground state with F -- 3 and F -- 4. We can see how, following optical pumping with AmF = +1, the non-absorbing sublevel mF --4 in the hyperfine F = 4 becomes increasingly populated with the increase of the pumping light intensity. Again, redistribution of the populations amongst the different Zeeman sublevels in the ground state can be obtained by means of a resonating r.f. field I-I1, with at least a component orthogonal to H0, but, contrary to the case of He 4, now we have, depending on the H0 value (Eq. (5.110)), as many resonant lines as possible transitions between sublevels. In some cases, the lines can be resolved and, according to the Breit-Rabi formula, a very precise field measurement can be done. This has been demonstrated with C s 133 for fields larger than about I mT [5.130] and is realized in potassium magnetometers in the range of the earth magnetic field [5.116]. In the latter case, a resonant linewidth A~0/~/around 100 pT (A~0 ~ 5 s -1) and sensitivity better than 0.1 pT are reported. The obvious limitation of a magnetometer exploiting narrow line response is its relative inability to track rapidly changing fields (i.e. over times shorter than about 1/A~0). In many practical instances, the individual resonance lines are broad enough to overlap and merge in a single unresolved structure. The intrinsic linewidths A~0/~ can be as narrow as I nT, but they become much larger under experimental conditions. Broad lines can actually provide an advantage in terms of the tracking capability of time-varying fields, but unresolved structures can lead to problems of measuring accuracy. The unresolved line shape is not symmetrical and, in particular, the position of its peak is a function of the sensor orientation with respect to the measuring field. This is the so-called "heading error", which is due to the fact that the amplitudes of the individual overlapping lines change in a different way when the component of H 0 along the optical axis is changed. In the limit where the direction of H0 is reversed, the symmetry is also reversed because this amounts to interchanging the signs of the magnetic quantum numbers. Alkali magnetometers are generally realized using the so-called autooscillating configuration, where resonance is revealed for its modulating effect on the transmission through the cell of an auxiliary polarized light beam directed orthogonal to the pumping beam. Let us suppose that H0 is aligned with the optical axis and consider the resonant precession around z of the magnetization M induced by the r.f. field H 1. The transverse magnetization component m (see Fig. 5.32) rotates at the Larmor frequency in the x - y plane and, in doing so, it periodically changes the angle it makes

260

CHAPTER 5 Measurement of Magnetic Fields r.f.

generator Rb lamp (~

I

) interference filter

lens

J

circular

polarizer",~

0=45 ~ photo detector hase ~er

output

FIGURE 5.48 Scheme of principle of the alkali-based autooscillating optically pumped magnetometer. At resonance the light beam intensity is modulated at the Larmor frequency. The signal at the output of the photodetector is re-injected into the r.f. coil after 90~ phase shift and the oscillation is self-sustained, provided the feedback loop gain is unity and the total phase shift is zero. The output signal amplitude varies approximately as the product sin 0 cos 0.

with the transversally directed light beam. Since light is absorbed to an extent depending on the angle made by the precessing moments with the light wave normal, a modulation of the transmitted light intensity at the resonance frequency occurs. Practical devices do not generally make use of two independent orthogonally directed light beams and, as shown in the schematic representation of Fig. 5.48, the same beam directed at 45 ~ to H0 is used for both pumping and monitoring. The signal collected by the photodetector at resonance, modulated at the Larmor frequency, is amplified, 90 ~ phase-shifted and re-injected in the r.f. coil. The 90 ~ phase shift is required because, at resonance, the transverse magnetization m lags 90 ~ behind the active rotating component of the r.f. field. Consequently, if the gain of the loop is unity and the sum of all phase shifts is zero, the system oscillation is self-sustained. While in the previously discussed servo-type magnetometer the output signal amplitude depends on

5.4 QUANTUM METHODS

261

the angle 0 made by H 0 with the pumping light propagation direction according to cos 20, in the single-beam autooscillating-type magnetometer it varies as the product sin 0 cos 0 because the secular pumping process varies as cos 0 and the Larmor frequency modulation varies as sin 0. The maximum signal is therefore achieved for either 0 -- 45 ~or 0 = 135 ~ In the latter case, however, wiring of the feedback loop must be interchanged because the product sin 0 cos 0 changes sign and the phase shift must correspondingly change sign. An alternative solution consists in making use of a dual cell system, placed back-to-back with a single light source at the center, which, however, requires strict geometrical tolerances [5.131]. Given this angular dependence of the signal, the response of the autooscillating magnetometer is affected by two dead zones, corresponding to field orientations close to the optical axis or a direction perpendicular to it. Either servoed or autooscillating, optically p u m p e d magnetometers represent, in summary, an ideal response to the need for accurate measurement and tracking of weak and very weak magnetic fields, like those of geological origin or the ones encountered in deep space. They can display sensitivities of the order of I pT and a fast response to transients, the latter being limited either by the transverse relaxation time T2 of the precessing spins or by the amplifier bandwidths. Their use in the laboratory is mainly associated with the generation of reference fields in the geomagnetic range [5.130, 5.132]. We have previously remarked how optical pumping in He 4 required the creation of a certain proportion of atoms in the metastable state 23S1, whose distribution among the three available Zeeman sublevels becomes inhomogeneous after radiative decay from the state 23p0. A very similar process can occur in the He 3 isotope, whose use in the previously described optically p u m p e d magnetometers can then be envisaged. However, the fact that He~ endowed with nuclear momentum (I = 1/2) implies not only hyperfine splitting of the atomic levels and, consequently, different Zeeman resonance frequencies with respect to He 4, but also the existence of an atomic ground state 11S0 forming a Zeeman doublet. Each metastable atom can exchange its metastability with a ground state atom by collision, under the condition of conservation of the angular momentum. In He 4, this has no special consequences because this atom has zero-momentum ground state and such an exchange leaves the incident and the emerging metastable atoms with the same momentum. In He 3 it may occur, instead, that incident and emerging ground state atoms have their magnetic quantum numbers differing by + 1, with the change u 1 simultaneously affecting the metastable atoms. Momentum transfer from the metastable to the ground state atoms can create a very large fraction of oriented nuclei and, at a pressure of 10-100 Pa, the net

262


oriented population can amount to 20-40% [5.133]. They are driven to resonance by a r.f. field at the Larmor frequency of 32.45 MHz/T, with resonance detected either by optical absorption monitoring or conventional nuclear induction methods. The two additional Zeeman resonances concerning the 23S1 metastable state can also be observed. In spite of the low density of the gas with respect to water (number of He 3 atoms per unit volume N h -~ 8.7 • 1022 m -3 at room temperature with a pressure of 200Pa vs. density of protons in water ---6.7x 1028m-3), the latter technique can provide an easily detectable signal. For example, for a 20% fraction of net oriented nuclei, which is easily obtained under geomagnetic fields, we calculate a magnetization M -- 0.2Nh/~h, where the J/T, of about helion (He 3 nucleus) magnetic moment is /d,h = - 1 . 0 7 4 6 7 x 10 -5 A/re. This is approximately the magnetization we would obtain in water under a field H0 ~- 2 x 104 A/re. Given the gaseous nature of He 3 and the weak interatomic interaction effects, very long spin-lattice and s p i n - s p i n relaxation times T1 and T2 are expected. They can vary in practice between very wide limits, depending on factors like the presence of field gradients, radiation damping and interaction with the cell walls, but values ranging between 103 and 105 s can be achieved [5.134]. A He 3 nuclear free-precession magnetometer with limiting transverse relaxation times ranging between 1 and 24 h and sensitivity of 0.1 nT has been demonstrated [5.134]. With the resonance only indirectly coupled to the pumping light, the problem of resonance frequency shifts due to various asymmetries and effects of the pumping light are largely avoided. Outstandingly accurate measurements of the nuclear free-induction decay in He 3 have been recently performed in order to determine the ratio between the helion and the shielded proton magnetic moments /d,h//d,~ [5.135]. NMR measurements in He 3 and pure water have been carried out, in particular, under the same field of 0.1 T, using the very same 25 m m diameter spherical cell for the two elements. These are interchanged without removing the cell from the magnet. The final result is ~h//d,~ -- -0.761786131, known with an accuracy of 4 parts in 109.

5.5 MAGNETIC FIELD S T A N D A R D S A N D TRACEABILITY Any magnetic field measurement has a meaning when it can be traced to the relevant base and derived SI units. Traceability requires the action of the National Metrological Institutes (NMIs), which have the mandate of developing, maintaining, and retaining custody of the standards

5.5 MAGNETIC FIELD STANDARDS AND TRACEABILITY

263

of measurements. Standards are traceable with stated uncertainties to the SI units, which can then be disseminated for general measurement and testing activities. The end users can thus relate their measurements to the SI units through an unbroken flow of calibrations originating in the NMI laboratories. These, in turn, engage in mutual comparisons with other NMIs, under supervision by the Bureau International Poids et Mesures (BIPM), by periodically reproducing the units and maintaining sound quality system principles [5.136]. Magnetic field (or, equivalently, magnetic flux density) standards were classically realized in the past as physical artifacts, that is, accurately designed and built solenoids or Helmholtz pairs, which were made traceable to the SI units of length, resistance, and voltage. Nowadays, when talking of a field standard, reference is usually made to a system combining a field source with a nuclear or atomic resonance device, provided with all the auxiliary setups required to stabilize the values of the involved physical quantities and minimize the effect of external interferences. The resonance devices play the role of intrinsic standard, realizing traceability to the SI unit of time via the resonance equation (5.82). The uncertainty of the whole standard is therefore made of two components, one associated with the consensus value of the gyromagnetic ratio ~, and the other associated with the practical realization of the standard. The latter is largely dominant and may be calculated according to the example reported in Section 10.4. An illustrative discussion on the realization of magnetic field standards by an NMI laboratory for the sake of dissemination of the SI field unit is given by Weyand [5.94]. This author shows, in particular, how NMR-based standards developed at the Physikalisch Technische Bundesanstalt (PTB) can cover a range of magnetic field values stretching between 10 ~T and 2 T, with relative uncertainties ranging between 10 -6 and 10 -4 (see Fig. 5.49). Various coil types (e.g. solenoids and Helmholtz pairs), characterized by accurate realization and low thermal coefficient, are used as field sources in the lower field range, up to about 100 mT. Electromagnets are used in the upper field range. In order to achieve the measurement and calibration capabilities described in Fig. 5.49, full use is made of specifically developed continuous-wave NMR magnetometers, based on the marginal oscillator technique, by which the lower measuring limit of present-day commercial NMR magnetometers of about 40 mT is extended down to 0.5 mT [5.85]. As remarked in Section 5.4.2.1, the sensing probe does not in this case contain pure water but a dilute CuSO4 aqueous solution, which implies a small shift of the resonance frequency. The amplitude of the modulating field is I ~T. The standard field sources used to cover the lower field range are calibrated by means of a free-precession magnetometer, pre-polarized in a conveniently strong field (e.g. 5 mT for

264


1E-4

nti~8~NMR

>, r

o

r

free inductiondecay

1E-5

> 0 rV

1E-6 #oH

(T)

.....

o'.1

. . . . . . .

'

FIGURE 5.49 Field range covered by the NMR-based magnetic field standards maintained at Physikalisch Technische Bundesanstalt and associated relative uncertainty. Solenoids and Helmholtz pairs are used as field sources up to about 100roT, while electromagnets cover the upper range of field strengths. Continuous-wave and free-induction decay NMR magnetometers are applied in the upper and lower field range, respectively. A laboratory-developed continuouswave NMR setup based on a marginal oscillator method permits one to overcome the low-field limitations of commercial devices (around 40 mT) and is applied down to about 0.5 mT [5.85] (adapted from Ref. [5.94]).

a measuring field/~0H--20 I~T). It is seen in Fig. 5.49 that the relative uncertainty associated with the free-precession based standard increases rapidly with decreasing of the field amplitude. This is due, on the one hand, to natural weakening of the signal-to-noise ratio and, on the other hand, to the decrease of the resonance frequency with the decrease of the measuring field strength, which implies progressively lower accuracy in its determination. The observation time is in fact limited by the transient nature of the experiment. The decay time is related to transverse relaxation and radiation damping (see Eq. (5.92)) and a reasonable measuring time (for example, a few seconds) requires sufficient homogeneity of the field across the probe. Possible alternatives to the free-precession NMR magnetometers for the calibration of low field standards are provided, for example, by optically p u m p e d and flowing water NMR magnetometers. Recent developments in the commercially available flowing water magnetometers have been announced [5.110]. With a very small sensing head (---10 m m 3) and a measuring range


265

extending from 1.4 ~T to 2.1 T, this instrument holds promise for improved and more generally available traceability to the SI units. In order to develop magnetic field standards with the very low uncertainties allowed for by the NMR techniques, it is imperative to achieve, especially at low and medium field strengths, shielding from the earth magnetic field [5.137]. To this end, calibration is performed by surrounding the coil with a large triaxial Helmholtz setup, supplied by three independent current sources, regulated in such a way to suppress the earth field components. The value of these components is around 40 ~T (vertical direction), 4 ~T (East-West direction), and 30 ~T (NorthSouth direction). With the setup shown in Fig. 4.8 (diameter 1.2 m), which displays a central spherical region of 10cm diameter with field homogeneity better than 5 x 10 -5, one can obtain, for example, cancellation of the earth field components down to about 20 nT by direct control of the supply currents. Earth field, however, is subjected to diurnal variations (of the order of 10-20 nT) and other uncontrolled environmental field sources are usually present. Consequently, for certain demanding low-field calibration requirements like those sometimes required with geomagnetometers, active external field cancellation by feedback is made. In the apparatus developed by Park et al. [5.138], schematically shown in Fig. 5.50, the standard solenoid, surrounded by a triaxial Helmholtz coil, is kept with its axis aligned in the East-West direction in a wooden building, far from power lines and other buildings. A series connected auxiliary triaxial Helmholtz coil, located at a distance of about 50 m in order to avoid interference with the solenoid, is used in association with an optically p u m p e d Cs magnetometer placed at its center. Any drift of the environmental field is detected by the atomic resonance magnetometer and converted into a current of proportional strength, which is injected into the main Helmholtz coils. A standard deviation of the resulting compensated field of the order of 0.1 nT is demonstrated against an actual drift of 19 nT over a period of I h. The standard solenoid, whose constant is determined by means of a C s - H e 4 magnetometer [5.132], is endowed with a number of supplementary windings, all laid on the same fused silica former of radius 10.1 cm and length 0.938 m, leading to uniformity better than 0.5 x 10 -6 over a 4 cm long central region [5.139]. A standard coil is characterized by a definite temperature dependence of its constant kH --p, oH/i, where i is the current, which must always be arranged as a function of the actual measuring temperature. An increase of temperature normally leads to a decrease of kH because of the thermal expansion of the former and the winding. Values of (1/kH)(dkH/dT) of the order of 10-5-10 -4 K -1 can be found in standard

266

CHAPTER 5 Measurement of Magnetic Fields Cs-He AMR magnetometer

Auxiliary compensation coil

Current source

l/

urc__ so e source

tL..[L l

,.

ii

~

"

I,,

Standard solenoi J

cs,

.

I

....

I ......

magnetometer ~

'

.

.

.

.

.

\

Main

compensation coil

FIGURE 5.50 Highly accurate standard for low strength magnetic fields, endowed with active nulling of the earth magnetic field. Compensation better than I nT is obtained by keeping the reference field source, a solenoid with axial field uniformity better than 0.5 x 10-6 over a 4 cm wide central region, within a compensating triaxial Helmholtz coil. Drift vs. time of the earth field is actively compensated by detecting it through an optically pumped Cs magnetometer, place d at the center of an auxiliary triaxial Helmholtz coil. The ensuing signal is converted into an adjusting current, which is injected in the main Helmholtz coil. The field generated by the standard solenoid is measured by means of a Cs-He 4 atomic magnetic resonance (AMR) magnetometer (adapted from Ref. [5.138]).

coils [5.140] and are sometimes compensated by use of an extra winding [5.132]. Stable and controlled temperature conditions are therefore required in any calibration procedure, with the actual coil temperature measured, for example, by means of a Pt resistor or determined by means of a separate measurement of its electrical resistance. It m a y happen that a certain coil cannot provide a sufficiently homogeneous field to be calibrated by means of the NMR method. If an NMR calibrated coil having suitable size is available, transfer of its coil constant can be envisaged by coaxially inserting the smaller coil within the bigger one and supplying the two coils with currents generating opposite directed fields [5.94]. By placing a sensitive field sensor (for example, a fluxgate magnetometer) in the central position of the coil axis, the current values are determined, via calibrated resistors,


267

which lead to zero field indication. The ratio of these values provides the ratio of the coil constants. The main source of error with this calibration procedure, which m a y be affected by a relative uncertainty of the order of 10 -3 , is related to the imperfect alignment of the coil axes. A possible and generally available alternative m e t h o d for transferring the constant kH from an N M R calibrated coil to a coil of u n k n o w n constant consists in comparing the readings of a Hall

R

1

T

"O

V2

L2

O

D' fM

O

w

Standard coil

1.02-

R = 30.-Q L = 12mH C = 50 pF

O E3

~= 1.01 O
> 1. Xm Sm ly

The effective field in the sample is then given to a good approximation by H m = Nlil/Im. If we pose Bm ~/.toM m and By ~ g0My, we find Mm/My = Sy/Sm. To remark that, whatever the method used to form it, a closed magnetic circuit is remarkably insensitive to external spurious fields, the larger the permeability, the higher the shielding effect. If we associate to the circuit a demagnetizing factor Nd and an intrinsic relative permeability/Xr, we obtain that the response to an external field is characterized by the apparent permeability: /Xr /~ar-- 1 - } - N d ( ~ r - 1)

9

(6.4)

Applied fields and spurious fields of equal strength produce induction variations in the material that are in the ratio /Xr//Xar--~ 1 + Nd/xr- For Nd = 0.05 and/Xr = 104 we find that gr//Xar ~ 5 X 102. We conclude this section by mentioning the subject of air flux compensation, to be treated in further detail in the following. The problem chiefly consists in the fact that, especially with thin laminations and films, the secondary winding may embrace a far larger cross-sectional area than the one occupied by the material and the detected flux includes then a contribution from a region where the induction is Ba =/x0H, while in the material we have B =/x0H +/-~0M. This extra contribution is, in general, automatically eliminated by connecting in series opposition the secondary winding and another winding with suitable t u r n - a r e a product, which

6.2 OPEN SAMPLES

295

is linked only to the induction Ba. With closed magnetic circuits, this process does not imply special difficulties because the field H is always perfectly defined and identical inside and outside the sample. In the Epstein frame the automatic air-flux compensation is actually carried out by means of a mutual inductor, so that only the polarization J =/z0M is left [6.6]. In other cases, both B and J can be obtained by combination of compensating coils and calculation [6.3].

6.2 OPEN SAMPLES We have previously remarked that the adoption of open magnetic circuits is conducive to homogeneous magnetization only in ellipsoids and spheres, a condition seldom attained in practical samples. More common specimen shapes are cylinders, parallelepipeds, and strips, for which the magnetization is not uniform and either the magnetometric or the fluxmetric demagnetizing factor approximations must be adopted (see Section 1.2). In a number of important cases, however, there are no specific advantages in closing the magnetic circuit, the flux closure is not required, or it is even impossible. A case in point is that regarding the measurement of the magnetic moment of a sample by force methods or by extraction. It turns out, at the end, that measurements with open samples are ubiquitously performed and accepted. We will introduce here some peculiar problem associated with the open sample condition, with the provision that details on techniques and methods will be discussed, within their appropriate context, in other parts of this book. Bulk soft magnets are frequently available as open specimens and are tested, as far as possible, enclosing them in a flux-closing yoke, more or less of the kind shown in Fig. 6.5 (permeameter method, see Section 7.1). One could actually shape the samples as ellipsoids and carry out a full J ( H ) characterization of them by making an exact determination of the demagnetizing field, but it is difficult to envisage easy application of this approach at low and medium induction values. Applied and demagnetizing fields would be in fact so close that the measuring accuracy in the determination of the effective field H = Ha - H d = Ha - N d M - "

Ha - (Nd/P,O)J

(6.5)

would be impaired. In addition, external spurious fields (e.g. the earth's magnetic field) might cause substantial errors in high-permeability materials. Measurements on open soft magnets must then be conducted in a shielded environment, obtained either with active field cancellation

296

CHAPTER 6 Magnetic Circuits

with large Helmholtz coils (see Fig. 4.8) or by enclosing the sample in a suitably large box made of Mumetal or similarly high-permeability alloys. A simple additional measure to reduce interference from the earth s field in samples of elongated shape is to align them with the east-west direction. Shielding by soft magnetic enclosures should obviously avoid relevant coupling of the shield with the sample and the field generating setup because it would introduce unwanted distortions of the field lines. The measuring accuracy is expected to increase on approaching the material saturation, where the applied field largely overcomes the demagnetizing field. Practical considerations suggest, however, to use cylindrical samples instead of ellipsoidal ones, with the secondary coil located around the mid-section and the effective field determined by use of the fluxmetric (also called ballistic) demagnetizing factor N(df) in Eq. (6.5). N(df) is either calculated by standard formulas [6.16] or it is determined by comparison with recorded data concerning precisely machined ellipsoidal samples. With the latter method, illustrated in Fig. 6.7, a measuring uncertainty lower than 1% is claimed, for example,

2.2

2.1

elli

2.0 Iv-

1.9

/

I

1.8 1.7

! t

1.6

'

Soft solid steel

r i

0.0

_ _

.

!

,-

--,

4.0x10 4

H

--

.

!

8.0x10 4

,

.

- r - -

'

1.2x10 s

(A/m)

FIGURE 6.7 Open sample measurements. The true normal magnetization curve of a soft steel specimen, shaped as a prolate ellipsoid (solid curve), is compared with the apparent curve of a cylindrical specimen (length 200 mm, diameter 10 mm) circumscribed to it. The ballistic demagnetizing factor of the cylindrical sample is experimentally determined by this comparison, which puts in evidence, on approaching the material saturation, a demagnetizing field at the sample midsection of the order of 3 X 10 3 A/m (adapted from Ref. [6.17]).

6.2 OPEN SAMPLES

297

in the determination of the high-field normal DC magnetization curve in 1 0 m m diameter and 2 0 0 m m long cylindrical steel samples [6.17]. No special difficulties would additionally be met for measuring the coercive field Hc. For this we only need to make the magnetization to recoil from the saturated state and record the value of the applied field at which the average magnetization in the sample, i.e. the average demagnetizing field, is zero. Such a condition can be verified by detecting the condition of zero stray field around the sample [6.18] (see Chapter 7). Bulk soft magnets have a relatively narrow field of applications, being chiefly used as DC flux multipliers. Most soft magnetic cores, being subjected to AC fields, are obtained by assembling sheets or ribbons, which, under certain circumstances, are tested as open samples. The magnetic anisotropy and the rotational hysteresis of magnetic laminations are, for example, typically measured in disk-shaped samples, for which we can take, as a first approximation, the demagnetizing factor of the oblate ellipsoid. Alternatively, the effective field is directly measured by use of a flat H-coil or a Chattock coil, provided the size of the sample allows for the use of such sensors. With extra-soft strip-like samples and unidirectional field, both flux-closed and open sample configurations bristle with difficulties. If the strip is wound as a toroid or even assembled to form an Epstein circuit [6.19], stresses build-up in the sample [6.20]. On the other hand, yokes may not provide the sought after magnetic short circuit, a faint remanent magnetization of it being detrimental to the meaningful determination of the intrinsic properties of the tested material. One might then resort to measurements on long open strips, but even in samples with high aspect ratio a substantial demagnetizing field correction by means of Eq. (6.5) is required if the material is very soft. Figure 6.8 illustrates the case of a near-zero magnetostrictive amorphous ribbon (length 202 mm, width 8.9 mm, thickness 19.6 ~m) annealed at 320 ~ under a saturating longitudinal magnetic field. This sample is known to have an intrinsic rectangular hysteresis loop with very low coercivity (Hc ~" 0.5 A/m) [6.21]. This loop can be recovered from the experimental one, obtained on an open strip sample (solid line, measurement performed on a east-west oriented strip contained in a large Mumetal shielding box), by calculating the effective field through Eq. (6.5). The value of the resulting fluxmetric demagnetizing factor (N~df~ = 1.25 X 10 -5) is about five times higher than the value predicted by Aharoni for a uniformly magnetized strip (X = 0) of equal aspect ratio [6.22] and closer (within a factor 2) to the prediction for high-permeability cylindrical samples of equal length and cross-sectional area [6.16]. This example illustrates that making accurate correction for the demagnetizing effects can be very difficult in practically shaped soft magnets, the softer

298

CHAPTER 6 Magnetic Circuits 0.8 C071Fe4B15Silo

,', ", i

amorphousstrip

0.4.

'i

!

o.o,

)

"--0.4"

-0.8:

J . i ,,..

-10

i" , ,,

w,

I , , , . ,

-5

k".,.

,.

0

, , , , , . . , .

Ha, H(A/m)

/= 202 mm

w = 8.9 mm

d= 19.6llm i,..,

5

,..r

:,

i.:

10

FIGURE 6.8 A high-permeability near-zero magnetostrictive amorphous strip

(length 202 ram, width 8.9 mm, thickness 19.6 ~m) annealed under a saturating longitudinal field develops a uniaxial magnetic anisotropy and is characterized by an intrinsic rectangular hysteresis loop (dashed line, Hc --" 0.5 A/m). This can be recovered from the sheared experimental hysteresis loop (solid line), which is measured, upon application of a uniform field, using a secondary coil localized on the sample mid-section. The effective field H = H a - N(dOMis calculated using the fluxmetric demagnetizing factor N (f) = 1.25 x 10-5.

the material the more approximate the calculation of the intrinsic magnetization curve. Aharoni's formulations for both fluxmetric and magnetometric demagnetizing factors [6.23] appear better suited to thin film structures, which are generally characterized by much lower permeability values than their bulk counterparts. Numerical methods may, at the end, be required to calculate the stray fields in open sample arrangements. Notice, however, that rough corrections for the demagnetizing field are totally acceptable in a number of cases. Weak magnets, either homogeneous or made of magnetic particles or second phase precipitates dispersed in a non-magnetic matrix, provide an obvious example where correction for the demagnetizing effect is of little or no relevance. We can equally content ourselves with approximate estimates for the demagnetizing field when testing hard and semi-hard magnetic thin films, for their aspect ratio is such as to generally make Hd small with respect to coercivity. It may also happen that these films have uniaxial anisotropy perpendicular to the substrate plane, in which case the demagnetizing effect is tightly knit to the intrinsic material properties and

6.2 OPEN SAMPLES

299

the conventional measuring methods are superseded by specific techniques for the measurement of the perpendicular magnetization (e.g. magneto-optic Kerr or Faraday methods, SQUID detection). It is to be stressed that the air flux compensation in open samples, if automatically accomplished as previously mentioned for closed magnetic circuits, does provide the material polarization J in the limit of low values of the demagnetizing coefficient only. With reference to Fig. 6.9, we assume that the compensating coil and the secondary coil, of turn-area NcSc and N2S2, respectively, are linked with the fluxes ~c = NcScla,oHa and ([')2 ~---N2S2(H,oHa -/z0Hd) q- X2Sml -- N2S2.H,oH q- N2SrnJ~ if Sm is the cross-sectional area of the sample. If N2S2- N~Sc, by connecting the two coils in series opposition we evidently obtain zero signal in the absence of the sample. In the presence of it, we measure, according to Eq. (6.5), the resulting flux:

$2 - 2Sm'(1 mNd )

(6.6)

The compensated flux is then proportional to the polarization in the sample as far as the demagnetizing field is negligible with respect to the magnetization M, a condition normally realized in strips, ribbons, and thin films. The use of a closed magnetic circuit, made of a cylindrical or parallelepipedic sample enclosed between the pole faces of an electromagnet (Fig. 6.5b), is the standard in the measurement of the J(H) and B(H) curves in permanent magnets [6.15]. Open sample measurement methods in hard magnets are, however, very often required or simply preferred for a number of reasons, ranging from speed of measurement to costs or nature of the available specimens. With conveniently large-sized cylindrical specimens, we can perform the conventional measurement of the induction in the material using tightly wound

FIGURE 6.9 Air flux compensation in a strip-like sample. The compensating coil and the secondary coil have equal turn-area products NcSc = N2S2 and are connected in series opposition, so that, in the absence of the sample, the total flux variation with changing applied field Ha is zero. In the presence of the sample, the resulting linked flux is proportional to the material polarization J (i.e. magnetization M), provided the demagnetizing factor Nd KK1.

300


secondary coils and either conventional or superconducting solenoids as field sources. On the other hand, a host of methods have been developed by which the magnetic m o m e n t of the whole sample, instead of the induction across a given cross-section of the sample, is obtained. They are characterized by high sensitivities, of the order of 10 -7 A m 2 (10 -4 emu) or better, and, consequently, they require small samples (for example, a 10 m m 3 specimen magnetized to 1 T is endowed with a total magnetic m o m e n t of about 8 x 1 0 - 3 A m 2 (8 emu)). Besides the classical force methods, with their extra-sensitive AC version (alternating gradient force magnetometer), inductive techniques have been developed, where the sample is made to play the role of a magnetic dipole, which is either vibrated, rotated, or displaced with respect to one or more sensing coils. Alternatively, one can keep the sample fixed and apply a high intensity pulsated field. When the specimen fulfills the dipole approximation, its magnetic m o m e n t can be determined by measuring the flux linked with a surrounding coil, which is related to it by a definite relationship of proportionality. Let us consider, as in Fig. 6.10, a point-like magnetic m o m e n t I n in the plane of a search coil of radius R and its equivalent current loop im of area a, such that m = a'im. The coordinates (x0, Y0) define the position of the magnetic moment. If we suppose, in a purely fictitious way, that a current is circulates also in the search coil, the two loops can be viewed as coupled circuits, characterized by a mutual inductance M [6.24]. They are linked through the fluxes ~ms = M'is (from the search coil to the small loop)

|

,

search coil

Z

! i i i

x

FIGURE 6.10 A small (ideally point-like) sample of magnetic moment m is represented as a loop of area a with a current im flowing in it (m -- a'im). The flux emitted by the loop which links with the search coil is 9 = k(xo,Yo).m, where k(xo,Yo) is the value of the coil constant associated with the loop coordinates (x0, Y0)- If m has components (rex, my, mz) and z is the axis of the search coil, then c~ = k(xo, yo).mz. A magnetic moment located at the center of a filamentary search coil of radius R provides 9 = #o(mz/2R).

6.2 OPEN SAMPLES

301

and ~sm = M'im (in the opposite direction). The search coil generates the magnetic induction Bz(x0, Y0), directed along the z-axis, in the point occupied by the magnetic dipole and is consequently characterized by a constant k(xo,Yo)= Bz(xo,Yo)/is. We can then also write ~ms = Bz(xo, yo)'a. It follows that M = [Bz(x0, yo)/is].a = k(xo, yo).a and the flux linked with the search coil eventually turns out to be CI)sm - -

k(xo, yo).m.

(6.7)

If the magnetic dipole is in the generic point of coordinates (x, y, z) and has components (rex, my, mz), the general relationship holds (I)sm

--

k(x, y, z).m = kx(x, y, z).mx + ky(x, y, z).my + kz(x, y, z).mz,

(6.8)

with ki(x, y,z) = Bi(x, y,z)/is. For a magnetic dipole located at the center of the coil, where k = ia,o/2R (see Eq. (4.4)), we obtain ~srn = (la,o/2R)mz, while, when a Helmholtz pair is used as search coil, we have from Eq. (4.20) ~sm = 0.7155(ia,oN/R)mz. This is just the flux variation we measure if we extract the sample from the coil. On the other hand, we know from Eq. (4.24) that if we connect the two coils of the Helmholtz pair in series opposition, we obtain a uniform gradient Ok(z)/Oz of the coil constant around the origin. This means that if we make the dipole to oscillate around this position, we can measure, according to Eq. (6.7), an induced voltage again proportional to the magnetic moment m. This is the principle, to be discussed in some detail in Chapter 8, exploited in the vibrating sample magnetometer. It was previously remarked that, if shielding during measurements is required, coupling of the shield with the sample and the useful field source might result in a distortion of the flux lines, eventually influencing the experiments to ill-defined extent. This effect is actually part of a general problem, always to be taken into account each time open samples are tested in the neighborhood of soft magnetic bodies. It can be illustrated by taking the case of a magnetic dipole of moment m brought in proximity of the flat surface of a magnetic body having relative permeability /Zr. It provides a schematic view of what happens, for example, when a permanent magnet sample is placed between the pole faces of an electromagnet. We obviously expect that, in response to the presence of the dipole field, the body will magnetize to an extent depending on the value of/Zr. This brings about a distortion of the field lines emerging from the dipole. The problem can be quantitatively understood, making use, for instance, of the scheme in Fig. 6.11 and

302


Hm.

rd'y l'41"lm ~

I" ~

~

dx

.....

" .---~

r

,

/lr> 1

(a)

/ ]1r > i

/ x, (b)

/dr < 1

(c)

FIGURE 6.11 Image effect. The field distribution around a magnetic dipole is perturbed in proximity of a magnetic medium. For a dipole of strength m this amounts to the presence of a fictitious dipole of strength m~= [(/~r - 1)/(/~r + 1)]m mirroring the real one. It can be demonstrated that with such an image dipole the continuity conditions for the tangential field and the normal induction component through the air-medium boundary are satisfied. With a ferromagnetic medium (/~r > 1), the image dipole is oriented like in (a) and (b). With a diamagnetic medium, it is oriented like in (c).

considering the continuity conditions on field and induction at the body surface. These regard the tangential component of the magnetic field (Hit(P)--HIt(P1)) and the normal component of the magnetic induction (B• ( P ) = B• (P/)), where we define with P and/Y two points adjacent to the surface, in the air and in the body, respectively. Let us therefore assume that a dipole of moment m is facing the flat surface of an indefinitely extended soft magnetic body. It is common experience that a magnet is attracted to a block of iron, as unlike charges attract themselves. One can therefore reasonably think that the field profile satisfying the previous continuity equations could be achieved by the combination of the dipole in the air and a fictitious dipole in the material, mirroring the real one. This can be demonstrated by calculating HII and B • in both P and P~. We consider then in Fig. 6.11 the field generated in P by the combined

6.2 OPEN SAMPLES

303

contribution of the real dipole of moment m (field Hm) and the notional image dipole of moment m ~(field Hm, ). Using standard formulas (see, for example, Ref. [6.25]) and noting the symmetrical position of m and m ~ with respect to the boundary, we obtain for the tangential field component and the normal induction component: 1 dxdy HII(P) = 4vr r 5 (m - m~),

I~o 3d 2 - r2 B• (P) --- 4vr ~ (m 4- m~),

(6.9)

where the distance r -- ~d 2 + d2 and the positive sign of B • (P), m, and m ! is conventionally associated with the direction of m and m ~ shown in Fig. 6.11a. In order to calculate the same quantities within the body at point P', we refer to a momentarily unknown moment m" (and the related field Hm") in place of m. If the relative permeability of the material is/d,r~ we obtain 1 d,.dy m", HI1(P~)- 4~r r 5

B

(19/) --

l

/d'0/d'r

4rr

3d2 - r2 m" ---7-"

(6.10)

We impose now the previous continuity conditions on HII and B • which provide a couple of equations in the unknown variables m~ and m': m - m~= m',

m + m~=/xr.m'.

(6.11)

By solving them, we find, in particular, that the image dipole is endowed with the magnetic moment:

nil ---- -~-rm- . 1 /J,r 4- 1

(6.12)

We see that in the presence of a high-permeability medium the flux lines emerging from a dipole of moment m are modified as if a mirror dipole of equal strength m/ = m were present within the material. This situation is depicted in Fig. 6.11a and b. It corresponds to the case of an open sample in the air gap of an electromagnet. To be remarked that, in such a case, the permeability of the pole faces tends to rapidly decrease on approaching the saturation of iron, so that, according to Eq. (6.12), the strength of the image dipole is correspondingly decreased. The distribution of the field lines of a sample having defined magnetic moment is consequently affected, together with the flux linkage with any measuring coil. Errors in measurements are consequently introduced. Equation (6.12) also shows that, if the medium is a perfect diamagnet (/d, r "- 0), m~= - m . Under the previous convention on the signs of m and m/, this means that the real and the image dipoles are oriented like in Fig. 6.11c. They have equally


304

directed tangential components and oppositely directed normal components. The image effect is therefore to be taken into account both w h e n the sample is placed within the pole faces of an electromagnet and inside a superconducting solenoid.

References 6.1. A. Hubert and R. Sch/ifer, Magnetic Domains (Berlin: Springer, 1998), p. 184. 6.2. EC.Y. Ling, A.J. Moses, and W. Grimmond, "Investigation of magnetic flux distribution in wound toroidal cores taking account of geometrical factors," Anal. Fis., B-86 (1990), 99-101. 6.3. IEC Standard Publication 60404-4, Methods of Measurement of the d.c. Magnetic Properties of Magnetically Soft Materials (Geneva: IEC Central Office, 1995). 6.4 T. Nakata, N. Takahashi, K. Fujiwara, M. Nakano, Y. Ogura, and K. Matsubara, "An improved method for determining the DC magnetization curve using a ring specimen," IEEE Trans. Magn., 28 (1992), 2456-2458. 6.5. H. B611,Handbook of Soft Magnetic Materials (London: Heyden, 1978), p. 64. 6.6. IEC Standard Publication 60404-2, Methods of Measurement of the Magnetic

Properties of Electrical Steel Sheet and Strip by Means of an Epstein Frame (Geneva: IEC Central Office, 1996). 6.7. IEC Standard Publication 60404-10, Methods of Measurement of Magnetic Properties of Magnetic Sheet and Strip at Medium Frequencies (Geneva: IEC Central Office, 1988). 6.8. J. Sievert, "Recent advances in the one- and two-dimensional magnetic measurement technique for electrical sheet steel," IEEE Trans. Magn., 26 (1990), 2553- 2558. 6.9. D.C. Dieterly, "DC permeability testing of Epstein samples with double-lap joints," ASTM Spec. Tech. Publ., 85 (1949), 39-62. 6.10. J. Sievert, "Determination of AC magnetic power loss of electrical steel sheet: present status and trends," IEEE Trans. Magn., 20 (1984), 1702-1707. 6.11. H. Ahlers, J.D. Sievert, and Qu.-ch. Qu, "Comparison of a single strip tester and Epstein frame measurements," J. Magn. Magn. Mater., 26 (1982), 176-178. 6.12. T. Nakata, N. Takahashi, K. Fujiwara, and M. Nakano, "Study of horizontaltype single sheet testers," J. Magn. Magn. Mater., 133 (1994), 416-418. 6.13. IEC Standard Publication 60404-3, Methods of Measurement of Magnetic Properties of Magnetic Sheet and Strip by Means of a Single Sheet Tester (Geneva: IEC Central Office, 1992). 6.14. H. Ahlers, A. Nafalski, L. Rahf, S. Siebert, J. Sievert, and D. Son, "The measurement of magnetic properties of amorphous strips at higher frequencies using a yoke system," J. Magn. Magn. Mater., 112 (1992), 88-90.

REFERENCES

305

6.15. IEC Standard Publication 60404-5, Permanent Magnet (Magnetically Hard) Materials. Methods of Measurement of Magnetic Properties (Geneva: IEC Central Office, 1993). 6.16. D.X. Chen, J.A. Brug, and R.B. Goldfarb, "Demagnetizing factors for cylinders," IEEE Trans. Magn., 27 (1991), 3601-3619. 6.17. H.R. Boesch, Accurate measurement of the DC magnetization of steel using simple cylindrical rods, Proc. Second Int. Conf. Soft Magn. Mater. (Cardiff, UK), 1975), 280-283. 6.18. IEC Standard Publication 60404-7, Method of Measurement of the Coercivity of Magnetic Materials in an Open Magnetic Circuit (Geneva: IEC Central Office, 1982). 6.19. A. Kedous-Lebouc and P. Brissonneau, "Magnetoelastic effects on practical properties of amorphous ribbons," IEEE Trans. Magn., 22 (1986), 439-441. 6.20. One is often interested, from the viewpoint of applications, in the final properties of a specific ring sample, once convenient thermal or thermomagnetic treatments have been carried out on it. Little interest would then be attached to the intrinsic magnetic properties of the material at start and their possible determination using open samples. 6.21. C. Beatrice, private communication. 6.22. A. Aharoni, L. Pust, and M. Kief, "Comparing theoretical demagnetizing factors with the observed saturation process in rectangular shields," J. Appl. Phys., 87 (2000), 6564-6566. 6.23. A. Aharoni, "Demagnetizing factors for rectangular ferromagnetic prisms," J. Appl. Phys., 83 (1998), 3432-3434. 6.24. It can be demonstrated that, whatever the coupled circuits, the coefficients of mutual inductance M21 -- ci)21//1 and M12 = ci)12//2, by which we denote the flux delivered in the coil 2 by a unitary current circulating in the circuit 1 and the flux delivered in the coil 1 by a unitary current circulating in the circuit 2, respectively, are equal (M21 -- M12 -- M). The demonstration of this statement (but not only this specific demonstration) is often called "reciprocity theorem". 6.25. D. Craik, Magnetism: Principles and Applications (Chichester: Wiley, 1995), p. 304.


