Characterisation of the Mechanisms of Magnetisation Change in Permanent Magnet … · HE mechanisms...

Characterisation of the Mechanisms of

Magnetisation Change in Permanent

Magnet Materials through the

Interpretation of Hysteresis Measurements

Simon Andrew Harrison BSc(Hons)

This thesis is presented for the degree of Doctor of Philosophy

of

The University of Western Australia

The Department of Physics

2004

Abstract

THE mechanisms by which magnetisation changes occur in magnetic materials may be inves-tigated by a variety of hysteresis measurements. During this study both alternating and

rotational hysteresis measurements were used to characterise the mechanisms of magnetisationchange in a number of permanent magnet materials.

Studies of the time dependence of magnetisation, remanent magnetisations and the dependenceof the reversible magnetisation on the irreversible magnetisation were undertaken. These stud-ies revealed that in sintered rare-earth iron magnets the magnetisation change is predominatelycontrolled by domain nucleation, with a lesser contribution from domain wall pinning within theboundary regions of the grains. Similar mechanisms control the magnetisation change in the largergrains of melt-quenched rare-earth iron magnets. In the single domain grains of the melt-quenchedmaterials incoherent rotation mechanisms control the changes of magnetisation. Magnetisationchange in MnAlC and sintered AlNiCo was found to be controlled by domain wall pinning withinthe interior of the grains of the materials.

Two devices were constructed for the measurement of rotational hysteresis. The first measuresthe angular acceleration of a sample set spinning in a magnetic field, from which the rotationalhysteresis loss may be determined. The second employs rotating search coils to make direct mea-surements of the component of magnetisation that contributes to rotational hysteresis loss duringthe rotation of a sample in a field. Both devices were found to produce data consistent with thatin the literature and to be useful for the characterisation of rotational hysteresis in permanentmagnet materials.

A simple model was used to examine the dependence of rotational hysteresis loss on various materialparameters. It was found that the value of the rotational hysteresis integral is dependent oninteractions and to a lesser extent distributions in anisotropy. This is contrary to assumptionscommonly made in the literature but consistent with published experimental data, which has beenreinterpreted.

Measurements of rotational hysteresis losses in the materials studied were found to be effectedby geometric demagnetisation effects. A method by which such data may be corrected for theseeffects is proposed. Following correction and consideration of the interactions within the materials,the rotational hysteresis data was found to be consistent with the characterisations performed inlinearly alternating fields.

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Acknowledgements

MY supervisors, Professor Robert Street and Dr. Steve Jones have devoted tremendousamounts of time and energy to assisting me in this project. I am very grateful for all

the pearls of wisdom they have tried to send my way.

The workshop staff in the Department of Physics at UWA have also given me a great deal of helpduring this work. In particular Budgie, Gary, Allan and Devo have always been willing to lend ahand or to tell me why something won’t work.

Robert Woodward, in the HiPerm laboratory, has shown me how to operate many pieces of equip-ment, saved me from many a bad decision and has generally been the one who knew when no oneelse did.

The HiPerm laboratory has been a fun place to work and I am grateful to all the people that havemade it so. In particular, the efforts of Greg Black in arranging laboratory home-brew tastingswere appreciated!

I have also spent considerable time during these studies in the Lions’ Cancer Institute. Thanks goto Jill, Liz, Mandy and Colin for all the help they have given me and for being such fun company!

Away from university I have been fortunate to have had the distractions of many fine friends.Thankyou especially to my housemates over the years for putting up with the highs and lows thatstudies such as these inevitably involve. Thanks for all the rounds of golf, the fishing trips, theBBQs, the games of beach cricket and the holidays that have made certain I saw the sunshine thatnever makes it into the basement of the physics building.

I am particularly grateful for the support and encouragement that my parents and family haveprovided during this long education.

Finally, throughout this seemingly endless work I have been blessed with the limitless love andsupport and assistance of my best friend, soulmate and wife Sarah. Thankyou sweetpea.

During the course of this work I received the financial assistance of both an Australian PostgraduateAward and and a Lions Cancer Institute Scholarship.

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Contents

Abstract i

Acknowledgements iii

1 Introduction: Characterisation of the Mechanisms of Magnetisation Change 11.1 The Aims of This Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The Mechanisms of Magnetisation Change . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Coherent Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Incoherent Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 Domain Wall Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.4 Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.5 Application of the Ideal Models to Real Materials . . . . . . . . . . . . . . 8

1.3 The Theory Behind the Methods of Probing the Mechanisms of MagnetisationChange in Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.1 Magnetic Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.2 Remanent Magnetisation Measurements . . . . . . . . . . . . . . . . . . . . 121.3.3 The Dependence of Reversible Magnetisation on Irreversible Magnetisation 161.3.4 Rotational Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4 Studies of the Mechanisms of Magnetisation Change . . . . . . . . . . . . . . . . . 221.4.1 Studies Using Magnetic Viscosity Measurements . . . . . . . . . . . . . . . 221.4.2 Studies of Magnetic Materials Using Remanent Magnetisation Measurements 241.4.3 Studies Using the Dependence of the Reversible Magnetisation on the Irre-

versible Magnetisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.4.4 Studies Using Rotational Hysteresis Measurements . . . . . . . . . . . . . . 29

1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2 Experimental Methods 332.1 Measurement Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1.1 Hardware for Measuring Alternating Hysteresis and the Time Dependenceof Magnetisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1.2 Hardware for Measuring Rotational Hysteresis . . . . . . . . . . . . . . . . 352.2 VSM Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2.1 General Measurement Regimes . . . . . . . . . . . . . . . . . . . . . . . . . 352.2.2 The Measurement of the Time Dependence of Magnetisation . . . . . . . . 362.2.3 The Measurement of Remanent Magnetisations . . . . . . . . . . . . . . . . 37

2.3 Analysis of the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.3.1 The Measurement and Calculation of Mrev and Mirr . . . . . . . . . . . . . 372.3.2 Determination of the Magnetic Viscosity Parameters from Conventional Vis-

cosity Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3.3 The Measurement and Calculation of η . . . . . . . . . . . . . . . . . . . . 452.3.4 The Measurement and Calculation of Rotational Hysteresis Properties . . . 49

3 The Samples Studied 53

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3.1 The Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.1.1 Rare Earth-Iron Permanent Magnets . . . . . . . . . . . . . . . . . . . . . . 533.1.2 AlNiCo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.1.3 MnAlC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.1.4 Sample Geometric Demagnetisation Effects . . . . . . . . . . . . . . . . . . 58

3.2 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2.1 Preparation of Sample Spheres . . . . . . . . . . . . . . . . . . . . . . . . . 603.2.2 Mounting the Sample Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3 Demagnetisation of the Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3.1 Thermal Demagnetisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.3.2 Rotational Demagnetisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.3.3 DCE Demagnetisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 Investigation of the Mechanisms of Magnetisation Change in Linearly Alternat-ing Magnetic Fields 654.1 Characterisation of the Mechanisms of Magnetisation Change . . . . . . . . . . . . 654.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2.1 SN1 and SN2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.2.2 MQ1 and MQ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2.3 MnAlC and AlNiCo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5 Measurement of Rotational Hysteresis and Resistivity by Angular AccelerationBased Methods 895.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.1.1 Measurement of Rotational Hysteresis Losses and Resistivity by AngularAcceleration Based Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2 The Dual Air Bearing Angular Acceleration Magnetometer . . . . . . . . . . . . . 905.2.1 The Mechanics of the Measurement . . . . . . . . . . . . . . . . . . . . . . 925.2.2 Measurement of Rotational Hysteresis Losses . . . . . . . . . . . . . . . . . 945.2.3 Measurement of Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.2.4 Measurement of Magnetisation . . . . . . . . . . . . . . . . . . . . . . . . . 955.2.5 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.2.6 Other Effects of Eddy Currents . . . . . . . . . . . . . . . . . . . . . . . . . 975.2.7 Heating of the Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.2.8 Sensitivity and Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.2.9 Uncertainties in the Measured Quantities . . . . . . . . . . . . . . . . . . . 101

5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.3.1 Rotational Hysteresis Loss Measurements . . . . . . . . . . . . . . . . . . . 1015.3.2 Resistivity Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.3.3 The Saturation Magnetisation of Nickel . . . . . . . . . . . . . . . . . . . . 108

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6 Measurement of the Magnetisation Vector During Rotational Hysteresis inIsotropic Materials 1116.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.2 The Rotating Sample and Coil Magnetometer . . . . . . . . . . . . . . . . . . . . . 114

6.2.1 Physical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.2.2 The Form of the Signals Induced in the Coils . . . . . . . . . . . . . . . . . 1176.2.3 The Method of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.2.4 Expressing the Data in Terms of Internal Field . . . . . . . . . . . . . . . . 1256.2.5 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.2.6 Characteristics of the Electromagnet . . . . . . . . . . . . . . . . . . . . . . 1276.2.7 Sensitivity and Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2.8 Uncertainties in the Measured Quantities . . . . . . . . . . . . . . . . . . . 1286.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.3.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.3.2 Rotational Hysteresis Measurements . . . . . . . . . . . . . . . . . . . . . . 129

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7 Modelling of Rotational Hysteresis 1417.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417.2 Description of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.3 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.4 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7.5.1 Verification of the Implementation of the Model . . . . . . . . . . . . . . . 1437.5.2 The Effect of Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1457.5.3 The Effect of a Distribution of Anisotropy . . . . . . . . . . . . . . . . . . . 1497.5.4 The Effect of the Geometry of the Anisotropy . . . . . . . . . . . . . . . . . 1507.5.5 The Effect of Time Dependence . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

8 Interpretation of the Field Dependence of Rotational Hysteresis Losses 1558.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8.1.1 The Effect of Sample Geometry on Rotational Hysteresis Losses . . . . . . 1558.1.2 Material Parameters Effecting Rotational Hysteresis Losses . . . . . . . . . 1578.1.3 Interpretation of the Rotational Hysteresis Integral . . . . . . . . . . . . . . 158

8.2 Correction of Rotational Hysteresis Measurements for Demagnetisation Effects . . 1588.3 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1598.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

8.4.1 Rotational Hysteresis Losses . . . . . . . . . . . . . . . . . . . . . . . . . . 1598.4.2 Interpretation of the Rotational Hysteresis Integral . . . . . . . . . . . . . . 164

8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

9 Conclusions and Suggestions for Further Study 1699.1 Suggestions for Further Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

A Derivation of Selected Equations 173A.1 Equation 8 in Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173A.2 Equation 9 in Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

B Description of the Other Samples Studied 177B.1 Ex1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177B.2 SrFe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178B.3 Co-γ-Fe2O3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

1. Introduction: Characterisation of the

Mechanisms of Magnetisation Change

THE ‘mechanisms of magnetisation change’, also referred to as ‘magnetisation reversal mech-anisms’ or ‘magnetisation processes’, are the manner in which the atomic or ionic moments

within a material alter their orientations so as to produce a net change of magnetisation. The im-petus for these changes of orientations may be provided by a change in the applied magnetic field,by the diffusion of impurities within a crystal lattice or by heat. The final state of the materialis that which minimises its macroscopic potential energy. When the individual moments withina grain or particle rotate collectively, maintaining a parallel alignment, then the magnetisationmechanism is said to be that of coherent rotation. Conversely when the moments rotate in suchmanner that they do not maintain parallel alignment during the rotation then the mechanism issaid to be incoherent. Where magnetic domains exist in a material then magnetisation changemay be mediated by changes of the domain structure. This may either occur by the rearrangementof the existing domains through domain wall motion, or by the formation of new domains withenergetically favorable orientation, referred to as nucleation.

Changes in magnetisation may be classified as either reversible or irreversible. Phenomonologicallyreversible changes are observed as those where the material reverts back to its original state afterremoval of the field which brought about the change. Irreversible changes are those caused bythermal fluctuations (magnetic viscosity) or the application of a field such that the induced changeremains after the field is removed. At the level of the individual magnetic moments reversiblechanges are associated with the rotation of moments away from an easy axis or the bowing ofdomain walls. Irreversible changes are associated with rotation where the moments flip from beingassociated with one easy axis to being associated with another, with the movement of domain wallsover pinning sites or with the nucleation and growth of reverse domains.