CHAPTER 7

Characterization of Soft Magnetic Materials

This chapter will review and discuss current methods in the determination of the DC and AC magnetization curves of soft magnets and the related physical parameters. Basically, this means that we will chiefly present applications of inductive measuring techniques to the characterization of laminations, ribbons, and bulk samples, including sintered powder materials. The specific problems associated with soft magnetic thin films will be briefly dealt with in this and in the next chapter. We have highlighted in the previous chapter some general problems regarding the geometry of the test specimen and the flux-closing magnetic circuit. In this chapter, we shall be more specific on the type of materials subjected to testing and the most frequently employed measuring arrangements. In particular, we shall pay special attention to the solutions endorsed by the technical committees responsible for the creation and revision of the magnetic measurement standards. We shall thus consider the testing arrangements employing toroidal, Epstein, single-sheet, single-strip, and bulk rod-like samples. They apply to the conventional conditions where magnetization and field are properly assumed to be in a scalar relationship. If magnetization and field (defined as average macroscopic quantities) are not collinear, further specifications are required in order to provide meaning to the measurement of the magnetization curves. These will be discussed to some extent in association with the testing of laminations under rotational magnetic field, which, although not yet sanctioned by a measuring standard, is the subject of increasing interest in the domain of computation and design of electrical machine cores. In defining and measuring the M(H) relationship in a magnetic material, we must specify whether we are looking at DC or AC properties. Actually, if we are to determine the magnetization curves, we necessarily have to change the strength of the applied field with time. Strictly speaking, we talk of DC curves when this change is accomplished in such 307

308

CHAPTER 7 Characterization of Soft Magnetic Materials

a way that every recorded M(H) point corresponds to an equilibrium stable microscopic configuration of the system. A rate-independent hysteresis loop is, for example, determined when the applied field is changed so slowly that the evolution of the system through successive metastable equilibrium states, occurring by means of Barkhausen jumps, becomes totally independent of the field rate of change. This is a somewhat ideal measuring condition because relaxation effects, due either to eddy currents or thermally activated processes, can make it difficult in practice to achieve a truly rate-independent M(H) behavior. The DC characterization can be accomplished by measuring the stray field emitted by an open sample as a function of H (magnetometric method), but the inductive method on a flux-closed configuration is largely preferred. Two basic inductive measuring procedures can be adopted. The first one, called ballistic or point-by-point method, consists in changing the field in a step-like fashion and determining each time the corresponding flux variations while the system is allowed to relax to a novel equilibrium state. In the second one, called continuous recording or the hysteresisgraph method, the field is slowly changed according to a suitable continuous law. These two experimental approaches do not always provide the very same results, reflecting the awkward definition of DC magnetization curve and hysteresis. When the previous conditions characterizing the DC behavior no longer apply, we fall into the general domain of AC testing. Basically, this implies that for a given induction rate/~, the applied field strength has to compensate for an additional counterfield related to/~, which is associated with energy dissipation phenomena. For the practically relevant case of soft magnetic laminations and ribbons, the chief source by far of energy losses is represented by eddy currents. Three overlapping domains of investigation are normally considered. The first one is of interest for power frequency applications and typically extends up to 400 Hz (the operating frequency of airborne transformers). A medium-to-high frequency region can then be identified, extending up to around 1 MHz, where the role of stray parameters must be duly accounted for in measurements. We eventually deal with the domain of radiofrequencies, where soft magnets (e.g. ferrites, garnets, thin films, and microwires) are finding increasing applications. However, while in the low and medium frequency range it is mainly the high-induction non-linear magnetization regime that is important, the small induction linear behavior is chiefly considered in theory and experiment at high frequencies.

7.1 BULK SAMPLES, LAMINATIONS, AND RIBBONS

309

7.1 BULK SAMPLES, LAMINATIONS, A N D RIBBONS: TEST SPECIMENS, MAGNETIZERS, STANDARDS

MEASURING

7.1.1 Bulk samples Metallic magnetic materials in bulk form are subjected to DC characterization only, because eddy currents already shield the interior of the core at very low induction rates. If the test specimens are shaped as toroids, their dimensions should conform to the rules given in Section 6.1 and the magnetic path length should be calculated accordingly. With ferrites and sintered or bonded metal particle aggregates, AC characterization can also be performed. The winding arrangement follows the usual rules, where the secondary winding is as close as possible to the specimen surface and the magnetizing winding is external to it, both being evenly laid around the core. Given the relatively large cross-sectional area of the test specimen, usually in the range 50-500 mm2; and the small diameter of the wire used for the secondary winding (0.1-0.2 mm), minor correction for the air flux is generally required, even at high fields. In some special cases (for example, when making measurements at high temperatures), the ring sample must be encased in a rigid container and there can be a substantial area included between the specimen and the windings. A correction must then be made for the extra flux detected by the secondary winding. If the cross-sectional areas of the sample and the secondary winding are A and A2, respectively, and the field is H, as determined by the measurement of the primary current, the sample induction B is obtained by subtracting, per unit turn, the flux contribution / ~ - #0H(A 2 - A ) to the measured flux. This can be easily accomplished by calculation, if A 2 is exactly known. Alternatively, a dummy specimen made of an identical empty container and identical windings can be used. Total air flux compensation is achieved in this case by connecting the primary and the secondary windings of tested and dummy specimens in series and series opposition, respectively. The resulting signal is then proportional to the polarization of the material J -- B -/z0H, from which B is easily retrieved if desired. Notice here that if testing at high field strength (i.e. very low permeability values) is performed, further compensation should be devised for the stray axial flux generated by the equivalent circular turn, with diameter equal to the mean diameter of the ring, formed by the primary winding and collected by the equivalent turn formed by the secondary winding (see also Section 6.1). If the magnetizing winding is made of one layer, compensation can be obtained by winding back one turn along the median circumference. Multiple

310


layers should be laid in pairs, with alternate layers wound clockwise and anti-clockwise around the ring. DC magnetic properties of bars, rods and thick-strip specimens, as obtained, for example, by Casting, forging, extrusion, hot rolling, powder compacting or sintering, are generally determined with the use of permeameters, soft magnetic structures of the type shown in Fig. 6.5a realizing a closed magnetic circuit. The yokes in a permeameter are preferably, but not necessarily, of the laminated type. In this case, high permeability Fe-Si or Fe-Ni laminations are employed, which are either U-bent and superposed to form a double-C structure schematically shown in Fig. 7.1 (as shown) or cut, stacked side by side, and assembled with staggered butt joints (Fig. 7.2a). If solid yokes'are used, they should be made of precisely machined soft iron or low-carbon steel. The two basic permeameter arrangements in use today, as recommended by the IEC 60404-4 standard [7.1], are qualitatively illustrated in Fig. 7.1. They both make use of double C-yokes and differ in the way the magnetic field is applied. In the so-called Type-A permeameter (Fig. 7.1a), the magnetizing coil is wound around the specimen, while in the Type-B permeameter (Fig. 7.1b) it is wound around the yoke. The latter solution, which was popular in the version offered by the Fahy permeameter [7.2], is adopted at present by commercial setups [7.3, 7.4]. The minimum recommended specimen length is 250 and 100 mm in Type-A and Type-B permeameters, respectively. Care must be paid to specimen clamping in the yokes in order to minimize the reluctance of the joints. The pole faces must be rectified and coplanar and, for tests on bars and rods, additional pairs of soft iron pole pieces, as shown in Fig. 7.2b, should be employed in order to closely accommodate the test specimen between the yoke pole faces. The flux sensing coil (3), centered on the specimen mid-section, has length between 10 and 50 ram. The experiments show that with typical samples, quite uniform induction is obtained over this length in both permeameter types. Figure 7.3 shows the dependence of B on the distance from the mid-section in a 2 mm thick, 21 mm wide, and 271 mm long soft iron bar (permeability of the order of 103 for H = 103 A/m), as determined by means of a localized few-turn secondary winding made to traverse the length of the specimen [7.5, 7.6]. This figure also shows correspondingly good uniformity of the effective field H, which can then confidently be determined using either a localized probe (i.e. a Hall device) or an H-sensing coil placed on the specimen surface. The Hall method is by far the quickest and simplest although some provision must be made in building the coils for admitting the small sensing head (either transverse or tangential) close to the specimen surface. The earth's magnetic field obviously combines with H in the region occupied by


250 mm

311

~1

(a)

350 mm' ,

I

(b) FIGURE 7.1 Permeameters for the characterization of bulk soft magnets, according to the IEC 60404-4 standard. In (a) the field is applied by means of a solenoid (1) wound on a former around the specimen (2), which is clamped between the two halves of the double-C laminated yoke (type A permeameter). The flux-sensing coil (3) has length between 10 and 50 mm and the tangential field is measured either by means of a Hall probe (4) or a fiat H-coil. In the Type-B permeameter (b) the magnetizing windings (1) are wound around the yoke and the sample (2) can be shorter (down to 100 instead of 250mm in Type-A permeameter). A compensated flux-sensing coil (3) can directly provide the sample polarization J. The field can be measured either by using a Hall probe (4) a fiat H-coil, or a Rogowski-Chattock potentiometer (5). It is assumed here that the yokes are of the strip-wound type, obtained by U-bending and stacking either grain-oriented Fe-Si or Ni-Fe laminations.

312


j

IIiI1~

i-"

~::

i.. ,,j.. 1--1,, I,

-v-

i I I

(b)

(a)

test specim~p,

N2A2 cl

ff=~ (c)

NcAc2 (d)

FIGURE 7.2 (a) Detailed view of a stacked C-yoke comer with staggered butt joints. (b) Pairs of soft polar pieces, housing the specimen (circular or square crosssection) in the region of contact with the yoke pole faces. They ensure a low reluctance path for the magnetic flux. (c) Determination of the tangential field by use of series connected fiat coils placed on opposite sides of the specimen. (d) Coaxial coil arrangement providing automatic air-flux compensation and secondary signal proportional to the sample polarization J. the probe and for this reason the sample is conveniently oriented along the East-West direction. The residual field is automatically subtracted by inverting the applied field polarity and averaging the obtained indications. Use of the Hall method with the Type-B permeameter is totally acceptable, provided there is negligible radial dependence of the field strength over the region occupied by the Hall plate. The two flat coils shown in Fig. 7.2c, placed on opposite sides of the sample and series connected, can be used for the inductive determination of the effective field. Alternatively, two coaxial coils of different diameters connected in series opposition are preferably employed with cylindrical and bar-like specimens. A magnetic potentiometer (Rogowski-Chattock) can also be applied with the Type-B permeameter (see Fig. 7.1b), under the condition of good contact of the coil end faces with the specimen surface and uniform turn density. It should be remarked, in any case, that the objective difficulty of accurately determining low field strengths

313


Type-A permeameter 1.04 O

v

:z-

1.02

Ht(x)/Ht(O)

s" :,, "" e -

l

~" 1.00

0.98

. . . .

-30

I

-20

.

.

.

.

I

-10

I_

. . . .

I

0

'

'

'

'

I

10

. . . .

I

20

'

""

'

'

'

:

30

x (mm)

Type-B permeameter 1.04

/

I

' O

v

I I1'

Ht(x) IHt(O)

,',J

I I /9 I I

~- 1.02

:Z-

v

\

0

I

\

.J~ .i~..--~ ~'~"~~

IK..

v

/

r

"~" 1.00

B(x) /B(O) 0.98

. . . .

-30

,

-20

. . . .

,

-10

....

'

'

0 x (mm)

'

'

I

10

'

'

'

'

I

20

'

'

'

'

30

FIGURE 7.3 Magnetic induction B(x) and tangential field Ht(x) measured as a function of the distance x from the mid-section in a soft iron bar (thickness 2.1 mm) tested in the two permeameter types shown in Fig. 7.1. The displayed quantities are normalized to their values at x = 0. Open dots: H ( 0 ) = 800 A/m. Full dots: H(0) = 8000 A/m. The data are taken from Refs. [7.5] (Type-A permeameter) and [7.6] (Type-B permeameter).

314


(say around a few A/m) makes the ring method more suitable than the permeameter method in the DC characterization of very soft magnetic bulk samples (for example, large-grained very pure Fe). Conversely, the permeameter method, allowing for the application of high fields, is preferentially employed on approaching the magnetic saturation. Concerning the problem of air-flux compensation, we can, as previously discussed for the ring setup, either obtain it by calculation, once we know the cross-sectional areas of winding and specimen and the tangential field Ht (generally assumed to coincide with the internal field H), or by automatic subtraction via coils connected in series opposition. Figure 7.2d provides a cross-sectional view of a secondary coil directly providing the sample polarization J q-compensated coils). An inner winding of turn-area N2A2 is series connected with two outer compensating windings of turn-areas NcA~I and NcAc2, which are, in turn, connected in series opposition. The flux linked with the outer coils, related to the shaded annulus in Fig. 7.2d, is cI)c = N c ( a c 2 - Ad)p,oH and totally compensates the air-flux linked with the inner winding if N2A2--Nc(Ac2-Acl). The flux globally linked with this triple-coil arrangement then becomes ~c - N2AJ, if A is the cross-sectional area of the specimen. It may happen that Nc and Ac2- Ad cannot perfectly satisfy the previous condition. In such a case, one may try to achieve Nc(Ac2 - Acl ) slightly larger than N2A2 and to fine adjust it connecting a resistance in parallel to the compensating coils. The Type-A permeameter can, at least in principle, be arranged in such way that the value of the effective field H is directly obtained by measuring the current il circulating in the magnetizing winding. This is the concept that has led to the development of the compensated permeameters, notable examples of which are provided by the Burrows [7.7] and the Iliovici [7.8] permeameters. What is compensated in these devices is the drop of the magnetomotive force occurring in the magnetic circuit outside the portion of the test specimen covered by the magnetizing winding. If this has length Im and the number of turns is N1, the effective field is then given by H = Nlil/lm. In order to achieve this condition, auxiliary magnetizing windings are employed, normally placed in proximity the yoke pole faces, and their supply current is suitably adjusted. Classical compensated DC permeameters appear rather obsolete nowadays, given the tedious point-by-point operations involved in the current adjustment and the present availability of quick and precise methods for the measurement of the effective field in the measuring region by means of sensors. An improved AC version of the compensated permeameter for use on single strips and sheets will be discussed below.


315

7.1.2 Sheet, strip, and ribbon specimens Soft magnets are applied for the most part in AC devices and for that reason they are generally produced as sheets and ribbons. To characterize them under a closed magnetic circuit configuration, we can, as discussed in Section 6.1, either build ring or Epstein frame samples, or resort to fluxclosure by a means of high-permeability large cross-sectional area yokes. Preparing a test specimen and measuring circuit is a delicate problem because we need to balance the ideal goal of determining the intrinsic magnetic behavior of the material with the practical constraints imposed by the necessity of having a reasonable sample size and geometry, a convenient arrangement of the testing apparatus, and reproducible measurements. Conventional magnetic laminations are usually delivered as 0.5-1.5 m wide sheets, from which testing samples must be cut. Rapidly solidified alloys are instead produced and tested as ribbons of variable width, from 1-2 to around 100 mm, and sometimes as wires. The latter have such a high aspect ratio that demagnetizing field corrections are irrelevant and open sample testing is appropriate, on condition that accurate shielding and East-West alignment of the specimen are provided. Tape-wound ring samples are the usual solution for testing amorphous ribbons. However, tape winding implies the buildup of stresses, half compressive and half tensile, and the magnetic properties of the sample become dependent, through magnetostrictive coupling, on the ring radius R. If Ey is the Young modulus of the material and d is the tape thickness, the maximum strain is ~ m a x - - d / 2 R and the corresponding tensile/compressive stress is O'max =

Eyd/2R.

(7.1)

A 30 ~m thick amorphous ribbon (Ey = 150 GPa) wound on a 2 cm diameter ring is subjected to a maximum stress of the order of 200 MPa. The correspondingly induced average magnetoelastic energy density Erae __ 3 As O.max~ where As is the saturation magnetostriction, can range between some 1 5 J / m 3 in Co-based alloy (As "-" 10 -7) and about 4.5 x 103 J / m 3 in Fe-based alloys (As ---30x 10-6). This brings about a drastic change of domain structure and magnetization curve with respect to the free ribbon, whose intrinsic behavior can possibly be measured only by using an open strip sample and applying the correction for the demagnetizing field (Fig. 6.8). However, we might be interested in the properties of the final ring sample, as they result upon convenient thermal and thermomagnetic treatments. These will generally be different from the properties of the free ribbon, even if the very same treatment sequence is applied in both samples, and, the complete stress-relief being difficult to

316


achieve without incurring some incipient crystalline transformation, somewhat dependent on the ring radius. With crystalline laminations, plastic straining can occur below a certain R value. A 0.23 rnm thick Fe(3 wt%)Si lamination (Ey -- 120 Gpa, yield stress cry --- 200 MPa) should be bent, for instance, over a radius larger than Rmin "" 70 mm in order to avoid permanent deformation. Of course, annealing can relieve both elastic and plastic straining, including the work hardening effect associated with strip cutting from the parent lamination but it can also permanently modify the structure of the starting material (for instance, by increasing the grain size) and, again, the measured properties may not be precisely the ones originally aimed at. An important point in tape winding, and common also to stacking, both in ring and Epstein specimens, is represented by the interlaminar insulation. This is provided by a few micrometer thick coating in the industrial Fe-Si laminations or, simply, by surface oxidation in lowcarbon steels and Fe-Ni alloys. Amorphous ribbons are uncoated and very little oxidized and interlaminar currents could potentially arise, in the cores. A number of studies-have actually shown that such currents do not have relevant effects [7.9], even if tension winding is applied in order to increase the packing fraction because of surface roughness. Tensionwound cores can nevertheless exhibit additional losses with respect to loosely wound or ribbon-coated cores after annealing due to the greatly increased number of shorts associated with bonding of adjacent laminations at contact points [7.10]. Note that shorts can also occur between the stacked Epstein strips if lamination cutting does not leave the strip edges completely burr-free. The accuracy of cutting should also be high regarding the geometrical tolerances, a maximum deviation of _ 1~ for example, being allowed, according to the standards, for the direction of cutting with respect to the rolling direction (RD) in grain-oriented alloys [7.11]. The Epstein test method is a widely accepted industry standard, characterized by a high degree of reproducibility, as shown by intercomparisons carried out by National Metrological Institutes and specialized industrial laboratories [7.12, 7.13]. Indeed, the reproducibility of measurements is central to the acceptance and assessment of a method as a standard because it attaches to the economic value of the material being characterized. For all its merits, including many years of solid experience by laboratories worldwide, the Epstein method has certain drawbacks, making its application difficult or not totally appropriate. For one thing, sample preparation, which can include stress-relief annealing after cutting the 3 0 m m wide strips, is time consuming and expensive. With high-permeability grain-oriented


317

laminations, stress-relief is mandatory, but in case of laser-scribed materials it is basically inapplicable because it would interfere with the physical mechanisms responsible for domain refining. Its use also appears questionable with ribbon-like samples, in particular amorphous alloys, which can be prepared narrower or wider than 30 mm. With large ribbons, longitudinal cutting cannot be envisaged. In addition, property degradation by magnetoelastic effects upon sample insertion in the frame can occur, which is difficult to control and impairs the measuring reproducibility [7.14]. If we also consider that the Epstein method does not provide absolute results, there are good reasons to look at the single plate test method, as schematically envisaged in Fig. 6.5a, as a practical and flexible alternative, where the absolute determination of the effective field is also possible. Of course, some working rule should be agreed on how to provide general consistency to the results obtained with different methods. Different kinds of single-strip/single-sheet testers (SST), such as those employing horizontal single, double, and symmetrical yokes, or the vertical single and double C-yokes, have been investigated in the literature and have been variously adopted in national and international measuring standards. An example of a double horizontal yoke magnetizer, as proposed by Yamamoto and Ohya [7.15], is shown in Fig. 7.4. The yoke frame is obtained by stacking 100 mm wide grainoriented Fe-Si strips, cut along RD, up to a thickness d > 11 mm [7.16] and the field strength is determined by means of a flat H-coil. Horizontal-type yokes lend themselves to quick operation, with automatic insertion and extraction of the sheet sample, and are therefore attractive from the viewpoint of quality control in the plant. Since strip specimens are in general wide (e.g. 200 mm), negligible effects from cutting stresses are expected and laser-scribed laminations can consequently be easily tested. Horizontal yokes with H-coils have found widespread acceptance in Japan and have been adopted in the JIS standards. The asymmetric structure of the yoke in Fig. 7.4 is conducive, however, to a systematic measuring error, as can appear in the vertical single C-yoke, which becomes relevant when the test plate is longer than the frame side (overhang effect) [7.17]. The overhang error chiefly arises because eddy currents are generated in the plane of the lamination by the magnetic flux leaving the sample and flowing into the yoke limbs and vice versa [7.18]. As schematically shown in Fig. 7.5, these currents give rise to an extra field He, which generates a systematic error in the H-coil signal, as well as extra losses, which are reflected in an overhang-dependent magnetic path length Im, if the field is determined through the measurement of the magnetizing current (MC method).

318

CHAPTER 7 Characterization of Soft Magnetic Materials 100 mm

800

500

(a)

testspec[,~n secondary winding

J

.~,

~

g winding

'

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

, . . . . . . .

L . . . . . . . .

L ~H-coil 9

~nn mm

(b)

FIGURE 7.4 Example of horizontal-type double-yoke single strip tester used for measurements on Fe-Si laminations. With this arrangement, relatively wide strips (e.g. 200 mm) can be tested, thereby making the error introduced by cutting stresses negligible. The air-flux compensation can be obtained by means of a mutual inductor (not shown in the figure). The effective field is measured over the region of maximum homogeneity of the magnetization by means of an H-coil (adapted from Ref. [7.15]). The overhang problem is basically eliminated by use of a symmetric horizontal yoke, where the test plate is sandwiched between two identical frames [7.16], or a double-C yoke. Note, however, that the single vertical C yoke is admitted, in spite of its asymmetry, by the MCbased ASTM standards, both with conventional steel-sheet laminations and the amorphous ribbons [7.19, 7.20]. It is also possible to get rid of asymmetry effects by using the double H-coil, as sketched in Fig. 7.5a. The double H-coil solution in SST is generally applied for the sake of accuracy in the measurement of the tangential field. This varies with the distance x from the sheet surface and the true value H can only be obtained in the ideal condition of an infinitely thin H-coil. Figure 7.5b provides an idea, for two different materials tested in a vertical C-yoke, of the variation of the tangential field H(x) on passing from the sheet surface to the inner surface of the magnetizing solenoid [7.21]. Remarkably, a linear increase of H(x) is found, which permits one to


test specimen "~

319

H-coils .~ H

/

He

(a) 15

10

v

!

x v

0

2

x(mm)

4

6

(b) FIGURE 7.5 (a) Eddy currents in the lamination plane associated with the flux flowing into and out of the yoke limbs. The eddy current field H e is the source of additional losses and of a systematic measuring error. This effect, which is equally observed in the horizontal-type yoke of Fig. 7.4, is basically eliminated either by using a synmletric yoke (e.g. vertical double C-yoke) or a double H-coil (adapted from Ref. [7.18]). (b) Increase of the tangential magnetic field with distance x from the surface of the sheet specimen. The effective field at the surface H can be determined by linear extrapolation to x -- 0 of the field strength values measured by two flat coils at different distances xl and x2 (from Ref. [7.21]).

extrapolate to x = 0 the e x p e r i m e n t a l values m e a s u r e d w i t h t w o coils of finite thickness p l a c e d at distances xl a n d x2. We find in p a r t i c u l a r H = H(0) =

x2H(Xl) - -

x 1H(x2)

.

(7.2)

X 2 -- X 1

A l t h o u g h H(Xl) a n d H(x2) can be affected, via the field He, b y the sheet o v e r h a n g , H is not b e c a u s e at the surface H e -- 0. It has b e e n s u g g e s t e d

320


that Eq. (7.2) can still be applied using a single H-coil if two measurements are performed, one of them with the coil lifted to a convenient distance from the test plate surface [7.22]. The H-coil method is, in principle, exactly what we need for the measurement of the magnetic properties of soft magnetic laminations because, by providing the value of the effective field directly, it does not require awkward assumptions regarding the magnetic path length. The condition of homogeneous sample magnetization must, of course, be fulfilled over the region covered by the B and H measuring coils. However, the application of this method to the magnetic measurement standards has been so far limited to Japan (JIS Standard H-7152 [7.23]). It is indeed difficult to envisage its general adoption in the industrial environment for a number of reasons: (1) The signal generated in the H-winding is usually small and prone to disturbances by interfering electromagnetic fields. (2) The turn-area calibration must be performed with maximum accuracy. Consequently, it requires a reference magnetic flux density source, traceable to the standards kept by the National Metrological Institutes. (3) The stability with time of the winding t u r n area, crucial to the measuring accuracy, is critical. It calls for a low temperature coefficient and a rigid non-aging structure. Experience shows that given the requirement of low coil thickness (typically around 1-2 mm), the latter is not easily obtained using a non-metallic former. In addition, the installed coil often becomes inaccessible to non-destructive inspection and adjustment. (4) The signal must be integrated in order to achieve the field H(t). The integrating chain can be the source of further instabilities and possible phase errors, especially relevant with analogic integration [7.24]. Consequently, if our key objective is to achieve excellent measurement reproducibility, besides coming reasonably close to the intrinsic material properties, it is acceptable to base the measurement of the magnetic field on the simple and accurate determination of the current circulating in the primary winding. This is just what we do with the Epstein frame, although we know, as discussed in the previous chapter (see Fig. 6.4), that in doing so we can incur in a systematic error. We have previously mentioned the concept of the compensated permeameter, where, in spite of the inhomogeneity of the magnetic circuit, the effective field on the measuring region is directly determined from the measured value of the primary current. Such a possibility arises because we are able to compensate for the effect of air gap and yoke reluctance through supplementary magnetizing coils, located near the pole faces of the yoke. Such compensation can be made automatic by means of a Rogowski-Chattock potentiometer (RCP) and a feedback circuit, as illustrated in Fig. 7.6 [7.25, 7.26]. If we consider in this figure

'~'i,../ o.o

-0.5 -1.0

X-stacked sheets

/!

.~

-41 ,o . . . . . .

(bl '4~o ........

o""

H (Nm)

i " , , , , ,

. . ,

2~o

.

,,

,i

400

Materials


329

phases. (3) In the (111) frame, three equi-occupied phases result at the end of mode I, which lasts until M / M s = 1/x/3. The remaining part of the process is covered by rotations (mode II). The same results regarding these three high symmetry directions could have been obtained by experiments on disks, cut parallel to the (100) or (110) planes, but for the additional complication related to the correction for the (isotropic) demagnetizing factor. If the magnetization curve along a generic direction is required, there are no equivalent frame structures to exploit, although we can still conceive, at least in principle, completely flux-closed arrangements in three dimensions [7.37], which can be assumed to provide intrinsic curves. In practice, we test crystals shaped as rods or strips, flux-closed at the ends, or disks. The domain wall processes, the rotations, and the equilibrium between the different phases, guided by the internal field, vector combination of the applied field and the demagnetizing field, now become dependent on the geometry of the specimen and the intrinsic curve cannot be retrieved. For sufficiently elongated rods and strips, the transverse demagnetizing coefficient is high enough to make the transverse magnetization component negligible, that is, M and H m (coincident with Ha in samples flux-closed at the ends) collinear. The magnetization curve along whatever direction can be correspondingly calculated, with mode I assumed to occur under zero coercivity, using N6el's theory [7.36]. An example of interest in applications is the one connected with the characterization of Fe-Si grain-oriented laminations in directions different from RD. Because of their high crystallographic perfection, these materials can be basically treated as single crystals. If tested as Epstein strips, as often done in the literature [7.38], they are expected to behave as N6el's rod [7.36] and the resulting magnetization curve beyond mode I can be predicted accordingly. The prediction along mode I instead requires pre-emptive determination of the intrinsic material properties of the lamination along RD and TD (the (100) and

FIGURE 7.10 Magnetic behavior of high-permeability grain-oriented Fe-Si laminations under application of a magnetic field making an angle 0 with respect to the rolling direction. The measurements have been performed under quasistatic conditions by means of a single-sheet tester [7.39]. In the narrow Epstein strip, the polarization J and the longitudinal field Hm are collinear. In the Xstacked sheet configuration (see inset), emulating an infinitely extended sample, the magnetization can acquire a substantial transversal component. (a) Normal magnetization curves for 0 = 30~ (b) Hysteresis loops for 0 = 75~

330


(110) directions in the single-crystal approximation) [7.39]. With larger strips and sheets, like those typically employed in a variety of SST, a certain component of the magnetization transverse to the applied field can arise and the measured M(H) curve becomes dependent on the width of the strip. Note, however, that the curves measured with cutting angles 0 = 0r 0--90~ and, beyond mode I, 0 = 54.7~ preserve their intrinsic character. By cross-stacking the strip specimens, complete flux closure in the plane of the lamination can be obtained [7.40] and the limit of the infinitely extended sample is emulated. We can talk here of intrinsic behavior of the material, which is equally predicted from analysis of the magnetization modes [7.39]. Figure 7.10 shows examples of hysteresis loops and normal magnetization curves measured in highpermeability Fe-(3 wt%)Si laminations, cut at different angles 0, under the Epstein strip and the cross-stacked (X-stacked) sheet configurations. The related measurements are conveniently performed employing one of the previously discussed SST. Recent trends in the development of magnetizers for soft magnetic laminations have favored a comprehensive approach to material testing, where the same setup is employed for measurements under one-(1D) and two-dimensional (2D) fields. These are generated by two- or three-phase supplied yoke magnetizers, which are designed to provide uniform fields, either rotating or alternating along a given direction, within a suitably large gap where the test plate is inserted. The applied field components are generally controlled by means of a feedback system in order to achieve a prescribed time dependence of the two components of the magnetization in the lamination plane (i.e. defined flux loci). These magnetizers, generally known under the name of rotational SST (RSST), have not yet been standardized and have mainly been developed on a laboratory scale. Consequently, in view also of their inherent complexity, there are in practice as many types of RSST magnetizers as the number of laboratories making use of them, as well as no general consensus on their optimal configuration. It is, therefore, not surprising that intercomparison exercises have shown poor reproducibility of the measurements under rotational fields [7.41]. Basically, the RSST magnetizers can be distinguished for the type of supply used (two- or three-phase), the type of yoke (vertical/horizontal, simple/double), and the specimen shape (square/ circular, with or without air gap, cross-shaped, etc.). The whole subject has been discussed quite extensively in the recent literature, especially in a series of lively workshops devoted also to some general problems regarding measurements in soft magnetic materials [7.42, 7.43]. Figure 7.11 provides a schematic view of the horizontal-type RSST magnetizer developed at PTB [7.44]. An alternating/rotating field is applied, by


331

Magnetizing windings

B-coils

sample

i

.........

.............

!

~..~/H-cods

..:i::. < .......................... 480 mm

,-

FIGURE 7.11 The PTB horizontal-type magnetizer for the 2D testing of soft magnetic laminations (upside and cross-sectional views) [7.44]. The laminated double yoke can provide either rotational or unidirectional field in the gap, according to the programmed time dependence of the current flowing in the exciting windings. The sample is an 80 x 80 r e _ I n 2 square, placed at the center of the double yoke and separated from it by a narrow (---1 mm) air-gap. The induction and the field are detected upon a 20 mm wide area, where both quantities achieve acceptable homogeneity. The two orthogonal B-coils are threaded through holes of 0.5 mm diameter. The corresponding multi-turn H-coils are wound on a thin former and placed on the lower sample surface. A couple of crossed RogowskiChattock coils can also be employed in place of the fiat H-coils.

suitably p r o g r a m m i n g phase and a m p l i t u d e of the currents circulating in the m a g n e t i z i n g windings, to a square specimen of 80 m m side, which is placed exactly in the m i d - p l a n e of the gap and is separated from the apex of the w e d g e - s h a p e d pole faces by a n a r r o w air-gap (~-1 mm). The two

332


components of the induction in the sample are detected by means of two orthogonal few-turn windings, which are threaded through 0.5 mm diameter holes drilled at a distance of 20 mm. This is the relatively small region upon which the uniformity of the magnetization suffices to provide acceptable measuring accuracy. Indeed, the notable distance between the pole faces, the leakage of flux in the orthogonal arm, and the non-ellipsoidal shape of the sample all combine to produce relatively poor homogeneity of the magnetization, but for such a limited area. To avoid hole drilling and the related localized work hardening, pairs of needle probes are sometimes used, by which the average value of the electrical field strength between the points of contact is measured [7.45]. Pole tapering and narrow air-gap both imply minimum power requirement in field generation. The air-gap cannot be too small, however, because it helps in achieving good homogeneity and waveshape control of the induction in the sample. By increasing its width, we also obtain that geometrical imperfections of sample and pole faces have little effect. Tapering is also associated with the rapid decrease in the field strength on leaving the sample surface (see Fig. 4.24). Consequently, it requires precise positioning of specimen and H-coils. The horizontal-type double yoke magnetizer is of simple construction and it has been adopted, with more or less significant variations (e.g. flat H-coil vs. RCP, tapered vs. untapered poles), by a good number of laboratories [7.46-7.49]. The more complex vertical double-yoke 2D magnetizer, sketched in Fig. 7.12, is sometimes preferred to the horizontal-type 2D magnetizer because the two orthogonal applied field components are generated by means of two nearly independent magnetic circuits [7.50, 7.51]. It is generally recognized, however, that with both these types of magnetizers it is difficult to control accurately the rotation of the magnetization in strongly anisotropic materials. This typically applies to grain-oriented Fe-Si laminations, which are very soft along RD, but quite hard along the direction making an angle of 54.7~ to RD (that is quite close to [001] and [111] in the individual grains, respectively). A three-phase 2D field source is expected to provide, at least in principle, better control of the flux loci, besides calling for less exciting power in each channel. An example of a 2D magnetizer with threephase supply and hexagonal test plate is shown in Fig. 7.13 [7.52]. An equivalent system is obtained in a simpler way by generating the field with the statoric core of a three-phase induction motor and placing a circular specimen in the mid-plane of the bore [7.53]. Figure 7.14 illustrates an example of the radial dependence of the magnetization in a non-oriented Fe-Si single disk of diameter 140 mm and thickness 0.50 mm, placed in a bore of diameter 200 mm and height 100 mm and subjected to a rotational field [7.54]. It is fair to say that the measured value of the tangential field


333

/ A sample~

(

/

H-coil

;gin FIGURE 7.12 Vertical-type double-yoke magnetizer.

mple A

nd H-sensors

FIGURE 7.13 2D field system with three-phase supplied magnetizer and hexagonal s}anmetry [7.52]. It is basically equivalent to the classical field source obtained by using the statoric core of a three-phase induction motor and a circular sample [7.53].

334

CHAPTER 7 Characterization of Soft Magnetic Materials 140 mm 2.0

.,.,

1.5

.

NO Fe-(3 wt%),a d= 0.50 mm i

oi

B-coils

I

,

~

~- 1.o cr ~

0.s

Disk sample

~ r I

0.0 ~ -60

-40

-20

0 20 x(mm)

40

60

FIGURE 7.14 Radial dependence of the magnetic polarization in a 140mm diameter Fe-(3 wt%) non-oriented Fe-Si disk placed in the bore of a three-phase statoric core of a rotating machine. The characterization of the material under 2D flux loci is performed over the central region of diameter 40 mm.

can, d e p e n d i n g on the actual value of the polarization 1, be s o m e w h a t different from the real one because it varies rapidly with the distance from the lamination surface. It is easily seen, however, that the m e a s u r e d energy loss is, u n d e r the usual condition of thin flat H-coil ( 1 - 2 m m ) or carefully realized RCP, not affected, at least in principle, from the distance of the coil from the sample surface. Let us consider the sheet sample, having demagnetizing coefficient Nd, and i m m e r s e d in a h o m o g e n e o u s applied magnetic field H a . An H-coil placed on the specimen surface will detect the tangential field H = H a - (Nd//Z0). J. The energy loss per cycle and unit volume can be expressed (see also Section 7.3) as W --

H. ~-~ dt,

w h e r e T is the period. Since

f ~ Nd ~ J. ~-~dt dj = /z0

0,

(7.8)


335

f

d

FIGURE 7.15 Rotating field and magnetization in the plane of an open test plate. The applied field H a leads the magnetization by the angle ~a. The field FI1 measured by a flat H-coil at a distance h 1 from the sample surface is H1 = H a - klJ and leads by the angle ~1. If the distance is increased to h 2 > h i , the field becomes H 2 - - H a - k2J , w i t h k 2 < k 1 and the leading angle becomes ~2 < ~1. The vector diagram shows that Ha sin ~a = /-/1 Sin ~1 = /-/2 Sin ~ , which implies that the measured loss does not depend, in principle, on the distance of the H-coil from the test plate surface.

we can equivalently write W --

dj

Ha" - ~ dt.

This is obviously u n d e r s t o o d as due to the demagnetizing field being in phase with the magnetization. It also means that whatever the distance h of the H-coil from the test plate surface, the phase shift between the detected field H(h) and J will change in such a w a y that the integral in Eq. (7.8) will not change at all. Figure 7.15 shows this clearly for the 2D case [7.55], where the vector J trails the m e a s u r e d field H(h) by an angle ~, which d e p e n d s on h in such a w a y that the quantity H(h)sin~(h)), proportional to the loss, is constant. The example reported in Fig. 7.16, which compares the hysteresis loops m e a s u r e d on the same lamination, first as a single strip in a closed circuit, then as a disk in a 2D yoke, both using a I m m thick flat H-coil, further illustrates the point. The loop m e a s u r e d on the single strip can be considered as intrinsic because the demagnetizing field is very small as is the increase of the tangential field versus distance from the strip surface (see Fig. 7.5b). Such an increase is m u c h faster with the open disk sample and the resulting m e a s u r e d loop appears sheared with respect to the previous one although, as expected, with the same area and coercive field. In practice, the previous

336

CHAPTER 7 Characterization of Soft Magnetic Materials N.O. Fe-(3 wt%)Si

1.0

t = 0.35 mm

S

f= 50 Hz

, ,..-

- -;-"

//

0.5 t-v r

o

N ._ i,..

0.0

t~ o t2_

I -0.5

-1.0

single strip

Field (A/m)

FIGURE 7.16 The same non-oriented Fe-Si lamination is tested as a single-strip in a closed yoke and as a 140 mm diameter disk in a 2D yoke (Fig. 7.14), in both cases using a I mm thick fiat H-coil. The corresponding hysteresis loops differ as regards to their shape, because of the correspondingly different dependence of the tangential field strength on the distance from the sample surface, but they have closely similar areas and coercivities.

relationships require that the magnetization in the sample is homogeneous, a condition only partially fulfilled. The property of conservation of the loop area is approximated to a more or less good extent, depending on the sample shape and the type of yoke employed.

7.2 M E A S U R E M E N T OF THE DC M A G N E T I Z A T I O N CURVES A N D THE RELATED PARAMETERS Ferromagnetic materials are complex physical systems, whose real defective structure brings about a manifold of metastable states responsible for stochastic microscopic behavior, hysteresis, and nonlocal m e m o r y effects. There is no simple w a y to define and unambiguously measure the DC magnetization curves and hysteresis loops of magnetic materials. It is understood that these represent the ratei n d e p e n d e n t J(H) behavior, the one related only to the sequence of values attained by the applied field and not to its rate of change. However,

7.2 MEASUREMENT OF THE DC MAGNETIZATION CURVES

337

the very same metastability leading to hysteresis can make the magnetization process prone to thermally activated processes. The method by which the applied field is changed with time might then interfere with the path followed by the system in the phase space and make somewhat elusive the concept of rate-independent magnetization process. Fortunately, in many practical soft magnets and at temperatures of interest for applications, thermally activated relaxation processes (after-effects), either due to diffusion of soluted atomic species (for example interstitial C atoms in Fe) or caused by thermal fluctuations, have little or negligible effect and a true rate-independent magnetic behavior of the material can be reasonably approached. Nevertheless, the intrinsic stochastic character of the magnetization process and, in metallic materials, residual eddy-current related relaxation effects can frequently cause, a certain lack of reproducibility of the J(H) curves, especially at low and intermediate polarization values. A conventional distinction between methods for measuring the DC magnetization curves is based, besides the way in which the time variation of the applied field is imposed, on how the sample magnetization is determined. We talk of magnetometric methods when J is induced from the measurement of the stray field generated by the open sample and of inductive methods when it is obtained by integrating the flux variation ensuing either from a variation of the relative position of open sample and sensing coils (vibrating/rotating sample, vibrating/rotating coil, extraction method) or from a variation of the applied field, both with open and closed samples. We have previously stressed, however, that soft magnets are seldom tested as open samples. Applied and demagnetizing fields can have very close values, the latter being also spatially nonuniform in ordinary test specimens, and the precise determination of the effective field is difficult. A notable exception, discussed in Section 7.1.3, is found with the two-dimensional testing of laminations, where methods for applying a rotational field with complete flux closure in two dimensions (for example, using cross-shaped specimens) do not ensure acceptably uniform magnetization in the sample [7.56]. The J(H) behavior of soft magnetic materials is then mostly determined by the inductive method using closed magnetic circuit configurations.

7.2.1 M a g n e t o m e t r i c m e t h o d s Among the open sample methods, the ones based on the measurement of forces and torques are rooted in the early history of magnetic measurements. An emblematic example comes from the experimental

338


setup developed by Lord Rayleigh in experiments on the low-field behavior of iron [7.57]. At the time there was debate as to whether the response of iron to feeble fields had a threshold or, as already hinted by Maxwell, it had first linear (elastic) then non-linear character. Lord Rayleigh was able to demonstrate that not only the initial susceptibility had a finite value, but that the magnetization depended quadratically on the field (the Rayleigh law). He was able to do so by means of a highsensitivity torque magnetometer, based on the mechanical action exerted by the stray field emerging from the end of the magnetized sample, a straight piece of iron wire, on a suspended needle. A null reading method was realized, where the needle was kept in its rest position by the compensating action of the field generated by a coil, coaxial with the wire, connected in series with the magnetizing solenoid. A measurement of this kind is characterized by a degree of sensitivity that is still challenging for present-time conventional inductive DC measurements on closed samples. Sensitivity is indeed the landmark property of the force methods that are frequently adopted for measurements on weakly magnetic materials (for example, with m o d e m versions of the Faraday balance) or the determination of very small magnetic moments (for example, by use of the alternating force gradient magnetometer), as we shall discuss below. However, if we are required to characterize the conventional bulk, sheet, or ribbon soft magnetic materials, inductive measurement methods, using the test specimens and magnetizers discussed in Chapter 6 and Section 7.1, are the rule. The magnetometric method can directly provide the total magnetic moment of the sample if this is placed at such a long distance from the field-sensing probe that it behaves as a dipole. At short distances, it is possible to relate the measured stray field with the material polarization by analytical formulation if the sample is ellipsoidally shaped. In all cases, the earth magnetic field and the field generated by the magnetizing winding contribute to the field sensor reading and some way must be found to suppress their interference. Remarkably, the magnetometric measurement techniques can find practical application as zero magnetization detecting methods. If we drive the sample, having regular ellipsoidal or cylindrical shape, to magnetic saturation and bring it back along the recoil curve, we can reveal the passage through the demagnetized state by sensing the zero stray field condition. At that moment, the applied field coincides with the coercive field, which is then measured, more easily and quickly than by determining the whole hysteresis loop and finding the passage of the curve through J = 0. Figure 7.17 illustrates recommended solutions for the magnetometric measurement of the coercive field in open samples. In the arrangement

7.2 MEASUREMENT OF THE DC MAGNETIZATION CURVES test specimen

~./'///////////////////////////~/////////////~ ~////////,,~

~

/

--t- .............. ,i~*~"~i;~ ................ ............. i ~/'/!!1/1111111/11111111111/I//!!1111/!/11/11/~,

339

field sensor

~,'///////////////////////////~ ~////////////////'~

.............

ti

(a) t East-West Hall plate

field sensors__._~ ...I

~i~";'~ .. ~.1,

i (bl

, (c)

FIGURE 7.17 Coercive field measured by detecting the zero stray field condition in open samples upon recoil from the saturated state. The symmetry of the arrangement makes the sensors immune to the field generated by the magnetizing windings. (a) The field sensor (Hall plate or fluxgate probe) is placed exactly at mid-point between two identical solenoids, one of which contains the test specimen. (b) Stray-field sensing is obtained either with a Hall plate, a s ~ e trically placed near the end face of the sample, or a coil vibrating along the axis of the solenoid. (c) Differentially connected probes are symmetrically placed outside the solenoid and detect the radial field emanated by the sample, which is placed off-center. Shielding against the ambient field is required if low-coercivity materials are to be tested.

s h o w n in (a) [7.58], the test specimen is placed inside one of two identical solenoids, which are designed to provide sufficient field strength to bring the material to saturation. Empirically, one can confidently assume that the material is, to all practical effects, saturated w h e n a 50% increase of the magnetizing field brings about less than 1% increase of the m e a s u r e d coercive field. The field sensor, a Hall plate or a fluxgate probe, is placed exactly at mid-point, where the fields generated by the two solenoids mutually neutralize. Once the specimen saturation is attained, the field is slowly decreased by decreasing the supply current i, which is then increased in the opposite direction till zero field reading is achieved. If this condition, corresponding to the vanishing of the stray field generated by the sample, is reached with a magnetizing current ic, the coercive field

340


is obtained as Hc = kHic,

(7.9)

if kH is the constant of the magnetizing winding. Notice that it is not required that the sample has ellipsoidal shape in order to define a demagnetized state, provided it is homogeneous and the applied field is uniform over its volume. Cylindrical test samples are normally used, for which the measured coercive field will correspond to a state of zero volume-averaged polarization. Of course, a coercimeter working on this principle can be employed for testing both soft and hard magnets, the latter case suffering, however, from obvious limitations in the available field strength. For soft magnets, measures must be taken to prevent the effect of the ambient magnetic field, which can affect both the field sensor reading and the sample magnetization. The simplest way to deal with this problem is to align the solenoid along the East-West direction and to make two measurements with inverted currents in the solenoids. For coercivities below 40 A / m , shielding against the environmental magnetic field is prescribed by the pertaining standard [7.59]. Active screening by triaxial Helmholtz coils or an other suitable combination of windings surrounding the measuring apparatus [7.58] is preferred to the use of magnetic shields made of high-permeability alloys because of the distortions introduced by the coupling of the test sample with the shield (image effect, see Chapter 6). The sensor in Fig. 7.17a is pretty far from the specimen and lacks sensitivity. The arrangements shown in Fig. 7.17b and c are then possibly adopted [7.59]. In (b), a Hall plate is placed horizontally at one of the ends of the sample so that it detects only the stray field component normal to the axis of the solenoid. Alternatively, a vibrating axial coil, again insensitive to the applied field, can be used in order to detect the zero stray field condition [7.60]. Finally, two differentially connected, ambient field compensating flux sensing probes are used in (c), symmetrically placed immediately outside the solenoid. The radial component of the stray field emanating from the sample is detected in this way. To achieve good sensitivity, the sample is placed offcenter. In all cases, an elongated sample (say with ratio length/diameter of the order of 5 or higher) is preferred because it is more easily saturated and the uncertainty in the coercivity determination due to shape effects is not significant.

7.2.2 I n d u c t i v e m e t h o d s When talking of DC (or, more appropriately, quasi-static) characterization of soft magnets with inductive methods, we tacitly assume that we are


341

exploring the J(H) relationship in such a way that time-dependent effects are irrelevant. We have previously mentioned, however, that conceptual and practical difficulties may arise in certain cases where various relaxation effects take place, having time constants comparable with the measuring times. When looking at the accuracy and reproducibility of the measurement of the DC hysteresis loops and magnetization curves, we should be aware of these difficulties, which often combine with relevant stochastic effects (Barkhausen noise), associated with the discrete character of the magnetization process. There are two typical ways to determine the magnetization curves under quasi-static conditions: (1) The magnetizing field strength is changed in a step-like fashion and the curves are obtained by a point-bypoint procedure. (2) The magnetizing field is changed in a continuous fashion, as slowly as reasonable to avoid eddy current effects (hysteresisgraph method). Ideally, the two methods should lead to the same results, but differences are often found. The point-by-point inductive determination of the magnetization curves was originally introduced to overcome many practical drawbacks in the torque method, which include the difficulty in performing absolute measurements, the need to use open samples, and the sensitivity of sample and sensing devices to the external fields and their fluctuations. What is detected with this technique is the transient voltage induced on a secondary winding by the step-like applied field variation &Ha~ which is integrated over a time interval sufficient to allow for complete decay of the eddy currents in order to determine the associated flux variation &~. Since we deal with flux variations, we always need to define a reference condition. This is normally identified with either the saturated or the demagnetized state. The latter can be attained either by bringing the sample beyond the Curie temperature and letting it to cool down in the absence of external field or by applying an alternating field (with no offset) whose amplitude is progressively decreased to zero, starting from a peak value so high as to attain technical saturation. The hysteretic behavior of the material also requires that the field history following the achievement of the reference state must be perfectly controlled. It is not uncommon that field transients are inadvertently applied (for example, switching off and on power supplies), subverting the prescribed orderly sequence of field values and leading to false results. If, after thermal demagnetization, a monotonically increasing field is applied, the material is brought along the so-called virgin magnetization curve. If this is done after cyclic field demagnetization, the initial magnetization curve is obtained. In some permanent magnets (e.g. nucleation-type magnets), these two curves can be somewhat different, because of different domain wall populations, and only

342


after thermal demagnetization can one confidently assume having achieved a reference state. In soft magnets, only minor differences can possibly occur, provided no complications arise from after-effects, and thermal demagnetization is seldom performed. An example of demagnetization process by decrease of cyclic field strength is presented in Fig. 7.18. Note that that the curve actually obtained in most cases and universally exploited in designing the magnetic cores of electrical machines and devices is the normal magnetization curve. This is defined as the locus of the tip points of the symmetric hysteresis loops extending from the demagnetized state to saturation. Again, for all practical purposes, the normal magnetization curve in soft magnets coincides with the initial magnetization curve. Let us now discuss how the normal magnetization curve and the symmetric hysteresis loops can be measured, under quasi-static conditions, by means of the previously introduced methods. Figure 7.19 shows the scheme of principle of the setup currently envisaged, according to the recommendations of the pertaining standards [7.1, 7.61], for the application of the point-by-point method. The primary circuit is designed in order to provide a discrete and defined cyclic sequence of field values through a stabilized DC power supply and a combination of two switches and two rheostats. In particular, the switch S~ is used to

1.5. GO Fe-(3 wt%)Si 1.0'

i

0.5i A

o.o, -0.5 -1.0

! 1' S

-1.5 -40

-20

0 H (A/m)

20

40

FIGURE 7.18 Demagnetization of a grain-oriented Fe-Si lamination by application of an alternating field of progressively decreasing amplitude at a frequency of 10 Hz.


,

343

I PG ]

B (T)

,

s

O S'

P

~'.S~'"

(G, p, FIGURE 7.19 Setup for point-by-point determination of normal magnetization curve and symmetric hysteresis loops in a closed sample (ring, Epstein frame, SST). The magnetizing current is provided by a power supply or rechargeable batteries. It can be inverted by the switch $1, changed in a step-like fashion using the switch $2, and regulated by means of the rheostats R1 and R 2. The normal magnetization curve is determined keeping the switch $2 closed, regulating R1 and switching $1 back and forth. The points belonging to the hysteresis loop are found, after having fixed the R1 value (that is the (Hp, Bp) coordinate), by regulating R 2 and acting on the switch $2. A supplementary circuit supplying the winding N3 c a n be used to generate a bias field Hb. By performing a demagnetization procedure at different values of Hb, the anhysteretic curve is obtained. Note that the magnetizing winding is, here as in all testing arrangements discussed in this book, uniformly distributed along the test specimen, in order to ensure full coupling with the secondary windings.

invert the direction of the magnetizing current and the switch S 2 to p r o d u c e step-like changes of its amplitude, according to the values set for R1 and R 2. The p o w e r s u p p l y is conveniently used as a voltagecontrolled current source, which ensures safe behavior u n d e r switching

344


in the output circuit. With typical sample arrangements, as recommended by the standards, and ensuing reactances, time constants in the current vs. time curves of the order of a few ms at most are expected. If, as in the example shown in the figure, a closed sample configuration with defined magnetic path length is realized (i.e. ring specimen, Epstein frame, SST), the field and induction values are determined by means of an ammeter in the primary circuit and a fluxmeter connected to the secondary winding, respectively. If the permeameter configuration is adopted (Figs. 7.1 and 7.4), either a Hall device or a second fluxmeter connected to the H-coil is employed for field reading. In the latter case, only field variations are determined and current reading is still required for setting specified field strengths. The fluxmeter calibration is done in the simplest way by using the same current source and a calibrated mutual inductor, as schematically illustrated in Fig. 5.8, and provision must be made for minimization and correction of drift. Crucial to the measuring accuracy of the magnetic induction is the use of a regular specimen, which is normally the case with industrial products, with precisely determined cross-sectional area A. With sheets and ribbons, the direct measurement of thickness and width is not recommended. One should instead calculate the value of A from knowledge of the sample mass and the material density. Preliminary to any measurement is sample demagnetization, which is carried out starting from a suitably high value of the magnetizing current and decreasing it in a continuous and slow fashion, while switching $1 back and forth (that is, inverting each time the current direction). The switch $3 is kept closed during this operation to maintain the flux integrator at zero. To find the normal magnetization curve, the current is initially increased from zero to a low value il, then $1 is inverted several times to achieve a steady cyclic state between the symmetric points (H1, B1) and (-H1, -B1). Once the cyclic stabilization is reached, $3 is thrown open and the flux variation A~I = 2N2AB1, where N2 is the number of turns of the secondary winding and A is the sample cross-sectional area, is measured each time the current is reversed. After the point (H1,B1) is determined by making the average of the two readings obtained from back and forth reversals, the current is increased to a suitable value i2 and the previous operation is repeated to find the novel point (H2,B2) , and so on. Of course, each time a step-like field variation AHa of the field is imposed, the corresponding flux variation A~ will be fully established over the sample cross-section with a certain time delay because of the rise and decay of the eddy-current generated counterfield. A classical calculation permits one to estimate the associated time constant ~. For example, in a lamination of relative


345

permeability/Zr, conductivity ~r, and thickness d, we obtain 7 " - /d,0/d,rO'd2/8,

(7.10)

which is found to be sufficiently small for making drift problems negligible with conventional fluxmeters. In the quite limiting case of a 10 m m thick pure Fe slab of relative permeability P'r- 103~ Eq. (7.10) provides ~'---0.15 s. Substantial immunity to drift (under the proper sequence of field steps) is the basic reason for the persisting interest in the point-by-point method (also called the ballistic method because of the earlier use of ballistic galvanometers as integrating devices), in spite of the apparent complication and tediousness of the measuring procedure. Notice that the noise developed because of arcing between switch contacts during the reversal of current could be a source of error in the fluxmeter reading. Mercury switches in place of knife switches are recommended for arc suppression. Once the cyclic state between a defined couple of symmetric points (Hp, Bp) and ( - H p , - B p ) has been stabilized, the point-by-point determination of the associated hysteresis loop can proceed [7.62]. We notice that having fixed the value of al and reached the upper tip point (Hp, Bp), the magnetizing current can be decreased from the peak value ip to zero, first by opening the switch $2 and increasing the value of R 2 and then by opening the switch $1. The remanence point R is reached in this way. If $1 is now closed in the reverse direction and R2 is decreased to zero, the lower tip point ( - H p , - B p ) is attained and we conclude that the whole hysteresis loop can be traversed by acting in the right sequence on $1, $2, and R 2. This is obviously done in a step-like fashion, but it is complicated by the fact that it is not convenient to run across the loop from the upper to the lower tip points by successive field steps AHa, while determining and summing up the corresponding flux variations A~. The uncertainty of the individual readings due to drift would sum up at each step, with the additional problem that if a fine subdivision of steps is required, sensitivity and noise problems can further impair the measuring accuracy. Consequently, the field step sequence is devised in such a way that any point on the loop is determined with reference to the tip points P and V. Two different sequences are followed to determine the points (like Q) included between P and the remanence R and the points (like S) included between remanence and the lower tip point P~. Consider starting, after stabilization of the cyclic state, from point P, which corresponds to the condition: switch $2 closed, switch $1 on the up position. R 2 is regulated to a convenient value, then $2 is opened, thereby generating a sudden decrease AHpQ of the field. The corresponding flux variation &(I)pQ is

346


recorded and from the calculated induction variation ABpQ =

A~pQ/N2A

(7.11)

the point of coordinates (Hp - AHpQ, Bp - ABpQ) is obtained. To improve the measuring accuracy and account for minor asymmetries in the measuring system, the operation is repeated starting from the tip point pi and the actually considered variation A(I)pQ is the average of the two readings (ABpQ Jr- ABp,Q,)/2. To find a novel point closer to remanence, the resistance R 2 is increased up to a point where the remanent induction BR at point R is attained by simply opening the switch $1. The points (like S) belonging to the half loop portion going from remanence to P~ are found, again starting from P, in two steps, after having regulated R 2 to a convenient value as usual. First, $1 is opened and the flux variation A(I)pR from P to R is measured. Then we open $2 and close $1 in the reverse direction, so that an additional negative field step is generated and the variation A~RS from R to S is detected and measured. The induction variation will be ABps -~ (~(I)pR q-

A~Rs)/N2A.

(7.12)

Again, ABps will be re-determined starting from the lower tip point P~ and the average of the two readings will be taken. The hysteresis loop measurement with the ballistic method is lengthy and involved, and, even if the previously described sequence can be at least in part made automatic using a programmable bipolar power supply as current source, it is scarcely suited to present-day industrial requirements. On the other hand, the point-by-point determination of the normal magnetization curve is accurate and simple to make, more so than with the continuous recording method where we have to determine a sequence of symmetric hysteresis loops and recover their tip points with the usual problems of control and compensation of the drift of the flux signal. Alternatively, the initial magnetization curve can be measured immediately after demagnetization by applying a field ramp, but drift may introduce uncertainties on the measured induction at high fields, while heat dissipation may pose substantial limits to the maximum achievable magnetizing current. Good accuracy is instead demonstrated in the high field region by the ballistic method due to the virtual absence of drift and minor Joule heating effects in the primary winding thanks to the very short integration time (Eq. (7.10)). Examples of normal magnetization curves detected in this way up to saturation in nonoriented Fe-Si laminations are shown in Fig. 7.20a. It is equally simple to apply the point-by-point method in the determination of the anhysteretic


347

GOFe-(3wt% ,,,~~

2.0 !N_OFe-Si!iminati'on__s - 13' ~.'_~

'"',-..., .....

(Hb, Jb)

/,

~lnl1"5 i Fe-(3"[~:/c 1 /,,,~LFe.(1 i~~ ~"~ ........ ........... wt%)Si--t

,/

-1 -1 (a)

;o3

io, ....

H (A/m) (b) 2.0 Anhysteretic

-60 -40 -20

0

20

40

60

H (A/ m)

o"

,.i~

o.5

0.0 ,

(c)

0

20

~

H (A/m)

40

60

FIGURE 7.20 (a) Normal magnetization curves J(H) in low-Si and high-Si non-oriented laminations determined by means of the point-by-point technique (ballistic method). The B(H) curve is also shown for the Fe-(1 wt%)Si alloy (dashed line). It coincides with the J(H) curve up to about 1.9 T. (b) Demagnetization process under bias field in a grain-oriented Fe-Si lamination and resulting anhysteretic curve (full dots). (c) Low-field behavior of normal and anhysterestic magnetization curves in the same lamination, as obtained by means of the ballistic method.

curve (or ideal magnetization curve). A n y point (Hb,Jb) of this curve is obtained by applying a bias field Hb and carrying out the demagnetization process a r o u n d it [7.63]. To apply Hb, a s u p p l e m e n t a r y w i n d i n g can be used, as s h o w n in Fig. 7.19, in association with an adjustable DC source and an inductance connected in series, which has the role of decoupling

348


the supplementary circuit from the imposed magnetization transients. From the physical viewpoint, it is assumed that the anhysterefic state (Hb, Jb) associates to a given field Hb the polarization value Jb realizing the condition of absolute minimum of energy, that is, the condition of thermodynamic equilibrium towards which the system would drift by thermally assisted processes if given the time to do so. In other words, it can be stated that the anhysteretic curve would characterize the response of the material to an applied field if only reversible processes could occur. It therefore provides a measure of the internal demagnetizing fields, provided rotations are not significant. It also represents an important piece of information in the physical modeling of magnetic hysteresis [7.64], besides having practical appeal in a number of applications (for example, in analog magnetic recording) [7.65]. An example of biased demagnetization ending in, a point (Hb, Jb) on the anhysterefic curve is shown in Fig. 7.20b. Figure 7.20c illustrates the significant difference existing at low fields between the normal and the anhysteretic curves, which at high fields, however, with the ending of the domain wall processes, become coincident. Incidentally, it also demonstrates that a demagnetizing procedure carried out on a soft open sample with relatively low demagnetizing coefficient in the presence of an unrecognized external field (for example, the earth's magnetic field) can actually magnetize the sample! After the demagnetization under a given bias field Hb, carried out as previously described for the unbiased condition, is completed, $1 is opened. The resistance R1 is then regulated to a low value such that when $1 is closed again in the up position, the current in the primary winding jumps to a high positive value, sufficient to generate, in combination with the bias field, a near-saturating field Hp. The ensuing flux variation, corresponding to a polarization jump of amplitude Jp - Jb, is recorded. Since the point (Hp,Jp) also belongs to the normal magnetization curve, which is independently determined, Jb is immediately obtained. By repeating this procedure for a conveniently large number of HB values, the whole anhysterefic curve is achieved. In its simplest realization, the continuous recording method employs a magnetizing current source, realized with a function generator and a power amplifier, and a flux measuring device connected to the secondary winding. The function generator is typically set to provide a triangular voltage waveform, with frequency as low as reasonable for reliable signal detection and handling. At such a low frequency, the combination of small signal and long integration times impose a tight control on the drift in the secondary circuit. Stable electronic components must be employed so that even if there is some residual drift, it can be corrected after analog-to-digital (ADC) conversion by linear numerical


349

compensation over a full period. In addition, the impedance of the primary circuit is mostly determined by the resistance of the winding and the current monitoring resistor so that due to the non-linear behavior of the magnetic core, the induction derivative in the sample has uncontrolled shape. The question on how low the magnetizing frequency should be in order to talk confidently of DC magnetization curves and hysteresis loops is intertwined with the problem of the control of the magnetization rate. If we exclude the case of insulating or near-insulating materials, like ferrites, and do not for the time being, consider the complications arising from diffusion or thermal fluctuation after-effects [7.66, 7.67], what we need is to drive the magnetization in the sample at a sufficiently low speed to avoid eddy current effects. This can be a demanding requirement, especially in bulk specimens. For example, let us make a rough estimation of the eddy current field by a classical calculation in a lamination of thickness d and conductivity cr subjected to the rate of change of the induction/~. We find in the lamination mid-plane

crd2 Heddy --- - ~ / ~ .

(7.13)

For a I m m thick iron sheet, magnetized at a frequency of 0.1 Hz between + 1.5 T, Eq. (7.13) provides Heddy = 0.75 A/m, which can be appreciated in large-grained good purity samples. In a rod sample of diameter D, the calculation provides Heady--crD2/~/16 on the longitudinal axis. For a 1 0 m m diameter iron cylinder cyclically tested between +1 T, it is required that the magnetization period is longer than 250 s for Heddy to become lower than I A / m . Notice that the time required to achieve, in the same sample, 99% decay of Heady when the same flux reversal is obtained by the ballistic method can be estimated (Eq. (7.10)) to be around I s or lower. This would be a good reason for adopting the point-by-point method in DC bulk sample testing, but, as we shall see later, it may occur that the specific nature of the magnetization process (e.g. domain wall nucleation vs. domain wall displacement) calls instead for the application of the continuous recording method. In the continuous recording method it is often required that/~ is held constant. Besides being an obvious reference condition for the investigation of the magnetization process (for example, with some further constraints, in Barkhausen noise experiments [7.68]), a controlled constant magnetization rate permits one unambiguously to define and minimize, according to Eq. (7.13), the role of eddy currents. Figure 7.21 reports the hysteresis loop determined between + 1.7 T in a 0.30 m m thick grainoriented Fe-Si lamination at a frequency f = 0.25 Hz. At such a low

350


1.5

GO Fe-(3 wt%)Si f= 0.25 Hz

~"~ f ~- /

1.0

:| /;

0.5 #~,

:://i

o.o

H

-0.5

V\

'I

-1.0

.

.

.

.

.

.

.

.

t k

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,

'

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I]' ~ t " ~ ( t )

-1.5 -80 '

(a)

'

'

-60 '

|

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'

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|

9

.

.

.

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9

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.

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.

.

.

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.

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.

.

.

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jl

(b)

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0

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I!

.

0 20 H (A/m)

f

0.0

' . . . .

~IL,,

,,,,i I r'.~, ~ . "

0.5

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9

. . . .

110 Time (s)

!

1.5

. . . .

2.0

FIGURE 7.21 Quasi-static hysteresis loop in a grain-oriented Fe-Si lamination measured at the frequency f = 0.25Hz under two different conditions: (1) Constant polarization rate of change (J = 1.7 T/s, solid line). (2) Constant field rate of change (H - 80 A / m s, dashed line). By the first condition, we closely approach the rate-independent (DC) hysteresis loop. In the second case, the additional dynamic loss contribution brings about enlargement of the loop, depending on the instantaneous value of the induction rate. The corresponding time dependence of dJ/dt over a half-period is shown in (b).

7.2 MEASUREMENTOF THE DC MAGNETIZATION CURVES

351

frequency, we can approach the rate-independent loop if B is kept constant. In the example reported in the figure we can estimate, using Eq. (7.13), Heddy ~'~ 0.04 A/m (vs. the coercive field value Hc --6.1 A/m) for the constant induction rate B = 1.7 T/s. Lack of control of/~ gives rise to additional rate-dependent loss contribution and the loop shape and area are modified. This is illustrated in Fig. 7.21a by the enlargement of the hysteresis loop occurring when, instead of the polarization, the time dependence of the applied field is controlled (dashed line). The sharp variation of the induction derivative along the loop (Fig. 7.21b), dictated by the strongly non-linear response of the material, gives rise to an additional dynamic loss contribution. Of course, any controlled B(t) waveshape (for example, sinusoidal) can lead to the rate-independent hysteresis loop provided it is always low enough to satisfy the condition that Heddy is negligible with respect to Hc. Another detrimental effect of uncontrolled B(t) behavior is apparent in Fig. 7.21b, where the peaked shape of the secondary signal strains the dynamic range of amplifiers and A / D converters, resulting in distortions and poor reproducibility of results. This problem is typically met in the characterization of circumferentially field-treated cores made of extra-soft materials (for example, amorphous, nanocrystalline, or permalloy ribbons), which exhibit near-rectangular hysteresis loops. Hysteresis loop tracers endowed with control of the induction rate are nowadays chiefly based on digital methods. They typically work on the principle of generating the magnetic field waveform by digital means ensuring the desired B(t) (i.e. J(t)) time dependence, besides handling the measuring procedure and recording of data by means of a computing unit. Systems employing analog feedback are still in use in some cases and sometimes preferred, under the condition that the employed electronic components have excellent thermal stability, where real-time control of the magnetization is important. Analog negative feedback in DC and lowfrequency hysteresis loop tracers is actually made difficult, as previously stressed, by the small value of the induced signal and the resistive nature of the primary circuit, which call for high amplification of the feedback chain and make the accurate control of the drift signal difficult. A number of apparatuses have been proposed in the literature, based either on the control of the flux derivative [7.69, 7.70] or the flux itself [7.71, 7.72]. Figure 7.22 provides a schematic description of the high-sensitivity, highstability analog electronic loop tracer holding constant induction derivative (i.e. triangular induction waveshape) developed by Mazzetti and Soardo [7.69]. The control of the magnetization rate in this device is accomplished by comparison of the flux derivative in the sample, detected by means of a separate feedback winding N3 and amplified by

352


~H(0 9

V'-"

b

1

---

Oomara,orp FIGURE 7.22 Block diagram of very low frequency hysteresis loop tracer with analog feedback imposing constant rate of change of induction dB/dt. Feedback is accomplished by comparison of the dB/dt signal with a reference rectangular waveform signal provided by a voltage comparator circuit and integration of their difference. Since the gain on the integrator I2 is very high, such a difference can be kept vanishingly small. Use of a high-performance low-noise low-drift DC amplifier A2 in the B chain is mandatory for good measuring accuracy over integration times of several minutes (adapted from Ref. [7.69]).

the low-noise DC-coupled amplifier A1, with a defined rectangular waveform generated by a voltage comparator circuit. The amplifier bandwidth is at least of the order of a few kHz to ensure stability in the closed loop feedback operation. The transitions between the two states of the comparator are driven by the passage of the output of the integrator I2 through two fixed values, which, for a symmetric loop, are equal and of opposite sign. I2 has a very large gain and the difference between the inputs a and b tends to vanish correspondingly. This implies that the voltage at the output of the DC amplifier A~ becomes constant and so does dB/dt, whose actual value can be changed by varying the resistance Re. The use of the extremely low-noise low-drift amplifier A2 in the secondary circuit is the key to good accuracy of the B(t) measurement over long periods. A drift value lower than 10 -8 W b / s is obtained at the integrator output, which amounts to an uncertainty lower than 1% with a flux rate of 10-6 Wb/s. A peculiar property of the devices accomplishing the control of the magnetization rate via negative feedback is that they can interfere with the microscopic mechanisms of the domain processes, to an extent


353

depending on the nature of the material, the b a n d w i d t h of the feedback chain, and its gain. It is observed in such cases that the Barkhausen noise is partially suppressed in the secondary winding to appear in the magnetizing current [7.71]. A classical example is represented by the so-called re-entrant loops, as typically observed in picture-frame Fe-Si single crystals, brought back and forth under tightly constant magnetization rate between the two saturated states [7.73], and in extra-pure, largegrained iron rings, as illustrated in Fig. 7.23. That the loop must be of re-entrant type follows from the fact that the nucleation of the domain walls requires a higher field H~ than the field Hc needed to drive the walls through the pinning centers in the material. It is clear that a different loop shape w o u l d be obtained instead using the ballistic method, by which

Purified iron

1.0

f

T = 1080 s

0.5

"0.... 4 I

0.0

I

-

I I I I I

-0.5

J

-1.0 , , , , l , , , l l , , , , l , , l ,

-20

-15

-10

-5

,

i

,

0

I

I

I

5

10

15

20

H (Nm)

FIGURE 7.23 Initial magnetization curve and hysteresis loop determined on hydrogen-purified Fe ring specimen 7 mm thick at constant rate of change of polarization by means of the loop tracer exploiting analog feedback shown in Fig. 7.22. The loop is traced in a time T -- 1080 s, in order to avoid eddy current effects, and displays a re-entrant shape, thereby revealing the existence of a threshold field Hn for the reversal of the magnetization, which is higher than the coercive field He. The feedback control lowers the magnetic field before reversal takes place and permits one to determine the value of Hc. In the absence of feedback, the hysteresis loop is expected to follow the trajectory described by the dashed lines (taken from Ref. [7.69]).

354


we would be unable to define the central portion of the loop, or applying a triangular field waveform. In the latter case, switching occurs once Hn is attained and from that point the magnetization reversal proceeds at a rate J(t) oc (Hn - Hc(t))~ where Hc(t) is the instantaneous value of the pinning field. The loop area is now increased (dashed lines in Fig. 7.23) because additional energy dissipation takes place during this transition, the instantaneous extra power loss term being Pdyn(t) -- (Hn - Hc(t))~l(t). Analog-feedback DC loop tracers are delicate setups, traditionally developed and applied in basic research, which have nowadays given way to fully computer controlled systems, where both field generation and signal treatment are digitally handled. There are two basic ways of implementing digital feedback. One consists in trying to emulate by computation the real time control of the sample magnetization realized by means of analog feedback, the other in programming the suitable time dependence of the magnetizing current by iterative augmentation of the input using an inverse approach. Computing requirements impose the basic limitation to the feedback chain bandwidth in real time control. The operations involved basically consist in: (1) Acquisition at given instants of time, separated by conveniently small intervals, and A / D conversion of a reference signal, describing the desired dB/dt waveform, and of the actual measured waveform. (2) Comparison of these two signals and computation, by means of a regulation algorithm, of the correct value of the magnetizing current, taking into account the composition of the primary circuit. (3) Digital-to-analog conversion and generation of the calculated current. It is expected that with the application of increasingly fast digital signal processor (DSP) cards, real-time digital control will gain general acceptance, both in commercial and laboratory setups. At present, the iterative method is most commonly applied to achieve the desired Jm(t) trajectory [7.74-7.76]. It can be realized by adopting the scheme shown in Fig. 7.24, which describes in summary the general structure of a computer-controlled digital hysteresis loop tracer. The operation of recursive digital control of J(t) starts with the generation of a function e(t), normally similar to the desired induction derivative dB/dt. The magnetizing winding is then supplied by a current ill(t) via a power amplifier used as a voltage amplifier with resistiveinductive load. Safe operation of the power amplifier in face of possible overvoltages at the output (which could turn up when working in current mode) is ensured in this way. The calibrated resistor RH provides a voltage drop uH(t) proportional to the magnetic field strength, which is detected and A / D converted, at least over one full period, together with the signal u2(t) -- -N2A dB/dt on the secondary winding. A two-channel digital oscilloscope or an acquisition card performing synchronous


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FIGURE 7.24 Basic scheme of a hysteresis loop measuring setup imposing a prescribed time dependence of the material polarization J (that is of the secondary voltage u2(t)) by means of a digitally controlled recursive technique. Air-flux compensation is automatically achieved by means of the mutual inductance Ma. After a first run with a sample e(t) waveform, from which an approximate J(H) relationship is obtained, the appropriate ill(t) function is computed and a novel e(t) function is generated. The process is iterated until the defined criterion for acceptance of the generated J(t) (e.g. the form factor) is met. A couple of identical DC-coupled variable-gain low-noise amplifiers is conveniently interposed between the H(t) and dJ/dt signal sources and the acquisition device. The diagram in (b), taken from experiments on non-oriented Fe-Si laminations, illustrates the behavior of the voltage signals in the primary circuit over a period once the iteration process leading to sinusoidal J(t) function is concluded.

356


acquisition can be used for this purpose. These devices are characterized by input impedance typically around 1 Mf~ or higher and do not load the secondary circuit. Multiplexing is not recommended as it introduces a time delay between channels. At the very low magnetizing frequencies involved with quasi-static hysteresis loop measurements, the sampling rate can be relatively low and high-resolution ADC converters can be employed. Typically, a few hundred kHz sampling rate in commercial digital signal analyzers is associated with 16-bit resolution and synchronous triggering over the two channels (interchannel delay and trigger jitter both lower than 10 -1~ s). A large number of sampled points per period (normally more than 103) makes negligible the error made in the measurement of the loop area, when the elementary time intervals have duration not commensurable with the magnetization period. It is also important, depending on the degree of control of the induction rate, when rectangular loops have to be measured. For maximum accuracy of the numerical integration, the trapezoidal rule or the Simpson's rule are typically applied. Normally, the secondary signal is small and needs to be amplified by a DC-coupled, variable gain, very stable low-noise amplifier. Commercial high-quality devices are usually endowed with RC filtering and may introduce small phase shifts. Consequently, both UH(t) and u2(t) should be passed through the same amplifier types (although usually with very different gains). The digitized and recorded signals are then stored into a PC (for example, via an IEEE 488 interface card), where integration is performed, the residual offsets and drifts are numerically eliminated and, in the absence of automatic compensation, correction for the air-flux is made. Thermal stability of the amplifiers is mandatory for meaningful linear drift compensation of the induction signal. Both H(t) and J(t) are thus calculated and the suitable adjustments on the magnetizing current strength are made in order to attain the desired Jp value. The time functions H(t) and J(t) can now be regarded as parametric representations of a hysteresis loop bearing a substantial similarity with the final loop. By changing H(t) we then expect that J(t) will be modified as dictated by the behavior of the function H(B). It is an easy matter to compute such a function, that is, the field H(t) depending on time in such a way that J(t) turns out to be identical to the desired one (for example, the one with constant J(t) value). A novel e(t) function can be calculated accordingly, programmed into the arbitrary function generator and delivered to the power amplifier. To this end, the equation of the primary circuit is considered, which can be written according to the scheme in Fig. 7.24 as

Ge(t) = uG(t) = UR(t) if- UL(t) -- (Rs + RH)iH(t) + N1A dJ

dt'

(7.14)


357

where G is the gain of the power amplifier, (Rs + RH) is the resistance of the whole primary circuit, including the winding resistance, A is the cross-sectional area of the sample, and the air-flux enclosed by the primary winding is disregarded. If the magnetic path length Im is defined, the current i(t) in this equation is related to the programmed field according to the usual equation ill(t)= H(t)N1/Im. In those cases where the air-flux contribution cannot be neglected, the voltage UL in Eq. (7.14) is better expressed as UL(t)= NI(A dJ/dt + At/t 0 dH/dt), with At the total area enclosed by magnetizing winding. An example of measured relationship between uG(t), UR(t), and uc(t) under controlled sinusoidal induction is reported in Fig. 7.24b for the case of a nonoriented Fe-Si lamination tested in an Epstein frame at a frequency of 0.5 Hz and peak polarization value Jp = 1.32 T. After the thus calculated e(t) function is generated, signal acquisition and calculation of H(t) and J(t) are repeated. Normally, the process cannot be concluded in a single step and e(t) is re-calculated and applied again. Iteration will proceed until the desired ](t) behavior is achieved, as objectively judged, for example, from the deviation of the actual value of the form factor from the theoretical one. Note that this feedback procedure, being based on analytical considerations regarding the loop shape, is in principle independent of the specific magnetizing frequency and is free of autooscillatory behavior. To improve the accuracy and speed of the feedback process, the reactive and the resistive terms in Eq. (7.14) should possibly be of comparable values. At very low magnetizing frequencies, this calls for increased mass of the specimen and reasonably low values of Rs and R H. It may occur in experiments that the hysteresis loop is of the reentrant type (Fig. 7.23) or that in order to emulate specific working conditions of magnetic cores in applications, complex time histories of J(t), including local minima, have to be considered. The previous feedback procedure becomes difficult to apply in such cases and it might be convenient to pose the whole problem under more general terms, where the input function e(t) is recursively calculated by introducing error terms proportional to the difference between actual and desired output values. A relationship between the values taken by e(t) upon successive iterations of the type

ek+l(t) = ek(t) + a(Bo(t) - Bk(t)) + ]3(/30(t) -/3k(t)),

(7.15)

where Bo(t) and /30(t) are the desired induction function and its time derivative, respectively, a and /3 are suitable constants, and k is the iteration order, can be envisaged in particular [7.77]. Mathematically,

358


Eq. (7.15) implements the search for the fixed point of the functional F(e(t)) -- e(t) + a(Bo(t) - B(e(t)) +/~(/~0(t) -/~(t)),

(7.16)

which, for suitable values of the constants c~ and/3, exists and is unique. An example of the application of Eq. (7.15) is illustrated in Fig. 7.25, showing the evolution of the polarization J(t) along the iteration process in the specific case of prescribed constant J(t) with local minima in a nonoriented Fe-Si lamination. The corresponding quasi-static hysteresis loop is endowed with minor loops. The foregoing considerations make clear that the measurement of the DC magnetization curves and hysteresis loops may have a somewhat elusive character. For example, even if we make negligible the role of eddy currents by magnetizing at extremely low speeds, time-phenomena may equally play a role because of the thermal, diffusive, or structural relaxation processes (after-effects). Specific anomalies, globally labeled as magnetic viscosity phenomena, can be found, for example, in pure Fe with faint concentrations of C or N at temperatures below 0 ~ Fe-Ni, Fe-Si and Fe-A1 alloys at the temperatures where atom diffusion occurs in times comparable to the measuring times [7.78]. In addition, however, when we are basically free from magnetic viscosity effects, there is no certainty that the point-by-point and the continuous recording methods should provide nearly identical results. The previously discussed reentrant loops is a somewhat extreme case, where large differences are found with the two methods and we can only speak of quasi-static behavior appropriately when J(t) is controlled. From a practical viewpoint, it is interesting to see to what extent the two methods can agree in applicative materials. The experiments show that in crystalline and amorphous laminations differences can be demonstrated, although comprehensive results on this point are not available. The reason for such differences is not clear, but it is understood that the way in which the field variation is imposed (continuous vs. discontinuous) can be expected to bring the system through different trajectories in the phase space and to slightly different end-points. Figure 7.26 provides two examples of comparison of the DC loop areas (i.e. energy loss per cycle) found with the continuous (J(t) = const.) and the ballistic methods in grain-oriented and non-oriented Fe-Si laminations, respectively. To accurately determine the DC energy loss W under continuous waveform magnetization, the hysteresis loop and losses are determined, under controlled sinusoidal induction waveshape, as a function of frequency and the extrapolation procedure to f = 0 normally adopted for achieving separation of the loss components is applied. There is a solid theoretical background to this


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FIGURE 7.25 (a) The prescribed time dependence with local minima of the polarization in a non-oriented Fe-(3 wt%)Si lamination (solid line) is attained upon a convenient number of iterations performed according to Eq. (7.15). The measurement is performed on an Epstein test frame by means of a digital hysteresis loop tracer like the one shown in the previous figure. The evolution of the J(t) waveshape vs. the number of iterations (1, 2, 5, and 12) is described by the dashed lines. At the end of the iteration process the quasi-static hysteresis loop with minor loops shown in (b) is obtained (courtesy of E. Barbisio and C. Ragusa).

360


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Wh,ball by a few percent. The method of loss separation can be applied, at least in principle, for the determination of Wh under a rotational field. It is difficult, however, to provide accurate energy loss figures at low frequencies because the field signal, determined either with a flat H-coil or a RCP, is very low. An alternative method consists in measuring the parasitic torque that is generated by the dissipation mechanisms during the rotation of the field. It is a very old method employing a sensitive torque magnetometer, providing the torque L(O) per unit volume as a function of the angle 0 made by the magnetization (or the field) with respect to a reference direction [7.80]. If we wish to make the measurement up to magnetic saturation, the classical setup with disc sample and rotating electromagnet can appropriately be used. The experiment consists in rotating the magnet by 360 ~ and recording the related L(O) behavior. L(0) is made of a reversible oscillating part, generated by the magnetic anisotropy, and a frictional irreversible contribution. The reversible torque averages out to zero upon integration over the whole period. The hysteresis rotational loss is then obtained as WRh = --

L( O)d O.