Characterisation of the mechanisms by which particular magnetic materials undergo magnetisationchange is essential to understand how the mechanisms affect the material properties and perfor-mance. Such an understanding allows better materials to be designed for particular applications.Various methods have been proposed to characterise the mechanisms of magnetisation change forparticular magnetic materials. Four such approaches are discussed in sections 1.3 and 1.4: themeasurement of magnetic viscosity; remanent magnetisation measurements; measurements of thedependence of reversible magnetisation on irreversible magnetisation; and measurements of rota-

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Characterisation of the Mechanisms of Magnetisation Change in Permanent Magnet Materials

tional hysteresis.

1.1 The Aims of This Work

The aims of this work are to make measurements of alternating and rotational hysteresis on arange of permanent magnetic materials and to interpret the data from those measurements interms of the mechanisms of magnetisation change occurring within each material. To this end itwill be necessary to develop apparatus suitable for the precise measurement of rotational hysteresis.Additionally, since alternating hysteresis is more widely studied than rotational hysteresis, thenthere would also be considerable advantage to be gained from an improved understanding of therelationship between the two types of hysteresis. Such knowledge may be of use in the design ofmaterials with particular magnetic properties, by providing an improved understanding of whatcontrols the magnetisation mechanisms in permanent magnets and the means by which thosemechanisms may be effectively characterised.

1.2 The Mechanisms of Magnetisation Change

Models of magnetisation may be divided into three categories: those that seek to provide a math-ematical formulation for the hysteresis, without regard to how changes of magnetisation are medi-ated; micromagnetic models; and models that postulate physical mechanisms for the magnetisationchange and evolve accordingly. Models in the first category, which includes the Preisach model[1, 2], are computationally less demanding and are commonly used as predictive tools once themodel parameters which most closely match a physical system have been determined. Micromag-netic modelling describes materials as interacting regions of heterogeneous magnetisation whichalter their orientation so as to minimise potential energy. This finite element approach may bevery successful in describing the distribution of magnetisation within complex structures. Modelsthat postulate specific mechanisms by which magnetisation changes are the most fundamental.They consider the change of orientation of individual spins within the distribution that forms thematerial. Knowledge of these mechanisms, and the factors controlling them, allows the propertiesof materials to be predicted a priori and customised according to their application. Here some ofthe more common mechanisms are described.

1.2.1 Coherent Rotation

During coherent rotation all of the spins in a region (or particle) of heterogeneous magnetisa-tion simultaneously and identically rotate so as to maintain mutual alignment at all times. Themagnitude of the net magnetic moment of each region, then, does not change, even though thenet magnetic moment of the material as a whole may change. Stoner and Wohlfarth were thefirst to report calculations for such a model [3]. They considered non-interacting single domain

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Introduction: Characterisation of the Mechanisms of Magnetisation Change

Figure 1.1: A Stoner-Wohlfarth particle - illustrating the definition of the parameters of the energyequation.

particles of uniform magnetisation throughout, which changed their magnetisation through themechanism of coherent rotation. Stoner and Wohlfarth considered the case of prolate and oblatespheroidal particles where uniaxial anisotropy arose due to their shape. Their theory, however, isequally applicable to particles experiencing uniaxial anisotropy arising from magnetocrystalline ormagnetostrictive effects.

In the case of a particle of volume V , spontaneous magnetisation Ms and uniaxial magnetocrys-talline anisotropy constant K, then the energy, E is given by

E(θ, φ,H) = KV sin2(φ− θ)−HVMs cos θ (1.1)

where H is the field, φ is the angle between the uniaxial easy axis and the field and θ is the anglebetween the magnetisation and the field, as shown in figure 1.1. Stoner and Wohlfarth showed thatit is only necessary to consider a single particle system in two dimensions [3]. The form of E vs θ,in different fields, for the particular case of φ = 30 is shown in figure 1.2.

The distinction between reversible and irreversible magnetisation under this formalism is illustratedin figure 1.2. Reversible changes correspond to changes in the orientation of the local magnetisa-tion vectors caused by movement of the energy minima. Irreversible changes correspond to localmagnetisation vectors moving from one energy well to another.

The magnetisation of the ensemble as a whole is determined by the relative population of the energywells as given by E(θ, φ,H). In the time dependent case the relative population of the metastablestates (i.e. orientations that correspond to a local minima in E but not the global minima) andthe stable states (i.e. the orientation corresponding to the global minima in E), is determined byactivation over the energy barrier, as approximated by Pfeiffer [4]:

Eact(H,φ) = E0

(1− H

HKg(φ)

)(0.86+1.14g(φ)

(1.2)

where E0 = KV is the energy barrier in zero field, HK = 2K/µ0MS is the anisotropy field, V the

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Figure 1.2: The energy wells experienced by a Storner Wohlfarth particle orientated at 30 to the field,at different field strengths. The difference between reversible and irreversible changes is illustrated.

particle volume, K the anisotropy constant and g(φ) = (cos2/3 φ+ sin2/3 φ)−2/3.

Such an approach is appropriate for all geometries of anisotropy and may be additionally modified toinclude the effect of interactions through the inclusion of mean field effects and/or local variationsin the interaction field [5]. Thus models of coherent rotation are able to exhibit much of themagnetisation phenomena seen in real magnetic materials, although quantitative agreement isuncommon as few materials change their magnetisation solely by coherent rotation.

1.2.2 Incoherent Rotation

Even at the time of the introduction of the model of coherent rotation it was clear that manymaterials did not conform to the theory. Stoner and Wohlfarth pointed out that materials withgrain dimensions sufficient to contain domain walls are able to change their magnetisation throughboth rotation and domain wall processes [3]. Equally, many materials composed entirely of singledomain grains or particles were observed to have much lower coercivities than that predicted bythe model of coherent rotation [6]. From an intuitive viewpoint magnetisation change exclusivelythrough coherent rotation is unrealistic. Elsewhere in physics state change is mediated by complexmechanisms (examples may be found in fluid motion, crystal formation etc.), since they oftenhave lower activation energies [7]. Thus both practical and theoretical considerations providedthe motivation for the development of theories of magnetisation change mediated by incoherentrotation.

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Incoherent rotation occurs when the individual atomic or ionic moments within a grain do notmaintain parallel alignment during rotation. A myriad of different mechanisms are possible, withthe actual mechanism determined by energy considerations. Such mechanisms involve non uniformrotation throughout a particle, as once particular regions have undergone rotation then neighbour-ing regions may be energetically more likely to do so because of exchange coupling. The mostfrequently cited mechanisms of incoherent rotation are those of fanning [8, 9] and curling [10].

Magnetisation Fanning

Jacobs and Bean have proposed that the change of magnetisation in single domain particles maybe considered in terms of a model of a chain of spheres[8]. They considered a mechanism wherebythe magnetisation of the system changed by opposing rotation of the moment of each successivesphere. This mechanism was referred to as fanning.

In two sphere chains and infinite chains symmetric fanning, where the moments of neighboringspheres rotate equal and oppositely, represents the lowest energy rotation process of magnetisa-tion change. In other finite chains, which are more representative of real elongated single domainparticles, the higher energy state of the end spheres promotes non symmetric fanning. Analysis ofdata from elongated single domain particles of iron [11], performed by Jacobs and Bean, suggestedthat nonsymmetric fanning provided a better representation of this actual case than either sym-metric fanning or coherent rotation. More generally the coercivities predicted by the model areapproximately a factor of 4 lower than those predicted by the Stoner Wohlfarth model and morein agreement with those observed in real materials.

Magnetisation Curling

The mechanism of magnetisation curling was first proposed by Kondorskij and developed by Shtrik-man and Treves [12]. In this model elongated particles are simulated as infinite length cylinders ofdifferent radii with rotation of the moments curling from the outer moments inwards. Like Jacoband Bean’s theory of magnetisation fanning this theory predicts lower values of coercivity thanthose predicted for coherent rotation. Again, this is more in line with experimental observations.The theory of Shtrikman and Treves is more general than that of Jacobs and Bean, however, asit allows for variation in the (radial) dimensions of the particles and predicts the correspondingchange in the magnetic properties.

In particular the model predicts that as the radii of the cylinders is reduced (akin to particlesbeing smaller) then coherent rotation becomes the energetically most favorable mecahnism of mag-netisation change. This makes intuitive sense and is consistent with measurements of γ-Fe2O3,which show that for particle sizes of less than 54nm to 124nm magnetisation reversal take place bycoherent rotation and by incoherent rotation above this [13, 14].

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1.2.3 Domain Wall Motion

The motion of domain walls in response to fields may be energetically or physically impeded bygrain boundaries, crystal defects [15], impurities [16, 17] and regions of inhomogeneous stress [18].Collectively these are referred to as pinning sites. When domain walls move in response to a fieldthen energy is required for the wall to pass over a pinning site. The requirement of this activationenergy means that domain wall motion over pinning sites is an irreversible process. In generaleach pinning site will require a different activation energy. It is the density of the sites and theprobability distribution of these energies that determines the characteristics of the domain wallmotion [19, 20]. If insufficient energy is available then the wall becomes pinned at the site and its’motion is prevented. A domain wall may be subject to the pinning of many sites simultaneouslyalong its length.

Weak Pinning

Weak domain wall pinning refers to cases where the unpinning of the domain walls from pinningsites occurs in a cooperative manner, with the domain wall breaking away from many sites simul-taneously. In this case the domain wall bows (bends) to its maximum energy geometry prior tobreaking free of the pinning sites. After the cooperative unpinning the domain wall reverts back toits unbowed state and moves through the material until it again becomes pinned. Gaunt has usedthe difference in the geometry of the domain wall immediately after it becomes pinned (planar)and immediately before a cooperative unpinning event (bowed) to calculate the volume, v, sampledby the domain wall prior to the irreversible unpinning event [21] as

vact = 31.0γb2

F0(1.3)

where γ is the domain wall energy, 4b is the maximum interaction range of a pinning site (i.e.how close a domain wall must be to be effected by it) and F0 is the maximum pinning force perunit area. This is the volume that is required to change magnetisation prior to the domain wallunpinning and so it may be considered as the activation volume, vact [22, 23].

Gaunt has similarly calculated the activation energy, Eact, required for unpinning to occur [21]:

Eact = 31.0γb2(

1− H

H0

)(1.4)

where H0 is the field required to unpin a domain wall in the absence of thermal activation.

Strong Pinning

In the case of strong pinning domain walls break away from a single pinning site at a time. Whena wall becomes pinned at multiple sites then the action of an applied field causes it to bow betweenthe pins until the bowing reaches a critical dimension and the domain wall breaks away from one of

6


the pins. In order for a steady state of rate of unpinning to be achieved Friedel has postulated thateach time this occurs the domain wall sweeps out a volume containing, on average, one replacementpin [24]; this is referred to as the Friedel Condition. A consequence of this is that if

β =3f

8πγb> 1 (1.5)

then strong domain wall pinning occurs, otherwise weak domain wall pinning occurs.

In the case of heat assisting the unpinning of domain walls in the strong pinning case Gaunt showedthat the activation energy required to break away from a single pinning site is [21]

Eact =43bf

[1−

(H

H0

) 12] 3

2

(1.6)

Similarly the activation volume (the volume swept out by the bow in the domain wall immediatelyprior to the unpinning event) is given by

vact = fbH

12

H320

[1−

(H

H0

) 12] 1

2

(1.7)

Equation 1.6 shows a different field dependence to equation 1.4 so measurements that are ableto give information concerning the form of the mean activation energy should be suitable fordistinguishing between the weak and strong pinning regimes.

1.2.4 Nucleation

Nucleation refers to the formation or liberation of a domain wall at a grain boundary. This isa necessary process for magnetisation change to proceed through domain wall movement in amaterial which is initially saturated and so contains no domain walls. Gau et al. have proposedthat nucleation processes may be a consequence of pinning sites being highly localised about grainboundaries [25]. This is reasonable since grain boundaries are likely to contain the greatest densityof crystalline defects; unnucleated domain walls may be considered as being pinned outside of thegrain.

The high densities of pinning sites about grain boundaries suggest that nucleation should occurthrough a cooperative unpinning effect. Ram and Gaunt have shown that by considering nucleationin this way it may be shown to be conceptually analogous to weak pinning by highly localisedpinning sites [26]. Furthermore they showed that the condition required for this highly localisedweak pinning is that of equation 1.5, which is that required for strong pinning. This again makesintuitive sense as both are highly localised processes.

Where the interior of the grain contains pinning sites then once nucleation occurs the propagation ofa domain wall through the grain is determined by the activation energy required to overcome thosepinning sites. If the initial activation energy required for nucleation is greater than that required to

7


overcome the pinning sites then the magnetisation change is controlled by the nucleation. Wherethe activation energy required for nucleation is less than that required to overcome the pinningsites then the magnetisation change is controlled by the domain wall pinning.

1.2.5 Application of the Ideal Models to Real Materials

Real materials seldom behave according to the ideal models described. In most cases it is unrealisticto assume that a single mechanism operates in a material; it is far more likely that multiplemechanisms will occur simultaneously. In general, then, measurements to determine the mechanismof magnetisation change in a material will at best indicate the dominant mechanism. The idealmodels presented here also all assume the absence of interactions between the grains or particlesthat constitute the material. Interactions may produce both magnetising and demagnetising effectsand may significantly alter the effective distributions of activation energies within a material. Suchlimitations should be remembered when considering results from the methods of characterising themechanisms of magnetisation change, as described below.

1.3 The Theory Behind the Methods of Probing the Mech-

anisms of Magnetisation Change in Magnetic Materials

The mechanisms by which a magnetic material undergoes magnetisation change may be deducedeither through direct observation, as in the case of the observation of domain wall movementusing Kerr, or Lorentz microscopy [27], or by indirect measurements of the magnetic response ofthe material to particular measurement regimes. Four such approaches subscribing the the latterregime are described here.

1.3.1 Magnetic Viscosity

The variation of magnetisation with time in a constant field was first observed by Ewing in 1885 andtermed ‘magnetic viscosity’ [28]. The phenomenon may be caused either by diffusion of impuritiesthrough a crystal lattice, causing displacements of domain walls, or by the thermal activation ofreversal processes. When the latter effect is responsible then the phenomenon has also been termedthe ‘irreversible aftereffect’ [29] or the ‘magnetic aftereffect’ [30]. Magnetic viscosity due to thethermal activation of reversal processes dominates the time dependent behaviour of permanentmagnets and of most materials used for magnetic recording. Its study is of interest as it may giveinformation regarding the distribution of the activation energies to magnetisation change, whichare intimately related to the mechanisms of magnetisation change.

Neel described the time dependence of magnetisation using a model that makes no assumptionsabout the mechanisms of magnetisation change. The thermal activation of reversal is simulated asbeing mediated by a fictitious fluctuation field, aligned with the applied field, and varying randomly

8


with time according to a Gaussian probability distribution function with a mean of zero and a rootmean square value of Hf [29]. Under this model the time dependence of magnetisation is given by

M(H, t) = M0(H) + χirr(H)Hf (H) ln(t) (1.8)

where M0(H) is the magnetisation at the start of the observation of the magnetic viscosity andχirr(H) is the irreversible susceptibility, which under the derivation is assumed to remain constantduring the measurement.

An alternative theory was independently proposed by Street and Woolley [22] who began by as-suming that in any material there exists a distribution of energy barriers which must be overcomein order for the magnetisation to reverse. The magnitude and distribution of these energy barriersto activation are field dependent. At a particular field the magnetisation as a function of time iscontrolled by thermal activation over these energy barriers and is given by

M(t) = Ms

∫ ∞0

f(∆E)(

2e−t

τ(∆E) − 1)d(∆E). (1.9)

where f(∆E) is the probability distribution function of particular activation energies and τ is atime constant given by the Arrhenius relation

1τ∝ exp

(− ∆EkBT

)(1.10)

as is appropriate for thermal activation.

Street et al. assumed the form of f(∆E) to be a slowly varying function over the range of energiesinvolved in the viscosity [31]. When this is the case then equation 1.9 may be approximated by

M(t) = M0 + S ln t (1.11)

where M0 is the magnetisation at the start of the viscosity measurement and S is known as the mag-netic viscosity parameter. The significance of this parameter has been reviewed by Wohlfarth[32].Equation 1.11 is of the same form as equation 1.8 and implies that

Hf =S

χirr(1.12)

when they are measured at the same value of irreversible magnetisation. The logarithmic nature ofthe change of magnetisation with time is the same as that derived by Neel (equation 1.8), despitethe different approach and assumptions used in its derivation.

Later work by Street et al. [31] highlighted the fact that since magnetic viscosity is a consequenceof the thermal activation of metastable states then the changes in magnetisation are irreversible.As such an expression for S in terms of χirr was derived which, in combination with equation 1.12

9


leads toHf =

−kBT

∂∆E∂Hi

∣∣∣Mirr

(1.13)

Showing that Hf is dependent on the rate of change of the activation energy with internal field.This is determined by the nature of the processes that are being activated and so equation 1.13provides the major basis for interest in measurements of magnetic viscosity.

El-Hilo et al. have noted that since the equations for activation energy of various mechanisms(equations 1.2, 1.4 and 1.6) may be expressed more generally in the form

Eact(H) = E0

(1−

(H

H0

)α)β(1.14)

then for the case of a single activation energy substitution into equation 1.13 gives

Hf =kBTH0

αβE0( HH0)α−1(1− ( HH0

)α)β−1. (1.15)

Allowing the variation of the fluctuation field with field to be directly related to the mechanism ofmagnetisation change.

By relating the fluctuation field to the magnetic energy within a material Wohlfarth has also shownthat

vact =kBT

µ0MSHf(1.16)

where vact is the activation volume related to reversal. This is also a basis for interest in magneticviscosity since the physical dimensions of an activation volume may also be related to the mechanismof reversal.

The theories of Neel and Street and Woolley both predict logarithmic time dependence of magneti-sation. Additionally they both assume that the irreversible susceptibility, χirr, does not changeduring the viscosity. This is not true as the time dependence of Mirr implies that χirr is also timedependent.

The problem of the time dependence of χirr was first solved by Estrin et al. when they proposeda new phenomenological theory of magnetic viscosity [33] by analogy with the theory of plasticdeformation of Hart et al. [34]. Estrin et al. proposed a new magnetic constitutive equation whereHi is a function of Mirr and its time derivative, Mirr. In differential form their equation is

dHi =(

1χiirr

)dMirr + Λd(ln Mirr) (1.17)

where1χiirr

=∂Hi

∂Mirr

∣∣∣∣ Mirr

(1.18)

10


and

Λ =∂Hi

∂ ln(Mirr)

∣∣∣∣ Mirr

(1.19)

The superscript i on the χiirr is to indicate that it is an intrinsic quantity (i.e. corrected fordemagnetisation effects).

The importance of this theory is that it specifically defines the irreversible susceptibility, χiirr, andsays how it should be measured (at a constant time rate of change of magnetisation) and so χiirris not dependent on arbitrary variations in the experimental procedure (namely arbitrary choicesof field sweep rate, the time spent at the constant field and the length of the measurement). Λ isnow the magnetic viscosity parameter.

Lyberatos and Chantrell have similarly considered magnetic viscosity from a constitutive equationof the form [35]

dMirr = Sid ln(t) + χiirrdHi (1.20)

where

χiirr =∂Mirr

∂Hi

∣∣∣∣ t (1.21)

and

Si =∂Mirr

∂Hi

∣∣∣∣ Hi

(1.22)

thus defining an alternative method of measuring χiirr and defining how Si should be measured.

Lyberatos et al. showed that this constitutive equation leads to the relation between Hf and Λ of

1Hif

= − 1Si

∂Si

∂Hi

∣∣∣∣ Mirr

+1Λ

(1.23)

and in doing so showed the consistency of the theories and that Λ is only equal to the fluctuationfield Hi

f when ∂Si/∂Hi|Mirr= 0.

A further consequence of the theory of Lyberatos et al. is that Hif may be derived from equation

1.20 as

Hif =

∂Hi

∂ ln(t)

∣∣∣∣ Mirr

(1.24)

which has previously been derived by other means [36, 37] and has been used extensively toexperimentally determine the fluctuation field (see for example [14, 38]). The attraction of equation1.24 is that it allows Hi

f to be determined with much less experimental uncertainty than doesequation 1.19. This is because equation 1.19 requires the measurement of the differences in ln( ˙Mirr)at constant values of Mirr and the experimentally determined values of the derivative have muchgreater uncertainty than those in t. This is discussed further in section 2.3.2.

When expressed in an approximate differential form equation 1.24 also provides an alternativedefinition for the fluctuation field. Hf is the change in field required to maintain the irreversible

11


magnetisation of a system unchanged during a time interval of ∆ ln(t) = 1 [39]. This is consistentwith Neel’s definition of Hf as being the field equivalent of thermal activation.

El-Hilo et al. have also considered the non-logarithmic time dependence of magnetisation by solvingequation 1.9 in terms of a series expansion [40]. They showed that

M(H, t) = M0(H) +∞∑n=0

Sn(∆ ln(t))n+1 (1.25)

and in the derivation determined that the higher order terms are related to the derivatives of thedistribution of energy barriers, f(∆E), with Sn being proportional to the nth derivative of f(∆E).Thus it was shown that deviations from equation 1.11 are caused by f(∆E) changing during thetime of measurement and that the greater the fit parameters Sn the narrower the distribution ofenergy barriers.

1.3.2 Remanent Magnetisation Measurements

Remanent magnetisation measurements may be of interest because they represent measurementswhere the total magnetisation is all irreversible, since they consist of measurements at zero field.Hence, if certain assumptions are valid (see section 2.3.1), remanent magnetisation measurementsmay be used as a means to resolve the reversible and irreversible components of magnetisation forany magnetic state of the material. In most cases characterisation of the reversible and irreversiblecomponents of magnetisation is of interest only for points along the initial and demagnetisationcurves. Measurements of the remanent magnetisations obtained by reducing the internal fieldto zero from points on the initial magnetisation curve are referred to as isothermal remanentmagnetisation (IRM) measurements. Measurements of the remanent magnetisations obtained byreducing the internal field to zero from points on the demagnetisation curve are referred to as DCdemagnetisation (DCD) measurements. The procedure for carrying out IRM/DCD measurementsis described in section 2.3.1.

Wohlfarth has derived a relation relating the DCD and IRM remanences, measured from the sameabsolute value of Hi, for particulate materials [41]. The Wohlfarth relation was derived for anon-interacting ensemble of Stoner-Wohlfarth particles (ie. uniaxial particles that reverse theirmagnetisation by coherent rotation). The relation is

Md(Hi) = 1− 2Mr(Hi) (1.26)

where Md(Hi) is the normalised DCD curve (normalised by the saturation remanent magnetisation)and Mr(Hi) is the normalised IRM curve.

Since the Wohlfarth relation is derived under the assumption that there are no interactions thendeviations from it may indicate the presence of interactions, although this is not necessarily the

12


case. Kelly et al. have modified the Wohlfarth relation to explicitly account for interactions:

∆M(Hi) = Md(Hi)− [1− 2Mr(Hi)] (1.27)

and proposed a new plot of ∆M(Hi) vs Hi, which gives a curve that may be representative of theinteractions in a system [42]. This plot is simply that of the deviations of the data points awayfrom the Wohlfarth relation. Positive ∆M(Hi) plots may indicate magnetising interactions (in-teractions that seek to align neighboring particles or regions), which are generally associated withexchange coupling. These plots usually consist of sharper peaks than the broader negative peaksassociated with dipolar coupling. In some unusual geometries dipolar coupling may produce posi-tive interactions [43], resulting in a broad positive peak in a ∆M(Hi) plot. Negative ∆M(Hi) plotsare interpreted as indicating demagnetising interactions, which are usually attributed to dipolarcoupling [44, 45], because the net effect of dipolar coupling in a random array is demagnetising. Insystems with particular structural features then negative exchange coupling can also cause nega-tive values of ∆M . Negative ∆M plots may also result from measurements on particulate systemswhere the field has not been corrected for demagnetising effects, because the demagnetisation factoris poorly known [46].