(7.17)

To improve the measuring accuracy, the torque is averaged upon both clockwise (c.w.) and counterclockwise (c.c.w.) rotation. Actually, the parasitic torque Lw also oscillates with 0 in an anisotropic material. To retrieve the Lw(0) behavior, we can make the difference for each 0 value between the c.w. and c.c.w, torques, thereby eliminating the reversible part, and dividing the result by 2 [7.81]. Laboratories engaged in the DC characterization of soft magnets can make their measurements traceable to the relevant base and derived SI units through accredited laboratories or by direct comparison with the National Metrological Institutes (NMIs). The NMIs provide a list of measurement capabilities with stated uncertainties as they result from

362


intercomparison exercises [7.82]. Few illustrative examples regarding the measurement of DC magnetic parameters in bulk, powder, and sheet soft magnetic materials are provided in Table 7.1. Achieving absolute calibration of the measuring setups and direct traceability to the SI units requires considerable effort and specialized equipment. For the sake of routine calibrations, reference samples can be used, with crosssectional area and number of turns appropriate to the ranges to be covered. Very pure Fe or fully decarburized and stabilized Fe-Si alloy samples can be employed for reference purposes. The material will be annealed for stress relief and stabilized against aging by applying, for example, prolonged thermal treatment at a temperature around 200 ~ Ni is not a truly soft material and is better used for the calibration of systems employed in the characterization of hard magnets. It should be stressed, in any case, that the intrinsic stochastic nature of the domain wall processes is responsible for relatively poor repeatability of the measurements in soft magnets at low polarization values, that is, below about J =Js/2, where the use of reference samples might not be totally satisfactory.

7.3 A C M E A S U R E M E N T S The normal operating conditions of soft magnetic cores in devices call for time-varying fields. Soft magnets then have commercial value when they are categorized according to a minimum set of magnetic properties determined under defined AC excitation. Technical difficulties and costs often limit the amount of information provided by manufacturers, while designers, who need to compare different materials in order to optimize their devices at reasonable costs, benefit from as large as possible an ensemble of significant material parameters, obtained by characterization of the material under both AC and DC fields. The need for measurements not necessarily limited to the base figures provided in the data sheets is therefore widespread and shared by research and industrial laboratories. It justifies efforts to present and discuss comprehensive measuring methods. As stressed in the previous sections, the determination of the intrinsic material behavior sometimes appears as an elusive goal. We can approach it to a reasonable extent, while preserving good measuring reproducibility, not only by applying tightly controlled measuring conditions, including the consensus rules dictated by the written standards, but also by understanding the physical problems lying behind the measured material properties. One can indeed observe a complex evolution of the magnetic phenomenology with the magnetizing

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frequency, which relates to a corresponding evolution of the relaxation processes, ranging from eddy current phenomena to various resonance effects occurring at radiofrequencies. In this section we shall treat the problems involved with the magnetic measurements under AC excitation in the light of the physical mechanisms of the magnetization process. A broad distinction will be made between measurements at low frequencies, where stray parameters have little or no influence (roughly speaking, up to a few hundred Hz), and measurements up to the MHz range (mediumto-high frequencies). Characterization methods at radiofrequencies will be summarized in the last part of this section.

7.3.1 Low and power frequencies: basic measurements Soft magnets employed in static and dynamic electrical machines are classified and marketed for their properties at 50 or 60 Hz and at such frequencies they are usually tested. This makes sense from a commercial viewpoint but it is rather unsatisfactory from the perspective of physical investigation or design of devices. Basically, one should aim at characterizing the material as a function of the exciting frequency, starting from quasi-static testing, under controlled (typically sinusoidal) induction waveshape, possibly assessing the whole phenomenology in the frame of physical theories [7.83]. The measuring setup schematically described in Fig. 7.24, exploiting the digital feedback procedure discussed in Section 7.2.2 (Eqs. (7.14)-(7.16)), is totally appropriate to this aim. With it one can perform precise and reproducible measurements of hysteresis loops and losses as a function of magnetizing frequency in soft magnetic laminations, powder cores, and ferrites. No special restrictions exist as to the induction waveform, thanks to digital feedback and use of a programmable arbitrary function generator, but modifications of the magnetizer configuration and hardware of the measuring setup are required on approaching the kHz range. We have described in Chapter 6 and in Section 7.1 the basic specimen configurations that can be devised for AC testing of soft magnetic materials: ring, Epstein frame, single-strip/single-sheet tester, and open samples. The permeameter arrangement with bulk specimens applies to DC characterization only. In all cases, we consider regular samples only, having defined cross-sectional area A, and a magnetic path length lm is identified. In the measuring setup illustrated in Fig. 7.24, the field is determined by measuring the current in the magnetizing winding. We can therefore make use of this circuit when testing rings, Epstein frames, and single-sheet assemblies. It basically fulfills the requirements of

7.3 AC MEASUREMENTS

365

the standards IEC 60404-2 [7.11], IEC 60404-3 [7.29], ASTM A804 [7.19], ASTM A343 [7.84], ASTM A912 [7.85], and ASTM A932 [7.20]. If flat H-coils or a RCP, placed on the sheet surface, are employed for the determination of the tangential field H(t) (Figs. 7.4-7.6), provision should be made for an additional acquisition channel in order to deal with the related signal, proportional to the time derivative of H(t). We will discuss this point to some extent in Section 7.3.3, while presenting measuring methods under rotational fields. The standard IEC 60404-2 deals with the measurements on steel sheets using the Epstein magnetizer from DC to 400 Hz. The standard IEC 60404-3 is devoted to the testing of single sheets (SST) at power frequencies. We have introduced them in Sections 6.1 (Figs. 6.3, 6.4, and 6.6) and 7.1.2 (Figs. 7.7 and 7.8). We have discussed to some extent there how the assumption of an a priori fixed value of the magnetic path length lm was reflected into a somewhat conventional determination of the magnetic parameters, namely the power loss, with ensuing discrepancies between results obtained on the same material by the two different methods. Procedures to reconcile such results in some specific cases have been considered (Section 7.1.2, Eqs. (7.5)-(7.7)). The basic provisions of the IEC 60404-2 and IEC 60404-3 standards and the specific features of magnetizer and specimens are summarized in Table 7.2. Whatever the specimen configuration, any digital hysteresisgraphwattmeter built according to the scheme of Fig. 7.24 can deliver complete information on the magnetic properties of the material over the appropriate range of magnetizing frequencies and defined induction waveforms: major and minor hysteresis loops, normal magnetization curve, permeability, apparent power, and power losses. All desired quantities are obtained in it by numerical elaboration after A / D conversion. Using high-resolution high sampling rate acquisition devices with synchronous sampling over the different channels, we can achieve excellent reproducibility of results [7.13]. It is of course possible, as envisaged in the standards, to employ physically different devices for the determination of the required quantities. The peak value Jp of the magnetic polarization can be obtained from the mean rectified value 0 2 o f the secondary, as provided by an average value voltmeter, according to the equation 02 =

4fN2AJp.

(7.18)

The standards prescribe that u2(t ) is sinusoidal (form factor, equal to the ratio between r.m.s, value and average rectified value, FF -- 1.1107 + 1%), but Eq. (7.18) is valid, in the absence of minor loops, whatever the secondary voltage waveform. The peak value of the magnetic field

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strength Hp is obtained by reading with a peak voltmeter the peak value/~H of the voltage drop across the calibrated resistor RH Hp-

N1 /hH Im RH"

(7.19)

The r.m.s, value H of the field strength can similarly be obtained by measuring /~H = RHi'H by means of an r.m.s, voltmeter (with accurate response in the presence of high crest factors). An r.m.s, voltmeter connected across the secondary winding, in parallel with the average value voltmeter, will equally provide u2, that is, the r.m.s, value of dB/dt. The maximum allowed uncertainties are + 0.2% for the average type and r.m.s, voltmeter readings and + 0.5% for the peak voltmeter reading. The specific apparent power, defined as N1 1 S -- ZHU2N2 ma

(7.20)

is immediately obtained. In this equation, the quantity ma -- 3lmA , with 3 the material density, is defined as the active mass of the specimen. It appears in place of the total mass to account for the fact that the magnetic path length lm can be different (as in the Epstein and SST testing) from the actual average length of the specimen. More complete information is retrieved, however, using the previously explained method of synchronous acquisition of the signals uH(t ) and u2(t), A / D conversion, and numerical computation of the desired quantities. We can thus obtain, at a given peak polarization value Jp, the hysteresis loop and its area (Eq. (1.29)) W=

H d J = f~ H(t) dJ(t) dt dt,

(7.21)

that is, energy loss per cycle and unit volume. The latter quantity or, equivalently, the specific average power loss per unit mass P=

ff~

dJ(t)

H(t) dt dt

(7.22)

is the base technical parameter used in the designation of the different material grades. Equation (7.22) can be derived under very general terms from Poynting theorem [7.83]. With a closed magnetic configuration, H(t) coincides with the applied field and under AC excitation it is the field existing at the specimen surface, where the eddy-current counterfield is zero. H(t) is then equal to the sum of the field required by the DC constitutive equation of the hysteresis loop and the additional field that must be applied at any instant of time in order to antagonize the eddy

368


current counterfield and preserve the same value J(t) of the polarization, averaged across the specimen cross-section. Let us consider the equivalent circuit in Fig. 7.27, where we have assumed, for the time being, that there are no leakage inductances and stray capacitances and that the resistance of the secondary winding is negligible with respect to the input resistance R 2 of the measuring instrument (either voltmeter, pre-amplifier, or acquisition device). We also assume that R 2 is so high that i2 ~ 0. The average power delivered by the generator into the magnetizing winding, purged of the ohmic losses in the winding resistance Rwl, is given, per unit of effective sample mass, by

Pw-- 11f~UL(t)iH(t)dt"

(7.23)

ma T

Since the available voltage on the primary circuit is ul(t) and not uL(t), the magnetic loss determination by direct application of Eq. (7.23) is possible only if the power dissipated in the winding resistance Rwl is calculated and subtracted from the loss measured on the primary circuit. This procedure is not desirable in general because the ohmic losses in Rwl are comparable to, or even higher than, the magnetic losses and they tend to change during the measurement because the winding temperature can change. The secondary voltage is therefore considered (virtual open circuit) and, since N2 u2(t ) --- _ ~ UL(t)'

by introducing it in Eq. (7.23), we obtain

Pw =

1 N1 N2 T11~u2(t)iH(t)dt"

(7.24)

ma

Equations (7.22) and (7.24) are equivalent because u2(t)= -N2A(dB/dt) and ill(t) = H(t)(Im/N1),the latter relationship implying that the magnetic circuit is closed and lm has a defined value. In fact, by substituting u2(t) and ill(t) in Eq. (7.24) we obtain Pwand, since

f I~ H(t) dB(t)dt dt-

B(t) = I~H(t) + J(t), we can also write T dH(t) d/(t) Pw=f lo[IZ~ +H(t) di dl(t) = f-~ I~ H(t) -~-dt=P,

(7.25)

]dt

(7.26)

7.3 AC MEASUREMENTS

369

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(b) FIGURE 7.27 (a) Equivalent AC circuit of the hysteresisgraph-wattmeter shown in Fig. 7.24. The magnetizing frequency is assumed to be sufficiently low as to permit one to neglect the effects of stray capacitances and leakage inductances of the primary and secondary windings. Rwl is the resistance of the magnetizing winding. The resistance of the secondary winding is negligible with respect to the input resistance R 2 of the secondary instrument. The e.m.f, u2(t) appearing across the secondary winding is related to the voltage uc(t) by the equation u2(t)= -(N2/N 1)uc(t). (b) Corresponding vector diagram (linear approximation) for nonnegligible load current i2(t) and imposed flux. The total magnetizing current im(t) (i.e. the field H(t)), phase shifted with respect to the flux ~(t) because of iron loss, is imposed and, consequently an extra-current i2~(t)= -(N2/N1)i2(t) must flow in the primary winding to counter the effect of i2(t) in the secondary winding. The total current ill(t) = ira(t) + i21(t) to be supplied in the primary winding accounts then for the additional power consumption in the load. The vector diagram is drawn here for N2/N1 = I and the proportions of i2 are somewhat exaggerated for the sake of clarity.


370

where the first term within square brackets integrates to zero over a period. We can therefore carry out the measurement of the power loss either by time averaging the product of primary current and secondary voltage or by calculating the hysteresis loop area. Equation (7.24) rests on the condition of negligible value of the secondary current i2(t), which is satisfied under normal measuring operations, where high input impedance signal conditioning devices (preamplifiers, acquisition cards, electronic wattmeters, etc.) are used. However, when old-fashioned electrodynamic wattmeters, especially if connected in parallel with an average value voltmeter and an r.m.s, value voltmeter, are employed, we might have to account for the additional power consumption in the instruments brought about by the circulatha.g current i2(t). Loading by the secondary circuit results, under rated flux, in an additional current i2/(t)--(N2/N1)i2(t) in the primary winding, as defined by the condition that the total magnetizing current ira(t)ill(t)- i2/(t) is imposed (see the vector diagram in Fig. 7.27b). The generated field, resulting from the currents circulating in the primary and secondary circuits, is then

N1 N2 H(t) - ~m ill(t) + -~m i2(t)"

(7.27)

By introducing it, together with the expression for the induction derivative dB u2(t) dt

N2A '

in Eq. (7.25), one obtains for the specific power loss P=

= -

H(t)- dt

,, ma

dt

u2(t)iH(t) d t -~2

0

(7.28)

u2(t)ia(t) d t "

The second term within square brackets is the power consumed in the secondary circuit, which must then be subtracted from the indication of the wattmeter in order to obtain the magnetic power loss. It may happen that AC testing is to be done on open sheet or strip samples. We know that in this case the effective field H is better measured, for instance by means of tangential H-coils, than calculated using the demagnetizing coefficient. If we are unable to determine

7.3 AC MEASUREMENTS

371

directly H, we might equally use the applied field Ha instead of the true Ha(t)in Eq. field involved in the problem. By posing (7.25), we find

H(t)=

P=

f f i"Ha(t) dJ(t) dt dt

(Nd/lzO)J

(7.29)

This equation is noteworthy, since it contains the applied field, which is the quantity directly under our control, and the polarization, which describes the magnetic state of the material under test. The magnetostatic energy is stored and released in a reversible fashion and averages out to zero over a complete hysteresis cycle. Four equivalent expressions for the specific power loss can then be implemented, according to Eqs. (7.22), (7.24), (7.25), and (7.29). To measure the specific power loss P, as well as the apparent power S, digital methods are nowadays the rule, both to set the magnetizing conditions (frequency, peak polarization value, and control of the induction (polarization) waveform) and to carry out all signal handling, computation, data storing and retrieval. Several examples are discussed in the literature [7.75, 7.76, 7.86-7.88] and commercial solutions are offered by instrument manufacturers [7.3, 7.89]. Regarding the digital control of magnetic field generation and induction waveform, this is in principle little dependent on the magnetizing frequency and the discussion given in Section 7.2.2 for the quasi-static case also applies to the AC regime. Significantly, the devised digital feedback methods, while being totally appropriate to satisfy the requirement of sinusoidal voltage u2(t) , are the ideal solution for emulating the non-sinusoidal induction waveforms expected in many applications, whether they are associated with distortions or are deliberately generated for specific needs. This is the case, for instance, for inductor cores used in switched mode power supplies or stator cores in variable speed motors supplied by means of pulse width modulated (PWM) voltages. Whatever the method of field control, the key device in the primary circuit is the power amplifier, which is required to handle signals having a large dynamic range, to be injected into strongly inductive loads. For measurements on Epstein frames up to 400 Hz, peak current values and voltages higher than 10 A and 100 V, respectively, might be required. As previously remarked, safe operation of the power amplifier can be obtained by using it in voltage mode, with a suitable power resistor in series with the magnetizing winding. Concerning data acquisition and the computation of P and S with the associated uncertainties (see also the related discussion in Section 10.4), a few basic points will be discussed here. Reference is made to the scheme of

372


Fig. 7.24a and the equivalent circuit of Fig. 7.27, and to a number of related studies dealing with this problem in the more or less recent literature [7.13, 7.24, 7.90-7.94]. (1) Field signal. The signal UH(t)----RHiH(t), proportional to the field H(t) in closed samples, is detected on a stable resistor calibrated against a standard. At low and power frequencies, anti-inductively wound manganin alloy wire resistors are appropriate, thanks to their near-zero temperature coefficient. Different kinds of power resistors (metal foil, metal film, and molded) can equally be employed, but their temperature coefficient must be checked and a heat sink made available. Calibration must be performed frequently and is indispensable to repeat it whenever thermal shocks due to uncontrolled current surges have occurred. Different contacts for current and voltage leads should also be adopted. The relative uncertainty on the resistance value can be kept around some 10 -4 , that is, negligible to all practical effects. If an H-coil is to be used, the related t u r n - a r e a product must be determined with the aid of a calibrated flux density source. It is typicall~ achieved with relative uncertainty in the range 1 x10 - 3 5 x 10 - o (Table 10.3), but its stability with time is critical and calibration must be frequently repeated. The range of the UH(t) signal is determined by the maximum J value required by the experiments and by the frequency. Exciting the material beyond the knee of the magnetization curve can soon strain both the power amplifier and the dynamic range of the acquisition device in the H-channel. If the measurement is made under sinusoidal polarization, as required by the measuring standards, only the fundamental harmonic UHI(t) will contribute to the power loss, since the products of the higher harmonics with the sinusoidal function u2(t ) will average to zero in the loss integrals (Eq. (7.24)). This brings about a reduced effective dynamic range of the signal conditioning devices because UHl(t) is only a fraction, increasingly smaller with increasing Jp, of the total signal to be handled in the H-channel. (2) Induction signal. The secondary signal is detected by means of 700 turns in the Epstein test frame and a convenient number of turns when using the SST, ring specimens, or open strips. Even by limiting the upper frequency to 400 Hz, a large dynamic range can result if the sample cross-sectional area remains unchanged. Figure 7.26 shows that the accurate determination of the quasi-static energy loss Wh requires that measurements are performed down to a frequency as low as 0.5-0.25 Hz, which implies more than 60 dB range in the induced voltage amplitude. To cope with it, it is convenient to interpose a DC-coupled low-noise amplifier with calibrated gain, variable in

7.3 AC MEASUREMENTS

373

a step-like fashion, between the secondary winding and the input of the acquisition device. An identical amplifier, with identically set upper cut-off frequency, is introduced in the H-channel in order to avoid any possible spurious phase shift between primary current and secondary voltage signals. Under normal conditions, the relative uncertainty of the uH(t) and u2(t) values related to gain and distortion of the pre-amplifiers is lower than 2 x 10 -3, while the imperfect compensation of the air-flux can provide a contribution to the uncertainty on u2(t) around 10 -3. A trivial and dangerous source of error comes from imperfect determination of the cross-sectional area A of the specimen. For example, a 1% uncertainty in the value of A, proportionally reflected in the value of Jp, can propagate into around 2-2.5% uncertainty in the power loss P and even 15-20% uncertainty in the apparent power S in a non-oriented material at 1.6 T and 50 Hz. The direct measurement of the strip/sheet gauge does not guarantee a sufficiently accurate determination of A. Amorphous ribbons, for example, which are inconveniently thin in this respect (typically 10-50 ~m), show a decrease of thickness going from the strip axis to the edges and their cross-section has more or less an elliptical profile. It is then recommended that mass m and length I of the test specimen are measured and that the cross-sectional area is calculated as A = m/81, with the density 8 known from composition or obtained by measurement (for example, with the immersion method). Mass and length of the specimen can be measured with very good accuracy (e.g. relative uncertainties lower than 10 -3 in Epstein strips) so that the major contribution to the uncertainty in the determination of the value of A comes from the measurement of the density. A relative uncertainty u ( A ) / A ~ 2 x 10 -3 can be achieved in Epstein specimens (see also Section 10.4). The case of non-oriented sheet steels, while having obvious industrial relevance, is made somewhat complicated by the variety of compositions associated with the different grades, which would often suggest direct measurement of the density. Such a measurement, however, may appear complicated and expensive in routine magnetic quality testing. We know that the electrical resistivity p and the density 8 are, within the usual compositional limits of the nonoriented alloys (random solutions), both monotonically dependent on the concentration of Si, A1, and Mn [7.95]. It has been accordingly verified by experiments that within the concentration limits for Si and AI: c(Si) .,

Jp= 1.5T

_go 200 O

Jp= 1.7T

(1)

>

1._

O3 c"

25 Hz in Brix's device). The measurement of the average parasitic torque provides the energy loss under pure rotational flux. Under general 2D magnetization process and AC exciting conditions, the magnetic field at the sheet surface H = Hy~ + Hx~ and the polarization J - Jy~ + Jx~ averaged over the sheet cross-section, are conveniently determined. A pair of flat H-coils, orthogonally placed on the sample surface across the region of homogeneous magnetization (Figs. 7.11-7.14 and 7.39) can be employed to detect the components Hy and Hx. Alternatively, a pair of RCPs covering the same region can be used [7.128]. The use of Hall sensors can also be envisaged. In this case, however, the measurement is somewhat localized and the active character of the device is a disadvantage. The polarization components Jy and Jx can be obtained either with B-coils or needle probes, as illustrated in Fig. 7.37. The measuring region, however, must be sufficiently large to encompass the structural inhomogeneities of the tested material. With grain-oriented laminations this is not easily achieved and averaging of the results obtained on a number of samples might be required. The directly detected signals are obviously proportional to the time derivative of the above quantities, which are then obtained by numerical integration (Fig. 7.39). In this way, we have all we need for the measurement of the 2D power losses. Under very general terms, it can be stated that the electromagnetic energy flowing in unit time into a given region of the lamination, bounded by the surface S~ is given by the integral - ~ s ( E X I-I).n dS~ where the product (E x I-I)~ with E and H the electric and magnetic field at the lamination surface, is known as the Poynting vector. By integrating the instantaneous value of the surface integral of the Poynting vector over a full period we obtain the energy loss W. Under normal measuring conditions, the edge effects are irrelevant and the energy streams only through the top and bottom surfaces of the lamination and we can conclude that the energy loss throughout the whole sample volume can be determined by means of a surface measurement of the electric and magnetic fields. These are exactly the quantities we obtain by means of

404


our needle probes (Fig. 7.37), which provide the voltages Vy = Eyl and Vx = Exl, and by the H-coils, from which the field components Hy and Hx are obtained after integration. With the symmetry of our problem, where the field is applied in the lamination plane and E and H do not have components along the z direction and are constant upon the measuring region of area l2, we obtain that the instantaneous power dissipated in the volume 12d, where d is the lamination thickness, is

P(t) = -21(E x H)I.I2 -- 212(EyHx - ExHy) = 2l(VyHx - VxHy),

(7.47)

from which the average power loss per unit mass is obtained as

PR-- 3dl21 TII~ P(t)dt" Thus, by needle contacts, we satisfy both the Poynting vector formulation of the loss and, under the condition l>> d, the need for precise determination of the induction value. However, the low signal level and the difficulty of avoiding linkage with stray flux often make it preferable to resort to the alternative method of detecting the induction derivative by windings threaded through tiny holes drilled at the distance l. The ofteninvoked detrimental effects related to local hardening by drilling can be safely avoided if this operation is carried out with care. In this way, the signals dBy/dt and dBx/dt become available. They are related to Ex and Ey, respectively, by the Faraday-Maxwell law and, by substitution in Eq. (7.47) with the appropriate sign convention, we obtain

PR-P+PRy =

1

dBy, 1 ~ x dt +Hy---~-~-)dt= --3Tf~(Hx~t+Hy~t)dt" (7.48)

The energy loss under 2D excitation can therefore be measured by summing the areas of the hysteresis loops taken along two orthogonal directions. In a perfectly isotropic material, subjected to a field Hp of constant modulus rotating with constant speed, the vector Jp has equally constant modulus and lags behind Hp by a fixed angle ~H. The resulting (Hx,Jx) and (Hy,Jy) loops are ellipses and the resulting specific rotational power loss is

PR-- 2-~~3HpJpsin ~H.

(7.49)

7.3 AC MEASUREMENTS

405

This condition is approximated at low inductions in ordinary nonoriented alloys (inset in Fig. 7.41). In particular, in the region of validity of the Rayleigh law, it is possible to predict theoretically the rotational hysteresis loss from measurements performed under alternating fields and suitable hypotheses on the statistical distribution of the grain orientations [7.129]. On increasing Jp, the actual anisotropic behavior of the material comes largely into play and both the amplitude Hp of the rotating field and the phase shift qOjHundergo fluctuations while trying to proceed, with the help of feedback, according to the desired time dependence (e.g. independent of time for circular polarization locus) of Jp and dOj/dt. The measured hysteresis loops then take the characteristic re-entrant shape shown in Fig. 7.41. Remarkably, there are instances where anisotropy is

1.5~ NO Fe'(3 wt%)Si f= 10Hz

l

1.0 0.5 '~ "~

0.0

0.5

-0.5 -1.0 -1.5-1000

-500

0

Hy (A/m)

500

1000

FIGURE 7.41 Hysteresis loops in non-oriented laminations measured under controlled circular polarization qp = const., dOj/dt = const.) at different Jp values by taking the components Hy and Jy of rotating field and polarization aIong the rolling direction over a period. The sample is a disk placed within a three-phase yoke system (Fig. 7.39). Hy and Jy are detected by means of a fiat H-coil and a few-turn B-winding, respectively, over a region of uniform magnetization in the sample. The pseudo-elliptic hysteresis loops obtained at low induction values are shown in the inset. The actual anisotropic behavior of the material leads to oscillations of the amplitude Hp of the field and of the phase shift {pbetween Hp and Jp, which are apparent from the shape of the loops.

406


sufficiently strong to bring Jp ahead of Hp during the rotation, which leads to typical buttonholes in the loops, signaling net energy release by the sample to the external world. This can add to the uncertainty of the determination of the area of the loops. Figure 7.42 illustrates the measurement of energy loss with circular polarization Jp=l.5T in 0.35 m m thick Fe-(3 wt%)Si non-oriented laminations via determination of the (Hx,Jx) and (Hy,Jy) hysteresis loops. These are actually the result of loop averaging upon clockwise and counterclockwise rotation of the applied field, by which spurious phase shifts, responsible for substantial errors in the measured loop area, are for the most part eliminated. Possible asymmetries of the measuring apparatus, especially misaligrunents of H-coils and B-coils, can detrimentally affect the loss-measuring accuracy if they are not suitably compensated. We see in the typical case reported in Fig. 7.42b (non-oriented alloy at Jp = 1.5T) that Jp lags behind Hp by an average angle around ~H "" 3~ This means that an all too comrnon 0.5-1 ~ misalignment of the windings can result in an intolerable 10-30% error in the measured loss. To recognize qualitatively the effect of compensation by c.w. and c.c.w, field rotation, we can try to evaluate, for example, the effect of a spurious misalignment between the Hx and Jx windings [7.130, 7.131]. ! If the error angle is A~x and the measured loss figures are PRcwxqp~Aqox)and P~Rccwx(Jp,A~px)for c.w. and c.c.w, rotation, the actual power loss PI~ = ~

Hx

dt

is related to the result of averaging by the equation i +P~ccw~PI~ = PRcwx ".

2cos(A~x)

(7.50)

Compensation is therefore effective under typical measuring arrangements, where A~px can be kept within 1~ and it can be safely assumed cos(A~x)= 1. It is generally acknowledged that it is difficult to perform acceptably accurate rotational loss measurements with the fieldmetric method discussed here beyond about 1.5 T in non-oriented Fe-Si laminations [7.41]. The chief limitation arises, as for the alternating case, from the corresponding rapid decrease of the average value of q0jH~which becomes dwarfed by fluctuations. In grain-oriented alloys already at low inductions, the problem is exacerbated, by the wild fluctuations undergone by both Hp and q0jH~ as demanded by the control of the polarization loci. Under such difficult conditions, a reasonable alternative solution is offered by the rate of rise of temperature method, as previously discussed

7.3 AC MEASUREMENTS

407

NO Fe-(3 wt%)Si f= 10Hz

1.5 1.o 0.5 0.0 --0.5

-1.0 Jp= 1.5T

-1.5 '

-

'

I

. . . .

I

. . . .

I

. . . .

. . . .

1500 -1000 -500

I

. . . .

500

0

I

. . . .

I

'

'

1000 1500

Hy, Hx (A/m)

(a) 12

NO Fe-(3 wt%)Si

/~

Jp=1.5T 10Hz

f=

o

8 o v

o

9CO

-(b

(/)

g 4

. . . . . . . . . . . . . . .

"~o~176

t13.

o

0

'

(b)

0.00

'

'

'

I

'

0.01

'

'

'

I

'

'

'

'

I

.

0.02 0.03 Time (s)

.

.

.

I

'

0.04

'

'

9

0.05

FIGURE 7.42 (a) Example of energy loss measurement under circular polarization locus in non-oriented Fe-Si, laminations. The areas of the hysteresis loops associated with the components of effective field and polarization along the two orthogonal axes x and y, made to coincide with RD and TD, respectively, are summed up to provide, according to Eq. (7.48), the energy dissipated in a magnetization period V W R q p , f ) = W R y q p , f ) + WRx(Jp,f) = P ( J p , f ) / f . (b) Because of anisotropy, the vector Jp is not always lagging behind H.. during rotation and @H Can be negative. The small value of the average phase sh~ft (horizontal dashed line) can make the accurate measurement of the loss difficult and a calorimetric approach can therefore be preferred if testing at higher Jp values is required.

408


to some extent and as schematically shown in Fig. 7.39. The signal provided by the thermal junctions or the thermistors in a given time interval, as defined by the requirement of operating close to the adiabatic regime, is directly proportional to the dissipated power. It is then proportional to the product J'~f"', where n can range between 1.5 and 2 and m is roughly around 1.5. Because of this power law dependence, it is difficult to apply the thermal method at low frequencies and low inductions. A reasonable approach to the whole problem would then call for application of the simpler and more informative fieldmetric method as far as the uncertainty of ~H is acceptably low and to make use of the thermal method in the high induction range only, provided a suitable overlap region is identified. Agreement of the measurements performed upon such a region would provide a stringent check of the accuracy of both methods. Of course, the expectedly imperfect control of the 2D magnetization pattern on approaching the magnetic saturation undoubtedly affects the accuracy of the thermal method, but, as already stressed, some kind of correction of the raw result (Eq. (7.46)) can be attempted. It should finally be remarked that, if the objective of the measurement is exclusively the determination of the rotational losses, sufficiently large DC fields are available, and accuracy is not a very stringent requisite, we could refrain from setting up complicated controls of the polarization loci and difficult signal handling by preparing a suitably small disk-shaped or cylinder-shaped sample, placing it within the polar faces of an electromagnet, setting it into spinning motion, and finally looking at the way damping is affected by the presence of the steady field. Basically, this means that as in the previously mentioned dynamic torque method [7.127], a measure is made of the average parasitic torque, now by rotating the sample instead of the magnetic field. With small diameter disks (e.g. 10-20 mm) the demagnetizing effect is sufficiently high to impose a nearcircular polarization locus even in highly anisotropic materials [7.132]. After setting the applied field to the desired value and bringing the sample to a spinning rate of 50-200 Hz, the drive is removed and the sample is allowed to spin freely under the restraining action of the mechanical and the eddy-current induced frictional torques. At the same time, the rate of change d~/dt of the decaying angular velocity is measured [7.133]. Such a measurement, usually performed by optical means, amounts to a determination of the parasitic torque Lw-- I dco/dt, where I is the moment of inertia of the sample, as a function of the spinning frequency f, ideally down to f = 0. The average torque at any given frequency provides then the rotational power loss PR -- 2Vr~w once the spurious mechanical parasitic torques are eliminated by making two identical measurements with and without the applied field.

7.3 AC MEASUREMENTS

409

The spinning sample technique appears attractive in those cases where the size of the sample is so small or its nature is so specific that the fieldmetric technique cannot be applied and it is desirable to measure the losses up to saturation. Contrary to the case where a rotating field is produced, saturation is not difficult to achieve by means of a DC source. Examples of rotational loss separations obtained by the spinning sample method have been reported by Cecchetti et al. in grain-oriented Fe-Si, non-oriented Fe-Si and amorphous ribbons [7.134, 7.135]. It is expected that this method can be applied to soft magnetic composites, where the necessarily bulk samples would offer too large a demagnetizing coefficient to any applied rotating field or, in general, to powder aggregates. Experiments on hard and semi-hard powders, where the theory of rotational hysteresis plays a major role in the physical modeling of the magnetization process, have been reported [7.136].

7.3.3 M e d i u m - t o - h i g h f r e q u e n c y m e a s u r e m e n t s There is an increasing trend towards the use of electrical machines and various types of devices over a wide range of frequencies and with a variety of supply methods, which call for the precise characterization of soft magnetic materials beyond the assessed DC and power frequency domain. There are indeed many kinds of excitation methods, which impose, within broad frequency limits, not only sinusoidal flux conditions, but also different types of rated voltage waveforms, with and without bias field, or pulsed magnetization (either with determined current or voltage pulses) [7.137]. While there are no special additional difficulties in setting up a system by which one can drive a particular type of excitation, many problems arise with the increase of the frequency. They can be faced by means of a rigorous approach to the measuring principles and their implementation in the testing operations. However, to adapt the conventional measuring setups to the characterization of soft magnets up to the MHz range might become a relatively complex task. We shall discuss here the main potential difficulties and problems arising with the increase of the magnetizing frequency, in the limit where electromagnetic propagation phenomena are still irrelevant, that is, up to frequencies where wavelengths are much larger than the size of the region occupied by the specimen. (1) The flux penetration in the test sample can be incomplete (skin effect) This effect can be evaluated by calculating the skin depth = ~/2//~0/~rCr~o~ that is, the depth in the sample where the induction value falls by a factor 1/e with respect to the induction value at the surface. Under these conditions, thickness-dependent instead of intrinsic

410


properties are measured. Figure 7.43 provides illustrative examples of induction profiles vs. thickness in non-oriented Fe(3 wt%) laminations (at 400 and 1000 Hz) and in amorphous and nanocrystalline tapes (at 1 MHz). The latter samples, endowed with very close resistivity values and the same thickness, are tested as strip-wound toroids, after having been optimized for high-frequency applications by means of annealing treatment under a transverse saturating field. They exhibit remarkably different profiles, which are due to their different permeability values (4.5 x 103 in the amorphous ribbon and 32 x 103 in the nanocrystalline sample). The skin effect can equally occur in the conductors and increase their AC resistance, often prompting the use of windings made of copper strips or multiconductor Lietz wire. It has troublesome consequences when it affects the calibrated resistor RH employed in the primary circuit as current probe. The frequency response of RH should therefore be verified before starting the measurements. Wirewound resistors are not recommended in general, while carbon resistors can display a flat response up to 10-20 MHz. Proximity effects, that is the interaction existing between closely placed conductors carrying an AC current, can also add to the skin effect in giving rise to an increase of the resistance of windings and leads. (2) The temperature of the sample can appreciably rise during the measurement. A 0.050 mm thick grain-oriented sheet, suitably developed for high-frequency applications, can display, for example, a power loss of about 500W/kg at 1.0T and 10 kHz. The sample temperature is correspondingly expected to rise at a rate around 1 ~ In a M n - Z n ferrite tested at 1 MHz and peak polarization Jp = 0.1 T, sample heating can proceed at a rate higher than 2 ~ Excessive heating is obviously detrimental to the measuring accuracy because the physical properties of the material can rapidly change with the temperature. This should be measured by placing a micro-thermocouple in contact with the specimen, which, in turn, should be suitably cooled, for example, by keeping it into an oil bath. If the specimen is encapsulated, the junction will be put in contact with the core material by making a small hole in the container. The method of single-shot acquisition and digital control of the measurement is the most appropriate when looking for minimum temperature increase because the time interval where the sample is excited is at a minimum. (3) The increase of the required exciting power P(t) = uc(t)iH(t ) with the frequency poses serious limitations on the achievable peak polarization value. Under most circumstances of high-frequency testing, the characterization of the material for Jp not far from the knee of the magnetization curve can only be done under pulse excitation [7.138]. To make an example, a M n - Z n ring of average diameter 50 mm and

7.3 AC MEASUREMENTS

1.0

"-

411

- I

.

.

.

.

-

. . . .

v

f= 1000 Hz

0.5

NO Fe-(3 wt%)Si d = 0.34 mm Jp=lT 0.0

'

'

I

'

'

'

'

I

'

'

'

'

I

'

'

'

'

I

'

-0.15 -0.10 -0.05 0.00 x(mm)

'

'

'

I

'

'

0.05

5

'

'

I

'

'

0.10

'

'

I

'

0.15

'

Jp = 0.005 T f= 1 MHz /

4

~o.

3

~'Q

2

1

Nanocrystalline =20 gm

/ -

/

Amorphous

-

,

I

-10

'

'

'

'

I

-5

'

'

'

'

I

0

~

_

'

'

'

'

I

5

'

'

'

'

I

10

x (l~m)

FIGURE 7.43 Examples of eddy current-induced profiles of the reduced peak polarization J (x)/Jp vs. distance x from mid-plane of 0.34 mm thick non-oriented Fe-(3 wt%)SiP laminations and 20 ~m thick amorphous and nanocrystalline ribbons. The profiles are calculated by an FEM technique taking into account the experimental DC B(H) hysteresis loop curve of the material. The amorphous and nanocrystalline ribbons have the same thickness and very close resistivity values, but the latter are endowed with a one order of magnitude larger permeability (courtesy of O. Bottauscio).

412


cross-sectional area A = 50 mm 2 requires about 100 V A peak power to be excited at Jp -- 0.1 T at the frequency of I MHz. The general trend is one of using small cores with few primary turns to limit the primary voltage. The number of turns of the secondary winding is equally small, but full coupling is to be ensured with the magnetizing winding, which must then be uniformly wound along the core [7.139]. As a rule, testing is made on ring-shaped specimens. If single strips are to be characterized, a fluxclosing yoke should be devised whose soft magnetic behavior associates with minimum skin effect. This condition may be satisfied by a double-C core made by assembling mumetal or amorphous tapes of as low a thickness as practical. It has been shown that with such an arrangement, where the field strength is determined either by means of an H-coil or by measuring the magnetizing current and adopting a magnetic path length equal to the internal yoke diameter, measurements on amorphous strips can be reliably carried out up to 100 kHz [7.140]. Note that it would be very difficult to overcome this frequency limit with H-coils, as their behavior is inevitably affected by self-capacitances. Measurements on open samples are obviously possible but they are not recommended in general. Besides the obvious correction for the demagnetizing field, they require certain precautions regarding the generation of the rapidly varying stray field in the surrounding milieu because of unwanted electromotive forces generated in metallic parts and conductors forming closed loops, which can eventually interact, through the generation of spurious fields, with the magnetization process. In the following, we will refer to the most common condition of closed samples (e.g. Epstein or ring specimen) with the field determined via the measurement of the magnetizing current. (4) Fast A / D converters are required to satisfy our requirement of single-shot signal acquisition and real-time analysis. This implies a certain limitation in signal-amplitude resolution and if we are to make a complete characterization starting from DC properties, an acquisition device (for example, a digital oscilloscope or a VXI system) with high resolution and relatively low sampling rate can be employed in the lower frequency range, to be substituted by a faster one with lower resolution at high frequencies. This makes sense because the contribution to the measuring uncertainty coming from the uncertainty in the phase shift ~p between the fundamental component of the magnetizing current and the secondary voltage (i.e. on the phase shift qOjH--('/r/2)- ~ between the fundamental component of the field and the polarization), whose minimization calls for a high number of sampling points and high amplitude resolution, is dominant only at low frequencies. The example shown in Fig. 7.44 of the evolution with frequency of the hysteresis loops

7.3 AC MEASUREMENTS ,

~

I

413 ~

~

i

I

,

,

,

I

,

,

,

nanocrystalline ribbon

Jp

0.004

I

~

~

,

I

,

,

~

I

~

~

,

I

,

,

14

0.002 o.ooo

-0.002 Hz

'

-0.004

-0.6-0.4-0.2

0.0 0.2 H (A/m)

0.4

0.6

70 nanocrystalline ribbon 60 50 "~ o v

40

"1-

30 20 10

I

1

10

100

Frequency (kHz)

1000

FIGURE 7.44 Evolution from quasi-static conditions to I MHz of hysteresis loops taken at Jp = 5 mT in nanocrystalline ribbons of composition Fe73.sCutNb3B9Si13.5 and corresponding behavior of the phase shift ~ I between the fundamental component of the field waveform and the polarization. The loops shown in the figure have been taken at the following frequencies: 100 Hz, 10, 50, 100, 300, 500 kHz, and 1 MHz.