The more general applicability of∆M(Hi) plots to continuous media has been investigated. TheWohlfarth relation has been shown by McCurrie and Gaunt to also be valid for cases of continuousmedia where the magnetisation reversal is dominated by nucleation, after which the domain wallrapidly sweeps through grain [47]. In the opposite case, where the magnetisation reversal is dom-inated by domain wall pinning (and the energy required for nucleation is very low), then for aninitially AC demagnetised system the relation between Md(Hi) and Mr(Hi) was shown by Bertottiand Basso [48] to be

Md(Hi) = 1− 2√Mr(Hi) (1.28)

This result was obtained from Preisach modelling, using a factorisable Preisach distribution whichis believed to represent domain wall motion [48]. However McMichael et al. have shown thatHenkel (Md(Hi) vs Mr(Hi)) plots and so ∆M plots are sensitive to the initial demagnetised stateused for IRM measurements [49, 50] and so it has not been established that the deviation fromthe Wohlfarth relation is a result of the magnetisation mechanism rather than the initial ACdemagnetised state.

The results of the micromagnetic modelling of McMichael et al. also demonstrated that Henkelplots, performed on bulk materials which reverse magnetisation through domain wall unpinning,deviate below the Wohlfarth relation when the IRM measurements are performed from an ACdemagnetised state, even in the absence of interactions [49]. Despite the absence of interactionsin their model, none of the demagnetised states examined by McMichael et al. produced Henkelplots in agreement with the Wohlfarth relation. This lead them to conclude that Henkel plots,performed on bulk materials that reverse by domain wall motion, may not be interpreted purely interms of interaction effects because both the initial demagnetised state and the reversal mechanismappear to effect the plot.

This conclusion is consistent with an alternative view of interpreting IRM and DCD curves proposed

13


by Donnet et al. [51] based upon modelling work by Mansuripur et al. [52, 53]. Donnet et al.interpreted differences in the evolution of the irreversible magnetisation during magnetisation anddemagnetisation in terms of differences in the mechanisms occurring on the two curves, rather thanin terms of interactions. This idea was developed further by Thomson and O’Grady who pointedout that the derivation of the Wohlfarth relation implicitly assumes that there is only a singlemechanism of magnetisation reversal occurring in the system, which is likely to be an unrealisticassumption in most real materials [54]. Instead they propose that IRM and DCD data may beinterpreted in terms of a two coercivity model, where the magnetisation change is controlled bytwo mechanisms which have different activation energies.

Thomson et al. consider the specific example of a material in which the mechanisms of magneti-sation change are reverse domain nucleation and domain wall movement past pinning sites, aspreviously considered by Donnet et al. [51] and Mansuripur et al. [52, 53]. They argue that in amaterial with these mechanisms then in an initially (thermally) demagnetised state domain wallsalready exist within grains so that the change of magnetisation as the material is magnetised isdue to domain wall movement. Hence the IRM curve may give information concerning domainwall movement in the material. Conversely in the saturated state no domain walls exist and sobefore magnetisation reversal can occur nucleation must take place. Thus the DCD curve may giveinformation about nucleation.

Specifically, the information that may be gleaned from the IRM curve is obtained by differentiatingthe data with respect to internal field, to give χirr, and normalising the resulting curve so thatthe area under it is unity. Thomson et al. follow the lead of Ferguson et al. [55] and refer tothe resulting curve as the ‘domain wall pinning energy barrier distribution’, although it is really adistribution of critical fields required for activation.

The information that may be obtained from the DCD curve is less accessible. Two extreme casesmay be considered. The first, where the mean energy required for nucleation is much less thanthe mean energy required for domain wall movement, will yield little additional information asnucleation will occur at low fields and the reversal will then be dominated by domain wall pinning.In this case the changes in magnetisation are effectively controlled by a single reversal mechanismand so the analysis of the ∆M(Hi) plot may be carried out as before - in terms of assessinginteractions. In the other extreme case, where the mean energy required for nucleation is muchgreater than the mean energy required for domain wall movement, then the magnetisation reversalwill be controlled by nucleation, as once nucleation occurs in a grain then the newly formed domainwall will rapidly sweep out the entire grain. In this case the nucleation activation energy (field)distribution may be obtained by differentiating and normalising the DCD curve.

Between these two extreme cases lie those cases where the energy required for nucleation is similarto the energy required for domain wall movement and the energy barrier distributions of theprocesses overlap. In such cases the information obtainable from the DCD curve is complicatedby the fact that both processes occur simultaneously during the reversal and so the energy barrierdistribution obtained by differentiating and normalising the DCD curve is a convolution of theseparate distributions relating to the nucleation and domain wall pinning.

14


When domain nucleation and domain wall unpinning occur at similar energies then the ∆M(Hi)plot may provide insight into the nature of the overlap of the energy barrier distributions of the twoprocesses [54]. If the domain wall pinning energy barrier distribution is at lower energies than thenucleation energy barrier distribution, and the overlap between the distributions is small, then the∆M(Hi) curve will be broad and will reach a relatively high maximum (the largest possible valueis ∆M = 2). This maximum is reached because in this case the IRM curve will reach its maximumvalue at a field magnitude in which the DCD curve has just started to decrease. Energy barrierdistributions and the resulting IRM/DCD curves and ∆M(Hi) plots for this case are representedin figure 1.3a, which is calculated for no interactions. The more overlapped the barriers become thelesser the maximum achieved by ∆M(Hi) will be and the more narrow the peak will become. Thisis because the IRM reaches a lesser magnitude before the field becomes such that the DCD changesstart to occur, bringing the total closer to the Wohlfarth relation. This trend is illustrated in figure1.3 part b and c. The extreme case, where the relative positions of the domain wall pinning energybarrier distribution and domain nucleation energy barrier distributions are such that the entiredistribution for nucleation is just lower that the entire pinning distribution, is when the peak inthe ∆M(Hi) plot vanishes. This is shown in figure 1.3d.

Under the specific model of nucleation and domain wall pinning controlling the magnetisationchange, the ∆M(Hi) plot is forced to be positive or zero (in the absence of interactions). This isbecause magnetisation changes on the DCD curve must take place at either the same or greaterfields than those on the IRM curve, as the changes in magnetisation are ultimately due to domainwall movement, even when the activation is initially controlled by nucleation. In other words, underthis model it is not possible for a material to be easier to demagnetise than it is to magnetise. Theinterpretation of Thomson et al. should, then, only be applied where the controlling mechanismsare believed to satisfy this condition and where the ∆M(Hi) plot is positive. The method is notsuitable for analysis of materials exhibiting negative values of ∆M(Hi), or plots of ∆M(Hi) whichare both positive and negative (see for example [50, 56]).

The possible effect of interactions should be considered when using the analysis of Thomson et al.to analyse data, which implicitly assumes that they play no part in the deviation of the data fromthe Wohlfarth relation. The equivalent assumption in the traditional Wohlfarth analysis is thatonly a single mechanism of magnetisation change exists, which may be equally invalid. It maybe the case that specific instances of oscillatory ∆M(Hi) plots are caused by combination of thecompeting effects of interactions and of the change of the controlling mechanism of magnetisationchange between the IRM and DCD curves. This would be consistent with the findings of Basso etal. who showed oscillatory ∆M(Hi) plots resulting from modelling of domain wall pinning withlocal and mean field interaction effects [56]. In general then, non zero plots of ∆M(Hi) may reflectboth interactions and changes of the mechanism of magnetisation change, with the analyses ofWohlfarth [57] and Thomson et al. representing the two extreme cases.

15


Figure 1.3: The form of ∆M produced by different relative distributions of the pinning field and nucleationfield. In these cases the pinning field is the lower broader distribution. The calculations assume nointeractions.

1.3.3 The Dependence of Reversible Magnetisation on Irreversible Mag-

netisation

Irreversible magnetisation changes are associated with the local magnetisation vector changingits orientation such that it changes the local energy well in which it lies. Reversible changes areassociated with the movement of the local magnetisation vector within a well, such that it remainswithin the well. Since the reversible magnetisation of any region in a material is associated with thecharacteristics of a single energy well then it is natural that the magnetisation will depend on thecharacteristics of the well itself. The irreversible magnetisation determines in which local energy

16


well the local magnetisation vector lies so the reversible magnetisation depends on the irreversiblemagnetisation. Della Torre et al. were the first to report this [58], their conclusion being supportedby calculations of the reversible and irreversible components of magnetisation around the hysteresisloop of an isotropic Stoner-Wohlfarth ensemble [3]. They showed that even in this simple systemthere was hysteresis in Mrev, indicating that Mrev depends on more than just the field. Similarcalculations have since been repeated by Cammarano et al. [59].

The dependence of Mrev on Mirr is not unique to Stoner-Wohlfarth particles. Consider a particlewith cubic anisotropy and a positive first anisotropy constant. If the magnetisation is initiallyaligned along one of the easy directions, ±x or ±y, and then a field is applied orthogonally tothese axes, in the positive z direction, then the magnetisation will rotate as the field increases(thus increasing the reversible magnetisation) until it is at 22.5 to the xy plane at which point itwill flip (irreversibly) and become aligned with the easy +z direction. This involves moving fromthe energy well of one of the ±x or ±y directions to the energy well of the +z direction. If weassume that the magnetisation was initially in the +x direction then just prior to this irreversiblechange Mirr = (+1,0,0)Ms and Mrev = (−.076,0,+0.383)Ms. Immediately after the changeMirr = (0,0,+1)Ms and Mrev = (0,0,0)Ms. A change in Mirr at constant H has caused achange in Mrev.

Cammarano et al. described the dependence of Mrev on H and Mirr phenomenologically as [59]

dMrev = χrevdH + ηdMirr (1.29)

where

χrev =∂Mrev

∂H

∣∣∣∣ Mirr

(1.30)

and

η =∂Mrev

∂Mirr

∣∣∣∣ H (1.31)

These equations define how the reversible susceptibility, χrev, should be measured (at constantMirr) and introduce the new parameter η which expresses the dependence of Mrev on Mirr (mea-sured at constant H). The method of measuring η proposed by Cammarano et al. is described insection 2.3.1. Measurements on real materials have confirmed the dependence of Mrev on Mirr.Cammarano et al. showed such a dependence in AlNiCo [59] and Crew et al. have shown similarbehaviour in NdFeB and SmCo [60].

Cammarano et al. proposed that the parameters χrev and η may provide insight into the mecha-nisms of magnetisation change occurring within magnetic materials [59]. This was the motivationfor the work of Crew, who computationally determined the form of η(H), measured along thehysteresis loop (see section 2.3.3), for various models of magnetisation processes [61]. Crew showedthat the form of η as a function of field was strongly dependent on the mechanism of magnetisationchange. For materials where the reversible magnetisation is caused by the rotation of the localmagnetisation vectors away from easy axes then η as a function of H is expected to be negativeand to monotonically increase in magnitude as the magnitude of H increases. This is illustrated

17


in figure 1.4. Modelling revealed that the form of this function, whilst being effected by interac-tions, was not sufficiently altered by them so as to make the general form indistinguishable fromthat produced by other mechanisms. In materials where the reversible magnetisation is caused bydomain wall bowing (such materials undergo irreversible changes of magnetisation by domain wallnucleation and domain wall unpinning) then η is expected to be positive in low fields, where thetotal area of domain walls is increasing with each irreversible change, and negative at higher fields,where the total area of domain walls decreases with each irreversible change of magnetisation. Theform of the η vs H curve expected for such mechanisms is shown in figure 1.5. Comparison of themeasured form of η with these curves, then, makes possible inferences concerning the mechanismsof magnetisation change occurring within a material.

The magnitude of the parameter η may also be related to the degree of texture of the material.Crew et al. showed that for a Stoner-Wohlfarth particle η at the critical field depends on thealignment of the particle: going from η = 0 when the particle has its easy axis aligned with thefield to η → ∞ when the particle is aligned at right angles to the field. Thus measurements ofanisotropic materials, performed with the field aligned with the macroscopic easy axis, would beexpected to yield η values of smaller magnitude than in similar isotropic materials or materialswith a lesser degree of texture. The original measurements of η in AlNiCo by Cammarano confirmthis [59], as do subsequent measurements of η in NdFeB by Crew et al. [62].

Figure 1.4: The form of η along the hysteresis loop predicted by Crew for materials in which the reversible

magnetisation is controlled by rotation.