414


and the phase shift ~H in an extra-low loss nanocrystalline ribbon demonstrates that the resolution features required for the A / D converters in order to appreciate the loop area are the more demanding the lower the frequency. These results, in particular, have been obtained by making signal acquisition by means of a 150 M sample/s digital oscilloscope having amplitude resolution decreasing, together with the sampled time window, from 14 to 8 bits. The record length is correspondingly decreased from 5000 samples to 150 samples minimum at 1 MHz. (5) With the increase of the magnetizing frequency above the kHz range, it becomes important to consider the role of stray inductances and capacitances. This is a most basic issue, the one making the real difference between low-frequency and high-frequency measurements, at least up to the radiofrequency domain, where the wavelength of the electromagnetic field becomes comparable with the dimensions of the test specimen. We need to reduce the effect of the stray parameters to the largest possible extent if we wish to determine the J(H) behavior as we do in the lowfrequency regime, without specific constraints regarding the non-linear behavior of the material. Under many practical circumstances, however, we might be specifically interested in the weak field response of the test sample, or simply in its behavior as an inductor, and correction for the effect of the stray parameters of windings and connecting cables can be attempted. While the effect of leakage inductances, that is, spurious flux linkages with windings and winding leads, can be generally kept negligible with respect to the flux generated by the magnetization in the material by closely fitting the windings on the magnetic core and by using short leads, the capacitance related effects require special consideration. Any winding is endowed with self-capacitance, which increasingly tends to drain current with increase in the frequency. This effect is reinforced by interwinding capacitance and capacitance in the connecting cables and at the input of the acquisition device. For example, a coaxial cable has a self-capacitance of the order of 50-100 pF/m, while the input impedance of a digital voltmeter or oscilloscope is typically around 1 Mf~ with a capacitance around 10-50 pF in parallel. The self-capacitances C1 and C2 of primary and secondary windings of the standard 700-turn Epstein frame can be of the order of 50-200 pF, much lower than the interwinding capacitance Co, which can be as high as 1000-3000 pF. Interposing an electrostatic screen between the windings and connecting it to the ground can eliminate the effect of Co, resulting, however, in a correspondingly enhanced value of C1 and C2. For magnetic sheet- and strip-testing beyond 400 Hz, this frame should be substituted, according to the Standard IEC 60404-10, by a 200-turn frame, which is deemed appropriate up to 10 kHz [7.141]. Because of the reduced number of turns, the effect of

7.3 AC MEASUREMENTS

415

interwinding capacitance can be minimized by laying out primary and secondary windings, which have the same number of turns, as a bifilar single layer with suitably spaced conductors. In this way, neighboring conductors are at the same potential and current leakage at high frequencies in the interposed dielectric is largely prevented. It is estimated that Co "~ 300 pF. To reduce both self and mutual capacitances of the windings and the related dielectric losses, not only must some space be allowed between successive turns, but also the dielectric material lying beneath the conductors must have low permittivity. Polysterene could be such a material. With ferrite cores, it is the test material itself that happens to be endowed with a high value of the dielectric constant, thereby favoring capacitive coupling between the neighboring winding turns. To limit this effect, a low permittivity dielectric tape should be wound beforehand on the core. Figure 7.45 provides an idea of the effect on the hysteresis loop, observed at 1 MHz in an M n - Z n ferrite ring, of a capacitance of 50 pF, equivalent to about a I m long connecting cable, inserted in parallel with the magnetizing winding (N1 = 5, N2 = 5, average ring diameter 30 mm). One can notice the tilting of the loop towards the second quadrant due to the fact that the measured primary current is the sum of the current leaking through the stray capacitance and the active magnetizing current and is consequently associated with an abnormal phase relationship with the magnetic induction. Figure 7.46a provides a qualitative description of the arrangement of connections and windings in a setup for the characterization of soft magnetic cores at medium and high frequencies. Bifilar single-layer windings are used and the connections are made by means of shielded cables. These cables should be short, as a rule, that is, of the order of a few centimeters at most in the MHz range to minimize the associated capacitances. In addition, the resistor R H should be connected to the magnetizing winding by a very short lead. Since the sample is flux-closed, there are no stray fields generated by it and the value of a H is not perturbed. Because N1 and N2 can be very low at high frequencies, the effect of coupling between the fictitious primary and secondary single turns located along the median circumference of the ring specimen could be appreciated. This effect should therefore be checked and possibly compensated. The setup in Fig. 7.46a is given a complete description in terms of lumped and stray parameters by the equivalent circuit shown in Fig. 7.47. Here, in particular, we have considered the self-capacitances (C1, C2) , the leakage inductances (Lwl, Lw2)~ and the resistances (Rwl ~Rw2) (primary and secondary windings), the interwinding capacitance Co, and the capacitances CH and Cj, which include the contribution of the cables and the input channels of the acquisition device. The value of RH~

416


0.006 " Mn-Zn ferrite ring sample 0.004 9 '

'

i

.

0.002

.

.

i

.

.

.

.

I

.

.

.

.

i

.

.

.

.

I

.

.

.

.

I

.

.

f

.

.

i

.

,

,

I-

.

o.ooo

-0.002

,.,/ C1- 50 pF

-0.004

t ,P

Jp=5 -0.006 f= 1 MHz ,

,

9 I

,

9 i

-1.5

,

I

. . . .

-1.0

I

,

-0.5

.

,

9 I

,

0.0

,

,

,

H (A/m)

i

,

0.5

,

,

.

,

1'.0

,

,

,

i

,

1.5

j

L

FIGURE 7.45 Hysteresis loop measured in a M n - Z n ferrite ring at 1 MHz with Jp = 5 mT before (solid line) and after (dashed line) insertion of a 50 pF capacitor in parallel with the primary winding. This capacitor emulates the effect of the stray capacitance introduced by a coaxial connecting cable about I m long. The primary and secondary windings are each made of five well-separated turns, wound in a bifilar layer. The tilting of the loop observed after insertion of the capacitor derives from the additional contribution of the current leaking through the capacitance to the measured primary current ill(t). Notice that the area of the loops (i.e. the energy loss) remains unchanged upon insertion of the capacitor.

wavfm, generatori ( ) power amplifier ~

1

b11~

"",~i

t

digital acquisition device

I' FIGURE 7.46 Schematic description of the setup for the characterization of soft magnets at medium and high frequencies. Primary and secondary windings are laid down as a single bifilar layer, with well-separated turns. A digital voltmeter or a digital oscilloscope can be employed for synchronous two-channel signal acquisition. The calibrated resistor RH is physically located close to the leads of the magnetizing winding and the shielded cables, which connect RH and the secondary winding with the acquisition device, are as short as possible. Notice that the lowpotential lead of the primary winding is separated from the ground by RH.

7.3 AC MEASUREMENTS

417 Co

G

i ~ R,-

T

=.L~ 1

Lwl

i

l:

UL1

lllI ILW2

c.I FIGURE 7.47 Equivalent circuit of the measuring setup in Fig. 7.46 taking into account the stray parameters: ul = voltage across the magnetizing winding; UL1= voltage balancing the primary e.m.f.; UL2 = -(N2/N1)UL1 secondary e.m.f.; u2=voltage across the secondary winding; C1,C2,Lwl,Lw2,Rwl,Rw2=self capacitances, leakage inductances, and resistances of primary and secondary windings; Cj, CH = capacitances of the connecting cables; Co = interwinding capacitance; R2 -- input resistance of the acquisition device.

typically ranging between I and 10 f~, is actually so small with respect to XcH = 1/~CH that we can safely disregard the role of CH and assume that the related voltage drop is always uH(t ) -- RHiH(t ). In fact, using a 20 cm long connecting cable and taking into account that with a digital oscilloscope we have typically R2 - 1 Mf~ with 10 pF capacity in parallel, we obtain XcH ~" 8 kf~ at 1 MHz. The basic question we pose here is how we can estimate the error introduced by the distributed parameters on the measured values of the power loss P and apparent power S. Both these quantities are experimentally determined, according to the base equations (7.20) and (7.24), starting from the current ill(t) supplied to the primary winding and the voltage u2(t ) appearing at the input of the acquisition device. In the presence of stray parameters, the quantities to be considered will be, instead, the magnetizing current im(t), resulting from the composition of ill(t) with the current i2~(t) (see Fig. 7.27b), and the voltage Uc2(t ) (see Fig. 7.47). The current ira(t) = (Im/N1)H(t) is then the current that flowing in the magnetizing winding in the absence of any secondary load, would

418


provide the field H(t) ensuring the rated induction B(t). If as is often the case, we disregard, the effect of the capacitance Co, we write for the current im(t) N2 im(t) - i l l ( t ) - icl(t)+ ~-~i2(t),

(7.51)

where icl(t) is the current leaking through the self-capacitance C1. To simplify the matter, we treat the material as a linear system, with the time dependence of field and induction described by phase-shifted sinusoidal functions. It is an acceptable approach because, on the one hand, we are often only interested in order of magnitude estimates of the errors deriving from the interference of the distributed parameters with field application and signal detection. On the other hand, a linear-like response of the material at increasing test frequencies is observed, due both to the prevalence of the classical eddy current loss contribution with respect to the domain-wall dependent loss contributions and the natural limitation on the achievable peak induction values. We have already introduced in Section 7.3.1 (Eqs. (7.27) and (7.28)) and illustrated with the equivalent circuit and the vector diagram in Fig. 7.27b the case where current is drained in the secondary circuit by the measuring instrument of input resistance R2. The correspondingly dissipated power AP must be subtracted from the measured loss Pmeasin order to obtain the actual power loss P in the material. With reference to Fig. 7.27a and b, no stray parameters being considered, we re-formulate Eq. (7.28) for the specific power loss under the assumption of sinusoidal time-dependent quantities.

P=

H(t) dt

is thus calculated by posing

H(t)-- Nlim(t)/lm and dB(t) dt

u2(t) N2A '

thereby obtaining p

~

~

~

1 N1

~

ma N2 imU2 cos qo--

~_~3Hp

Bp sin ~H

1 Xl~ ~ ma N2 IHU2 COS q012-- ma R2 -- Pmeas - hP,

(7.52)

7.3 AC MEASUREMENTS

419

where ~12 is the phase shift between ill(t) and u2(t)~ the measured power loss 1 Xl~ ~ ma N2 ZHU2 COS q~12

Pmeas - -

and the power dissipated in the load is &p-

1 522 ma R 2 "

Also, the measured and actual values of the specific apparent power (Smeas and S, respectively) differ because of the current circulating in the resistive load. We define them, based on Eq. (7.20), as ~ ~ N1 1 Smeas - - I H U 2 ~ ~ ~

S --

N1 1 l'mR2 ~ ~

N2ma

(7.53)

N2ma

and, being in this case

N2

im(t) -- ill(t) + ~ / 2 ( t ) , we obtain, under the general assumption i 2 'x

" ',

',,

I I

l

II

o...

9

...........~ 0.9 T

0.98

,~o

I

10

_

.8 T

~,

. . . .

0 x (mm)

(a) 1.00

I

--:~x Oo

,, ~ I, I I X , , .

Soft Fe C-core DO= 180 mm

oo omm

0.92 . . . . . . . . . , . . . . . . . . . . . . . . . . . . . -20 -10 0 (b) z (mm)

Z

l I Il

10

",1' -)---)!t---I ~:::r ~ ' J V ,uoHg

-)" x

20

FIGURE 8.5 Behavior of the reduced value of the axial flux density on the gap along the axial coordinate x (a) and the radial coordinate z (b) predicted by FEM calculation on the model C-core electromagnet shown in Fig. 4.23. The iron core is assumed without hysteresis and its magnetization curve up to saturation is known. Tapered (/3 = 54.74 ~ and untapered pole faces are considered, with gap length lg, core diameter Do, and pole cap diameter Dg as shown in the figure. The comparison is made for two values of the magnetomotive force. For N i = 25 x 103A we obtain /~0Hg(0, 0)---0.9 T, with both fiat and tapered pole faces. With N i = 125 x 103, tapering brings about 40% increase of/z0Hg(0, 0) (from 1.8 to 2.5 T) at the cost of reduced uniformity of the field in the gap.

486

CHAPTER 8 Characterization of Hard Magnets

power and cost of the apparatus, roughly scaling as D~. Notice that, the permeability of permanent magnets being very low, there is no shaping of the field by the sample, as is always the case with soft magnets (see Section 6.1), and the field generated by the electromagnet must necessarily be highly uniform. Let us consider the permanent magnet specimen inserted in the gap of the electromagnet and let us assume that the iron core is far from saturation. We can reasonably state, in view of the large differences between cross-sectional areas and permeabilities of sample and soft core, that the latter is a short circuit for the magnetomotive force generated by the specimen and that the magnetomotive force Ni exerted by the windings appears all across the specimen. An ideal magnet placed between the pole pieces of the electromagnet is perfectly flux closed and, as predicted by FEM calculations in the example provided in Fig. 8.6 (where the magnet saturation polarization Js -- 1 T has been assumed), the effective field on the sample H - - H g is zero (i.e. the demagnetizing field disappears) and the remanent state is attained (Br--Jr = Js). If the electromagnet is excited, the field appearing across the specimen is uniform and equal to H - - H g ~ Ni/lg as far as the permeability of the iron core is high. The superposition principle applies and the measured induction in the sample is B = Js-/z0Hg. Once the magnet polarization becomes balanced by the induction generated by the electromagnet, the coercive point Hcs is attained. The distribution of the flux lines in the neighborhood of this condition is described in Fig. 8.6b. A real magnet will actually suffer a decrease of its polarization, which implies that Hcs will be lower, but the homogeneity of field and polarization will be preserved. This will continue to be the case even when the field H is increased beyond the intrinsic coercive field Hcl, up to the point where the knee of the magnetization curve of the iron core is left behind and the saturated state is approached, at least, for wedgeshaped poles, in the region immediately behind the apex of the pole. Under these conditions, not only the field generated by the electromagnet will be increasingly non-uniform, but the soft core will progressively repel the flux lines generated by the magnet under test. The field distribution around the sample will then approach that of an open magnet immersed in the field generated by the electromagnet. In a real magnet, both polarization and effective field will be nonhomogeneous and the measuring accuracy will be impaired. Figure 8.7 provides a view of this effect through FEM calculations on the model system discussed in the previous figure. (2) Sample arrangement and signal detection. The test specimen is normally shaped, according to the recommendation of the IEC and ASTM

8.1 CLOSED MAGNETIC CIRCUIT MEASUREMENTS

\

.i"" B - -rT,c

/"

..... ~ ....

. .... j l ....

(a)

!!i .........

.,..,..-- - . . . . . .

I,

,,,... . . . .

.

-

487

.............. .......-................

i
1.2H1 are in the relationship (BH)max2 < ( H 2 ) 0"02454 (BH)maxl- H1 "

(8.4)

Alternatively, the same condition is applied to the coercivity ratio After having accurately centered the cylindrical sample surrounded by the secondary coil between the pole faces of the electromagnet, displaced the moveable pole and gently pressed it against the sample end faces, the magnetic field is brought to the maximum value Hp, known to satisfy Eq. (8.4), by applying a voltage ramp to the bipolar power supply via the programmable function generator. This can be followed by a triangular voltage function lasting one period. At the same time, the signals from search coil and Hall sensor (or H-coil) are detected, sampled, and A / D converted and the major hysteresis loop with all the relevant parameters is calculated. The supply can be used in either current mode or voltage mode. The former is more efficient, but the latter is to be preferred because it ensures protection against possible faults in the output circuit. Good information on the nature of the magnetization process (e.g. domain nucleation vs. domain wall pinning) comes from the initial magnetization curve. Its measurement requires that not only the sample, but also the iron core be fully demagnetized. By switching off the supply of the electromagnet at the end of a measurement run and taking off the specimen, a residual field in the gap is detected, the higher its value the narrower the gap. For a 10-20 mm gap, useful for typical samples, this

HcB2/HcB 1.


495

field can be of the order of 10-20 k A / m , sufficient to miss or perturb a portion of the initial magnetization curve. This occurs even in rare-earth magnets when a decoupled soft phase is present. The best way to cope with this problem is to demagnetize the iron core by putting the pole faces in contact and applying the usual procedure. Since the magnetization rate must in any case be very low, in order to always guarantee full penetration of the field in the core, and the whole process can last more than 30 min, one could alternatively adjust the pole pieces to the right distance and then inject a slowly increasing demagnetizing supply current. This is eventually regulated to the value corresponding to zero indication by a Hall sensor placed in the gap. If the test magnet can be saturated within the electromagnet, the demagnetization procedure can be done with the specimen in place. Many alternations are required to complete the process reliably. To demagnetize the specimen outside the electromagnet, a pulsed field source of the type described in Section 4.3 can be employed. For rare-earth magnets, a difficulty could arise because of eddy currents. The source should then be designed in order to provide a high initial peak field value and conveniently low oscillation frequency. In order to refine the process, a number of repetitions can be made, each time decreasing the charging voltage by a convenient factor. In this case, there can be considerable heat production in the solenoid and the sample temperature must be checked. Thermal demagnetization can be safely applied on ferrites, while it requires caution in all other cases. Rare-earth magnets, in particular, may suffer from uncontrolled structural modifications, not least oxidation, on temperature rising above the Curie point. Figure 8.11 provides an example of initial magnetization curve and hysteresis loop obtained in a 10 mm high anisotropic Ba ferrite cylinder of diameter 20 m m using an electromagnet with untapered pole faces of diameter Do = 250 mm. The sample was demagnetized via an oscillating discharge in a pulsed field solenoid with the maximum value of the peak field 2400 k A / m (flux density ---3 T), oscillating period around 5 ms, and time constant ~"= 18 ms. A triangular voltage waveform was generated, starting from zero value, in a burst-like fashion over 5/4 of a period T = 150 s and supplied to a 5 kVA DC-coupled power amplifier. The dJ/dt and dH/dt signals, achieved by means of a permcard, were amplified by means of two identical DC-coupled low-noise 100 Mf~ input impedance amplifiers and A / D converted via a 14-bit digital oscilloscope. The full magnetic characterization of the extra-hard rare-earth magnets cannot be carried out by means of closed circuit methods. There is little room for these methods in basic research on extra-hard materials. However, the determination of the parameters defining quality and domain of application of industrial products is generally possible,

496

CHAPTER 8 Characterization of Hard Magnets 1.5

Barium ferrite in electrom T= 150 s

I

1.0

0.5 .

.

.

j"-

.

0.0

i

Z

I-" v

..~ -0.5

-1.0

-1.5 I

'

I

'

I

'

I

-800 - 6 0 0 - 4 0 0 - 2 0 0

'

'

I

0 200 H (kA/m)

'

I

"

400

'

'1

600

'

I

800

FIGURE 8.11 Initial magnetization curve and J(H) and B(H) hysteresis loops measured in an anisotropic Ba ferrite permanent magnet using a closed magnetic circuit and the hysteresisgraph method. The test specimen was demagnetized by oscillating discharge in a pulsed field source (maximum field 2400 kA/m). The full hysteresis loop was traced in a time interval T = 150 s. according to the recommendation of the measuring Standards, because the material can be driven along the demagnetization curve in the second quadrant by means of an electromagnet generated field. Of course, other field sources must be employed to bring the material into the saturated state before starting the travel along the demagnetization curve. Magnetic flux densities up to 9 T (~-6400 k A / m ) are usually produced using NbTi superconducting solenoids at 4.2 K. Pulsed field solenoids capable of producing transient fields up to about 8000 k A / m over a useful volume by capacitor discharge are commercially available. Used in the LR mode (see Section 4.3), they can saturate most practical rare-earth magnets. Because of possible lack of flux penetration in the test sample during application of the transient field, it is good empirical practice to ensure that the field strength remains higher than about 3000 k A / m for a time interval longer than 10 ms. Once saturated, the sample is brought into the gap of the electromagnet, where the iron core is demagnetized, and it is inserted in the search coil, which is kept axially centered by means of a suitable holder. It is understood that the electromagnet is of the tapered


49V

type, possibly with Fe-Co pole caps, and the supply system is adequately sized. Immediately before insertion of the sample in the coil, acquisition of the flux derivative starts. It continues over the successive steps of the process, which consist in the closure of the pole pieces over the sample and the application of a voltage ramp to the power supply in order to provide as high as possible a field in the same direction of the magnetization in the sample (the "forward direction"). When the peak field value Hp is reached, the corresponding magnetization value, calculated by integrating the signal collected through the search coil since start and accurately eliminating the drift, is recorded. All these operations are carried out by digital methods and computer control. We can reasonably assume that, since the materials to which this procedure applies have all very high Hcj value, only a small fraction of the magnetization is irreversibly lost under the influence of the demagnetizing field and that most of it is recovered by remagnetizing the sample to the maximum field Hp. This conclusion is expected to apply both to conventional magnets and nanostructured spring magnets. One can prove it empirically by repeating the experiment with samples having different aspect ratios. It is in any case recommended that the height-todiameter ratio of the test specimen be greater than 1. In conclusion, it is expected that, once Hp has been reached, the magnetizing current is decreased to zero, and finally it is reversed down to the symmetric - H p field, that the whole second quadrant along the limiting curves B(H) and J(H) is covered. The absolute determination of the B(H) and J(H) curves via closed magnetic circuit and the hysteresisgraph or ballistic method can be made traceable to the base SI units by means of a suitable calibration procedure which involves: (1) Measurement of the cross-sectional area A of the cylindrical specimen via the measurement of the diameter with a calibrated micrometer. Alternatively, A is obtained by a measurement of mass, density, and length. Traceability is made to the standard of length in the first case and to the standards of length and mass in the second case. (2) Calibration of the turn-area product of compensated coils and H-coil. It is made using a standard field source (solenoid or Helmholtz coil), supplied by a precisely known AC current. Traceability is made to voltage and resistance standards (for the supply current) and the frequency standard (for the generated field, via NMR measurement). (3) Calibration of the Hall setup for the measurement of the magnetic field. To this end, field sources (solenoids and electromagnets) calibrated through NMR measurements are used. Again, traceability is made against the frequency standard. (4) Calibration of amplifiers and acquisition setup by means of reference sources and voltmeters, ensuring traceability to the voltage

498


standard. In order to avoid the burden of absolute calibration, reference Ni samples, if available, can be conveniently exploited [8.6]. On the whole, the measurement of the technical parameters of the permanent magnets, which are associated with the behavior of the B(H) and ](H) curves in the second quadrant, suffers from a certain lack of reproducibility if compared with measurements made on typical soft magnetic materials. This is confirmed by the results of intercomparison exercises, such as those on Fe-Si laminations mentioned in Chapter 7 (Fig. 7.8) and Chapter 10 (Figs. 10.2 and 10.4 and Table 10.2) and those on N d - F e - B specimens reported in Fig. 8.12 [8.10]. From the statistical analysis of the 1700 1600

E 1500 1400 1300

Nd-Fe-B '

I

'

I

2

'

I

4

6

'

I

'

I

'

I

8 10 Laboratories

'

I

12

14

'

I

16

240 co

E

230

~ 220 E

~ 210 200

Nd-Fe-B '

I

'

I

.

'

I

6

'

;

'

Laboratories

'

,'2

'

1'.'1'6

FIGURE 8.12 Intercomparison exercise on the measurement with the closed magnetic circuit method of the intrinsic coercive field Hq and the maximum energy product (BH)max on Nd-Fe-B magnet specimens (15 mm diameter, 5 mm height cylinders). Each point represents the best estimate of the measured quantity made by each laboratory. The calculated relative standard deviation of the results around the reference value (the unweighted average), calculated after having excluded the outliers, turns out to be 4.7% for Hcl and 3.6% for (BH)max (adapted from Ref. [8.10]).

8.2 OPEN SAMPLE MEASUREMENTS

499

latter results, one finds, for example, that the best estimates on the intrinsic coercive field Hc! reported by the different laboratories fluctuate around the reference value (the unweighted average) with a relative standard deviation cr = 4.7%. For the maximum energy product (BH)max cr = 3.6% is obtained. It is not surprising that materials like rare-earth magnets, having extreme and somewhat unstable magnetic behavior that is strongly temperature dependent and influenced by magnetic viscosity effects, exhibit appreciable scattering of the measured properties from laboratory to laboratory. According to the reported ensemble of results, this conclusion can be drawn for both the closed magnetic circuit and the open magnetic circuit methods.

8.2 O P E N SAMPLE M E A S U R E M E N T S Hard magnets have their properties influenced by the demagnetizing fields, but they are only slightly sensitive to the environmental fields, which can disrupt or falsify measurements made on open soft magnetic strips and sheets. Consequently, testing hard magnets as open samples is not only methodologically correct, but it opens a good, wide scenario in terms of novel measuring techniques and flexibility as to the type of materials to be characterized and the size and shape of the test specimens. Open samples reveal their presence through the stray field they emit and their state of magnetization can be determined through, besides the previously mentioned torque and force methods, the measurement of such fields. Homogeneous and ellipsoidally shaped samples have uniform magnetization, which can be determined by measuring the field in a point close to the sample, for example in a point belonging to one axis of the ellipsoid. In this case, in fact, stray field and magnetization are related by closed analytical expression. Search for the zero stray field condition upon the application of an increasing reverse field on a previously saturated sample leads to the determination of the intrinsic coercive field Hcl. For a non-ellipsoidal sample, the condition of zero average sample polarization is detected. The measuring arrangements shown in Fig. 7.17 and the procedure described in detail in the Standard IEC 60404-7 [8.18] can be applied to both hard and soft magnets, as discussed in Section 7.2.1, with the limitations imposed by the relatively low field strengths which can be generated. With the field-sensing probe placed at a sufficiently large distance from the sample, only the dipolar field is detected and the magnetic moment can be determined, again providing the volume-averaged polarization. In all cases, if the measurement is made in the presence of an applied field, methods must be devised

500


for compensating its effect. In permanent magnets, a field can easily result much larger than the field generated by the test specimen at the probe location. For example, the field generated on its axis by a 5 m m x 5 m m cylindrical permanent magnet with polarization 1 T at a distance of 20 m m is of the order of 1.5 k A / m , orders of magnitude lower than the field to be applied to bring it into such a magnetic state.

8.2.1 Vibrating sample magnetometer The flux linked with a sensing coil placed at a certain distance from an open sample subjected to an intense magnetizing field can be seen as the sum of a main contribution due to such a field plus a perturbation originating from the sample. We are interested in measuring such a perturbation. An effective and simple way of separating it from the background is a sort of AC magnetometric method, where linkage of the sensing coil with the signal generated by the sample is made to vary rapidly with time, all the rest remaining unaltered. This can be obtained by imparting a vibrating motion to the sample, so as to produce an AC signal while making a DC characterisation. Any background constant flux is automatically filtered out and signal optimization can be pursued if some degree of flexibility exists in the amplitude and frequency of the oscillation and the arrangement of the sensing coils. The popular vibrating sample magnetometer (VSM) is based on this principle [8.19]. It is a general purpose, high-sensitivity magnetic moment measuring device, not only perfectly suited to the characterization of permanent magnets and recording media, both in advanced research and routine industrial testing, but also applicable to weakly magnetic and paramagnetic materials. Its sensitivity can be made very high and a lower limit for measurable magnetic moment is typically around some 10 -9 A m 2 in commercial setups. This limit can be extended, by optimization of the coupling between sample and sensing coils, down to a noise floor of the order of 10 -12 A m 2 (10 -9 emu) [8.20]. On the other hand, the upper limit for the measurable moment can be of the order of 0.1 A m 2 and higher (for example, the magnetic moment of a 7 m m diameter iron sphere is around 0.3 A m2). To see how the variable magnetic field generated by the oscillating sample links to the pickup coils, one may start from the equivalent dipole of moment m and the known formulation for its field, from which the flux threading the coils and its time variation upon the dipole vibration are calculated. A more elegant and simpler way of treating this problem can be pursued by invoking the reciprocity principle [8.21]. We have discussed it in Section 6.2. In essence, this principle amounts to the well-known theorem stating that the flux mutually linking


501

two coils is independent of which one carries the current, i.e. a unique mutual inductance coefficient M -- M12 -- M21 can be identified. By virtue of the equivalence between magnetic dipole and current loop, it can be generally stated that the magnetic flux 9 originating from a dipole of moment m located in a point of coordinates (x, y, z) and threading a coil (or a system of coils) characterized by the constant k(x, y, z) = B(x, y, z)/is, the ratio between the flux density B(x, y, z) generated by the coil in this point when a current is circulates in it, is given by the equation = k(x, y, z).m -- kx(x, y, z).mx + ky(x, y, z).my 4- kz(x, y, z).mz

(8.5)

(already presented as Eq. (6.8)). A dipole moving with velocity t will therefore induce in these coils the instantaneous voltage

u(t)-

d(I) dt - (1/is).grad(B.m).r.

(8.6)

For a voltage to be generated upon vibration of the dipole, it is required that the gradient of the flux component along the direction of m that would be generated by the fictitious current is circulating in the coil be different from zero. A simple coil arrangement, ensuring defined value of this gradient together with symmetry properties, is obtained with the series opposing pair shown in Fig. 8.13a, already introduced in Section 4.1.3. They generate an axial field Bx(x) always passing through the zero value at the origin, where the gradient is maximum and attains a value depending on the ratio between the radius of the coils a and their distance d. When d - - a , they form the so-called inverse Helmholtz pair and the induction derivative at the origin is

Nis dBx(0) dx - 0"8587/~~ a 2 if N is the number of turns in each coil. Because of the antisymmetrical configuration of the coils, the field derivative contains only even terms in the development around the origin. By placing the coils at the distance d --- x/3a, the third order term in the field disappears and the linearity is improved. With the magnetic moment m directed along the axis of the coil pair, as shown in the figure, and moving with velocity • Eq. (8.6) takes the form d u(x, t) = ~x (mkx(x)).• = m.gx(X).•

(8.7)

502

CHAPTER 8 Characterization of Hard Magnets z

T.a

I

(a)

w

m w-

I d

1.01

r

gx (x)

1.00

d = .v/3a

-4

0.99

-2

i

...... ""

i

~

k (b)

-0.4

-0.2

0.0

x/a

0.2

0.4

2

.,, //J"

4

)

Cc)

FIGURE 8.13 Relative sensitivity function vs. displacement of a small sample of magnetic moment m along the x axis of a thin-coil pair for different values of the ratio between intercoil distance d and coil radius a. The coils are connected in series opposition. Curve 1: d = a (inverse Helmholtz pair). Curve 2: d - x/3a (maximum homogeneity of the sensitivity function) (Eq. (4.24)). Curve 3: d = 1.848a. (Graphics in b), (adapted from Ref. [8.22]). (c) The sensitivity function averages out to zero over a region of the order of 4a.

w h e r e k x ( x ) - Bx(x)/is. The function

gx(X) is called reduced radius a different

dkx(x) dx

(8.8)

sensitivity function. Figure 8.13b s h o w s the b e h a v i o r of the sensitivity function gx(x)/gx(O) for the o p p o s i n g coil pair of as a function of the r e d u c e d distance from the origin for three values of the coil interdistance d. As p r e v i o u s l y r e m a r k e d ,


503

the maximum homoj~eneity around the origin of the sensitivity function is obtained with d -- ~/3a. Figure 8.13c shows that this comes at the expense of a certain reduction in the maximum value of the function attained at the origin. The vanishing of the derivative Ogx(X)/Ox at the origin is evidently due to the symmetry of the coil arrangement. In general, one looks for sensing coil arrangements making the point at the origin a saddle point for the sensitivity function because, in the neighborhood of such a point, the signal is insensitive to the first order to the sample position. For a small-amplitude vibration of the magnetic moment around the origin, we can safely assume gx(X)-~ gx(O). If, for example, the time-dependent amplitude for an oscillation in the x direction is x ( t ) - - X o sin ~ot, the sinusoidal voltage u(t) = mgx(O).k = mgx(O).XooJ cos o~t

(8.9)

is induced in the coils. The symmetry conditions on the coil arrangement leading to a saddle point can have a large number of solutions. If the field is applied by means of a superconducting solenoid, the vibration is necessarily exerted along the axial direction (x-axis) and the series opposition coil pair we have just discussed is commonly employed. In practice, however, the filamentary coil approximation may not apply and one should make use of numerical methods in coil design. Recall that general rules for the realization of the Helmholtz condition with thick coils have been discussed in Section 4.2. When the field is applied by conventional electromagnets or by permanent magnet sources, transverse sample vibration (z-axis) is generally adopted. In such a case, the previous Eq. (8.7) becomes d u(z, t) = -~z (mkx(z)).• = m.gx(Z).•

(8.10)

with kx(z) = Bx(0, 0, z)/is and the sensitivity function gx(Z) dkx(O,O,z)/dz. Some of the principal coil arrangements employed in VSM with transverse vibration are sketched in Fig. 8.14 [8.22]. They employ two, four, and eight identical coils, respectively, and are endowed with the symmetry properties making the point at the origin a saddle point. Notice that all these arrangements, like the previously discussed inverse coil pair, are of the compensating type, making the measured signal insensitive to variable external fields. The marked arrows indicate the way in which the signals from the coils have to be added. The four-coil configuration is known as the Mallinson set [8.21], while with the eight-coil arrangement one can intercept the components of the magnetic moment along x, y, and z. In particular, with

504

CHAPTER 8

Characterization of Hard Magnets

X

Ha

@ |

1

h

i

! ~

(a)

(b)

1.2

2-coil

x

1.0

@ ...."-i ..... (c)

d

~5 o

0.8

1

Y" Z" "4-coil ~

x

.__. 0.6 (D

rr 0.4 x

0.2 0.0 -0.3 -0.2 -0.1 0.0 0.1 x/a, y/a, z/a (d)

Y 0.2

0.3

FIGURE 8.14 Examples of saddle point coil arrangements in a VSM for vibration perpendicular (z-axis) to the direction of the magnetic moment (x-axis). The sample is attached to a non-magnetic vibrating rod. The arrows marked on the coils identify the way in which the signals from the coils have to be added. The four-coil system is the so-called Mallinson's set [8.21], while with the eight-coil configuration and a different set of connections, the component of the magnetic moment along the y axis can also be detected [8.23]. The diagram (d) shows the behavior of the relative magnetometer output (i.e. the sensitivity function gx(Z)) for transverse vibration when the position of the vibrating sample is displaced along one of the reference axes x, y, z. The calculations are made assuming small coil size with respect to the intercoil distances (small coil approximation) and the relative output is normalized to the unit turn-area of the coil system. The upper and lower families of curves for the two-coil arrangement are obtained for flat coils (d = 1, h - - 0 ) and long coils (d -- 1, h -- x/~), respectively. Four-coil arrangement: the dashed lines are obtained with dx = 1, dz = 0.389, while the continuous lines correspond to d.,. = 1, dz = 0.678. Eight-coil system: d.,. = dy = dz = 1/~/2. This system exhibits, by virtue of its cubic symmetry, identical response along the three axes (adapted from Ref. [8.22]).


505

the connections made as indicated in the figure, it detects the mx component, while with a different set of connections it can be made to detect one of the other two components [8.23]. The diagram in Fig. 8.14 provides an overview of the response of these different coil arrangements, with m directed along the x-axis, calculated under the assumption of a coil size small with respect to the intercoil distance [8.22]. In particular, the behavior of the relative magnetometer output, that is, the behavior of the associated sensitivity function, is calculated as a function of the position along the three reference axes of the transversally vibrating sample in the neighborhood of the saddle point. It is generally observed that increased flatness of the sensitivity function comes at the expense of its maximum value, that is, of the voltage output of the pickup coils. To increase the output value, one could increase the number of turns as far as this implies a gain with respect to the associated increase of the Johnson noise, which goes proportionally with the total coil resistance. The discussion so far on the prediction of the sensitivity function for the various sensing coil arrangements has been based on the assumption that the test specimen can be assimilated to a point-like dipole. If this is not the case, we shall have to consider the variation of the sensitivity function and, in case of non-ellipsoidal samples, of the magnetization over the sample volume. If we take the simple case of the inverse coil pair in Fig. 8.13a and a sample of volume V oscillating along the x-axis, the instantaneous induced voltage will be obtained by generalization of Eq. (8.7) by integrating it over the sample volume. If we assume that the center of gravity of the homogeneous sample has coordinate x, we obtain

Ux(X, t) -- ~v Mx(r' x)gx(r' x)dr3"•

(8.11)

where r identifies a generic point within the sample. For ellipsoidal samples, Mx is uniform and Eq. (8.11) reduces to

mf

Ux(X, t) = -~

v gx(r, x)dr3.x,

(8.12)

where, in place of the sensitivity function at point x, we have an average over the sample volume, shifted by a distance x from the origin. It can be induced from Fig. 8.13c, showing the extended behavior of the sensitivity function gx(X), that a long sample in the x direction generates a low signal and practically no signal at all when its length is a few times the coil radius. An interesting consequence of this fact is that,

506


since ~o gx(x)dx = 0, the vibrating rod is not expected to contribute to the measured signal. To apply the vibrating sample principle in a permanent magnet measuring setup we basically need a stable and rugged vibrating assembly, a digitally driven magnetic field source (an electromagnet or superconducting solenoid), a lock-in amplifier for the voltage induced in the pickup coils, an auxiliary signal source synchronous with the frequency of oscillation of the sample, to be exploited for precisely driving the vibration amplitude, a field sensing system and, if required, a temperature controller. A computer is used for general control of the measuring procedure and for analysis of the results. A measuring arrangement implementing these requirements is schematically represented in Fig. 8.15. We summarize here the basic operations performed with a system like this one and a few general problems associated with the VSM method. (1) Field generation and control. The measurement of the magnetic moment of the test specimen as a function of the field strength can be performed by continuous variation of the field with time, as conventionally carried out with the hysteresisgraph method (sweeping mode). The voltage simultaneously induced in the pickup coils is simultaneously detected and processed to determine the magnetic moment. However, in order to improve the signal-to-noise ratio, averaging of the signal must be performed with a reasonable integration time of the order of I s or longer. The lower the signal, the longer the integration time and the larger the lagging of the magnetic moment with respect to the field. This problem can be addressed by generating the field in steps. After each step, the measurement of the magnetic moment takes place. Because of the non-linear behavior of the material, it might be desired to regulate the step amplitude along the hysteresis loop in order to make homogeneous the vertical resolution. This can be done by means of a real-time feedback procedure, based on the reading of the field in the gap by the Hall sensor, comparison with the target field value, and generation of the appropriate magnetizing current via a suitable algorithm. An interfaced DC source, driving a bipolar power supply connected to the magnetizing winding, is used to this purpose. A complete hysteresis loop can be traversed in several minutes, a far longer time than with the conventional hysteresisgraph method. In addition, the maximum available field in the gap of the electromagnet is lower when employed in the VSM mode because space must be allowed for the vibrating rod and the pickup coils. The latter, in particular, should be placed close to the sample for obvious sensitivity reasons [8.20] but, at the same time, keeping them at some distance from the pole


507

P C - Vibration control, field control, processing.

I

DC

Vibrating

Ref. magnet

head

to the

oven / cryostat

source

/

Bipolarpower supply

_

Ref. coils Sample

Pickup coils'[

i

p

.

1

I

.

g rod

/ .

.

.

/

Pickup coils

Hall plate

FIGURE 8.15 Scheme of vibrating sample magnetometer using an electromagnet as field source. The law of variation of the magnetizing current with time is defined by software, implementing real-time control of the field strength by means of feedback, driven by continuous reading by an interfaced Hall unit. The current is generated by means of a bipolar power supply driven by an interfaced DC source. The voltage induced in the pickup coils is amplified by means of a lock-in amplifier, whose internal reference signal, driven via a computer-controlled procedure relying on the signal generated by the vibrating reference magnet, is used to feed the power amplifier supplying the vibrator. Tracing a complete hysteresis loop can take several minutes.

508


faces helps in reducing the image effect (Fig. 6.11). In this way, the field lines emerging from the sample and intercepted by the coils are the least distorted by such an effect. (2) Test sample arrangement and vibrating system. A permanent magnet test sample most commonly comes shaped as a 2 - 3 m m diameter sphere. With the usual size and arrangement of practical pickup coils, during the measurement it lies in a region of uniform sensitivity function. The spherical geometry is obviously the ideal one because it guarantees uniform magnetization in the sample and accurate retrieval of the effective magnetization curve after correction for the demagnetizing field. It is also very easy to prepare small spheres using the method of random grinding mentioned in Section 8.1. Other shapes, even irregular ones, can be tested as well. For cylinders and parallelepipeds, the measurement will provide the average polarization in the sample and the magnetometric demagnetization coefficient will be used for field correction [8.24]. With magnetic tapes and thin films, the disk geometry can be conveniently adopted and the associated demagnetization factor will approximately correspond to that of the oblate ellipsoid with the same axes length. For prismatic thin films, the magnetometric demagnetizing coefficients calculated by Aharoni could be adopted [8.25]. Notice, in any case, that the combination of small thickness-todiameter ratio and hard magnetic properties make even an approximate correction for the demagnetizing effect acceptable in many cases. The test sample is firmly held in a small container, which is screwed into the end of the vibrating rod. If the material is anisotropic, grinding a small sphere out of the bulk wipes out any visible information on the direction of the macroscopic anisotropy axis. We can recover it by letting the spherical sample freely orient itself in a sufficiently high field. For a precise determination, we can search for the orientation of the sphere in a VSM setup associated with the maximum value of the remanence. Figure 8.16a provides an example of hysteresis loops measured in a 3 m m diameter sphere of anisotropic Ba ferrite along different directions in a defined plane. Successive directions are tested and the preferential axis is determined by making the sample rotate first around a generic axis, to identify a plane containing the anisotropy axis, then around an axis perpendicular to such a plane. We see here how one can easily and advantageously exploit the open sample geometry of the VSM method and the spherical shape of the test specimen for obtaining physical information on the properties of the material, which are difficult to obtain with the conventional closed circuit method. Preliminary to the measurement, the sample must be centered to find the saddle point of the sensitivity


0.4

509

Anisotropic bariu'm f ~

Isotropic strontium ferrite

0.4

,/f

0.2

0.2

i

/

G" ~, 0.0

o.o

t

J i

t !

-0.4 84 ~

(a)

/

-0.2

-0.2

-~ooo -5oo

o

H (kA/m)

5oo

J

-0.4

.....

,

~ooo

,

|

. . . .

|

(b)

Anisotropic bariurn ferrite - , ~ ; ~

0.4

.

-1000 -500

.

.

.

.

.

.

.

|

. . . .

0 500 H (kA/m)

|

,

,

1000

__

Closed

0.2

, magnetic ' !

circuit

o.o ! ! !

--0.2

!

| !

'!

'

| !

~ _ j t r

-0.4-

-- ~:....

_

.

(c)

i i#

_ --

.

.

000 9

.

.

|

.

-500

.

.

.

.

.

.

0

.

H (kA/m)

~VSM !

.

500

.

.

.

1000

FIGURE 8.16 (a) Hysteresis loops measured with a VSM on an anisotropic sintered BaFe12019 sample, tested as a 3 mm diameter sphere. The preferential direction (0~ and the distribution of the easy axes around it on a plane, induced from the value of the remanence, are determined by making the sample to rotate first around a generic axis, to identify a plane containing the anisotropy axis, then around an axis perpendicular to such a plane. (b) The same experiment made on nominally isotropic SrFe12019 spherical sample provides nearly coincident hysteresis loops. (c) The hysteresis loop taken along the preferred direction is compared with the loop obtained on the parent bulk sample with the closed magnetic circuit and hysteresisgraph method.

function gx(Z) (transverse vibration), to avoid a n y d i s t u r b i n g a s y m m e t r y r e g a r d i n g the mechanical action of the charged pole faces on the sample, a n d to ensure reproducible m e a s u r e m e n t s . It is a s s u m e d that the coils h a v e a l r e a d y b e e n set in place before centering a n d they have

510


been tightly locked to the pole caps. The centering operation consists in starting the vibration, applying a sufficiently high DC field, and making mechanical regulations on the position of the sample along the x-, y-, z-axes (as defined in Fig. 8.14), in order to eventually leave the sample in the saddle point. This amounts to finding the position where the signal induced in the coils is minimum for displacements along the x-axis and is maximum for displacements along the y- and z-axes (e.g. Mallinsons's coils). Notice that at the end of this operation the sample is left at some remanence point. It might therefore be necessary to demagnetize it before starting the measurements. Notice also in Fig. 8.16c the comparison between the hysteresis loops obtained by the VSM and the closed circuit method in the previous anisotropic Ba ferrite. The coercive field appears to be slightly lower in the VSM determined loop. Since the time taken to traverse the whole loop is much longer in this case (about 20 min vs. about 100 s), it is plausible to attribute such a difference to the thermal fluctuation aftereffect. A fundamental requisite of the vibrating system regards the stability of frequency and amplitude of the oscillation imparted to the sample. To this end, the vibrating head is supplied by a reference signal generated by the lock-in amplifier and suitably amplified. Mechanical and electrical effects may, however, cause a drift in the performance of the vibrator. For this reason, a reference signal is generated in a pair of coils by a permanent magnet attached to the vibrating rod at a distant position from the measuring pickup coils, it is amplified and compared by software to the target signal. Any difference is numerically compensated and the driving signal of the lock-in amplifier is modified in order to recover the programmed vibration frequency and amplitude. There are no special restrictions as to the frequency of vibration, provided it is far from any mechanical resonance frequency of the apparatus. It is also useful to keep it incommensurate with the line frequency. Since frequency and amplitude of the vibration go hand-in-hand, it is required that their product does guarantee useful signal amplitude. Typically, f ranges between few Hz and some 100 Hz, with the sample oscillating from around 0.1 mm to a few mm. (3) Calibration, sensitivity, and noise. The calculation of the sensitivity function gx(Z)~ either in closed form or by means of a numerical procedure, does in principle provide the means for an absolute measurement of the magnetic moment. In practice, the burden of an absolute approach is not worth the expected resulting low accuracy. Consequently, calibration by comparison with the measured magnetization of a standard nickel sphere at a defined high field value is the rule (Fig. 8.1). However, this procedure also has certain limitations: (1) The measurement may be more or less


511

affected by the image effect, which is automatically taken into account by means of the comparative calibration procedure. However, the image effect depends on the permeability of the pole faces, while the calibration factor for the measured magnetic moment is determined for the defined value of the field applied to the Ni sphere (i.e. of the permeability). Such a factor is therefore expected to change slightly with the strength of the field in the gap, for example along the hysteresis loop. (2) If the test sample and reference Ni sphere are very different in size, calibration may be affected by non-flatness of the sensitivity function. (3) The comparative calibration with the Ni sphere is based on the dipole approximation. If non-spherical samples are to be tested (for example, particulate media and thin films) and their size is larger than the size of the Ni sphere, calibration can introduce appreciable systematic errors [8.26]. The most direct way to cope with this problem would consist in the absolute calibration of reference Ni samples as far as possible similar to the actual non-spherical test specimens. The maximum sensitivity of commercial VSM setups is around 10 - 9 A m 2 (10 - 6 emu). This is totally appropriate for the majority of testing requirements in bulk permanent magnets and recording media. Weak magnets and paramagnets can equally be tested, while the high value of the demagnetizing coefficient of the test specimen makes the VSM method unsuitable for the characterization of soft magnets. Foner has actually shown that, since the signal intercepted by the coils is expected to increase approximately inversely as the cube of the distance between the detection coils and the sample, while the noise decreases by decreasing the size of the coils, a very large sensitivity gain can be obtained by simultaneously decreasing the coil size and the coil distance from the sample [8.20]. Ultimately, a change in magnetic moment around 10-12A m 2 (10 -9 emu) should be detectable. In practical setups, small intercoil distances introduce certain complications. For example, all operations regarding sample insertion and centering become more difficult, the sensitivity function gx(Z) becomes sharper, and calibration uncertainties due to different size of sample and Ni standard are enhanced. In any case, maximum control and reduction of the background signal and noise must be sought, an increasingly difficult task with increasing sensitivity. In general, the following sources of background signal and noise are expected to play a role: (1) Signal from the paramagnetic or diamagnetic sample holder and, if any, from the sample substrate. In case of thin films or hard disks, the contribution from the substrate can be even larger than the contribution from the magnetic material. It must be subtracted by making measurements with and without the magnetic layer. (2) Interference from sources of

512


electromagnetic fields in the neighborhood of the apparatus and within the apparatus itself. The coils are intrinsically insensitive to external variable fields, but compensation is never perfect. (3) Vibrations of the pickup coils. Spurious signals can be induced in the coils if they vibrate in the applied field because this, besides being very high, can also be slightly inhomogeneous. Vibrations are principally caused by mechanical coupling between the vibrating assembly and the coils. They represent a troublesome problem because the ensuing signal, having the same frequency as the signal induced by the vibrating sample, cannot be filtered out by the lock-in amplifier. The simplest way to minimize the effect of vibrations is by locking the coils to the magnet. An additional countermeasure consists in interposing vibration dampers or even active antivibration elements between the vibrating head and the rest of the apparatus [8.27]. (4) Johnson noise generated in the coils. A few thousand turns of small diameter copper wire (0.1 m m or less) are normally used in each coil, which implies a resistance R around 100 12 and higher. Johnson noise has a white spectrum of density ~(f) = 4kTR, with k the Boltzmann's constant and T the absolute temperature. If the measurement bandwidth is hf, the associated r.m.s, voltage is uj = 4 k T R A f .

The degree of reproducibility of the measurements made on permanent magnets using the VSM method is comparable with that expected for measurements made with the closed magnetic circuit method. This has been demonstrated by means of an interlaboratory comparison, where both methods have been applied [8.10]. For the VSM-based comparison, Ni spheres of diameter 2.79-2.99 m m have been circulated among the laboratories. We have mentioned the results concerning the closed magnetic circuit in Fig. 8.12. A similar analysis is presented for the VSM tested N d - F e - B magnets in Fig. 8.17. If outliers are excluded from the analysis it is found that the best estimates made by the different laboratories of the intrinsic coercive field Hc! fluctuate around the unweighted average value with a relative standard deviation cr = 3.7%. For the maximum energy product (BH)max, it is found cr-- 2.3%. For investigation on anisotropic materials, information on the magnetization value and hysteresis behavior only in the direction of the applied field may provide too limited information. The effective field can, in fact, be very different not in only in magnitude, but also in direction with respect to the applied field, and the simultaneous measurement of the components of the magnetization along the field direction and orthogonal to it (the x- and y-axes in Fig. 8.14) is generally required. The simplest way to achieve vector measurement is by adding a set of pickup coils in the direction orthogonal to the field direction. It is also possible to


513

2500

Nd-Fe-B

--, 2000

E

:1::~ 1500

1000

9

'

I

'

2

I

6

'

I

Laboratories

9

I

8

10

'

9

I

12

240 E

220 200

E

~-

18o 160 140

Nd-Fe-B '

I

2

'

I

4

'

I

6

'

Laboratories

I

8

'

I

10

'

I

12

FIGURE 8.17 (a) Intercomparison exercise on the measurement with the VSM method of the intrinsic coercive field Hcj and the maximum energy product (BH)max on N d - F e - B magnets. The measurements have been performed on spheres of diameter around 3 mm circulated among the laboratories. Each point represents the best estimate of the measured quantity made by each laboratory. The calculated relative standard deviation of the results around the reference value (the unweighted average), calculated after having excluded the outliers, turns out to be 3.6% for Hcl and 2.3% for (BH)max (adapted from Ref. [8.10]).

detect either mx or my with the very same coil assembly, p r o v i d e d the connections between the individual coils are modified in a suitable way. Four-, eight-, and twelve-coil configurations have been devised on purpose, as discussed in detail by Bernards [8.23]. Vector m e a s u r e m e n t s are frequently used in the investigation and characterization of recording media. If a small disk-shaped thin film or particulate media sample is placed with its plane coincident with the (x, y) plane and its hysteresis loop is d e t e r m i n e d as usual along the direction of the applied field (x-axis), we

514


have an indirect method for measuring the anisotropy field by taking loops at different angles of rotation of the sample around the z-axis (Fig. 8.18a). If the magnetization component My is simultaneously determined together with the component Mx, we obtain the instantaneous

Sample /

/~

g Easy axis

i',, Pickup coil (a)

Easy ax"

/.//"

(b) FIGURE 8.18 (a) Schematic view of a vector VSM setup employing two orthogonal sets of pickup coils and circular test sample (not in scale). The disk-shaped sample is vibrated along the z-axis. Under sufficiently high field I-I,, the anisotropy constant of the material can be determined by measuring the component My of the magnetization as a function of the angle 8. (b) In perpendicular/oblique-anisotropy recording media (uniaxial anisotropy), there is interest in determining the magnetic properties as a function of the angle made by ~ with respect to the normal to the sample plane. To this end, the test plate is placed at the center of the gap, with its plane perpendicular to the (x, y) plane, and it is rotated by an angle cr about the z-axis. The demagnetizing field Ha is normal to the sample plane.


515

value of the torque per unit volume exerted by the field "r =/z0M • Ha -- la,oMyHa ~..

(8.13)

At equilibrium, this torque is balanced by the torque due to the magnetic anisotropy. In fact, for sufficiently high applied field strength, the residual parasitic torque due to domain wall processes is negligible and iMI = Ms. In the case of uniaxial anisotropy, the energy per unit volume arising from the rotation of the magnetization vector by an angle 0 with respect to the easy axis is EA = K1 sin20 (disregarding higher order constants) and the associated torque is "rA = (dEA/dO)~.. At equilibrium, ~'A = ~'H, that is la,oMyHa = 2K1 sin 0 cos 0, and the orthogonal component My oscillates when the sample is made to rotate around the z-axis. The measured maximum value (My)max, attained when 2 sin 0 cos 0 = 1, then provides

K1 -= tzo(My)maxHa.

(8.14)

There are cases where the easy axis does not lie in the plane of the film or tape test sample. This occurs, for example, in thin films for perpendicular recording (e.g. C o - C r layers), or in obliquely evaporated metal tapes. In these materials, one is often interested in investigating the dependence of the magnetic properties on the angle made by the applied field with the normal to the sample plane [8.28-8.30]. To this end, the sample plane, placed perpendicular to the (x,y) plane, is rotated about the z-axis (Fig. 8.18b). For each angle c~made by the normal to the sample plane with respect to the x-axis (i.e. the direction of the applied field Ha), the magnetization components Mx and My are measured and the components M• normal to the sample plane, and MII , lying within the plane, are obtained as M• -- Mx cos a - My sin a and MII = Mx sin a 4- My cos cr The problem with this measurement is the existence of a large demagnetizing coefficient Nd associated with the M• component, which is close to 1. The demagnetizing field Hd -- - N d M z combines with the applied field to provide the effective field Heff -- H a 4- H d. If the angle c~is kept fixed and the applied field amplitude is cyclically varied, the hysteresis loops (Mx, Ha) and (My, Ha) are measured. They do not evidently reflect the intrinsic magnetic properties of the material and a correction for the demagnetizing field must be applied. However, for a fixed value of the angle c~,the magnetization M changes in magnitude and direction with the value of the applied field and so does the effective field Heft . One can estimate the intrinsic magnetic behavior of the tape along the direction of the applied field by calculating the effective field Heff,x = H a - - H d cos o~ and taking the loops (Mx,Heff~x) and (My, Heft,x) [8.29]. Bernards and Cramer have developed a method where the angle made by

516


Heft with the sample normal is maintained constant along the measurement [8.31]. This is achieved by simultaneous variation and adaptation in small steps of the value of the applied field and the sample orientation. In this way, the magnetization components Mx and My are measured as a function of Heff and the related hysteresis loops (Mx, Heff) and (My, Heff) are obtained. Most VSM apparatuses in industry and research make use of an electromagnet as a variable field source. The limitations on the maximum available field in the gap are somewhat stronger with respect to the closed circuit arrangement because of the space required for the coils. The image effect is also a disturbing factor with narrow gaps. It turns out that full investigation of the high-coercivity high saturation field magnets, like SmCo5 and the N d - F e - B based compounds, is out of reach of conventional electromagnet-based VSM apparatus. The obvious solution to the requirement of very high fields is provided by the superconducting solenoids. It is a relatively complex solution, characterized by high running costs, because it requires continuous refrigeration at the liquid helium temperature. It is normally unsuitable to industry needs, but it is fundamental to many physical investigations. A superconducting solenoid is, in principle, the same device using copper wire at room temperature. In practice, things are rather more complex, both regarding the manufacturing of the solenoid and the operating procedures for the generation of fixed or slowly variable fields. A superconducting solenoid is realized by winding a superconducting cable on a former (made, for instance, of aluminum or stainless steel), upon which it is firmly clamped with the help of special impregnating resins, contrasting the strong Lorentz forces tending to expand radially and compress axially the solenoid (see Fig. 4.12). The superconducting cable is made of a very large number of superconducting filaments immersed in a resistive matrix (normally copper). The filaments are made of either NbTi, by which practical maximum fields around 7000 k A / m (~-9 T) are reached at the boiling point of He (4.2 K), or the more expensive and brittle Nb3Sn, normally providing maximum fields of the order of 16,000 kA/rn (---20 T). These compounds are Type II superconductors. In them, the Meissner effect (the disappearance of the superconducting state engendered by a magnetic field) takes place gradually above a certain critical field through the creation of an increasing number of vortexes, i.e. flux tubes enclosing a flux quantum. The vortexes are subjected to a force by the flowing supercurrent and, by rearranging themselves in the cross-section of the conductor, they create electrical fields, i.e. resistive phenomena and energy dissipation. With the use of a large number of micrometer-sized superconducting filaments, the motion of vortexes is restrained. Safe


517

operations are carried out by keeping the superconducting solenoid at 4.2 K, far from the critical temperatures of NbTi (Tc = 9.5 K) and Nb3Sn (To = 18 K). The solenoid is housed in a cryostat, a vessel shielding it from heat transfer by conduction, convection, and radiation, where it is immersed in the liquid He bath. The sample is contained in a variable temperature insert (VTI) placed in the bore of the solenoid. Figure 8.19 provides a simplified view of a VSM magnetometer using a superconducting solenoid field source [8.32]. This setup, developed at the Laboratoire Louis N6el, is specifically designed for high-sensitivity measurements (resolution 2 x 1 0 - 1 ~ 2) in the temperature range 1.4-300 K, with the variable field supplied either in sweeping or stepping mode. The pickup coils are of the axial type, as imposed by the geometry of the field source. In order to compensate the spurious flux variations due to vibrations, two sets of coils connected in series opposition are used. They have the same area-turn product, but obviously different sensitivity functions, so that the strong reduction of the background signal comes at the cost of a partial reduction only of the signal generated by the vibrating sample and the signal-to-noise ratio is improved. The vibrating rod, connected by C u - B e springs at the top and bottom of the VTI, is acted on by the oscillating force generated on a permanent magnet affixed to the rod by two AC-fed superconducting coils connected in series opposition, which generate an AC gradient at the position of the magnet. In order to minimize the AC field possibly generated in the measuring area, a second pair of coaxial coils of larger diameter and same area-turn product is employed and the whole vibrator is shielded. The sample is held at the center of a Perspex slab, which, by extending far from the pickup coils, does not generate any additional signal (see Fig. 8.13c). The background signal, measured without the test sample on the rod, is reproducible and corresponds to an equivalent magnetic moment of 1.5 x 10 -8 A m 2. It is precisely subtracted taking into account its weak dependence on field strength and temperature. A superconducting solenoid used at 4.2 K is a perfect or nearly perfect diamagnet. Any magnetized sample placed inside it will have its field lines distorted, according to the notional idea of the image dipole (Fig. 6.11). The main factors affecting the image effect in a superconducting solenoid are the ratio 2a/D of the pickup coil diameter to the solenoid bore diameter and the ratio A of the superconducting volume to the total volume of the windings. In a Type II superconductor, A decreases with increasing the field strength and so does the image effect. Figure 8.20 shows the dependence of the normalized VSM output on the field strength measured by Zieba and Foner in a N b - T i 9 T magnet working at 4.2 K [8.33]. The experiment was made using a field-independent

518

CHAPTER 8 Characterization of Hard Magnets Magnet

Spring

Shield

S

Vibrating coils

Vibrating rod j

Pickup coils

Sample

\ Superconducting solenoid ~.fJ-.f]

Spring

He bath

He exchanger

FIGURE 8.19 Example of VSM setup using a superconducting solenoid field source. The axial pickup coils are compensated by concentric coils connected in series opposition having same area-turn product and far lower sensitivity function. The vibration is generated by a couple of AC-supplied superconducting coils connected in series opposition, which create an AC force on a magnet affixed to the vibrating rod. The vibration frequency is 14 Hz and the peak-to-peak oscillation amplitude is 4 mm (adapted from Ref. [8.32]).


519

magnetic moment, obtained by a small solenoid carrying a precisely known constant current, and pickup coils with maximum-homogeneity interdistance d - - 4~a and diameter 2a = 0.66D. With zero applied field and the magnet in the superconducting state, the measured magnetic m o m e n t is nearly 8% lower than the actual value. It tends then to increase by increasing the superconducting current because of the decrease of the factor K The change speeds up on approaching the critical field (around 9000 k A / m (11 T) at 4.2 K). From a practical viewpoint, we do not perform absolute measurements and the measuring uncertainty is only due to the field dependence of the image effect, if calibration is made with reference Ni sample. A complete correction as a function of the applied field can be made by testing a calibrated magnetic moment, obtained by means of a current-carrying small coil, which is field independent.

0.94t Nb-Tisolenoid

o

"o

.N_ 0.93o Z

0.92 '

I

2000

'

'

'

'

I

4000

'

Field(kA/m)

'

'

'

I

'

'

'

6000

FIGURE 8.20 Normalized output of VSM using Nb-Ti superconducting solenoid as a function of the applied field. A precisely known magnetic moment, realized with a small solenoid carrying a constant current is measured with axial pickup coils of diameter 2a and interdistance d = x/~a. The diameter of the solenoid bore is D--3.03a. After magnet cool down, the image effect makes the measured magnetic moment nearly 8% smaller than the actual one. On increasing the field, the image effect decreases because magnetic flux creeps in the superconducting material and the output signal increases (adapted from Ref. [8.33]).

520


Coey and co-workers have realized an interesting development in VSM setups through an apparatus exploiting a permanent magnet-based variable field source [8.34]. As schematically shown in Fig. 8.21, the basic elements of this source are a pair of nested Halbach's cylinders made of N d - F e - B . We have illustrated this structure and the working principle of the Halbach's cylinder in Section 4.4 (Figs. 4.19-4.21). Thanks to the exceptionally high value of the anisotropy field (Hk -- 8 T in Nd2Fe14B), which make these magnets close to ideal magnets, any building block of the cylinder is transparent to the field generated by itself and the other blocks, and a stable high field normal to the cylinder axis is obtained in the

Vibrator" Y

Hallbachcylinders

Hallplate

~.

I //J //

Sample

Pickup coils FIGURE 8.21 Permanent magnet based vibrating sample magnetometer developed by Coey and co-workers [8.34]. A field Ha of fixed direction and variable amplitude is generated in the cavity by making two nested Halbach's cylinders made of Nd-Fe-B to rotate in opposite directions. Each cylinder generates a field of strength H0 in the bore and the resulting field varies with the angle of rotation a as H(a)= 2H0 cos a, making a full period for a 360~ rotation of the cylinders (adapted from Ref. [8.34]).


521

shell cavity. Since the generated field strength is H - - Brln r~ /~0 rg with r0 and rg the radius of the cylinder and the radius of the cavity, respectively, two nested cylinders endowed with the same ratio ro/rg generate the same field strength on their axis. In the setup shown in Fig. 8.21, these two cylinders, realized in practice by means of an octagonal structure, are rotated in opposite directions in small successive steps by means of a pair of motors, with a belt and pulley mechanism. The corresponding fields, of equal modulus H0, rotate with them. It is easily seen, looking at Figs. 4.20a and 4.21a, that the resulting field in the cavity has a fixed direction and strength depending on the angle a covered by the counter-rotating cylinders, according to the equation H ( a ) = 2H0 x cos a. This field then oscillates between +2H0 through a 360 ~ rotation of the nested cylinders. In the setup realized by Coey et al., the outer and inner cylinders have outside diameter 108 and 52 ram, inside diameter 52 and 26 mm, and height 115 and 65 mm, respectively. The resulting device turns out to be extremely compact, with a mass around 20 kg. A conventional electromagnet, providing approximately the same field strength with the same degree of homogeneity, has typically a mass around a few hundred kilograms. Each cylinder generates a field H0 = 470 k A / m and the peak value of the total field is therefore 940 k A / m , less than half the maximum value achievable in electromagnet-based VSMs. Considering that the field strength follows a logarithmic law on the ratio ro/rg and that the diameter of the cavity cannot shrink below obvious values, there is little room for increasing the maximum available field strength by increasing the volume of the cylinders.

8.2.2 Alternating gradient force magnetometer Increasing trends towards the miniaturization of devices and the development of artificial structures with faint magnetic moments can impose demanding requirements in terms of measuring sensitivity. The VSM is in general the preferred solution for the determination of low magnetic moments, down to the some 10-8-10 -9 A m 2, since it combines ruggedness and a relatively simple measuring procedure with good accuracy, solid experience in many laboratories, and availability of reliable commercial setups. It may happen, however, that magnetic moments lower than the typical VSM noise floor have to be determined. For example, one might be interested in following the basic magnetization process in isolated particles. A BaFe12019 single particle of about 5 ~m has

522


a moment of the order of 5 x 10 -11 A m 2, far below the VSM sensitivity. Besides the SQUID magnetometers, with their well-known problems of high running costs and impractically long measuring times, a solution to very low magnetic moment measurements is provided by the alternating gradient force magnetometer (AGFM), also called the vibrating reed magnetometer (VRM). It is realized as a sort of inverted VSM. We can imagine, in fact, energizing the pickup coils in a VSM by means of an AC current. These generate a non-uniform alternating field in the gap, so that a moment-bearing sample placed between them becomes subjected to an oscillating force. The sample displacement could then be revealed, at least in principle, across the vibrator. Under general terms, we can express the force acting on a magnetic dipole of moment m subjected to an inhomogeneous field H as F = V(m./.~0H).

(8.15)

The component of the force along the x-axis is therefore given by

Fx = ~

[

0Hx 0Hy 0/-/z ] mx--~x + m y ~ + m~~

(8.16)

and totally analogous expressions are obtained for the Fy and F~ components. With the typical configuration of the VSM pickup coils shown in Figs. 8.13 and 8.14, the field generated in the gap has defined symmetry properties. The force it exerts on the dipole depends, according to Eq. (8.15), on the value taken by the gradient of its components Hx, H v, and H~ along the reference directions. This force is then proportional to the sensitivity function of the coils. If we assume, in particular, that the field is generated by a pair of identical coils connected in series opposition, like those described in Fig. 8.13, and that the dipole of moment m is placed at the center and directed along the x-axis, we find that the force acting on it is obtained from Eq. (8.15) as

F = Fx - ~ m ~.OHx Ox

(8.17)

For symmetry reasons, Fy = i~om(OHx/Oy) and Fz = I.~om(OHx/OZ)are equal to zero on the (y,z) midplane. With a sinusoidal current i ( t ) - io sin ~ot supplying the coils, a sinusoidally varying force Fx(t)- F0 sin ~ot is applied to the sample. For small oscillations around the center, we obtain, according to Eq. (8.8), F0 = mgx(O)io, with gx(O) the value of the sensitivity function at the origin. A VRM was originally developed by H. Zijlstra in order to investigate the magnetic behavior of single micrometer-sized hard magnetic particles [8.35]. This device was based on the idea of placing the sample on the tip of a micro-cantilever beam, in turn located at


523

the center of a pair of AC supplied coils, connected in series opposition and 2 m m spaced. The typical maximum field gradient value obtained at the sample location is of the order 4000 A / m m m -1. Under these conditions, an oscillating force, as given by Eq. (8.17), is applied to the beam tip, which is made to vibrate. The ensuing deflection amplitude can be largely magnified by bringing the system made by cantilever and the sample to resonance. For sufficiently small oscillation amplitudes, the resonating system, characterized by a quality factor Q, behaves linearly and the peak vibration amplitude is x0 = XDcQ, if XDC is the deflection suffered by the beam tip when subjected to the same maximum force F0 under static conditions. A 20 m m long golden wire with diameter 38 p~m, cemented at one end to a brass bar and endowed with a Q factor around 100, was used as a resonating beam in Zijlstra's setup. The oscillation amplitude of the beam tip was measured by observing with a microscope the stationary image of the deflected reed, as obtained by stroboscopic illumination. A modified version of Zijlstra's magnetometer, developed by Roos et al. [8.36], is schematically illustrated in Fig. 8.22a. Again, a golden wire reed (diameter 18 ~m, length 10mm) is used, but the mechanical vibration is converted into an electrical signal by means of a bimorph piezoelectric plate (20 m m x 1.5 m m x 0.5 mm), connected to the reed by means of glass fiber (length 70 mm, diameter 150 ~m). The sample is stuck at the tip of the reed using wax and it can be oriented under a DC field by heating the wax. The signal provided by the bimorph (resonance frequency around 62Hz), proportional to the magnetic moment of the sample, and the signal proportional to the slowly varying magnetizing field, measured by means of a Hall device, are then used to trace the hysteresis loop behavior. Given the microscopic size of the sample, a substantial negative contribution to the measured magnetization comes from the diamagnetic golden wire, which must then be subtracted from the total signal. A large improvement in the quality factor of the resonating system, which depends on the viscosity of the surrounding gas, can be achieved by enclosing the reed assembly in an evacuated holder. This is demonstrated by the behavior of the output signal vs. the vibration frequency for different air pressures shown in Fig. 8.22b. Under vacuum conditions (P -- 1 Pa), a quality factor as high as Q = 550 is obtained and a magnetic moment measuring sensitivity of the order of 10-13A m 2 is ultimately achieved. However, the use of stiffer reeds can be envisaged, so that the effect of air damping can be strongly decreased and vacuum can be dispensed with [8.37]. General use of an AGFM setup in the laboratory requires a certain degree of ruggedness, relative ease of operation in mounting and substituting the sample and elements of the resonating-detecting

524


Lock-in

Oscillat~ Magnet

(31ass fiber"',,, Piezoceramic

Sample

Hall probe

(a)

1500 >:::L -~ 1000 t-

v

Q.

WW-d

0

40 Pa

Pa

500

,

(b)

//1

,

,

i

.

61.5

.

.

.

i

,

62.0 62.5 Frequency (Hz)

FIGURE 8.22 Scheme of the alternating gradient force magnetometer using vibrating reed, developed by Roos et al. [8.36]. The reed vibration is transmitted to a piezoelectric element by means of a glass fiber. The non-homogeneous AC magnetic field is created in the gap of the electromagnet by a couple of coils connected in series opposition (diameter 3 mm, distance 2 mm). They are supplied at the resonant frequency of the reed, whose oscillation amplitude is amplified by the quality factor of the resonating system. The hysteresis behavior of the material is obtained by plotting the signal proportional to the DC field, provided by a Hall sensor, and the signal proportional to the sample magnetization, provided by the piezoelectric bimorph and amplified by the lock-in amplifier. The measuring sensitivity is strongly affected by the viscosity of the surrounding gas and can be increased by decreasing the gas pressure. This is demonstrated in (b) by


525

assembly, possible application over a range of temperatures, and some kind of calibration procedure. Flanders has developed two kinds of AGFMs which appear basically to satisfy these requirements. Figure 8.23a shows the arrangement of sample and support system, which incorporates the piezoelectric element, in the horizontal gradient setup [8.38]. The vertical gradient AGFM is shown in Fig. 8.23b [8.39]. Let us consider the first instrument. Here, the horizontal AC field gradient is generated along either the x- or the y-axis, depending on the specific coil arrangements. With the magnetic moment m directed along x, the force Fx ensuing from the x-directed gradient is given by Eq. (8.17), while with the y-directed gradient it is Fy = Ixom(OHx/Oy). For a cantilever rod of thickness d, length l, density 3, and Young modulus Y, the fundamental resonance frequency is

foc-- 27r -g l

"

If a mass ms is fastened at the end of the cantilever of mass mc, the resonance frequency decreases to the value fos ---focx/[mc/(mc + 4.2ms)] = focx/-R. The complete cantilever system (bimorph plus extension with fastened sample) then resonates at the frequency fo =

fo~

,

(8.18)

~oc/fob)2 'q- 1/R where fob is the fundamental resonance frequency of the free bimorph. Extensions made of either glass, plastic, or copper have been used in this device, with lengths varying between 10 and 70 m m and thicknesses between 0.12 and 0.6mm, depending on the material employed. Experiments have been carried out at frequencies in the range 1 0 0 H z - l k H z . The output voltage V0 generated by the bimorph, detected by metallic contacts at the clamped end of the bimorph, is supplied to a high impedance pre-amplifier, feeding the lock-in amplifier. This voltage can be calculated and is approximately given,

the behavior of the voltage measured at the output of the lock-in amplifier around the resonance frequency. The quality factor of the resonating system passes from Q = 70 (P - 105 Pa, atmospheric pressure) to Q = 190 (P = 40 Pa) and Q = 550 (P -- 1 Pa) (adapted from Ref. [8.36]).

526


Clamp

zT

Contacts

Ceramic mount Rubber~

Piezo bimorph

ro~

~'~ i / kx\\\\\\\\\\\\\\\x~

..I] II

~1211Jklh2/,/,2/2,1d~ I I /

Glass/quartz Glass

.._ 11D e r

- - - ' l l =] J~

Extension

1

0, where r is

562

CHAPTER 9

Measurement of Intrinsic Magnetic lJroperties

r////.4

IN.

I !

NF-a-

j'l|

]|

_..L_

Gradient coils ~

!

d

'

~

Sample

~

Magnetizing coil

~

Z

i

i !

(a)

Y

Gradient coils

x~

iiiiiii

O

Hx r

(b)

. . . . . . .

i

,

,i

FIGURE 9.7 Schematics of vertical-field (a) and horizontal-field (b) Faraday magnetometers. One can realize with them the measurement of the magnetic moment m of a sample through the measurement of the pulling force Fz exerted on it by a magnetic field gradient. In the first case, the vertical magnetizing field Hz and the gradient OHz/OZ are generated by a solenoid and by identical windings connected in series opposition, respectively. The resulting force on the sample is Fz = la,omz(OHz/Oz). In the second case, the field Hx is generated by the electromagnet and the gradient OHx/OZ is provided by the coils placed on the pole faces. The downward pulling force is Fz = p,omx(OHx/OZ). The set of gradient coils used with the electromagnet is shown in some detail, and the circulating currents are indicated. The scheme shown in (b) is adapted from Ref. [9.26].

9.1 SPONTANEOUS MAGNETIZATION AND CURIE TEMPERATURE

563

the radial coordinate, is proportional to r and results, according to Eq. (9.8), in a force tending to pull the sample away from the solenoid axis. In order to achieve very high fields and gradients, superconducting solenoid and coils have been employed [9.21, 9.22]. Bitter solenoids have also been used. Interestingly, it has been verified in the latter case that it is possible to dispense with the gradient coils. In fact, the generated magnetizing field can be so large that its own axial gradient may suffice to achieve the desired moment measuring sensitivity [9.23, 9.24]. In this case, the sample is placed near or in correspondence to the end plane of the solenoid, where the maximum value of OHz/OZ is attained. If maximum sensitivity is not required, there is convenience in placing the sample immediately beyond the end of the solenoid [9.23]. In fact, on leaving the interior region of the solenoid, the radial dependence of the field intensity is reversed and so is the resulting lateral force on the sample, which, instead of being pulled away from the solenoid axis, is automatically centered. In order to gain easier access to the measuring region, the horizontalfield type Faraday magnetometer with electromagnet field source can be employed. In the classical device of this type, the electromagnet is provided with polar caps having a specific profile, by which the generated field becomes endowed with a suitable gradient OHx/OZ along the vertical axis [9.25]. Such gradient is dependent, however, on the field strength, and measurements at low fields necessarily imply low gradients, an undesirable limitation when dealing with ferromagnetic samples, for instance. On the other hand, very high gradients are not required with ferromagnets, where independent adjustment of main field and gradient are appropriate instead. This objective can be reached by adopting an arrangement as shown in Fig. 9.7b, where coils placed in the gap of the electromagnet make possible the desired independent regulations [9.26]. The set of coils adopted in this specific case realize a scheme similar to Mallison's set, for which the behavior of the gradient OHx/Oz is given in Fig. 8.14d. If we pose my = mz = 0, Eq. (9.6) becomes

0Hx

Fx = I~omx 0---~'

0HI

G = ~omx 0---y-'

0HK

F~ = ~omx O---z-"

(9.9)

Again, the symmetrical arrangement of the coils makes the gradients OHx/OX --- OHx/Oy - 0 at the rest position of the sample. Only the vertical force F~, pulling the sample downward when the currents circulate in the coils as shown in Fig. 9.7b, is different from zero. On the other hand, if the sample is slightly displaced from the center of symmetry along the x-axis, a lateral force due to the magnetic image will arise, which is roughly proportional to the displacement [9.27].

564

CHAPTER 9 Measurement of Intrinsic Magnetic Properties

To simplify the generation of the field gradient, the use of a pair of identical strips in the gap has been suggested [9.28]. The strips are parallel to the pole faces and are oriented along the z-direction. They bear the same current, thereby producing zero field and non-zero field gradient at the center of the gap, i.e. at the sample position. Shull et al. have used such a field gradient generating setup in their Faraday magnetometer, by which they have performed absolute measurements of the magnetic moment of pure Ni spheres (m = 3.47x 10 -3 A m 2) [9.19]. Absolute determination of m with stated expanded relative uncertainty of 3 x 10 -3 has been obtained by making a direct measurement of the generated field gradient.

9.2 M A G N E T I C

ANISOTROPY

Magnetic materials seldom behave isotropically. The classical Heisenberg exchange interaction -JqSi.Sj is isotropic, but an anisotropic exchange term, bonding the spins more tightly when they point along certain directions, can exist. In addition, the spins individually interact with the crystalline field, which, being endowed with the symmetry properties of the host lattice, provides preferential orientations for the exchangecoupled spins. This property can be phenomenologically described by expressing the associated energy term by means of a function provided with suitable symmetry properties. Thus, the crystalline magnetic anisotropy energy in a cubic crystal is appropriately defined through a polynomial series in even powers of the direction cosines 6r 6r and OZ3 made by the direction of the magnetization with the cube edges. The relevant equation is Ea -- K1(cr 2 cr2 q- c~2c~2 q- cr cr

q- K2cr 2 ~2202

(9.10)

higher order terms being normally of little or no relevance. K1 and K2 are constants characterizing a given material and are expressed in J / m 3 (we consider here and in the following only quantities related to the unit volume). Their amplitude and sign determine the directions along which the anisotropy energy is minimum. In Fe crystals, it is K1 > 0 and K1 >> IK21, which makes the (100) axes the m i n i m u m energy directions for the magnetization (easy axes). In hexagonal crystals (e.g.h.c.p. Co), the energy is expressed as Ea = K1 sin 20 + K2 sin 40,

(9.11)

where 0 is the angle made by the magnetization with the c-axis. With K1 > 0 and K2 ~ - K 1, the c-axis is an easy axis, a condition met at room

9.2 MAGNETIC ANISOTROPY

565

temperature in Co and Ba ferrite single crystals. Polycrystalline materials combine the crystallographic orientations of the single grains in a variety of ways. The resulting texture translates into more or less pronounced magnetically anisotropic behavior, which, however, is not univocally related to the distribution of the orientations. Besides crystallographic texture, demagnetizing fields, stresses, and various effects of directional atomic ordering can induce anisotropic effects. In these cases, we have to deal in general with uniaxial anisotropies. For most purposes, the dependence of the uniaxial anisotropy energy on the angle 0 between magnetization and easy axis is described by the equation: Ea -- Ku sin20.

(9.12)

Demagnetizing fields are the source of shape anisotropy, which inevitably affects all non-spherical samples. With ellipsoidal samples, the demagnetizing field Hd = - N d M is homogeneous and the constant Ku can be exactly defined in terms of the difference between demagnetizing coefficients pertaining to the minor and major axes, respectively. If Xda and Ndc are such coefficients and 0 is the angle between the uniform magnetization M and the major axis, we obtain the magnetostafic energy as Ems--(~/2)M2(Nda cos2Oq-NdcSin20). We thus define the shape anisotropy energy as Ea--(IJ, o/2)M2(Nda- Ndc)Sin20 = Ku sin20. If a tensile/compressive stress is applied to a sample and the ensuing magnetoelastic energy can be described by means of an isotropic magnetostriction coefficient As, the stress anisotropy energy turns out as Ea = -~Aso"sin20-- Ku sin20. The stress axis is an easy axis if the product AsO" is positive (e.g. tensile stress and positive magnetostriction) and a hard axis if this product is negative (e.g. compressive stress and positive magnetostriction). In the latter case, the plane normal to the stress axis is an easy plane. A straightforward measurement of magnetic anisotropy can be performed by determining the magnetization curve up to saturation along different directions. This method implements the definition of anisotropy energy as the difference in energy required to saturate the material along different axes. If the easy axis is known or it has been identified and the magnetization curves have been obtained along both such an axis and the direction under investigation, one makes the difference of the areas included between these magnetization curves and the J-axis (Fig. 1.6b). It is understood that all other energy terms have changed negligibly on passing from one direction to the other. This method is made somewhat complicated and often unreliable by a number of factors: (1) Measurements on closed magnetic circuits might require

566


complex flux-closing arrangements in two or even three dimensions to cope with the presence of the demagnetizing field pointing along a direction different from the direction of the applied field [9.29]. Measurements on open samples should preferably be made on spheres, but the huge shearing of the curves can make the determination of the difference of areas imprecise. (2) There is an hysteresis effect, which should be eliminated by considering the anhysteretic curve or, at least, the median curve between ascending and descending branches of the hysteresis loop. (3) The approach to saturation may be affected by the presence of various defects, inclusions and localized stresses to an extent depending on the specific investigated crystallographic direction. Anisotropies can be measured in a direct fashion using a torque magnetometer. Spherical or disk-shaped samples are subjected in it to a slowly rotating field, whose strength must be sufficient to eliminate the domain walls and drag the magnetization without discontinuities along a whole 360 ~ turn. Let us consider the scheme shown in Fig. 9.8a, where the applied rotating field H a makes, at a given instant of time, an angle ~0 with the easy axis (taken as the reference direction) in a sample with uniaxial anisotropy. The magnetization Ms makes the angle 0 with the easy axis when the torque per unit volume: ZH =/zoMsHa sin(~o- 0),

(9.13)

due to the field is balanced by the intrinsic torque ZK = -OEa/O0 due to the anisotropy. This follows from the minimization condition OE/O0 = 0 imposed onto the total energy: E = -/zoMs'Ha q- Ea,

(9.14)

sum of the anisotropy and field interaction energies. With Ea given by Eq. (9.12), it is rK = Ku sin 20.

(9.15)

At equilibrium, TH = rK = Z and, according to Eq. (9.13), the angle 0 is related to ~ by the equation: 0 = ~0- sin -1

~(~o) . /z0MsHa

(9.16)

The applied field appears in Eq. (9.13) instead of the effective field H - - H a - N d M s . The demagnetizing field, being co-linear with the magnetization, cannot in fact contribute to the torque. The experiment requires that H a is larger than a threshold value, beyond which totally reversible rotation of the magnetization can occur. Such a value is


567

Easy axis

,

Ha

(a) 1000

/

r(~)

500 04

E

I

z

~1 ~:/2

(1) '-I 0 I--

-500

K u = 1000 J / m 3 Ha = H k

(b)

~ ~

/ i II / iIII ~ /11

-1000

FIGURE 9.8 Torque in a sample with uniaxial anisotropy. (a) A slowly rotating applied field Ha exerts a torque on the saturation magnetization Ms in a spherical or disk-shaped sample, dragging it along a complete 360 ~ turn. The field torque per unit volume of the sample TH = p,oMsHa sin(~o - 0) is balanced at any time by the intrinsic torque ZK = -OEa/OO due to magnetic anisotropy. ~0 and O are the angles made by Ha and Ms with the reference axis, respectively. (b) From the measured oscillatory torque ~(~) (solid line), the desired behavior of the torque 9(O) (dashed line) is obtained. The example reported here refers to uniaxial anisotropy. Both curves attain the same peak value, which coincides with the value of the anisotropy constant Ku.