18


Figure 1.5: The form of η along the hysteresis loop predicted by Crew for materials in which the reversible

magnetisation is controlled by domain wall bowing.

1.3.4 Rotational Hysteresis

Rotational hysteresis is a hysteresis that may occur whenever M and H are not parallel. It occursin magnetic materials when they are rotated within a field and results from competition betweenthe competing torques acting on the magnetisation due to the field, the local anisotropy wells andany other energy required by the material to activate magnetisation change processes.

Rotational hysteresis is manifest as a torque acting on a material given by

τ = µ0(M ×Ha) (1.32)

where τ is the torque acting per unit volume. In the case of single crystals the measured form of τvs the angle of rotation, θ, at high fields may be related to the strength and geometry of anisotropyand so the type of magnetocrystalline anisotropy present in the material may determined [63, 64].

In isotropic materials the torque is independent of θ, once the field has rotated through at least215[65]. In either case the work done per unit volume by this torque, in a single revolution of thefield (or of the sample in a stationary field), is given by Wr, where

Wr =∫ 2π

0

τdθ

=∫ 2π

0

µ0MHa sinφdθ (1.33)

19


Figure 1.6: The extrapolation of the linear region of Wr vs 1/H data to determine the effective anisotropyfield.

where φ is the angle between M and Ha.

Various methods of using rotational hysteresis measurements to determine the anisotropy fielddistribution of polycrystalline or particulate materials have been proposed. Flanders et al. haveproposed a method involving the decomposition of the Wr vs Ha curve into the sum of theoreticalcurves of Wr vs Ha for Stoner-Wohlfarth particles of different anisotropies [66]. The assumptions ofthe absence of interactions and magnetisation change by coherent rotation only are implicit in thismethod. Elk et al. have proposed a method in which the rotational hysteresis torque is measuredin planes oblique to the initial saturation direction or the easy axis of the material [67, 68]. Suchmeasurements may be used to determine both the degree of texture of the material and, where thedistribution is monotonic, the anisotropy field distribution, from measurements performed over arange of field strengths.

Paige et al. have suggested that a single parameter, HK , that represents the effective mean of theanisotropy field distribution, may be a more appropriate measure of the anisotropy contribution tocoercivity in real materials [65]. Their approach makes use of the fact that the rotational hysteresisfalls to zero above the anisotropy field, irrespective of the magnetisation change mechanism [9].They propose that the root of the extrapolation of the linear rising region of the Wr vs 1/H curvemay be interpreted as the mean of the anisotropy field of the material, as shown in figure 1.6.This approach has subsequently been utilised by several authors [69, 65] and has been examinedin comparison with other methods by Uesaka et al. who found that, for their model, it producedsimilar results to the alternative approaches [70].

20


The use of rotational hysteresis loss measurements for characterising the mechanisms of magneti-sation change occurring in materials was pioneered by Bean and Meiklejohn [71] through the useof a dimensionless integral calculated from data of Wr vs H:

RH =1µ0

∫ ∞0

Wr(H)Ms

d1H

(1.34)

where the value of RH depends on the mechanism of magnetisation change and the degree of textureof the material. Bean and Meiklejohn claimed that the value of RH was essentially independentof interactions. This claim has been subsequently relied upon by many authors (see for example[72, 73]), despite observations that this may not be the case [74, 75].

Jacobs et al. calculated the theoretical value of this integral for models of chains of spheresreversing by fanning and coherent rotation. They found that the value of RH was significantlydifferent when the magnetisation change occurred by coherent rotation (0.42) than it was whenit occurred by fanning (1.54). This lead them to conclude that the parameter was suitable forexperimentally determining which model more accurately portrayed real materials. Furthermorethey suggested that where more than one mechanism of magnetisation change is available then thevalue of RH should be a linear combination of the appropriate RH values for each mechanism witheach component taken in proportion with its volume fraction.

Since the initial calculations of Jacobs et al. the value of the rotational hysteresis integral hasbeen calculated by other authors for various models of mechanisms of magnetisation change. Suchvalues are presented in table 1.1. The range of values in the models of Ishii and Sato [76] and ofShtrikman and Treves [10] are caused by variation in the model parameters.

Rotational hysteresis loss caused by the domain wall mechanisms has not been as widely studied.Elk has performed calculations of the rotational hysteresis loss for a model of combined domainwall pinning and coherent rotation [77]. The results of this work suggest that where the mechanismof magnetisation change is purely domain wall pinning then RH ' 3.2, for an isotropic polycrys-talline material, and RH ' 3.6 for an aligned material (with the measurements performed in arotational plane containing the easy axis). Elk found that as the pinning field was increased andso rotation became more favorable then the value of RH decreased towards 0.4, in agreement withthe predictions of Jacobs and Luborsky that the value of RH would be a linear combination of thevalues for the distinct processes [9].

The theoretically predicted values of RH in table 1.1 suggest that if only the simple models ofcoherent rotation[3], fanning[9], curling[12] and domain wall pinning[21, 77] are considered thenthe measured value of RH may be useful in distinguishing between the mechanism of magnetisationchange in real materials. Where the extended models of Ishii et al. [76, 78] are included, however,then the ability to make such distinctions is somewhat diminished. Additionally, the presence ofmultiple mechanisms of magnetisation change in real materials will alter the value of RH , in themanner described by Jacobs and Bean, making the interpretation of its value less clear still. Theidea of unambiguously characterising the magnetisation mechanisms in real materials through thedetermination of a single dimensionless parameter is perhaps a bit too ambitious.

21


Model Reference Alignment RH

Coherent rotation Stoner and Wohlfarth[3, 9] aligned orrandom-in-plane 0.42

isotropic 0.38

Chain-of-spheres Jacobs and Luborsky[9] aligned or(fanning) random-in-plane 1.54

isotropic 1.02

Chain-of-touching-spheres Ishii and Sato[76] aligned 0.98 - 4.00(fanning) isotropic 0.67 - π

Infinite length cylinders Shtrikman and Treves[12] aligned 0.42 - 4.00(curling) isotropic 0.42 - π

Domain wall pinning Elk[77] aligned ∼3.7isotropic ∼3.2

Table 1.1: The calculated values of the rotational hysteresis integral, RH , for various models of themechanisms of magnetisation change

1.4 Studies of the Mechanisms of Magnetisation Change

1.4.1 Studies Using Magnetic Viscosity Measurements

Givord et al. have reported a study of magnetic viscosity in sintered NdFeB, with Hf beingdetermined as both a function of field and of temperature [79]. They found that at a giventemperature Hf was essentially independent of field but that its value was strongly temperaturedependent. Under the assumption, then, that Hf was independent of field they were able to derivean approximate expression for the mean activation energy barrier for reversal in the material:

∆E(T ) ≈ kBT(

25− Hc

Hf

)(1.35)

and an expression for the mean intrinsic coercive field, H0 (measured at t = 0):

H0(T ) ≈ ∆E0(T )Hf

kBT(1.36)

where Hc is the measured coercivity, which is negative.

Another significant result of this study was that Givord was able to show that in sintered NdFeB theactivation volume, determined by equation 1.16, was proportional to the cube of the domain wallwidth, δ. This implies that a reverse domain of minimum size δ3 must be created before reversalcan take place, which is consistent with a model of reverse domain nucleation as the controllingmechanism of magnetisation change. Such an interpretation was given further credence by a later

22


study by Givord et al. comparing sintered and melt-quenched NdFeB [80]. The proportionalityva ∝ δ3 was found not to hold in the melt-quenched NdFeB, consistent with the fact that in thesematerials most of the grains are below the size required to support a domain wall and so domainnucleation is not possible.

Other studies of magnetic viscosity in NdFeB materials have been reported. Givord et al. havestudied the dependence of Hf on the magnetic history of the material [81] in melt-quenched Nd-FeB. They found that if the angular dependence of the coercivity obeys a 1/ cos θ law then thedependence of Hf on the initial magnetising field is consistent with the observed dependence of thecoercivity. This in turn implies that Hf varies with the orientation of anisotropy axes. In contrast,a study of the angular dependence of Hf in sintered NdFeB, by Street et al. [82], produced noevidence of this. Street et al. found that Hf in samples of sintered NdFeB, of varying degrees ofalignment, was essentially independent of the degree of texture and of the applied angle betweenthe field and the easy axis. It is possible that the difference in results of these two studies may alsobe a reflection of the different mechanisms of magnetisation change between the melt-quenchedand sintered materials.

A study of the mechanisms of magnetisation change in sintered NdFeB by Ferguson et al. utilisedboth remanent magnetisation measurements and magnetic viscosity measurements [55]. Fergusonet al. showed that in the sintered material the energy barrier distribution for demagnetisation(obtained by differentiating the Mirr curve - see section 1.3.2) was strongly temperature depen-dent but that this was not the case for the initial magnetisation curve. This was interpreted asevidence of differing mechanisms controlling the initial magnetisation and the demagnetisation. Bysubstituting the form of the energy barrier for strong domain wall pinning (see section 1.2.3) intoequation 1.13 Ferguson et al. were able to relate χirr and S by

χirr =fb

kBT

√1−

√H

HA

1√HHA

S. (1.37)

Substitution of the measured values of S into equation 1.37 produced curves in excellent agreementwith that obtained from the remanent magnetisation measurements, over all of the temperaturesexamined, suggesting that strong domain wall pinning controls the demagnetisation in sinteredNdFeB.

Crew et al. have argued that the utility of Λ (and so Hf ) in terms of characterising the mechanismsof magnetisation change in bulk materials is limited as it is a complex average over an ensemble ofgrains in the material [83, 84]. This is necessarily the case for any measurement of any macroscopicproperty of a bulk material made up of grains with distributions in size, orientation, compositionand anisotropy. At any given field, a number of grains with different ∆E(Hi) functions may bethermally activated and so change their magnetisation. Hence the derivative of ∆E(Hi) in equation1.13 is a complex average of the activation energy functions of different grains at different fieldsand so Λ is also a complex average. Crew et al. argue that this makes its’ interpretation obscure asthe functional form of Λ with field cannot be compared with the known form for different modelsof magnetisation change unless the distribution of parameters in the material is known a prioriand can be taken into account.

23


El-Hilo et al. have considered the variation of Hf during reversal in models of granular systems[39]. They showed that constant values of Hf may be indicative both of weak domain wall pinningor of rotation in the case that the standard distributions of the grain volumes and anisotropy aresimilar or very narrow. They have thus suggested caution when interpreting the significance of Hf

being constant with field in a particular material. The form of Hf predicted by their model is alsoseen to vary between increasing or decreasing during reversal depending on the relative width ofthe distributions. Thus these results are consistent with the cautions of Crew et al..

In materials used for magnetic recording the distributions in material parameters are frequentlynarrow compared to those in bulk materials. This in turn means that the distribution of energy bar-riers to reversal, f(∆E) in equation 1.9, is narrow as is evident by the highly non linear logarithmictime dependence frequently observed in such materials (see for example [54, 85]). Measurements ofHf in magnetic recording media, then, have allowed for more definite interpretation of its variationin terms of the mechanisms of magnetisation change.

Bottoni has reported a study of the effects of interactions on the measured value of the magneticviscosity parameter [86]. The measured form of S, as a function of field, was seen to vary as thepacking density of a sample of iron particles was changed. The interactions between the particleswere assessed by remanent magnetisation measurements and ∆M plots. The data suggest thatdemagnetising interactions serve to lower the observed values of S. To remove the effect of theinteractions on the determined values of the activation volume, va, Bottoni proposed that aninterpolated (in this case) or extrapolated value of va should be used, such that the area of the∆M plot is zero. In the case of the iron particles this value of va was found to be consistent withthe volume of a single sphere in the chain-of-spheres model, which previous studies have suggestedis the mechanism of magnetisation change in the material [87, 88].

1.4.2 Studies of Magnetic Materials Using Remanent Magnetisation Mea-

surements

Remanent magnetisation measurements have been used by many authors to characterise the in-teractions in magnetic materials and their use is now standard practice (see for example [45, 86]).∆M plots provide a convenient means of studying such deviations and have been predominantlyutilised in studies of the interactions in particulate media [89]. They have, however, also been ap-plied to permanent magnet materials. Folks et al. have reported a study of interaction mechanismsin melt quenched NdFeB using ∆M plots[90]. They interpreted the positive forms of ∆M as beingindicative of exchange interactions between grains. However the IRM measurements were madefrom thermally demagnetised states so it is not clear that the same mechanism was occurring onboth the magnetising and demagnetising curves, particularly in the larger grain materials, and sosuch an interpretation is uncertain.

The effects of the initial demagnetised state on IRM data, as discussed in section 1.3.2, are displayedin both the results of modelling [91, 92] and in data obtained for real materials. El-Hilo et al.have studied the effect of the initial sample demagnetisation process on ∆M plots [93, 94]. They

24


showed that the plot is effected by the direction in which measurements are made relative to theAC demagnetisation direction and is also effected by the field decrement size in the case of DCcyclic erasure. Pfeiffer and Schuppel have observed similar effects in the IRM data for bariumferrite powders [95]. They point out that a DC demagnetised state is highly anisotropic in thefield direction as 50 vol% of the particles (the particles with lower critical fields) have previouslyswitched and the other 50% have not. Similar arguments can also be applied to AC demagnetisedstates.

Demagnetising fields also effect the form of Henkel and ∆M plots, as they may be considered tobe analogous to a negative mean interaction field. Samwel et al. have investigated the effects ofdemagnetising fields on remanent magnetisation measurements [96]. They proposed that remanentmagnetisation measurements performed against applied field may be corrected by modifying thefield by

Hintern = Ha − βM (1.38)

where β is the slope of the hysteresis loop at coercivity. This method, then, assumes that theslope of the hysteresis loop at the coercivity is solely due to demagnetisation and so β = D,which will only be true for materials with perfectly square hysteresis loops in the absence ofgeometrical demagnetising effects. This is recognised by Samwel et al., who suggest the method isprobably only suitable for use on perpendicular recording media, which have very square hysteresisloops [96]. They also showed that even in these cases the corrected form of the Henkel plots and∆M plots is very sensitive to small errors in the measurement of β. They concluded that “it isimpossible to determine the position of the Henkel curve with any precision” in samples where thedemagnetisation coefficient is not well known.

Quantities other than ∆M have been defined as indicators of interactions [57, 91]. The remanentcoercivities, Hr and H ′r, are defined by Md(Hr) = 0 and Mr(H ′r) = 1

2Mr(∞). The coercivity factor,CF , is defined as CF = (Hr −Hc)/Hc and the interaction field factor is IFF = (H ′r −Hr)/Hc.The remanent coercivity is the reverse field which must be applied to an initially saturated samplein order to DC demagnetise it. The IFF is considered to be representative of the overall strengthof interactions in a sample [87, 88] and has been shown to provide similar information as |∆M |max[97]. The parameters have been calculated for various models without interactions and so deviationfrom them in real systems is sometimes attributed to the presence of interactions [92].

Kato and Tsutsumi have reported a study of the physical interpretation of ∆M(H) [98]. Theybased their work on three assumptions: 1) that the only interactions in a system may be representedas a mean field, kM , which is directly proportional to the magnetisation; 2) that the anisotropyfield distribution is the derivative of the IRM curve in the absence of interactions and 3) that themechanism of magnetisation change is such that the particles (or grains) completely switch theirmagnetisation when the effective field is equal to the anisotropy field. From these assumptionsthey derived an expression of the form

∆M(H) ≈ −2k[MDCD(−H)−MIRM (H)]PDF (HK = H) (1.39)

so ∆M(H) may be interpreted as being approximately the product of the anisotropy field distri-

25


bution with twice the difference between the interactions in the demagnetisation process and themagnetisation process. Kato and Tsutsumi compared their theoretical form of ∆M(H) with theexperimentally determined forms for Co-γ-Fe2O3 and Ba ferrite and found reasonable agreementwhen the form of the anisotropy field distribution was assumed to be Gaussian. Such an approachis interesting as it allows ∆M(H) to be interpreted not only in terms of (mean field) interactionsbut also in terms of the width of the anisotropy field distribution.

The alternative interpretation of remanent magnetisation measurements, in terms of switching fielddistributions, has been performed by Pfeiffer and Schppel in barium ferrite powders [95]. Theydefined the first approximation to the switching field distribution (SFD) as the derivative of theIRM curve and developed refinement procedures to account for the effect of thermal fluctuations[95]. They point out that in the absence of interactions, and when only a single magnetisationmechanism takes place in the material, then the coercivity may be found in terms of the SFD by∫ ∞

0

SFD(HK)µ(Hc

HK

)dHK = 0 (1.40)

where µ(h) is the primitive hysteresis loop for the particular mechanism of magnetisation change.µ(h) itself has a coercivity of h = 1. Using the SFDs obtained from IRM measurements Pfeifferand Schppel calculated coercivities for two barium ferrite samples, assuming the Stoner Wohlfarthform of µ(h). They found excellent agreement with the measured coercivity of substituted bariumferrite (Fe being partially substituted by Co and Ti) but poor agreement with unsubstituted. Thiswas interpreted as evidence of coherent rotation being the mechanism of magnetisation change insubstituted barium ferrite but not in the unsubstituted material. In these cases the calculatedcoercivities were much lower than that expected for a single Stoner Wohlfarth particle with ananisotropy field equal to the mean of the SFD. Pfeiffer and Schppel point out that this is consistentwith the Gerlach principle which states that in systems of particles with a range of coercivities thelower coercivities tend to dominate [99, 100].

The methods of characterising the magnetisation reversal mechanisms used by Thomson andO’Grady in their studies of Tb-Fe-Co alloy films [101] and Tb/Fe multilayers [54] have been previ-ously described in section 1.3.2. Thomson and O’Grady found that in both systems large values of∆M in the thinner films seemed to indicate that domain nucleation was the controlling mechanismbut in thicker films domain wall pinning became the dominant process. Their work also showednegative values of ∆M in the thicker Tb/Fe multilayer films. As explained in section 1.3.2 thespecific two coercivity model of Thomson and O’Grady allows only positive values of ∆M becausedomain wall motion cannot proceed (in the demagnetisation case) until nucleation has taken place.In this case it is likely that the negative values of ∆M are a result of magnetostatic interactions,which may even serve to obscure any positive contributions to ∆M due to competition betweenthe mechanisms.

Bissell and Lyberatos have taken a different approach to interpreting remanent magnetisationmeasurements by examining the form of the reversible magnetisation, rather than the irreversiblemagnetisation [102]. By analogy with the irreversible case, and as a result of studies of a Stoner-

26


Wohlfarth system, they proposed a plot similar to ∆M(H), which may be expressed as

∆Mrev(H) = MDCDrev (−H)− 2M IRM

rev (H) +Malignedrev (H) (1.41)

where Malignedrev is the value of Mrev measured from positive remanence with the field increased until

saturation. In this case all of the values are normalised to their value at saturation. These plots weregenerated for eleven different magnetic recording materials and their form seen to be similar to thecorresponding ∆M(H) plots. Bissel and Lyberatos concluded, however, that in terms of assessinginteractions they were less sensitive than ∆M(H) plots and so less useful. Additionally they notedthat they had observed materials with very similar irreversible properties but distinctly differentreversible properties. It was proposed, then, that measurements of reversible magnetisation maybe useful in investigating the mechanisms of magnetisation change.

1.4.3 Studies Using the Dependence of the Reversible Magnetisation on

the Irreversible Magnetisation

Studies of the mechanisms of magnetisation change using measurements of the dependence ofthe reversible magnetisation on the irreversible magnetisation, as characterised by η, have beenreported by Crew et al. [103, 60]. Crew interprets the measured form of η vs Hi, along the initial anddemagnetisation curves, in terms of mechanisms of magnetisation change in the manner describedin section 1.3.3. Using this procedure Crew et al. characterised the magnetisation mechanisms inisotropic and anisotropic NdFeB [103] and in Sm2(Co,Fe,Cu,Zr)17[60]. The results of these studiesare consistent with previous characterisations of the magnetisation reversal mechanisms and in linewith expectations based on knowledge of the microstructure of the materials studied.

In optimally melt-quenched NdFeB (MQ1) Crew et al. showed that the form of η vs Hi wasconsistent with that produced by incoherent rotation [103]. Such a conclusion is consistent withthe findings of Folks et al. [90], who estimated that 94% of the grains in MQ1 have dimensions lessthan that required to contain a domain wall. Similarly in sintered NdFeB, which contains largemulti-domain grains, the form of η vs Hi was interpreted in terms of a model of magnetisationchange in which the domain walls are locally pinned in the grain boundaries and after unpinningrapidly sweep across the grain. Such a model is consistent with in-field magnetic force microscopyobservations of sintered NdFeB reported by Babcock et al. [104] but different from the modelof reverse domain nucleation followed by rapid propagation through the grain suggested by otherauthors [105, 90]. Crew explains this difference by pointing out that Livingston has noted thattrue nucleation is almost indistinguishable from localised pinning in the grain boundaries.

The controlling mechanism of magnetisation change in Sm2(Co,Fe,Cu,Zr)17 has been well char-acterised as being domain wall pinning [106, 107]. Investigations of the form of η vs Hi alongthe hysteresis loop in this material have also been performed by Crew et al. [60] and found tobe consistent with magnetisation reversal controlled by domain wall pinning. Further to this themeasurements of Crew et al. are able to determine that only the grains with the lower averagepinning strength (there are two populations with average pinning strengths differing by a factor of

27


about 2 [107]) contribute significantly to the reversible magnetisation. Crew et al. note that theirmeasured form of η showed no linear correlation with the measured form of χirev, in disagreementwith the measurements and calculations of Emura et al. in a study of exchange spring magnets[60, 108].

The results of Emura et al. [108] indicate that η is proportional to χirev, which would seem toindicate that measurements and calculations of η are not useful in characterising the mechanismsof magnetisation change. The materials in which they perform their measurements, however, havesteep open recoil loops, indicating that irreversible changes of magnetisation occur during the recoiland so the DCD method of determining Mrev is unsuitable (see section 2.3.1). Recognizing this,they performed measurements of Mrev using a modified χrev method (see section 2.3.1) based uponearlier work by Cornejo et al. [109]. During the development of this method η is derived as

η(Hi) =η(0)

χirev(0)χirev(Hi) (1.42)

hence defining it to be proportional to χirev. The value of η(0), on the right hand side of equation1.42 is determined by the approximation

η(0) ≈ MRrev

MR −MRrev(1.43)

where MRrev is the remanent value of the reversible magnetisation. Under the DCD method this isdefined to be zero but under this approach a coercive field HCrev (at which MRrev = 0) is defined,corresponding to the maximum value of χirev. MRrev is then obtained by integrating χirev fromthis field to Hi = 0.

By analogy with results of their work using a Moving Preisach Model (MPM) [110, 111] Emura etal. define

η = kmχirev (1.44)

and so by combination with equations 1.42 and 1.43

km =MRrev

(MR −MRrev)χirev(0)(1.45)

where km is the moving parameter of the Preisach model, giving a measure of the mean interactionfield (as a factor of M) and so a measure of long range order in a material.

The different nature of η exhibited in the works of Crew et al. [60] and Emura et al. [108] reflect thedifferent definitions that each use for Mrev and Mirr and should not be interpreted as invalidatingthe approach of either author. Crew uses a conventional definition for Mirr, analogous to that usedby Wohlfarth [57], which he points out is not a thermodynamic irreversible quantity, as an increasein Mirr does not imply an increase in entropy. Cornejo et al. claim that under their definitionMirr and Mrev are thermodynamic quantities because of their modification of the χrev methodaccording to the MPM, which has itself has been shown to be thermodynamically consistent [112].The fact that under this regime Mrev vs Hi + kmM is single valued and monotonic is evidence ofthis [109, 108].

28


1.4.4 Studies Using Rotational Hysteresis Measurements

The earliest studies into the mechanisms of magnetisation change using rotational hysteresis mea-surements were conducted by Bean and Meiklejohn [71, 113] on single domain Co-CoO particlesand by Jacobs and Luborsky[9] on electro-deposited elongated single domain (ESD) particles ofFe-Co. The rotational hysteresis loss was measured as a function of the field strength and therotational hysteresis integral, given by equation 1.34 calculated. The values determined showedexcellent agreement with that predicted by the chain of spheres model with fanning[8] (see section1.2.2), particularly in the case of the FeCo particles, leading to the conclusion that the particleschange their magnetisation by this mechanism.