568


obtained imposing the additional condition of stability on the second derivative O2:Ea/O02 and it turns out to be the value of the so-called anisotropy field Hk = 2Ku/la,oMs. Figure 9.8b illustrates the behavior of the torque curve ~(~) measured with a field slightly larger than Hk and its transformation into the "r(0) curve, the one achievable with saturating fields, using Eq. (9.16). The value of the anisotropy constant Ku coincides, according to Eq. (9.15), with the maximum of the torque curve. This transformation procedure is general, but requires careful application because it might easily generate errors in the following harmonic expansion of the torque curve, if this is required to reveal higher order anisotropy terms. When possible, it is appropriate to apply a sufficiently large field H a in order to have a small difference ~ - 0 and consequently linearize the transformation given by Eq. (9.16). In many experimental circumstances, the applied field strength is indeed so high that 0 = ~, no transformation is required, and the anisotropy is directly provided by the measured curve. Notice, however, that the measured anisotropy may not be observed to saturate with the applied field. Kouvel and Graham, making experiments on Fe-Si disk-shaped single crystal samples (thickness 0.3 mm), ascribed this effect to the presence of residual edge domains, depending in a complicated way on the direction of the field [9.30]. These are expected to introduce small variations of the magnetization amplitude as the saturating field rotates in the plane of the disk, resulting in an apparent small extra torque. There also appears to be a more fundamental problem of anisotropy of the saturation magnetization, as demonstrated by Aubert in Ni single crystals [9.31]. Figure 9.9 provides an experimental example of field dependence of the anisotropy constant K1 in the saturation region in Fe-Si single crystals, determined in spherical samples [9.32]. This effect is attributed to the orientation dependence of the saturation magnetization. Notice the steep increase of the measured K1 value in the field region going up to Ha ~ Ms~3. It is related to the progressive disappearance of the Bloch walls and their contribution to the magnetization process. A classical realization of the torque magnetometer is shown in Fig. 9.10. Here, the field is obtained by means of an electromagnet, installed on a rotating platform. The sample, located at the center of the gap, is either a disk or a sphere. It is held solidly in place in the interior of a quartz tube, which is connected by means of a taut suspension (e.g. a tungsten wire) to a rigid non-vibrating structure. A weight attached at the bottom of the sample provides stabilization against lateral movement. The whole assembly is kept in vacuum making it possible to perform measurements as a function of temperature whilst avoiding the disturbing action of air convection. On top of the quartz tube and integral


569

_

40

_

~

Fe-(3.4 %)Si T = 373 K

30 co

iI !

20

o

T-v

v

Fe-(6.1%)Si T = 293 K

iI

E

v--

10 Hk I ~ 9

'

I--4

i I;

500

'

'

'

'

'

I0'00

'

'

'

....

i ....

1500

H a (kA I m)

FIGURE 9.9 Magnetic anisotropy constant K1 in two Fe-Si single crystals of different Si content as a function of the applied field H a. The measurements have been performed by means of a torque magnetometer on 5 mm diameter spheres. The steep increase of K1 with the field for H a < Ms~3, where applied field and demagnetizing field have nearly same strength, is due to the gradual disappearance of the Bloch walls. Once magnetic saturation is attained, K 1 shows a feeble increase with Ha, which is attributed to the anisotropy of magnetization (adapted from Ref. [9.32]).

with it, there are a mirror and a multiturn coil, which are part of a servo system keeping the sample firmly in place against the torque applied by the rotating field (automatic force-balancing method). When the field starts rotating, the sample tends to follow because of the action of the torque ~H (Eq. (9.13)). The collimated light beam, striking the mirror and evenly reflecting at rest position into a dual photocell, generates unbalanced photocurrents, which are amplified by the high-gain DC amplifier and injected into the multiturn coil attached to the tube. This coil, being immersed in the gap of a permanent magnet, imparts a restoring torque to the assembly, the higher the gain of the amplifier, the tighter the balancing of TH. One can thus take the value of the current circulating in the coil as a measure of the magnetic torque and record it together with a signal proportional to the angle ~ in order to recover the anisotropy curve shown in Fig. 9.8b.

570

CHAPTER 9

Measurement of Intrinsic Magnetic Properties

V//////Z Tungsten wire

/

Permanent magnet

~",, S js

s

,~

",

Photocells

DC a

Acquisition setup

Sample

J

,

,

i

)

1

s/SS"

FIGURE 9.10 Schematics of a torque magnetometer for magnetic anisotropy measurement. An electromagnet on a rotating platform provides large and uniform field over 360 ~ to a disk-shaped or spherical sample, placed in the center of the gap. The sample is solidly held within a quartz tube, which is suspended at the upper end by means of a taut wire and is provided with a suitable stabilizing weight at the bottom. A servo system keeps the sample in a fixed position during rotation of the field, by injecting a suitable current in a multi-turn coil integral with the tube. This is immersed in the field of a permanent magnet and provides a balancing torque to the tube. The current circulating in the coil provides a measure of the torque and is recorded together with the angle r correspondingly made by the field with the reference direction.


571

Torque magnetometers of this type are somewhat cumbersome to use, but they are rugged devices, displaying long-standing performances (the same unit has been in use for 35 years at IEN), in association with good sensitivity and wide dynamic range. With the development of Halbach's cylinders, made of rare-earth-based permanent magnets (Section 4.4), it has become possible to dispense with large electromagnets in the generation of the rotating field (though only below Ha---800 kA/m), resulting in compact devices [9.33]. In general, one can measure with the same apparatus either the large torque offered by Fe single crystals or the faint torques associated with induced anisotropies in soft amorFhous alloys. Typical measuring ranges are between 10 -8 and 1 0 - ~ N m . Especially sensitive devices have been built for measurements on thin films, where torque measurement capabilities may range between some 10 -5 and 10 -12 N m [9.34, 9.35]. Limits to the sensitivity may come from defective centering of the sample, unwanted shape anisotropy effects due to imperfect sample preparation, parasitic torques deriving from the possible presence of metallic parts in the region invested by the rotating field, and noise and instabilities in the servo system. The background signal is determined by means of a measurement made without the sample. Calibration can be performed by means of a reference sample, but absolute calibration is also possible. One way to do this is by measuring the torque exerted by the field Ha on an artificial dipole created by means of a current-carrying loop of known t u r n - a r e a product [9.36]. With this procedure, measurement of the field at the sample position is required. Another method consists in measuring the torsional coefficient of the taut suspension by means of a separate experiment and in determining the current generated by the servo system when, in the absence of field and sample, the assembly itself is rotated by an angle, imposing a known torsional moment to the suspension. The measurement of the magnetic torque can provide unambiguous quantitative information on the magnetocrystalline anisotropy of single crystals and on uniaxial induced anisotropies. A general approach to the analysis of the torque curves -r(q~) is based on a Fourier development, where one can express their periodic dependence on the angle qJ made with respect to a suitable direction as 00

9(~) = Y. An sin(nq~),

(9.17)

n=l

where one can take q~-= 0 in the case of uniaxial anisotropy. In cubic crystals, 4~is connected to the direction cosines appearing in Eq. (9.10). In this way, the coefficients A~ can be related to the anisotropy constants K1,

572


K2, and Ku. The application of the Fourier analysis requires unskewed torque curves or, better, the curves obtained under saturating fields, where the magnetization is practically aligned with the field. Let us consider the simple example of uniaxial anisotropy. We have already shown that when the anisotropy energy can be expressed as Ea -- Ku sin 20, the anisotropy constant Ku is provided by the peak value of the torque curve (Fig. 9.8b). If the energy Ea is instead expressed as in Eq. (9.11), with K2 non-negligible with respect to K1, we need to analyze the whole curve. By making the derivative OE/O0and developing it according to Eq. (9.17), we get

K2

"r(0) = (K1 + K2) sin 20 - -~-sin 40 = A 2 sin 20 + A4 sin 40.

(9.18)

Analysis of the curve provides the coefficients A 2 and A 4 and the anisotropy constants are therefore obtained as K1 =

A2 q- 2A4,

K2 = - 2A4.

(9.19)

When applied to polycrystalline materials, the torque analysis can reveal the presence of dominant textures, but it cannot unambiguously provide quantitative information on them without associated X-ray diffraction investigation. Figure 9.11 illustrates the case of a torque curve measured on a non-oriented Fe-(3 wt%)Si lamination. The sample is made of a disk of diameter 15 m m and thickness 0.35 m m and the applied field is of the order of 106 A / m . The curve fits into a (110) [001] texture, seemingly occupying a volume fraction of the sample around 20%. The X-ray ODF analysis does demonstrate the existence of such a texture, though in lower proportion than suggested by the magnetic experiment. This can be understood by considering that the torque results from the combined effects of several crystallographic components, whose volume fractions are not proportionally reflected on the torque curve. It is known, for example, that the contribution of the (111) planes, being related to the small second order constant K2 only, is totally masked by the contributions of other textures. The measurement of the magnetic anisotropy made according to the scheme shown in Fig. 9.8a, where the rotating magnetic field is sufficiently large to drag the magnetization reversibly along a 360 ~ period, but not so large to keep 0----~0, can be made without resorting to the direct determination of the torque with a magnetic balance. We see in Fig. 9.8a that the phase shift ~o- 0 between H a and Ms can be viewed in terms of component of the magnetization M l perpendicular to Ha. Since


573

'

I

'

I

Non-oriented Fe-(3 wt%)Si

4000

@

O\

9

o~ :

@

• ood •

2000 eo

E E z

OI

@

i

~ : Ol"

(~ I

0

p9

I

b I 9

O" 0

q~ ,

I

0

~

9

,

aj

P

I

l- -2000

......

b

I

t ,? -4000 0

,

I

45

i

I

90

J

I

135

,

180

e(~ RD FIGURE 9.11 Magnetic anisotropy torque curve in a non-oriented Fe-(3 wt%)Si lamination. Points represent the experiments. The fitting line results from Eq. (9.10) with dominant (110) [001] texture. X-ray diffraction ODF analysis shows that this is only partially the case and puts in evidence the semi-quantitative information on the texture of polycrystalline materials conveyed by magnetic torque experiments.

M• = Ms sin(~0- 0), we write Eq. (9.13) as 7(~o) =/~0M• Ha

(9.20)

and we conclude that, once the applied field is known, the torque is indirectly obtained by the m e a s u r e m e n t of the normal c o m p o n e n t of the magnetization M• This m a y represent an excellent m e a s u r i n g solution in those cases where the material is very hard, like the rare-earth-based magnets, m a k i n g ~0 - 0 seldom negligible d u r i n g the rotation [9.37]. Since we p u t ourselves in a condition where the magnetization components can be directly measured, which is not generally the case with torque balances, we can exploit the simultaneous determination of M• and Mli to obtain 0 = ~o- t a n - l ( M •

that is, the -r(O) torque curve.

(9.21)

574


We have shown in Fig. 8.18a an arrangement for the measurement of the torque through the measurement of M~ using a vector VSM and we have briefly discussed its operation in Section 8.2.1. With this method, two pairs of sensing coils are mounted along two orthogonal axes x and y, the field is directed along x, and the disk-shaped or spherical sample is rotated stepwise around the vibration axis z. The orthogonally placed coil pairs gather the signals proportional to M • and MII = Mx, respectively, and the torque -r(0) is eventually obtained via Eq. (9.20), after measurement of the field strength Ha. For uniaxial anisotropy, as described by Eq. (9.12), the constant Ku is provided by the peak value (M• x as Ku = p~(M• If the field, instead of the sample, is rotated, adopting, for example, the compact solution offered by Halbach's cylinders [9.38], we need to detect both the magnetization components Mx and My and the field components Hax and Hay. With reference to Fig. 9.8a, we obtain that M• and MII are related to the measured quantities Mx and My by the equations:

M• = Mx cos ~ - My sin ~,

Mll = My cos ~p+ Mx sin ~p.

(9.22)

Substitution of M• in Eq. (9.20) provides the expression for the torque: 9( ~ ) = a 0 ( M x H a y

-

MyHax),

(9.23)

the relationship between 0 and ~ being given by Eq. (9.21). Use of the vector VSM in torque measurements does not require special modifications of the conventional setup, but for the addition of an orthogonal set of coils and some extra electronics and software. Alternative methods may nevertheless be considered. For example, an extraction magnetometer with suitably modified sets of orthogonal coils has been employed [9.37] and a great deal of activity was carried out in the past using the rotating sample magnetometer [9.39, 9.40]. In the latter case, the voltage generated in the pickup coil positioned to sense M I is proportional to the derivative of the torque curve. The measurement of the anisotropy is often identified with that of the anisotropy field Hk. We have previously stated, that in a uniaxial system with energy given by Eq. (9.12), Hk -- 2Ku/la,oMs happens to be the field at which the magnetization starts to follow the rotating field in a fully reversible fashion. On the other hand, Hk also represents, according to the Stoner-Wohlfartti model, the nucleation field for coherent reversal of the magnetization in single domain particles, with Ku taken to derive from both intrinsic and shape effects. If we subject an ensemble of independent Stoner-Wohlfarth particles to a rotating field Ha, we thus expect that the nucleation field will be just the Ha value corresponding to the vanishing of


575

the irreversible processes, that is, of the energy loss. Note that, this continues to be true even for randomly distributed non-ideal StonerWohlfarth particles and the rotational loss measurement (that is, of the field at which it disappears) can therefore be taken as a good measure of the particle anisotropy [9.41]. The experiments on the rotational loss show that single-particle features can be retrieved to some extent from experiments on particle aggregates. It has actually been demonstrated in theory and experiment that investigation of the magnetization curve of a polycrystalline ferromagnet can lead to the determination of the anisotropy field of the individual crystals. The idea is from Asti and Rinaldi [9.42], who took at face value the fact that a single crystallite brought to saturation along a hard direction exhibits a magnetizing curve with a discontinuity at the saturation point. This can easily be demonstrated if, for example, we calculate the magnetization curve in a uniaxial crystal along a direction orthogonal to the easy axis. It is sufficient to take the expression for the

1.0

0.8 t'-

=

0.6

~ 0.4 % BaFe

0.2

0.0

J

0.0

015

.

.

.

.

.

.

1.'0

.

.

I

1.5

H/H k

FIGURE 9.12 Experimental behavior of the second derivative d2M/dH 2 along the magnetization curve in a polycrystalline Ba ferrite sample. H is the effective field H = H a - NckM, where Ha is the applied field and Nd is the demagnetizing coefficient, d ' M / d H 2 exhibits a cusp exactly at the position of the anisotropy field Hk = 2(K1 + 2K2)/M s (adapted from Ref. [9.42]).

576


anisotropy energy Ea given by Eq. (9.11) and minimize the total energy E - -/~0Ms.H + Ea to obtain the curve M(H). Having a definite slope, this attains the saturated state at a finite field value, specifically the anisotropy field value Hk = 2(K1 + 2K2)/M s. The derivative d M / d H thus presents a step-like discontinuity at H = Hk, which transforms into a cusp upon making the second derivative d2M/dH 2. Asti and Rinaldi showed that this singularity is not lost when such a crystal is immersed in a sea of randomly oriented grains. They demonstrated this for non-interacting grains, but the experiments revealed that also in dense aggregates a cusp showed up in correspondence with the anisotropy field upon successive derivations of the magnetization curve. Indeed, it is difficult to observe such a singularity in soft magnets, where magnetostatic interactions appear to be very effective in the face of the anisotropy energies. The determination of the anisotropy field in polycrystalline hard magnets using the singular point detection (SPD) technique is best accomplished by analyzing the curve obtained in transient fashion by means of a pulsed field magnetizer (Section 8.2.4) [9.43]. Figure 9.12 provides an example of experimental behavior of d2M/dH 2 along the magnetization curve in a polycrystalline BaFel2019 sample [9.42]. The cusp in the second derivative occurs, as predicted, in correspondence with the anisotropy field. It is verified that this always occurs, independent of the specific distribution of orientation and size of the grains, even when the size is larger than required for achieving the single domain state.

aefeyences 9.1. S. Chikazumi, Physics of Ferromagnetism (Oxford: Oxford University Press, 1997), p. 274. 9.2. H. Zijlstra, (Amsterdam: North-Holland, Experimental Methods in Magnetism, 1967), vol. 2, 182. 9.3. R. Pauthenet, "Experimental verification of spin-wave theory in high fields," I. Appl. Phys., 53 (1982), 8187-8192. 9.4. T. Holstein and H. Primakoff, "Field dependence of the intrinsic domain magnetization of a ferromagnet," Phys. Rev., 58 (1940), 1098-1113. 9.5. H. Kronmfiller, "Micromagnefism in amorphous alloys," IEEE Trans. Magn., 15 (1979), 1218-1225. 9.6. E.M. Chudnovsky, "Magnetic properties of amorphous ferromagnets," J. Appl. Phys., 64 (1988), 5770-5775. 9.7. P. Szymczak, C.D. Graham, Jr., and M.R.J. Gibbs, "High-field magnetization measurements on a ferromagnetic amorphous alloy from 295 to 5 K," IEEE Trans. Magn., 30 (1994), 4788-4790.

REFERENCES

577

9.8. M.K. Wilkinson and C.G. Shull, "Neutron diffraction studies on iron at high temperatures," Phys. Rev., 103 (1956), 516-524. 9.9. A. Ferro, G. Montalenti, and G.P. Soardo, "Temperature dependence of power loss anomalies in directional Fe-Si 3%," IEEE Trans. Magn., 12 (1976), 870-873. 9.10. A. Arrott, "Criterion for ferromagnetism from observations of magnetic isotherms," Phys. Rev., 108 (1957), 1394-1396. 9.11. T.R. McGuire and P.J. Flanders, "Direct current magnetic measurements," in Magnetism and Metallurgy (A.E. Berkowitz and E. Kneller, eds., New York: Academic Press, 1969), p. 123. 9.12. J.E. Noakes and A. Arrott, "Initial susceptibility of ferromagnetic iron and iron-vanadium alloys just above their Curie temperature," J. Appl. Phys., 35 (1964), 931-932. 9.13. S. Arajs and R.V. Colvin, "Ferromagnetic-paramagnetic transition in iron," J. Appl. Phys., 35 (1964), 2424-2426. 9.14. S. Arajs, "Paramagnetic behavior of nickel just above the ferromagnetic Curie temperature," J. Appl. Phys., 36 (1965), 1136-1137. 9.15. J. Sievert, H. Ahlers, S. Siebert, and M. Enokizono, "On the calibration of magnetometers having electromagnets with the help of cylindrical nickel reference samples," IEEE Trans. Magn., 26 (1990), 2052-2054. 9.16. J. Crangle and G.M. Goodman, "The magnetization of pure iron and nickel," Proc. Roy. Soc. Lond., A321 (1971), 477-491. 9.17. S. Foner, A.J. Freeman, N.A. Blum, R.B. Frankel, E.J. McNiff, Jr., and H.C. Praddaude, "High-field studies of band ferromagnetism in Fe and Ni by M6ssbauer and magnetic moment measurements," Phys. Rev., 181 (1969), 863-882. 9.18. M. Liniers, J. Flores, F.J. Bermejo, J.M. Gonzalez, J.L. Vicent, and J. Tejada, "Systematic study of the temperature dependence of the saturation magnetization in Fe, Fe-Ni, and Co-based amorphous alloys," IEEE Trans. Magn., 25 (1989), 3363- 3365. 9.19. R.D. Shull, R.D. McMichael, L.J. Swartzendruber, and S.D. Leigh, "Absolute magnetic moment measurements of nickel spheres," J. Appl. Phys., 87 (2000), 5992-5994. 9.20. L. Petersson and A. Ehrenberg, "Highly sensitive Faraday balance for magnetic susceptibility studies of dilute protein solutions," Rev. Sci. Instrum., 56 (1985), 575-580. 9.21. A.M. Stewart, "The superconducting Faraday magnetometer: error forces and lateral stability," J. Phys. E: Sci. Instrum., 8 (1975), 55-59. 9.22. D. Zhang, Ch. Probst, and K. Andres, "A novel and sensitive Faraday-type magnetometer for the field range from 0 to 12 T," Rev. Sci. Instrum., 68 (1997), 3755-3760.

578


9.23. P.J. Flanders and C.D. Graham, "Magnetization measurements using the field gradient of a high-field solenoid," Rev. Sci. Instrum., 50 (1979), 1564-1566. 9.24. G. Felten and Ch. Schwink, "Design of a Faraday magnetometer in Bitter coils," J. Phys. E: Sci. Instrum., 13 (1980), 487-488. 9.25. R.D. Heyding, J.B. Taylor, and M.L. Hair, "Four-inch shaped pole caps for susceptibility measurements by the Curie method," Rev. Sci. Instrum., 32 (1960), 161-163. 9.26. R.T. Lewis, "A Faraday type magnetometer with an adjustable field independent gradient," Rev. Sci. Instrum., 42 (1971), 31-34. 9.27. A.M. Stewart, "Prediction of lateral instabilities in the Faraday magnetometer," J. Phys. E: Sci. Instrum., 2 (1969), 851-854. 9.28. R.D. Spal, "Production of uniform field gradients for magnetometers by means of current-carrying strips," J. Appl. Phys., 48 (1977), 1338-1341. 9.29. A. Hubert and R. Sch/ifer, Magnetic Domains (Berlin: Springer, 1998), 184. 9.30. J.S. Kouvel and C.D. Graham, Jr., "On the determination of magnetocrystalline anisotropy constants from torque measurements," J. Appl. Phys., 28 (1957), 340-343. 9.31. G. Aubert, "Torque measurements of the anisotropy energy and magnetization of nickel," J. Appl. Phys., 39 (1968), 504-510. 9.32. J.D. Sievert, "Anisotropy of energy and magnetization of iron-rich Si-Fe alloys," J. Magn. Magn. Mater., 2 (1976), 162-166. 9.33. B. Comut, S. Catellani, J.C. Perrier, A. Kedous-Lebouc, T. Waeckerl6, and H. Fraisse, "New compact and precise magnetometer," J. Magn. Magn. Mater., 254-255 (2003), 97-99. 9.34. EB. Humprey and A.R. Johnston, "Sensitive automatic torque balance for thin magnetic films," Rev. Sci. Instrum., 34 (1963), 348-358. 9.35. M. Tejedor, A. Fernandez, B. Hemando, and J. Carrizo, "Very simple torque magnetometer for measuring magnetic thin films," Rev. Sci. Instrum., 56 (1985), 2160-2161. 9.36. M.J. Pechan, A. Runge, and M.E. Bait, "Variable temperature ultralow compliance torque magnetometer," Rev. Sci. Instrum., 64 (1993), 802-805. 9.37. E. Joven, A. del Moral, and J.I. Arnaudas, "Magnetometer for anisotropy measurement using perpendicular magnetization," J. Magn. Magn. Mater., 83 (1990), 548- 550. 9.38. O. Cugat, R. Byme, J. McCaulay, and J.M.D. Coey, "A compact vibrating sample magnetometer with variable permanent magnet flux source," Rev. Sci. Instrum., 65 (1994), 3570-3573. 9.39. P.J. Flanders, "Magnetic measurements with the rotating sample magnetometer," IEEE Trans. Magn., 9 (1973), 94-109.

REFERENCES

579

9.40. P.J. Flanders, "A Hall sensing magnetometer for measuring magnetization, anisotropy, rotational loss and time effects," IEEE Trans. Magn., 21 (1985), 1584-1589. 9.41. G. Bottoni, D. Candolfo, A. Cecchetti, and F. Masoli, "Ratio of the rotational loss to hysteresis loss in ferromagnetic powders," IEEE Trans. Magn., 10 (1974), 317-320. 9.42. G. Asti and S. Rinaldi, "Singular points in the magnetization curve of a polycrystalline ferromagnet," J. Appl. Phys., 45 (1974), 3600-3610. 9.43. R. Gr6ssinger, Ch. Gigler, A. Keresztes, and H. Fillunger, "A pulsed field magnetometer for the characterization of hard magnetic materials," IEEE Trans. Magn., 24 (1988), 970-973.


CHAPTER 10

Uncertainty and Confidence in Measurements

The ideal objective of any measurement is the determination of the true value of a measurand. The real objective is to make the most accurate estimate of this true ~/alue because no measuring operation can exist without an error. Consequently, a measurement has a meaning if, having defined a measuring method and a measuring procedure, it provides the best estimate of the value of the measurand and a related uncertainty, the latter representing the degree of dispersion of the results around such an estimate. At the core of the concept of measurement lies the principle of reproducibility, which implies the possibility to compare results obtained at different times and in different laboratories. It is not only a vital requirement of any scientific investigation, but it also responds to practical needs in various fields, such as industrial production and quality control, commerce, law, health, and environment. In order to make meaningful comparisons, it is necessary that measuring uncertainties be treated through a consistent approach. Although the subject is very old, general consensus on the procedures to be followed for expressing the uncertainty has only been reached in recent times, under the initiative of the Comitf International des Poids et Mesures (CIPM), the highest authority in the field of metrology. Through the active cooperation of the National Metrological Institutes and various international organizations, the International Standard Organization (ISO) undertook the task of preparing a Guide to the Expression of Uncertainty of Measurement, which was eventually published in 1993 [10.1]. We will refer to this Guide in the following.

10.1 E S T I M A T E O F A M E A S U R A N D MEASURING UNCERTAINTY

VALUE AND

Measuring a physical quantity is a very common activity in everyday life and the concepts of measurement accuracy and repeatability do not 581

582

CHAPTER 10 Uncertainty and Confidence in Measurements

require special competence to be appreciated. It is intuitive to recognize that some kind of stochastic behavior inevitably affects any measuring operation, be it some gross evaluation performed through our senses or some sophisticated measure made by specialists in the laboratory. If we go somewhat deeper into the problem, we can easily verify that, by repeating the very same measurement m a n y times under identical conditions, scattered values of the measurand are found. Once ordered according to the customary histogram or frequency representations, these do provide the idea of an underlying probability distribution function [10.2]. Such an idea was made quantitative a long time ago. By denoting with x the generic value of the measurand subjected to direct determination, it can be shown that, for example, if the condition of stationarity is satisfied, the probability of finding it within a prescribed interval (x, x + dx) is given by the normal distribution function:

(X- ~)2) exp 2~2 dr(x) dx -~r2 ~

"

(10.1)

f(x) is a symmetric function, peaked at the mean value x = ~, and satisfies the normalization condition y~-oof(x) dx --- 1./~ is also the most probable value of x and is identified with the true value of the measurand, with the meaning that this term has in a statistical sense. It would be the result of a perfect measurement and cannot be known. If the outcome of a measurement is x, the difference 8 = x - ~ is defined as the measurement error, again an unknowable quantity, cr2 is the second-order m o m e n t about the mean:

~

oo

o .'2 - -

(x -

~)2f(x)

dx

(10.2)

and is called variance. Its square root cr provides a measure of the dispersion of the measured values around the true value and is called

standard deviation. Historically, the normal function was proposed by Gauss in order to represent the error distribution in the astronomical observations. It is the idealized distribution function associated with a truly stochastic variable. Any reading or measurement of this variable can be thought of as affected by m a n y small contributions of r a n d o m sign and amplitude, which are generated by a large n u m b e r of sources of influence. The central limit theorem [10.2] ensures that, in a case like this, the values taken by the variable closely follow a normal law, whatever the distribution function of the contributing variables. It is therefore understood that fix) can be

10.1 ESTIMATE OF A MEASURAND VALUE

583

assumed of normal type [10.3]. Any practical measuring operation, carefully performed and corrected for any possible bias, can only approximate the generation of a truly Gaussian process and what one achieves, in general, is an estimate of the true value of the measurand. If n independent observations of the measurable quantity x are performed, providing the values X(1),X(2)~...~X(n)~ the best estimate is given by the arithmetic mean:

YC--

~.k

=1 F/

x(k) ,

(10.3)

where the individual outcomes X (i) differ because of random effects. In the limit n--* oo~ it is expected that ~ =/~. In reality, it is difficult in most instances to fulfill the condition of stationarity for a sufficiently long time and a convenient number of repetitions is chosen according to specific conditions imposed by the problem under testing. From a sample of measurements, one can make an estimate of the variance cr 2 of the whole population of the possible values of the measurand by defining an experimental variance s2(x(k)). This characterizes the dispersion of the measured values around ~:

11 82(x(k)) = Yk=l

(x(k) --

~)2

n- 1

'

(10.4)

together with its square root, the experimental standard deviation s(x(k)). Notice that the number of degrees of freedom v = n - 1 is used in the definition of the experimental variance in Eq. (10.4). In fact, of the n terms (x (k) -Yc)~ only n - 1 are independent. Since the experiment provides the value ~ as a best estimate of the true value of the measurand, we wish to know how good such an estimation is. ~ is itself a random quantity and, according to Eq. (10.3), its variance and standard deviation are cr2(Yc) = cr2/n and cr(~)= cr/x/~ , respectively (the law of large numbers). The best estimates of 02(~) and cr(~) are

$2(~) __

S2(x(k))

__

n

ylkZ=l (x(k) __ ~)2 n(n - 1) (10.5)

S(x(k)) s(~)

-

-- .. I ~k=l (x(k) --

~

n ( n - 1)

~)2

584


the experimental variance and the standard deviation of the mean, respectively, s(~) is also called the standard uncertainty u(yc) of the best estimate of the measurable quantity x u(~) = s(~)

(10.6)

and the corresponding variance is u2(x)~-s2(x). According to this definition, the standard uncertainty u(~) is a parameter providing a quantitative evaluation of the dispersion of values that can be reasonably attributed to the measurand. By making repeated measurements of the same quantity, stochastic effects are thus revealed and can be quantified through the standard uncertainty. There are, however, further sources of uncertainty, whose contribution remains constant while the measurements are repeated. They can derive from the environmental conditions (e.g. temperature, humidity, and electromagnetic interference), calibration and resolution of the equipment, peculiarities of the electrical circuit (e.g. thermoelectromotive forces), drifts and distortions, inaccurate assumptions about constants and parameters to be used in the data treatment procedure, and personal errors. The related uncertainty is traditionally classified as systematic, in contrast with the random uncertainty, associated with repeated measurements. It is recognized, however, that such a classification applies to a specific measurement only, because what is random in a measurement can become systematic in a further measurement at a different level. For example, an instrument calibration made in a standard laboratory will report the combination of random and systematic components as a single value of the total uncertainty. A laboratory at a lower hierarchical level, making use of this calibrated instrument, will introduce this value as a systematic effect in the derivation of its uncertainty budget. Systematic effects are expected to produce a bias on the random distribution of the x (k) values obtained upon repeated measurand determinations. This bias can be quantified and corrected for a good proportion, as schematically illustrated in Fig. 10.1, but a residual contribution to the uncertainty of the measurement, having a systematic origin, is nevertheless expected to remain. This can be evaluated by judicious appraisal of all available information on the physical quantity being measured, the measuring procedure and the measuring setups, previous knowledge on the subject, etc. Notice that, in some instances, the correction for the bias can be estimated to be zero, without implying that the associated uncertainty contribution is also zero. The method by which an uncertainty contribution deriving from systematic effects is obtained is defined as a Type B evaluation method. For repeated measurements, we speak of a Type A evaluation method of the uncertainty. These definitions

10.1 ESTIMATE OF A MEASURAND VALUE

0.201

585

-",,,,

0.10

o 0.00. (a)

i\ ~

95

1O0

105

x

m,L

,

bias

i--

12

i _ e

A

x

v

i

Z

(b)

85

90

....

i ....

95 x

i ......

100

105

FIGURE 10.1 (a) Normal distribution function f(x) for the probability of finding the value of a measurand in the interval (x,x + dx) (Eq. (10.1)). x is assumed to be a truly stochastic quantity. The mean value /~ is defined as the true value of the measurand, an ideal and unknowable quantity, cr is the standard deviation. (b) Independent repeated measurements generate numbers that, arranged in a histogram, emulate the normal distribution. The raw data are shown on the left, with their mean ~ and the related standard deviation s(2) (not in scale). The standard deviation of the mean is equivalently called standard uncertainty u(~) (see Eqs. (10.5) and (10.6)). After correction for the systematic effects (somewhat exaggerated here for clarity), the best estimate of the true measurand value is characterized by a combined uncertainty uc(x), including the uncertainty on the correction.

586


are recommended by the ISO Guide [10.1]. In any case, all components of the uncertainty, be they evaluated with A or B methods, are described by the same statistical methods, characterized by probability densities with variances and standard deviations, and are treated and combined in the same way. For the Type A uncertainties, the probability densities are obtained from observed frequency distributions, while for the Type B uncertainties one makes use of "a priori" probabilities. When applying the B method, an assumption is made regarding the distribution function of the measurand values. If derived from a calibration certificate, this distribution is conveniently assumed as being of the normal type. In other cases, it is only possible to estimate upper and lower bounds x0- and x0+ for the values that x can take in a specific measurement. With no further knowledge on how these values are distributed, one can reasonably assume that they are equally likely to belong to any point of the interval (Xo-,Xo+). The variance and the uncertainty associated with the expectation value ~ = ( x 0 - + x0+)/2 of this rectangular distribution are expressed, posing x 0 - + x0+ = 2a, by the equations: U2(~ ) __ (X0+ -- X 0 _ ) 2

12

10.2 C O M B I N E D

a2

= -~-,

a

u(~) = - ~ .

(10.7)

UNCERTAINTY

In the usual case one does not make a direct determination of the measurand, but a certain number of input quantities are sampled or considered, to which the measurand is related by a functional relationship. Let the functional relationship between the output quantity y and the input quantities Xl,X2, ...,XN be y = g(xl~x2, ..., XN).

(10.8)

If the best estimates of the input quantities a r e Xl~ x2~-..~ XN~ we write for the best estimate of the output quantity:

9 = g(x~,X2,

"", IN).

(10.9)

The identificationof the input quantities is a crucial step of the whole process of uncertainty determination. They can include, besides the quantities subjected to direct measurement, the bias corrections to suggested by the specifically considered measuring procedure. The problem then becomes one of determining variance and uncertainty of from knowledge of the same quantifies for ~i,x2,...,xN, taking into

10.2 COMBINED UNCERTAINTY

587

account possible correlation effects. To this end, we assume that the function g and its derivatives are continuous around the expectation value Y- A Taylor series development, truncated to the first order, provides y-- y = ~ ~ i=1

(10.10)

(Xi - YCi)

for small intervals (X i -- YCi). The square of Eq. (10.10) is

(y_~)2=~.

N (0g)2 N-1 N OR OR ~ ( XOXj ~ (Xi_YCi)2q_2 y . y . ~OXi i=1 i=1 j=i+l

i -- YCi)(Xj -- YCj). (10.11)

By interpreting the differences appearing in Eq. (10.11) as experimental samples and taking the averages, we can express the variance of the output estimate as a combination of the variances u2(xi) and covariances U(YCi,YCj) of the input estimates, according to the law of propagation of

uncertainty: N (Og)2

uc(y)= ~22

i=1

~

N-1N OgOg /,/2(~i)+ 2 y. y. OXi ~u(Yci,Yq). OXi i=1 j=i+l

(10.12)

-

uc(y) is called combined variance and its square root Uc(y) is the combined standard uncertainty. The partial derivatives in Eq. (10.12) are called sensitivity coefficients and by posing ci -~ Og/Oxi we can rewrite Eq. (10.12) as N N-1 N 2Uc(y) = y c2ua(xi)-}-2 y~ y . CiCjU(Xi~YCj). i=1 i=1 j-i+1

(10.13)

Equation (10.13) is of general character and combines variances and covariances of the input quantities irrespective of the type of evaluation method (either A or B) employed in their derivation. An input estimate Xi can be associated with both Type A and Type B uncertainties and the related variance in Eq. (10.13) is written as U2(Xi) = U2A(YCi)-~- U~(YCi).

(10.14)

Notice that the output quantity y is associated in many cases with an approximately normal distribution function, although the distribution functions of the input quantities can be far from normal. One can, in fact, linearize the functional relationship (10.8) around the best estimates of the input quantities by means of a Taylor development truncated to

~8


the first order (Eq. (10.10)). The output distribution is then provided by the convolution of the input distributions and, according to the central limit theorem, it can be approximated by a normal distribution, the higher the number of repetitions and the input quantities, the better the approximation. The case where a dominant Type B component of uncertainty exists, with distribution different from normal, is an exception to this rule. 2- -2 In many cases, it is useful to resort to the relative variance Uc(y)/y and the relative standard uncertainty uc(Y)/9. Remarkably enough, if it occurs that the functional relationship relating the output quantity and the input independent quantities has the general form y = m.xPl.x p2, ...,XPNN, with m a constant coefficient and Pl,P2, ...,Pn either positive or negative exponents, we can express the relative variance as

2Uc(y)

N U2(Xi) ~2 __~p2 L-~ 9 i=l Xi

(10.15)

The sensitivity coefficients are sometimes evaluated by experiments by determining the variation of the output quantity y upon a small variation of an input quantity xi, the other input quantities being kept fixed (see Eq. (10.10)). If the input quantities are uncorrelated, the covariance u(Yci,ycj) is zero and the combined uncertainty reduces to

Uc(Y) = i ~'i=1r

(10.16)

A measure of the degree of correlation is provided by the value of the coefficient: u(~i, ~j) r(Yci,Ycj) = u(Yci)u(Ycj),

(10.17)

which varies from 0 to 1 on going from uncorrelated to completely correlated input variables. In the latter case, the combined standard uncertainty becomes

N UcQ~) --- ~ CiU(YCi). i=1

(10.18)

For a Type A evaluation of the uncertainty, the covariance of two correlated input estimates (2i, x]) can be experimentally evaluated by forming the cross-products (xl k)- Yci).(x~k)- 2]) at each repetition and

10.3 EXPANDED UNCERTAINTY AND CONFIDENCE LEVEL

589

averaging them according to the equation: i! u(2i,Yq) = s(2i, 2j)= n(n - 1) y (xlk) -- Xi)'(X~k) -- ~j)"

(10.19)

k=l

For a Type B evaluation, critical analysis of the available information on the reciprocal influence of the input estimates should be carried out. If, for example, it is known that a variation Ai of xi produces a variation &j of ~j, the correlation coefficient can be roughly estimated as r(Yci, Ycj) ~ u(Ycj)a i "

(10.20)

Two input quantities (Xi,Xj) can have their correlation originating from a common set of independent and uncorrelated quantities z,,,. Let gi(Zl, ...,ZM) and gj(zl, ...,zM) be the functional relationships associated with xi and xj, respectively. We can write for the covariance of the best estimates (2i, ~j): M Ogi Ogj U2(~,n). U(YCi'YCJ) = Y" OZ,,, OZ,n llZ--1

(10.21)

10.3 E X P A N D E D U N C E R T A I N T Y A N D C O N F I D E N C E LEVEL. WEIGHTED UNCERTAINTY The discussion of the derivation of the uncertainty budget carried out in the previous sections illustrates the great merit of the procedure recommended by the ISO Guide, which permits one to combine in a consistent way all the contributions to the uncertainty, derived either from repeated measurements or through "a priori" probabilities. The information conveyed by the measurement can then be confidently collected in two parameters: the best estimate (experimental mean) and the combined uncertainty. Except for special cases, the probability distribution of the output quantity, being the convolution of the input distributions, is approximately of normal type (central limit theorem). If z is a quantity described by a normal distribution function, characterized by an expectation value/~ and a standard deviation or, and we define a confidence interval + kcr around/~, by integrating the distribution function over it we will achieve a corresponding confidence level p (the included portion of the area of the distribution). With coverage factors k = 1, 2, and 3, the confidence levels are p = 68, 95.5, and 99.7%, respectively. Let y be a quantity, defined as in Eq. (10.8), subjected to measurement and characterized by an experimental best estimate 9 and

590


a combined standard uncertainty Uc(y). We wish to determine the coverage factor k identifying an expanded uncertainty U = kuc(~) , that is, an interval 9 - U --< y - 9 + U, to which the true value of the measurand is expected to belong with a given high confidence level p (e.g. 95%). With knowledge of U, the result of the measurement can be declared in the form: Y = 9 + U.

(10.22)

We do not actually know the expectation value /~ and the standard deviation or of the output distribution, but only the best estimate 9 and the standard uncertainty uc(~). We then consider the quantity t-

y- y

(10.23)

Uc(9)

and its probability distribution function ~ t ) . The integration of q~(t) over a certain interval ( - t p , +tp): f+t~ q~(t) dt

(10.24)

P = d-tp

provides the confidence level for the expanded interval U = t~Uc(~)= kuc(9). In fact, the condition ( - t p ~ t ~ tp) is equivalent, according to Eq. (10.23), to the condition (9 - tpUc(9) ~ y ~ 9 + tpUc(9)). When y is a single quantity subjected to direct measurement and its best estimate 9 = ~ is obtained by means of a series of n independent repeated measurements, q~(t) is described by the Student distribution function (t-distribution): ~b(t) -

~

1

F((v + 1)/2) ~+1)/2, F(u/2) (1 + ta/v) -(

(10.25)

where the properties of the F function are known and v = n - 1 is the n u m b e r of degrees of freedom, q~(t) reduces to the normal distribution in the limit v---+ oo, a condition already well approximated for v --- 50. For the general case where y is a function of two or more input quantities, Eq. (10.25) can be used only as an approximation by introducing an effective number of degrees of freedom /jeff in place of v. /jeff can be calculated in terms of the degrees of freedom/ji of the input uncertainty contributions u2(9) c2ua(xi)~ under the assumption of independent input estimates. It is provided, in particular, by the Welch-Satterthwaite formula: =

4-

/Jeff =

uc(y) 4-

ui (y) /ji

,

(10.26)

where u~(9) = (y../N_au 2i (y)) - ~_. While for the Type A evaluation, z,i = n - 1 , the degrees of freedom in the Type B evaluation can be estimated on


591

the principle that the more reliable the standard uncertainty ui(y) the higher ~'i. For the usual case where an "a priori" probability is taken, the uncertainty is completely defined and z,i--* oo. Tabulations are available where, for given ~'eff values, the coverage factors k = tp associated with a confidence level p are provided [10.4]. In most cases, a confidence level p---95% is deemed adequate. The corresponding coverage factors are provided, for different values of ~'eff, in Table 10.1. An application of the concepts discussed in this section is given in Appendix C (Example 2). The procedure for providing the result of a measurement can therefore be summarized as follows: (1) The measurement process is modeled and the mathematical relationship y = g(xl,x2, ...,x N) between the measurand y and the input quantities Xl,X2, ...,XN is expressed. These quantities also include possible bias corrections. (2) The best estimates xl, x2,..., XN of the input quantities are made. (3) The standard uncertainties u(xi) of the input estimates are found. Type A evaluation is applied for input estimates obtained by means of repeated measurements and Type B evaluation for all other kinds of estimates. If there are correlations between input quantities, covariances are considered. (4) Using the functional relationship y = g ( x l , x 2 , . . . , X N ) , the best estimate 9 of the measurand is made. (5) The combined standard uncertainty Uc(y) is calculated by combination, with the appropriate sensitivity coefficients, of the variances and covariances of the input estimates. (6) The expanded uncertainty U = kuc(9) is determined, with the coverage factor k typically ranging from 2 to 3 for a confidence level of 95%. The actual value of k depends on the effective number of degrees of freedom Veff, calculated by means of the Welch-Satterthwaite formula, and are found in generally available tabulations. The result of the measurement is eventually declared a s y = ~/+ U.

TABLE 10.1 Coverage factor k as a function of the effective degrees of freedom Veff

for a confidence level p = 95.45%. Ueff k

1 13.97

2 4.53

3 3.31

4 2.87

5 2.65

7 2.43

10 2.28

20 2.13

50 2.05

oo 2

k is provided by the Student distribution function and coincides, in the limit veff~ o% with the value provided by the normal distribution.

592


It may happen that the same quantity is measured by means of different methods or in different laboratories and the problem arises of combining the results in order to obtain the most reliable estimate of the measurand value. It is usually held that, being the different estimates normally associated with different uncertainties, a comprehensive estimate based on the data generated by the whole ensemble of experiments is best obtained by means of weighted averaging. Let us assume that M independent experiments, made of a convenient number of repeated measurements, have produced the best estimates Yl~ Y2~'" "~ YM and the related uncertainties uc(yl),Uc(92),...,Uc(gM). We look for a weighted estimate of the type M

= ZgilJi

(10.27)

i=1

having minimum variance. The weight factors gi must satisfy the condition: M

Z gi = 1.

(10.28)

i=1

We then write the variance of ~ in terms of the variances of the estimates Yl,Y2, ...,YM

M U2(~) = Z gi2 Uc(Yi) 2 i=1

(10.29)

and find the set of factors gl,g2, ...,gM minimizing u2(~) [10.5]. With the use of the Lagrange multiplier k, the variance can be written as u 2 ( Y ) = Z gi2 Uc(Yi) 2 - + ,~ 1 - Z g i i=1 i=1

(lO.3O)

and the minimization conditions

0u2(#) Ogi

= 2giu2(gi)- X = 0

(10.31)

provide the weight factors k

gi = ," cl,y "

(10.32)

as a function of the multiplier JL This is eliminated through the normalization constraint (10.28) and the factors gi are thus obtained as


593

a function of the input variances:

U2(~ti)

g~ =

~.iM1

.

(10.33)

1 U2(~/i)

The weighted mean and the associated weighted variance follow from Eqs. (10.27) and (10.29): y.iM1

Yi

u2(yi)

~t ~-

1

,

(10.34)

2 Uc(Yi)

1

u2(~)

M =

1 u2(yi) "

(10.35)

According to these equations, the smaller the uncertainty associated with a result, the higher its role in determining the reference value ~ and the uncertainty u(,0). A confidence interval can be identified with ~ + U r e f = ~t + ku(~t), where the coverage factor k is taken from the t-distribution for v = M - 1 (see Table 10.1). An example of intercomparison of magnetic measurements is shown in Fig. 10.2. Laboratories find the most stringent test of their measuring capabilities in the comparison exercise. At the highest level, the national metrological laboratories organize key comparisons as a technical basis for establishing the equivalence of measurement standards and the mutual recognition of calibration and measurement certificates. The degree of equivalence of each national measurement standard is expressed quantitatively by two terms: its deviation from the key comparison reference value and the uncertainty of this deviation at 95% level of confidence [10.6]. The assumption of the weighted mean (10.34) as the reference value is considered appropriate if the collective measurements are consistent and they can be treated as part of a homogeneous population. Discrepant results often arise in intercomparisons and special approaches have therefore been proposed to deal with the problems, including politically sensitive issues, raised by the presence of inconsistent data [10.7, 10.8]. It is clear that meaningful comparisons can be pursued only where all laboratories follow a common approach to the evaluation of the measuring uncertainty, such as the one provided by the ISO Guide [10.9].

594


3.2

t_ - Uref

3.1

cL 3.0

2.9

Laboratories

FIGURE 10.2 Six different laboratories perform the measurement of magnetic power losses on the same set of non-oriented Fe-Si laminations with the SST method [10.10]. They report their best estimates and the related extended uncertainties as shown in this figure. Analysis shows that the result provided by laboratory 6 is to be excluded, because it largely fails the consistency test provided by the calculation of the normalized error (Eq. (10.36)). The reference value Pref solid line) and the expanded uncertainty Uref (delimited by the dashed lines) are then obtained by re-calculating Eqs. (10.34) and (10.35) with the results of laboratories 1-5.

Let us analyze, as an example regarding magnetic measurements, some results derived from an intercomparison of magnetic power losses in non-oriented Fe-Si laminations [10.10]. Six laboratories (i = 1, ..., 6) provide, as shown in Fig. 10.2, their best estimates Pi of 50 Hz losses at 1.5 T (full dots) and the associated expanded uncertainties (at 95% confidence level) Ui = ku~(Pi). These data are all included in a preliminary determination of the weighted mean (reference value Pref) and the expanded weighted uncertainty Uref~ according to Eqs. (10.34) and (10.35). The consistency of the reported uncertainties with the observed deviations of the best estimates Pi from Pref is then verified. To this end, the normalized error

Eni

=

IPi - Prefl ~/U2..}_Ur2;

(10.36)

10.4 TRACEABILITYAND UNCERTAINTY

595

is considered [10.11]. When the dispersion of the individual estimates is in a correct relationship with the correspondingly provided uncertainties, it is expected that Eni < 1 [10.12]. In the present case, the reported (P6, U6) values largely fail to satisfy this condition (En6--3.1), due to both unrecognized bias and unrealistic uncertainty estimate. They are consequently excluded from the analysis. Pref and Ure f are then re-calculated by means of Eqs. (10.34) and (10.35), providing the results reported in Table 10.2. It should be stressed that the estimated expanded relative uncertainty of the reference value Uref~re1 is of the order of 1%, typical of this kind of measurement. It is also noticed that the result of laboratory 5 is not completely satisfactory because IP i - Prefl is higher than the related expanded uncertainty: (10.37)

U(Pi - Pref) -- k~/u2(Pi) if- u2(Pref) 9

10.4 T R A C E A B I L I T Y MAGNETIC

AND UNCERTAINTY MEASUREMENTS

IN

Measurements are indispensable for the manufacturing and trade of products and for any conceivably related research activity. They need to be traceable to the relevant base and derived SI units, that is, related to the corresponding standards through an "unbroken chain of comparisons, all having stated uncertainties" [10.13]. Industrial and research laboratories can achieve traceability for a specific kind of measurement through accredited laboratories or directly to National Metrological Laboratories (NMIs). The mission of NMIs is to ensure that the standards are the most accurate realization of the units, so that these can be disseminated to the national measurement network. To ensure this calibration flow, national calibration and accreditation systems have been developed. The NMIs engage in extensive intercomparisons of standards, organized either by the regional metrological organizations (e.g. EUROMET and NORAMET) or the Consultative Committees of the International Committee for Weights and Measures (CIPM). Supervision of the intercomparison activity is carried out by the Bureau International des Poids et Mesures (BIPM), which has the task of ensuring worldwide uniformity of measurements and their traceability to the SI units (Fig. 10.3). Physical standards for magnetic units, traceable with stated uncertainties to the base SI units, are maintained in several NMIs and used for dissemination to measurement and testing laboratories [10.14, 10.15]. Illustrative examples of magnetic standards and calibration capabilities

596

c5

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~b

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0

0

0

0

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qa

x

0

b, lii 0 ,,,...

0 0

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l>

0

.,ii

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.,i,,,z

.~i~

Ii,I

".n

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0 .4.a

~,

II

~:

~

II

9~

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,4.,

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v

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O~

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r~

Uncertainty and Confidence in Measurements

I

I

o oo

L~b 04 0

xo o

i c5

i c5

Lr) L~ ~0 t~ O'b O~ t~. ~11 ~0 brb 0 0 0 0 0

I

c~c~c~c~c~

I

0 I..~ L~ I..~ 0 O0 C~ L~ ~'~

0 LO 0 LCb 0 C~b t'~ ,~ XO t~b 0 0 0 0 0

c5c5c5c~c5

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0 0 0 0 c : )

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c:5 c:5 ~-~ c:5 ~-~

CHAPTER 10

b

bO

I

b~

bO

bO

v

bO

0 0


597

W......................................... I BIPM

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--~

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NMI

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Accredited

Laboratory .

.

.

.

.

.

.

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.

.

.

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Calibration& TestingLab. FIGURE 10.3 Traceability chain in measurements. The calibration and testing laboratories can relate their measurements to the SI units through a flow of calibrations starting from the National Metrological Laboratories (NMIs). The NMIs perform mutual comparisons of their standards, under supervision by the Bureau International Poids et Mesures (BIPM). developed by NMIs are given in Table 10.3 [10.6, 10.16]. Examples of recent NMI intercomparisons of magnetic measurements, regarding DC and AC flux density and apparent p o w e r / p o w e r loss in electrical steel sheets, are reported in Refs. [10.17, 10.18]. The importance for industrial customers of measurement traceability and calibrations, as ensured today by the NMIs, is easily appreciated for magnetic measurements. For example, a magnetic steel producer can ensure the quality of the grainoriented laminations delivered by one of its plants only through periodic calibrations of its magnetic equipment, traceable to an NMI laboratory. Since a large plant can produce 105 ton/year of this high-quality material, worth around s 10s, the economic impact of traceable magnetic measurements is apparent. It should be stressed that when ferromagnetic (or ferrimagnetic) materials are characterized, several factors can detrimentally affect the reproducibility of the measurements. For one thing, the magnetization process is stochastic in character and strongly affected by the geometrical properties of the sample and magnetic circuit. In addition, time effects are ubiquitous, either due to aging or various types of relaxation effects, and many alloys (e.g. rapidly quenched magnetic ribbons) display an intrinsically metastable behavior. Measurements must then be carried out under tightly prescribed conditions, such as those defined by written standards, and by means of accurately calibrated setups in order to achieve

598

O

t~

t~

Z

~ t~

t~

g~

t~

t~

o

x

?, X

T

?

'i'

? X

?

I

? X

I

X

?

X

IN

X

T o

?

L~

y

?

X

? X Y

I

m 0

I 0

? 0

I ~

, 0

4-

o -,~ ,-..,

~

r~

x

b,,,

x

~

~

~

x

~

x

L~

8

r~

",,0

Y

Z

~

4-1


g~ •

D

~

z

r,,,2

10.4 T R A C E A B I L I T Y

r

? x

~

AND

I

O r

o

cf] .i.a

UNCERTAINTY

~2

?

x

?

I

?

e:a3

I

u3

[Pi- PrefI. The same condition is satisfied by ( S i - S r e f ) / S r e f , which is found to be in the 1% range. It has been previously emphasized that intercomparisons have meaning if the different laboratories follow a uniform approach to the measuring uncertainty. The ISO Guide provides such an approach. It therefore appears appropriate to discuss a few illustrative cases of correspondingly obtained uncertainty determination in magnetic measurements.

10.4.1 Calibration of a magnetic flux density standard A Helmholtz pair is prepared to serve as a magnetic flux density standard in the range 2 x 10-4T ~ B-----4 x 10-2T. Each winding is made of 2210 turns, with average radius r - 0.125 m, and is provided with a supplementary 200-turn coil, improving the uniformity of the generated axial field. The Helmholtz coil constant kH - - B / i is obtained by the simultaneous measurement of the magnetic flux density in the center of the pair and the circulating current. B is detected by means of a lowfield NMR probe operating in the range 0.034-0.121 T. By placing the Helmholtz pair within a triaxial Helmholtz coil system, the earth's magnetic field is compensated to the level of 0.02 ~T. The calculations show that inside the active volume of the probe (11 m m x 6.5 mm), the produced field is homogeneous enough to permit safe establishment of the resonance conditions. It is observed, in particular, that the maximum relative variation of the field amplitude in this volume is --- 1.5 x 10 -5. The current i (of the order of 1.6 A for B --- 0.036 T) is determined by detecting the voltage drop Vacross a standard resistor R = I f~ (four point contacts). Twenty repeated acquisitions are made over a time span of 10 s, short enough to keep the temperature increase of the windings due to Joule heating within z~T---0.4 ~ The coil temperature is controlled immediately before and after the determination of B by measuring the resistance

10.4 TRACEABILITY AND UNCERTAINTY

601

E ~ NMI1

15.

Jp= 1.7T f = 50 Hz

rrrrrl NMI 2 NMI 3

10.

52

.5

-1.0

-d.5

9 o~o

"

!

0~5 ' 1.0

9

1.5

( ~'mref)/mref (%)

(a) 15.

Epstein test frame, Jp = 1.7T f= 50 Hz

~NMI, r~lT[ NMI2 NMI3

10. ,.,., .,_.

- 3. . . . - 2

(b)

"1

'

6

m

"

i

"

~

"

~

( S i- Sre f) / Sre f ( % )

FIGURE 10.4 An intercomparison of power loss P and apparent power S carried out by three NMIs in 15 different types of grain-oriented Fe-Si laminations tested by the Epstein method identifies, for each lamination, the reference values Pref and Sref. The relative deviations (Pi - P r e f ) / P r e f and (Si - S r e f ) / S r e f of the best estimates from the reference values are in the 0.5 and 1.0% range, respectively (adapted from Ref. [10.18]).


602

of the winding. B and V are s i m u l t a n e o u s l y detected and the ratio k H -is calculated. The best estimate is written as

(B/V)R

w

kH = (B/V)R + (~(kH)B -4- (~(kH)V q- ~(kH)R,

(10.38)

w h e r e the first term on the right-hand side is the experimental mean, as p r o v i d e d b y Eq. (10.3) w i t h n = 20 a n d (~(kH)B, (~(kH)v, and (~(kH)R are the bias c o m p e n s a t i o n terms associated w i t h field reading, voltage reading, a n d s t a n d a r d resistance value. The bias terms are d e e m e d negligible a n d w e find, from averaging, kH --0.01952636 T/A. The relative c o m b i n e d u n c e r t a i n t y is provided, according to Eq. (10.16), as a function of the Type A a n d Type B i n p u t uncertainties b y the expression: w

uc(kH) kH

i

u2(kH) +

U2B(BB)A U2B(V)} u2B(R)

~2

e 2

R2

(10.39)

,

as s u m m a r i z e d in Table 10.4. The Type A uncertainty, evaluated for n - - 2 0 , is uA(kH)/kH = 1.5X10 -6. The Type B uncertainty

UB(B)/B

10.4 Uncertainty budget in the calibration of a standard magnetic flux density source

TABLE

Source of uncertainty

Distribution Divisor Relative Sensitivity Degrees of function uncertainty coefficient freedom

Magnetic field reading Voltage reading Standard resistor (calibration) Repeatability Combined relative standard uncertainty Expanded uncertainty (95% confidence level)

Rectangular

x/3

2.1 x 10 -5

1

oo

Rectangular Normal

~/3 2

1.2 x 10 -5 2.5 x 10 -5

1 1

oo oo

Normal Normal

1

1.5 X 10 -6

1

-

3.4 x 10 -5

-

19 137

6.8 x 10 -5

Coverage factor k=2

The source is realized by means of a Helmholtz pair (average winding radius r = 0.125 m) energized by a maximum current of 1.6 A. The calibration is performed via a low field NMR probe at B = 0.036 T. The Helmholtz pair is placed in a field-immune region (residual B--- 20 nT), at the center of a triaxial Helmholtz coil setup providing active compensation of the earth's magnetic field. The maximum field gradient in the active NMR probed central volume 11 mm x 6.5 mm of the standard coil is 3 x 10 -7 T / m m at B = 0.036 T. The result is kH = kH + U = 0.01952636 + 1.33 x 10 -6 T/A.


603

associated with the B reading is the sum of two main independent contributions: (1) spatial inhomogeneity of the generated field and uncertainty in the position of the NMR probe; (2) temperature variation and uncertainty on its value during the measuring time. Regarding contribution (1), we assume a rectangular distribution with upper and lower bounds differing by 2a(1)/B--4x10 -5 and we estimate from Eq. (10.7) u~)(B)/B -- 1.2 x 10 -5. Contribution (2) is estimated on the basis of a series of measurements around room temperature (e.g. between 19 and 30 ~ With measuring temperature fluctuation limits +AT = 0.2 ~ the rectangular distribution of half-width a (2)/B = 3 x 10 -5 is evaluated, leading to U(B2)(B)/B-- 1.7 • 10 -5. Consequently, UB(B)/B= 2.1 x 10 -5. The voltmeter specifications provided by the manufacturer give, in the employed 10 V scale, a 1 year accuracy of 12 p p m of reading + 2 p p m of range. For a 2.3 V read-out on this scale, this corresponds to a semi-amplitude of the distribution a = 4.8 x 10 -5 V. The corresponding relative uncertainty is UB(V)/V = (a/x/3)(1/V)--1.2• 10 -5. The calibration certificate of the standard I f~ resistor provides a 2or uncertainty of + 5 0 p p m . It is then UB(R)/R=2.5xIO -5. The relative combined standard uncertainty is obtained by Eq. (10.39) as uc(kH)/kH = 3.4 X 10 -5 and the expanded uncertainty at 95% confidence level is, with coverage factor k---2, U/fcH = 6.8• 10 -5. We eventually write the Helmholtz coil constant at the temperature T -- 23 ~ as kH = kH + U -- 0.01952636 + 1.33 • 10 -6 T/A.

10.4.2 D e t e r m i n a t i o n of the D C polarization in a ferromagnetic alloy We wish to determine the normal magnetization curve of a non-oriented Fe-(3 wt%)Si lamination with the ballistic method. We want to know, in particular, the uncertainty associated with the determination of the polarization value J at a given applied field. Testing is made, according to standards, on half longitudinal and half transverse Epstein strips on a rig provided with compensation of the air flux. Each point of the curve is obtained, after demagnetization, by switching the field several times between symmetric positive and negative values and eventually recording the flux swing 2 h ~ = 2NSJ, where N is the number of turns of the secondary winding (N = 700) and S is the cross-sectional area of the sample, by means of a calibrated fluxmeter. The measurement is repeated five times, always using the same procedure, and the arithmetic mean h ~ is obtained. The best estimate of the polarization value for

604


a given applied field H is A~ J= ~ 4- 3(J)d q- ~)(J)a if- 3(1)s q- 3(J)T,

(10.40)

where 3(J)d, 8(/)a, 3(/)T, and 3(J) s are the bias corrections for fluxmeter reading, residual air flux, temperature, and sample cross-sectional area, respectively. The relative combined standard uncertainty of the polarization value is then expressed through the Type A and Type B contributions (see Eq. (10.15)) as Uc(j[) / U2(A(I )) U2(~-~)d u2(A(I))a U2(~-~)T u2(S) . (10.41) i -- V ~-~2 if- A(I)2 -}- A ~ 2 if- A ~ 2 -} ~ $2

Eight strips (two on each leg of the frame), nominally 305 m m long and 30 m m wide, are tested. The sample cross-sectional area S is determined by the precise measurement of the total mass 8m and the strip length l as S = m/431, assuming the nominal material density 8 = 7650 k g / m 3. It is found S = 2.9529 x 10 -5 m 2. The variance u2(S) is obtained by combination of the variances associated with m, 1 and 3. The uncertainty of the value of 8 is by far the largest and we write

S2

~

(~2

(10.42)

Based on the data provided by the steel producer, it is assumed for 8 a rectangular distribution of semi-amplitude a - 25 k g / m 3. Since UB(3)a/x/-3, we obtain u2(S)/S 2= 3.6• -6. Again, we consider the bias compensation terms equal to zero and from the five repeated measurements, it is obtained, for H - 80 A / m , A ~ - 0.02013 V s, i.e. J = 0 . 9 7 3 8 T . The associated Type A uncertainty is found to be UA(A~)/A(I) = 2 x 10 -3. The fluxmeter, calibrated by means of a standard mutual inductor, is assigned a relative uncertainty (1or) on the employed scale (r/A(I) - 4 x 10 -3, from which UB(A(I))d/A(I) = 4 x 10 -3. The uncertainty for the uncompensated air flux is estimated UB(A(I))a/A(I ) = 5 X 10 -4, while any contribution to the uncertainty of the measured polarization value related to temperature is deemed negligible in these alloys and UB(A(I))T/A(I ) ~" 0. It should be stressed, however, that this last term could become very important in some hard magnets (e.g. N d - F e - B alloys) to which the present discussion clearly applies. The combined and expanded uncertainties are thus obtained by means of Eq. (10.41), as summarized in Table 10.5. This specific result is expressed as J = j + u = 0.9738 + 1.00 x 10 -2 T.


605

T A B L E 10.5 Uncertainty budget in the measurement with the ballistic method of the magnetic polarization in non-oriented Fe-(3 wt%)Si laminations

Source of uncertainty

Distribution Divisor Relative Sensitivity Degrees of function uncertainty coefficient freedom

Normal 1 Fluxmeter reading and drift Rectangular x/3 Air-flux compensation Rectangular x/3 Cross-sectional area of the sample Rectangular x/3 Sample temperature 1 Repeatability Normal Combined relative Normal standard uncertainty Expanded uncertainty (95% confidence level)

4 x 10 -3

1

co

5X

10 - 4

1

co

1.9 X

10 - 3

1

co

1

co

1

4 20

---0 2X

10 - 3

4.9 X 10 -3

1.03 x

10 - 2

-

Coverage factor k=2.1

The uncertainty components are specified in Eq. (10.41). The measurement is performed on a point of the normal magnetization curve, under an applied field H -- 80 A/m, with eight strips inserted in an Epstein rig. The result is J = j + U = 0.9738 ___1.00 x 10 - 2 T.

10.4.3 Measurement of power losses in soft magnetic laminations Soft magnetic materials are p r o d u c e d and sold for use p r e d o m i n a n t l y in energy applications and have their quality classified according to their p o w e r loss figure. The precise and reproducible m e a s u r e m e n t of the p o w e r losses in these materials is industrially significant and is required in m a n y application-oriented research investigations. Specific measurem e n t standards have therefore been developed and u p g r a d e d over the years [10.19-10.21]. Inter laboratory comparisons have been carried out to validate these standards, settling to a broad extent the m e a s u r e m e n t capabilities of metrological and industrial laboratories. Critical to the appraisal of the reproducibility and degree of equivalence of the measurements p e r f o r m e d by different laboratories is the correct determination of

606


the measurement uncertainty. This is quite a complex task because many possible contributions have to be taken into account, as thoroughly discussed in Ref. [10.22]. Unduly optimistic or pessimistic evaluations are not infrequent, as revealed by the analysis of intercomparisons (see Fig. 10.2). We shall discuss here a largely simplified approach, focused on the testing of soft magnetic laminations at power frequencies, by considering only the most relevant contributions to the uncertainty and assuming that the signal treatment is performed by digital methods (see also Section 7.3). Let us therefore express the average magnetic power loss per unit mass at the frequency f as

Ps = ~ H dB = ~

/fH(t)---d--~dt ,

(10.43)

where H and B are applied field and induction in the sample, respectively, and 3 is the material density. Equation (10.43) can equivalently be written in terms of the current iH in the primary circuit and the secondary voltage VB as

f NH fl/d VB(t)iH(t)dt (10.44) 3 NB~,*S Jo having posed H(t) = (NH/~,*)iH(t)and VB(t) -- NBS(dB(t)/dt), NH and NB Ps

_

being the number of turns of the primary and the secondary windings, respectively, s the magnetic path length and S the sample cross-sectional area. We assume that the measurement is performed under sinusoidal induction waveform (i.e. sinusoidal secondary voltage VB(t)) and we consequently write 1

Ps-- 3r

N H

NB

1

~

~

RH VBVH1cos qG

(10.45)

where RH is the resistance value of a calibrated shunt in the primary circuit. 17B and 17m are the rms values of the secondary voltage and of the fundamental harmonic of the voltage drop on the shunt, respectively, which are phase shifted by the angle ~. Let us thus consider a possible approach to the evaluation of the uncertainty in the specific practical case of grain-oriented Fe-(3 wt%)Si laminations, tested by means of an Epstein frame and a digital wattmeter. It is assumed that testing is made at peak polarization Jp--1.7 T and frequency f = 50 Hz, with automatic air-flux compensation. We can therefore assume VB(t)-NBS(dJ(t)/dt) = 2~fNBSJp sin tot. A number of repeated determinations of P are made, each time disassembling and assembling the strips in the same order. Sixteen strips, 305 mm long and 30 mm wide, are used


607

(four in each leg of the frame) and the cross-sectional area of the resulting sample is determined measuring the total mass m (8 = 7650 k g / m 3) and the length of the strips. It is obtained that S--3.3569 x 10 -5 m 2. The voltages VB(t) and Vm (t) are amplified and fed by synchronous sampling (e.g. 2000 points per period, interchannel delay lower than 1 x 10 -9 s, trigger jitter ~- 10-1~ -11 s) into a two-channel acquisition setup and A / D converter and Eq. (10.44) is computed by means of suitable software. By denoting with Pmeas the result of such a calculation, we express the best estimate of the power loss as P -- PmeasF(AJp)F(AFF)F(AT)"4- 8(P)VBa q- 8(P)VBg q- 3(P)vH1

(10.46)

+ 8(P)a + 8(P)s + 8(P)~,

where the first term on the right-hand side is the mean, made over the repeated measures, of the values of Pmeas times the correction factors F(AJp), F(AFF), and F(AT). These factors account for the differences AJp, AFF, and AT between the actual and the prescribed values of peak polarization, form factor of VB(t) (FF = 1.1107), and temperature T (23 ~ respectively. They are recorded and automatically multiplied by Pmeas with each measurement repetition. Experiments [10.23-10.25] suggest the following approximate relationships:

( ,p )18 jp+Ajp'

F(T) =

(1-5 x 10-4AT),

F(AFF)=

( ) 1.8 1 + fl 1.1107 + AFF 1.1107 (10.47)

where fl is the ratio between the dynamic and hysteresis loss components at the measuring frequency. The bias correction terms ~(P)VBa, (~(P)VBg, 8(P)vH1 , 8(P)a, 8(P)s, 8(P), are assumed to have zero value and non-zero uncertainty and are associated with air-flux compensation, gain and offset of the secondary voltage channel, gain of the field channel, material density, cross-sectional area, phase shift between VB(t) and VHl(t), respectively. With the employed measuring setup, the contributions to the uncertainty deriving from frequency setting, synchronization error during signal acquisition, standard shunt resistor in the primary circuit, and quantization of the signal by the A / D converters are deemed negligible. The latter might become important at very high inductions, where the peak amplitude of the fundamental harmonic of the field (i.e. VH1) reduces to a small fraction of the peak field amplitude, with ensuing reduction of the effective dynamic range of the field channel.

608


Based on the foregoing discussion a n d Eq. (10.16), w e express the relative c o m b i n e d s t a n d a r d u n c e r t a i n t y of the p o w e r loss m e a s u r e m e n t as

u~(P) P = ~ u2(p)~2 + u2(VB)a ~ +

u2(VB)g V-----~ +

U~_II(VH1) V21

U~(~)T +2

32

U~(q0) +qo2tan2cp qo2 (10.48)

TABLE 10.6 Uncertainty budget in the measurement of the magnetic power losses at 50 Hz and 1.7 T peak polarization in a grain-oriented Fe-(3 wt%)Si lamination (Eq. (10.48)) Source of uncertainty

Distribution function

Divisor Relative uncertainty

Sensitivity coefficient

Degrees of freedom

Air-flux compensation Gain and offset of B channel Gain of H channel Sample cross-sectional area Material density Phase shift between VB and VH1 Repeatability Combined relative standard uncertainty Expanded uncertainty (95% confidence level)

Rectangular

x/3

2 x 10-3

1

oo

Rectangular

~/3

2 x 10-3

1

OO

Rectangular

x/3

1 x 10 -3

1

oo

Rectangular

x/3

1.5 x 10 -3

1

oo

Rectangular Rectangular

x/3 ~/3

1.5 x 10 -3 4 x 10 -3

1 co qotan~ - 0.7 oo

Normal Normal

1 -

I x 10 -3 5.5 x 10 -3

1 -

-

-

11 x 10 -3

6 1536

Coverage factor k=2

The measurement is performed by a digital wattmeter, using an Epstein test frame. For this specific case, the phase shift ~ between secondary voltage and fundamental component of the primary current is 43~. The result is expressed as P = P + U = 1.176 + 13 x 10 -3 W/kg.

REFERENCES

609

2.5 2.0

Ni/

!o/o

1.5

:2)

1.0 0.5

.0

O~o_o,=0~o~o=g=Q=0=Q,,0=8~8~/~fo/O/GO

0,0

.

.

.

.

i

0.5

.

.

.

.

i

.

1.0 Jp (T)

.

.

.

I

1.5

.

.

.

.

2.0

FIGURE 10.5 Expanded relative uncertainty and its dependence on peak polarization in the measurement of the magnetic power losses at 50 Hz in grainoriented and non-oriented Fe-Si laminations (PTB laboratory [10.18, 10.22]). where it is assumed, according to Eq. (10.42), that the uncertainty of S is chiefly due to the uncertainty of 8. Notice also that we have treated as independent all the input quantities appearing in Eq. (10.48). This is a crude approximation. In particular, for a given Jp value, VB and ~ are expected to correlate and the associated covariance should be considered, via Eqs. (10.17) and (10.20), in Eq. (10.48). Table 10.6 provides, u n d e r s u c h an approximation, the details of the calculation of uc(P)/P and the related expanded uncertainty U. For the specific measurement reported in this example, we find P = P + U - 1 . 1 7 6 + 11x10-BW/kg. Notice that uc(P)/P is observed to increase rapidly on increasing Jp towards saturation, as illustrated for non-oriented and grain-oriented laminations in Fig. 10.5 [10.18, 10.22]. This is chiefly ascribed to the previously stressed detrimental effect of reduced dynamic range in the field channel and the phase error. In fact, for high J~ values, 17B and VH1 are nearly in quadrature and the term ~2 tan2q0(u2(qo)/~2) in Eq. (10.48) tends to diverge.

References 10.1. BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML, Guide to the Expression of Uncertainty in Measurement (Geneva, Switzerland: International Organization for Standardization, 1993).

610


10.2. In repeated measurements one may find from time to time seemingly inconsistent outcomes (outliers). A good appraisal of the kind of measurement being performed and expertise should guide the person making the experiment in accepting or rejecting these outcomes. 10.3. A. Papoulis, Probability, Random Variables and Stochastic Processes (Tokyo: McGraw-Hill Kogakusha, 1965), p. 266. 10.4. C.F. Dietrich, Uncertainty, Calibration and Probability (Bristol: Adam Hilger, 1991), p. 10. 10.5. S. Rabinovich, Measurement Errors (New York: AIP, 1995), p. 195. 10.6. BIPM, Mutual Recognition of National Measurement Standards and of

10.7. 10.8. 10.9.

10.10.

10.11. 10.12. 10.13. 10.14. 10.15.

10.16.

10.17.

Calibration and Measurement Certificates Issued by the National Metrology Institutes (S6vres: Bureau International des Poids et Mesures, 1999), http:// www.bipm.fr / BIPM-KCDB. M.G. Cox, "A discussion of approaches for determining a reference value in the analysis of key-comparison data," NPL Report CISE 42 (1999). J.W. M(iller, "Possible advantages of a robust evaluation of comparisons," BIPM Report 95/2 (1995). S. D'Emilio and F. Galliana, "Application of the GUM to measurement situations in metrology," in Proc. 16th IMEKO World Congress (A. AfjehiSadat, M.N. Durakbasa, and P.H. Osanna, eds., Wien: ASMA, 2000), 253-258. J. Sievert, M. Binder, and L. Rahf, "On the reproducibility of single sheet testers: comparison of different measuring procedures and SST designs," Anal. Fis. B, 86 (1990), 76-78. EUROMET, "Guidelines for the organisation of comparisons," (EUROMET Guidance Document No. 3). R. Thalmann, "EUROMET key comparison: cylindrical diameter standards," Metrologia, 37 (2000), 253-260. ISO, International Vocabulary of Basic and General Terms in Metrology (Geneva, Switzerland: International Organization for Standardization, 2000). A.E. Drake, "Traceable magnetic measurements," J. Magn. Magn. Mater., 133 (1994), 371-376. M.J. Hall, A.E. Drake, and L.C.A. Henderson, "Traceable measurement of soft magnetic materials at high frequencies," J. Magn. Magn. Mater., 215216 (2000), 717-719. R.D. ShuU, R.D. McMichael, L.J. Swartzendruber, and S. Leigh, "Absolute magnetic moment measurements of nickel spheres," J. Appl. Phys., 87 (2000), 5992-5994. "Intercomparison of magneticflux density by means ofj~'eld coil transfer standard," (EUROMET Project No. 446, Final Report, 2001).

REFERENCES

611

10.18. J. Sievert, H. Ahlers, F. Fiorillo, L. Rocchino, M. Hall, and L. Henderson, "Magnetic measurements on electrical steels using Epstein and SST methods. Summary report of the EUROMET comparison project no. 489," PTB-Bericht, E-74 (2001), 1-28. 10.19. IEC Standard Publication 60404-2, Methods of Measurement of the Magnetic

Properties of Electrical Steel Sheet and Strip by Means of an Epstein Frame (Geneva: IEC Central Office, 1996). 10.20. IEC Standard Publication 60404-3, Methods of Measurement of the Magnetic Properties of Magnetic Sheet and Strip by Means of Single Sheet Tester (Geneva: IEC Central Office, 1992). 10.21. IEC Standard Publication 60404-10, Methods of Measurement of Magnetic Properties of Magnetic Sheet and Strip at Medium Frequencies (Geneva: IEC Central Office, 1988). 10.22. H. Ahlers and J. Liidke, "The uncertainties of magnetic properties measurements of electrical sheet steel," J. Magn. Magn. Mater., 215-216 (2000), 711-713. 10.23. G. Bertotti, F. Fiorillo, and G.P. Soardo, "Dependence of power losses on peak induction and magnetization frequency in grain-oriented and nonoriented 3% Si-Fe," IEEE Trans. Magn., 23 (1987), 3520-3522. 10.24. F. Fiorillo and A. Novikov, "An improved approach to power losses in magnetic laminations under nonsinusoidal waveform," IEEE Trans. Magn., 26 (1990), 2904-2910. 10.25. A. Ferro, G. Montalenti, and G.P. Soardo, "Temperature dependence of power loss anomalies in directional Fe-Si 3%," IEEE Trans. Magn., 12 (1976), 870-872.


APPENDIX A

The SI and the CGS Unit Systems in Magnetism

Magnetic measurements are generally traceable to the SI unit system. This means that reference can be made, for any measured quantity, to the standards of the base units maintained by the National Metrological Institutes, a feat accomplished by connecting the primary laboratory with the end user through a chain of comparisons, having stated uncertainties. SI (Sist6me International d'Unit6s) has been adopted and updated through successive CGPM (Conf6rence G6n6rale des Poids et Mesures) resolutions, starting with the task assigned by the 9th CGPM in 1948 to the CIPM (Comit6 International des Poids et Mesures) "to study the establishment of a complete set of rules for units of measurement and to make recommendations on the establishment of a practical system of units of measurement suitable for adoption by all signatories of the Convention du M6tre" [A.1]. SI is an absolute system, which means that the corresponding base units are invariant with time and space and all other units can be derived from them through definite mathematical relationships. In particular, a unit {y} of a quantity y is given in this system in terms of the base units {Xl},{x2}, ..., {x,,}, by an expression of the type: {y} =

(A.1)

with c~,/3, ..., v either positive or negative integers and no numerical factors other than 1. The SI units are then said to form a coherent set of units. SI is based on seven base units: meter (m) for the length, second (s) for the time, kilogram (kg) for the mass, ampere (A) for the electric current, kelvin (K) for the temperature, candela (cd) for the luminous intensity, and the mole (mol) for the amount of substance. The base units relevant for electromagnetic phenomena are m, s, kg, and A. The "vexata quaestio" of unit systems in electromagnetism is rooted in the development of the CGS system in the 19th century, which accompanied over several decades the development of the electrical sciences and their broadening impact on industry and society [A.2]. 613

614

APPENDIX A The SI and the CGS Unit Systems in Magnetism

With CGS all electrical and magnetic units are expressed in terms of three base mechanical units: cm, g, and s. Two different unit systems, however, exist, according to whether electrostatic or magnetostatic base equations are used. The electrostatic units (e.s.u.) are defined starting from Coulomb's law, where the interaction force F between two electrical charges (Q1, Q2) at a distance r is expressed by the equation: F = k I Q1Q2 r2 .

(A.2)

By taking the adimensional proportionality constant kS = 1, the dimensions of the electric charge are [M1/2L3/2T-1] and the corresponding unit is 1 statcoulomb = 1 dyn cm. With another choice of the primary equation, where electrodynamic forces are described, the electromagnetic units (e.m.u.) are defined. The force per unit length F/L mutually exerted by two infinitely long straight wires carrying the currents il and i2 is 9

1 / 2

9

F _ k'2 il i2 -7,

.

--

7

(A.3)

where r is the distance between the wires. With the constant ks/= 1, the dimensions of the current are [M1/2L1/2T -1] and those of the electric charge, which is the time integral of the current, are [M1/2L1/2]. It is therefore apparent that the electric charge has different dimensions in the e.s.u, and the e.m.u, systems, the ratio between them having the dimension of a velocity. This duality is a drawback and, to compound it, the mixed CGS (or Gaussian) system has been introduced, where the e.s.u, units are used for the electric quantities and the e.m.u, units are used for the magnetic quantities. The Gaussian system has been instrumental in the development of the electrical sciences; it has been largely applied in the past literature, and is still applied to some extent in papers and textbooks on magnetism and magnetic materials. But most of its units have inconvenient size with respect to practical engineering needs and objective complications arise from the multiplicity of notations. Consequently, since their official adoption by the British Association for the Advancement of Science in 1873, the absolute Gaussian units have been used in association with practical electrical units, such as the volt, the ampere, and the ohm. In 1901, G. Giorgi, by remarking that the watt could equally be the absolute unit of mechanical and electric power, the latter being the product of voltage and current, proposed a system where a fourth independent electrical quantity was associated with the base mechanical units m, kg, and s [A.3]. By abandoning a purely mechanical system, dimensional factors had evidently to be introduced in the laws of force between charges and between currents, but now all the practical

APPENDIX A The SI and the CGS Unit Systems in Magnetism

615

electrical units slipped coherently into this system. Giorgi's proposal won immediate recognition, but it took several decades for general and official acceptance to be achieved, following thorough discussions by International Union of Applied and Pure Physics (IUPAP), International Electrotechnical Commission (IEC), and the Consultative Committee for Electricity (CCE) of CIPM. In 1946, CIPM approved the adoption of the meter, kilogram, second, ampere system (MKSA) [A.4]. The ampere was thereby defined as "that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular crosssection, and placed I m apart in vacuum, would produce between these conductors a force equal to 2 x 1 0 - 7 N of force per meter of length". MKSA was eventually sanctioned, with the addition of the K and the cd as base units, by the 10th CGPM in 1954 as practical system of units of measurement [A.5]. According to this definition of the ampere, the proportionality constant in Eq. (A.3) has definite dimensions and the value 2 x 10-7 F _ /z0 2 ili2. L 4~r r

(A.4)

The magnetic constant ~0 (formerly called "magnetic permeability of free space") has therefore an exact value by definition/z 0 -- 4rrx 10 -7 N / A 2. The unit of electric charge is now 1 C - - 1 A s and the constant kJ in Coulomb's law (A.2) is no more adimensional F-

1 Q1Q2 47r80 r 2 "

(A.5)

The electric constant 80 is related, according to the solution of Maxwell's equations for the propagation of the electromagnetic field in vacuum, to /z0 and the speed of light c: c = (80/z0)-1/2.

(A.6)

In 1983, the 17th CGPM re-defined the meter, assigning the speed of light the precise value c -- 299 792 458 m / s [A.6]. In force of this decision, the electric constant 8o takes, according to Eq. (A.6), the exact value 80 = 8.854187817 x 10 -12 F/m. The SI unit system is strongly recommended by the international metrological and standardization organizations and is increasingly applied in magnetism. However, a massive body of literature exists which makes use of CGS and a substantial resilience to SI, dictated either by attachment to routinary use of old units or by sheer distaste of the awkward redundancy of fields in free space generated by SI, is observed in some areas (for example, in the field of permanent magnets).

616

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