The information that may be imparted by the value of the rotational hysteresis integral in partic-ulate materials has been extensively studied by Bottoni et al.. He has published a study of theinfluence of magnetic interactions on the reversal mechanism in both pure and surface modifiedγ-Fe2O3 [72]. This study employed ∆M plots to determine the level of interactions in samplesof varying dilution. The mechanisms of magnetisation change were characterised by rotationalhysteresis measurements and the angular dependence of coercivity. In both the pure and surfacemodified γ-Fe2O3 it was shown that as the inter-particle interactions increase then more particlestend to change their magnetisation more coherently.

The effect of magnetic anisotropy on the form of rotational hysteresis loss has also been examinedby Bottoni [114]. Bottoni measured the rotational hysteresis loss in isotropic particulate systemsof γ-Fe2O3, CrO2, Ba ferrite and α-Fe with varying concentrations of particular dopants, which areknown to effect the magnetocrystalline anisotropy of the particles. These measurements showeda strong positive correlation between the strength of the magnetocrystalline anisotropy and boththe peak value of Wr and the field at which the rotational hysteresis loss vanishes, Hmax. Bottoniinterprets the peak in Wr as the energy barrier to rotate the magnetisation out of the easy axisand Hmax as the maximum anisotropy field. A lesser positive correlation was observed betweenthe anisotropy and the field at which the rotational hysteresis loss begins and consequently therewas an indirect positive correlation between the anisotropy and the width of the Wr vs Ha curve.These correlations led Bottoni to conclude that anisotropy is the main factor effecting rotationalhysteresis.

Subsequent work by Bottoni et al. examines the effect of changes in the magnetisation mech-anisms on rotational hysteresis losses. Data similar to that presented in the earlier study [114]is reinterpreted in terms of altered mechanisms of magnetisation change [115]. The plots of Wr

vs H are redone with Wr being divided by the anisotropy constant, K, and the field being nor-malised in terms of the anisotropy field, HK . As the concentration of dopants is increased thenthe magnetisation mechanisms are believed to become more incoherent. The resulting curves, forvarying concentrations of dopants, are consistent with this, showing qualitative agreement withthose predicted by Shtrikman and Treves [12] for varying degrees of incoherent rotation (fanning).

Yet more convincing evidence of the dependence of the rotational hysteresis loss on the mechanismsof magnetisation change is seen in Bottoni’s study of the evolution of the rotational hysteresis inte-

29


gral with changing microstructure of initially amorphous ferromagnets [116, 117]. Bottoni is able toshow a smooth monotonic decrease in the value of RH as the sample is heat treated to successivelyhigher temperatures, causing formation of single domain crystallites and a corresponding decreasein the amount of amorphous phase. The decrease in RH (in reference [117] from ∼ 3.7 to ∼ 1.4)is consistent with the ratio of the volume of the amorphous phase (in which domain wall motionoccurs) to the volume of the crystalline phase (in which incoherent rotation occurs) decreasing.Such an interpretation is given further credence by measurements at temperatures above the Curietemperature of the amorphous phase yielding a large decrease in RH for a partially heat treatedmaterial and a much smaller decrease for a more completely treated material, which has a muchlesser relative volume of amorphous phase [116].

Studies of rotational hysteresis losses in permanent magnet materials seem to have produced lessinsightful results. Wyslocki et al. have reported such measurements in sintered NdFeB [118];AlNiCo [119]; NdFeCrB [120]; melt quenched NdFeB [121]; UFeAlSi and ScFeSi [122, 123]; andFeAlC [124]. In all of these papers comparison is made between the experimentally determinedvalues of the rotational hysteresis integral, RH , and the theoretical values arising from the Stoner-Wohlfarth model of coherent rotation, the Jacobs-Bean model of fanning and the Shtrikman-Trevestheory of magnetisation curling. Given that these models all concern magnetisation change in singledomain particles their application to multi-domain materials, as in the case of sintered NdFeB [118],NdFeCrB [120] and UFeAlSi [122], is probably not appropriate. The conclusion in each case thatthe value of the rotational hysteresis integral is indicative of magnetisation reversal by an incoherentmechanism is to be expected.

Consideration of the microstructure of a material under study is, then, necessary before conclu-sions about the mechanisms of magnetisation change based on the measured value of the rotationalhysteresis integral may be made. The study of the mechanisms of magnetisation change in AlNiCoperformed by McCurrie and Jackson [75] takes such an approach. Consideration of the microstruc-ture of the material (obtained elsewhere [125] by electron microscopy) suggested that comparisonwith the Shtrikman-Treves theory of curling may be appropriate. From the electron micrographsthey calculated an approximate value of the reduced radius of S ≈ 2. From rotational hysteresismeasurements they determined the value of the rotational hysteresis integral, RH , to be 1.35, whichcorresponds to S = 1.8, in reasonable agreement with their previous determination. This, togetherwith the value of RH , led them to conclude that the mechanism of magnetisation change in thesmall grained AlNiCo magnets studied was magnetisation curling.

1.5 Conclusions

The theory pertaining to and the use of time dependence measurements is far more developed inthe area of particulate media than in that of permanent magnets. El-Hilo, O’Grady and Chantrellin particular have been prolific in developing such theory (see for example [39, 35]). The narrowerdistributions of energy barriers in recording media has allowed for clearer characterisation of themechanisms of magnetisation change occurring within particulate media by examination of the

30


variation of Hf or the activation volume (by equation 1.16) during magnetisation reversal [126, 127].In bulk permanent magnets the distribution of material parameters contributing to the distributionof energy barriers to reversal are likely to be broader and interaction effects more significant. Thevariation of Hf in such materials, then, is likely to be less and its form more complex to predict.Reports of Hf increasing [81], decreasing [128] or remaining constant [62] during reversal in NdFeBare evidence of this. Nevertheless measurements of time dependence in permanent magnets havebeen successful in terms of measuring activation volumes that are consistent with the mechanismbelieved to control magnetisation change within the materials [79, 129].

The waiting time technique of analysing magnetic viscosity (equation 1.24) has been shown byLyberatos et al. to be equivalent to the method of Estrin et al. [35] and has been widely usedfor measurements in the field of magnetic recording media. This method allows for much lessuncertainty in the measured values of Hf , as explained in section 1.3.1, chapter 2 and by O’Gradyet al. in [126]. Despite this the method has not been widely used in studies of magnetic viscosity inthe field of permanent magnets. The method is equally applicable to such materials and so shouldbe preferred to the alternative approaches of analysing magnetic viscosity in permanent magnetsthat generate much greater uncertainty.

The use of ∆M(H) plots to characterise interactions is now so well established as to be acceptedpractice. When interpreting such data, however, care should be taken to consider the possiblecompeting effects of a change of the magnetisation mechanism between the initial magnetisationand the demagnetisation cases. Interpretation in terms of interactions is only possible where themechanisms during magnetisation and demagnetisation are the same or have very similar criticalfield distributions. In materials where magnetisation change is controlled by a combination ofdomain wall pinning and domain nucleation then ∆M(H) measured from both AC and thermallydemagnetised states may provide insight into the relative importance of each mechanism, ratherthan of interactions [54, 101]. Such an approach relies on the assumption that domain walls exist inthe thermally demagnetised state but not in the AC demagnetised state and requires that ∆M(H)must be positive. The presence of both positive and negative values of ∆M(H) suggest the presenceof interactions and makes the interpretation of either the mechanisms or interactions less certain.

Information regarding the mechanisms of magnetisation change may also be obtained from studiesof the reversible magnetisation. This is due to the fact that the reversible magnetisation of aparticle or grain depends on the nature of the energy well within which the magnetisation vectorresides, which is directly related to the mechanism of magnetisation change. The study of Mrev

conducted by Bissell et al. suggests that the dependence of Mrev on field alone is insufficient to offermuch insight [102]. Subsequent theories of the dependence of the reversible magnetisation on theirreversible magnetisation, and in particular the results of Crew et al. [103], seem to provide a firmbasis on which to characterise the mechanisms of magnetisation change, although the significanceof η has not been widely recognised.

The extensive work of Bottoni et al. in interpreting rotational hysteresis measurements in magneticrecording media (see for example [115, 117]) provides strong evidence that such measurements mayprovide insight into the mechanisms of magnetisation change in such materials. Similar such mea-

31


surements in permanent magnet materials, however, appear to have provided less insight (see forexample [121]). All of the reported values of the rotational hysteresis integral in multi-domain per-manent magnets have been significantly less than the high values (∼ 3 [77]) expected for materialsin which domain wall processes control the magnetisation change (see for example [120]). It is notclear what role interactions and distributions in material parameters may play in determining therotational hysteresis properties in such materials. Indeed the effect of interactions on rotationalhysteresis in general has received very little attention. It is possible that re-interpretation of thedata of Bottoni [72, 117] in terms of the effects of interactions on the value of RH may providesome insight.

Given that each of the approaches to characterising the mechanisms of magnetisation changedescribed here may yield distinct information concerning the properties of a material under study,it seems prudent that such studies should be performed using a combination of the methods. Inmany cases the interpretation of the data obtained using such methods is somewhat subjective andso studies employing more than one method may offer greater insight (see for example [101, 88]).Additionally combining the alternating field approaches with that of rotational hysteresis may allowfor further insight into the unclear relationship [130, 131] between the two types of hysteresis.

32

2. Experimental Methods

THE methods of characterising the mechanisms of magnetisation change described in chapter1 call for quite specific measurement regimes. The implementations of these regimes are

described in this chapter. Three measurement apparatus were primarily used during this study: aVibrating Sample Magnetometer (VSM), a Dual Air Bearing Angular Acceleration Magnetometer(DABAAM) and a Rotating Sample and Coil Magnetometer (RSCM). The latter two deviceswere designed and constructed as part of this work and so are described in detail in chapters 5and 6. This chapter, then, primarily discusses those measurements performed using a VSM. VSMmeasurements were used to characterise the reversible and irreversible components of magnetisationin the materials studied and to investigate their interrelation. Measurements were also performedto study the time dependence of the magnetisation. The measurement methods involved in suchstudies are described in this chapter together with a discussion of the uncertainties in the measuredvalues.

2.1 Measurement Hardware

2.1.1 Hardware for Measuring Alternating Hysteresis and the Time De-

pendence of Magnetisation

The VSM Setup

The VSM unit was an Aerosonics Limited Vibrating Sample Magnetometer which was mounted sothat the samples vibrated along the axis of a 5 Tesla Oxford Instruments superconducting solenoid.The VSM measurements were performed using a vibrational (lock-in) frequency of 66.667 Hz and ashort lock-in amplifier time constant of 16 ms, so as to allow the time dependence of magnetisationto be observed. The superconducting solenoid was made of multifilamentary NdTi and Nb3Sn wirecontained within a liquid He/N2 cryostat, consisting of a liquid He vessel surrounded by liquidN2 and layers of super-insulation in an evacuated jacket. The solenoid was driven by a OxfordInstruments Bipolar Power Supply, capable of delivering up to 120 A, with a very high degree ofstability.

33


Measurement Control

Both the VSM unit and the solenoid power supply were controlled remotely by means of two waycommunications through RS232 serial ports with a Personal Computer. The controlling softwareused during this project was custom written since software to conveniently perform all of themeasurements made during this project was not available. An advantage of the new softwarewas an improved efficiency in the code, which allowed for higher sampling rates (∼17 Hz) duringconventional viscosity measurements than had previously been possible using the existing software.This lessened the experimental uncertainties in the magnetisation measurements. Additionally thesoftware allowed for long combinations of measurements to be performed unattended (often overa period of ∼36 hours) with DC cyclic erasure (DCE) demagnetisation performed automaticallybetween different types of tests. For example, a sample measurement could be set to consist of asequence of tests such as the following: DCE demagnetisation, followed by an IRM test, followedby DCE demagnetisation, followed by a Wa test (measurement of successively larger symmetrichysteresis loops), followed by a DCD/Conventional viscosity test and then finally measurement ofthe demagnetisation curve at a variety of field sweep rates.

The PC control of the VSM and power supply allowed for measurements to be performed withfeedback control of the applied field according to the sample moment. This it made to possiblemeasure to specified values of internal field, Hi, or sample magnetisation. The ability to measureto particular values of Hi was used in the DCD/IRM tests to ensure the remanences were measuredat true points of Hi = 0, rather than Ha = 0. Advantage was taken of the ability to measure tospecified values of M during the DCE demagnetisation procedure, which consisted of cycling thefield between values yielding successively smaller magnetisations, until the remanent magnetisationwas below a specified value.

Calibration of the VSM

Calibration of the VSM system required separate calibration of the field of the solenoid and of themoment measured by the VSM. The field was calibrated by programming the power supply withthe appropriate current vs field characteristics, as specified by the manufacturer of the solenoid(Oxford Instruments). The values of the field were verified as being correct by measurement witha longitudinal Hall probe, which had been independently calibrated by the probes’ manufacturer(FW Bell) and against a Walker reference magnetic moment. The moment measured by the VSMwas calibrated prior to every measurement, to ensure that any effects of drift in the electronicswere minimised. For this purpose a US National Bureau of Standards nickel sphere (standardreference material 772) was used. The calibration was performed at 298 K in an applied magneticflux density of 1 Tesla, yielding a sample moment of 3.552 × 10−3 Am2.

34

Experimental Methods

Uncertainty in VSM Measurements

The VSM hardware is capable of resolving magnetic moment to 1 part in 32 768. In practice,though, the resolution is almost never as good as this because of uncertainty introduced by theamplification and electronic drift. Folks has previously assessed the uncertainty in measurementsof magnetic moments in the VSM used during this work and has estimated it to be ±0.3% [132].When combined with the uncertainty in sample volume (see section 3.2.1) this implies that theuncertainty in measurements of magnetisation is ±0.9%.

The uncertainty in the values of applied field generated by the superconducting coil of the VSM arevery low and contribute negligible uncertainty to the quantities measured during this work. Theuncertainty in the values Hi, however, are more significant since they include the uncertainties inthe demagnetisation factor and sample magnetisation. Since the uncertainty in M is scaled by Dthen values of Hi typically have uncertainty of ±0.3%.

2.1.2 Hardware for Measuring Rotational Hysteresis

Rotational hysteresis measurements of magnetic samples may be made by three quite distinct meth-ods. The magnetisation vector may be measured using vector search coil techniques; the torqueacting on the sample can be measured using a torque magnetometer[118, 63]; or the rotationalhysteresis energy loss from a sample may be measured as it is rotated in a field[123].

Since the means of measuring rotational hysteresis were not available at the University of WesternAustralia at the commencement of this study, two new apparatus for the purpose were developed.The first of these devices, the Dual Air Bearing Angular Acceleration Magnetometer performsdirect dynamic measurements of rotational hysteresis loss. This device is described in chapter 5.The second device, the Rotating Sample and Coil Magnetometer (RSCM), uses search coils tomeasure a samples’ magnetisation vector during rotational hysteresis. This device is the subjectof chapter 6. The two devices were found to produce data in excellent agreement with each other,with measurements performed elsewhere and with that presented in the literature.

2.2 VSM Measurements

2.2.1 General Measurement Regimes

In general, when a quantity is being measured at various points on the hysteresis loop then themeasurement may either be performed at successive fields during a single cycle of the loop or themeasurement may be performed at successive values of field with just one measurement performedfor each complete cycle of the hysteresis loop. These two measurement regimes are referred to asMSL (Multiple Steps per Loop) and SSL (Single Step per Loop).

35


The advantage of the SSL regime over the MSL regime is that it removes the possibility of thehistory of previous measurements effecting the current measurement. This is because all mea-surements are made from the same initial state - saturation for measurements made along thedemagnetisation curve and the same demagnetised state for measurements made along the initialmagnetisation curve. The disadvantage of the approach is that it is far more time consuming sincejust a single measurement is performed for each complete cycle of the hysteresis loop or each sampledemagnetisation procedure. When super-conducting magnets, such as is in the VSM, are used thistime may represent a significant financial cost because of cryogen consumption. Where measure-ments are made from a thermally demagnetised state then the SSL regime is all but impossiblesince it would be necessary to remove the sample between measurements for demagnetisation in avacuum furnace.

During this work all measurements of magnetic viscosity, recoil curves and remanences made alongthe demagnetisation curve were performed using the SSL regime. This ensures that each individualmeasurement is independent of the rest. Measurements made along the initial magnetisation curvewere performed using the MSL regime because the SSL regime was impractical and risked changingthe microstructure of the materials by repeated thermal demagnetisation.

Use of the MSL regime for measurements made from the initial magnetisation curve is consistentwith the majority of such measurements reported in the literature (see for example [133]). It does,though, raise the possibility that IRM measurements made under such a regime may be unsuitablefor comparison with the DCD measurements in, for example, a ∆M plot. Measurements madefrom a demagnetised state, though, are less effected by the MSL regime than those made along thedemagnetisation curve since the majority of the material is in a much lower energy state on theinitial magnetisation curve than on the demagnetisation curve. In all of the materials measuredduring this work the magnetic viscosity and the area enclosed by recoil loops was found to be verysmall for measurements performed from the initial magnetisation curve, suggesting that very fewirreversible changes to magnetisation occur during the individual measurements. Thus in thesematerials it is safe to make comparisons between measurements made on the initial magnetisationcurve under the MSL regime with those made on the demagnetisation curve under the SSL regime.

2.2.2 The Measurement of the Time Dependence of Magnetisation

Conventional measurements of magnetic viscosity are made by maintaining the applied field ata known constant value and measuring the magnetisation of the sample as a function of time.Such measurements are usually performed from points on the demagnetisation curve of the majorhysteresis loop or from points on the initial magnetisation curve [132], although measurements fromother states have also been reported [134, 135]. In this work all conventional viscosity measurementswere made from the demagnetisation curve and the initial magnetisation curve.

Often other measurements, such as DCD/IRM (see section 2.2.3) or χrev measurements (see section2.3.1), are performed at the end of each measurement of magnetic viscosity, before moving on tothe next viscosity measurement. This is convenient and allows the waiting time involved in the

36


viscosity measurement to reduce the time dependence of magnetisation during the subsequentrecoil curve measurement. This is discussed further in section 2.2.3. During this work conventionalmeasurements of viscosity were combined with IRM and DCD measurements.

When using a VSM to make conventional (constant applied field) measurements of magnetic vis-cosity then the measurement is typically performed over a time period of 60 - 120s (see for example[82, 62]). Measurements over longer time intervals are of limited additional value as the mag-netisation change at longer times is so small that it may approach the noise level of the system.Conversely measurements over shorter periods may lack statistical significance. During this workall conventional measurements of magnetic viscosity were performed over a period of 120s.

2.2.3 The Measurement of Remanent Magnetisations

Remanent magnetisations may be measured from any initial magnetic state simply by reducing(recoiling) the internal field, Hi, to zero and then measuring the magnetisation. Such measurementsare of interest since if all of the magnetisation processes that occur along the recoil to zero arereversible then they may provide a measure of the irreversible component of magnetisation inthe initial state (this assumption is discussed further in section 2.3.1). When performed frompoints on the initial magnetisation curve such measurements are referred to as isothermal remanentmagnetisation (IRM) measurements. From points on the demagnetisation curve they are knownas DC demagnetisation (DCD) measurements. In the DCD case the remanences measured areanalogous to ID(H) and in the IRM case to IR(H) as defined by Wohlfarth [57] and employed inHenkel plots. Both IRM and DCD measurements are commonly used as a measure of irreversiblemagnetisation along the initial magnetisation and demagnetisation curves (see for example [129]).

During this work remanent magnetisations were measured from points on the initial magnetisationand demagnetisation curves following the measurement of 120 s of viscosity in a constant appliedfield. The recoil loops to zero field were measured with the field sweeping at the maximum rateallowed by the magnet, so as to minimise the effects of time dependence during the recoil. Thefield was stopped at zero internal field, using feedback from the VSM. Both the lower and uppercurves of the recoil loops were measured so as to allow their reversibility to be assessed. Typically180 recoil loops were measured along the demagnetisation curve with a field spacing of about 1 ×104 Am−1 in the NdFeB samples. The IRM measurements were performed with wider spacings(by a factor of 2) to avoid measurements being effected by previous recoil loops.

2.3 Analysis of the Data

2.3.1 The Measurement and Calculation of Mrev and Mirr

In the analysis of magnetic hysteresis or magnetic viscosity it is necessary to separate the total mag-netisation into that associated with irreversible changes, Mirr, and that associated with reversible

37


Figure 2.1:An example of the small recoil loop measurements used in the χrev method.

changes, Mrev. Two methods for resolving Mtot into Mrev and Mirr are commonly employed inthe literature: the χrev method [59] and the DCD/IRM method [133].

The χrev Method of Resolving Mrev and Mirr

The χrev method involves the measurement of small recoil loops along the initial magnetisationcurve or hysteresis loop, as shown in figure 2.1. Such recoil loops would typically be performedover a field range of about 8 kAm−1 for measurements of a high coercivity material such as NdFeBand over a lesser range for lower coercivity materials. For a recoil loop performed from a field ofH, Tebble and Corner define the reversible susceptibility, χrev(H), as the mean slope of the recoilloop. A more rigorous definition of the reversible susceptibility is that it is the mean slope of therecoil curve measured with a small reversing field increment ∆H in the limit ∆H → 0 [59]. Thisimplies that χrev should be obtained through the measurement of successively smaller recoil loops,from the same field, but in practice this is seldom done but rather χrev at a particular point onthe hysteresis loop is obtained from a single small recoil loop.

Once χrev(H) is known then the value of the reversible magnetisation is obtained by integration ofχrev(H) from an initial field, Hinitial, to the field of interest, Hmeas, as shown by equation 2.1.

Mrev(Hmeas) =∫ Hmeas

Hinitial

χrev(H)dH (2.1)

The assumption that Mrev (Hinitial) = 0 is implicit in equation 2.1. Hinitial, then is usually taken

38


to be the field at which Hi = 0 (either at the remanence or in a demagnetised state) as this iswhen Mrev = 0.

Once Mrev(H) is known then Mirr may be found simply by subtracting Mrev from Mtot, as shownby equation 2.2:

Mirrev(H) = Mtotal(H)−Mrev(H). (2.2)

Since the χrev method provides a local measure of reversible (and irreversible) magnetisation thenit is particularly useful when measurements are performed on materials where the recoil loops aresteep and transcribe an area, such as in exchange-spring materials [136]. In such materials thehysteresis and the large change in magnetisation along the recoil curves [108, 136] indicate thatirreversible processes occur during the recoil and so a local measure of irreversible magnetisationis necessary [111, 110]. The materials studied in this work have relatively flat recoil loops thatenclose small areas, meaning use of the χrev method to determine Mrev and Mirr is not necessary.

The χrev method is unsuitable for determining η, as discussed in section 2.3.3, and involves theassumption that the reversible magnetisation increases monotonically with field, since χrev is alwayspositive. Della Torre et al. have shown that this is not the case in even the Stoner Wohlfarth model[58]. Similarly Crew has reported that it is not the case in sintered NdFeB [61, 62]. Thus the χrevmethod has not been used during this work but instead the IRM/DCD method has been preferred.

The IRM/DCD Method of Resolving Mrev and Mirr

Since a remanent magnetisation (M at Hi = 0) is, by definition, irreversible then an IRM or DCDmeasurement represents a measurement of irreversible magnetisation. If an IRM or DCD recoilloop is measured from an initial field of H, then the irreversible magnetisation, Mirr(H), may bedefined as being the remanent magnetisation at the end of the recoil loop, where Hi = 0. This isillustrated in figure 2.2.

Such an interpretation involves the following assumptions:

1. That no irreversible changes to magnetisation occur along the recoil loops.

2. That the reversible changes of magnetisation along the recoil loops are not time dependent.

These assumptions amount to the same thing since the time dependence of reversible magnetisationonly arises because of its dependence on the irreversible magnetisation. Thus if no irreversiblemagnetisation changes occur along the recoil loops then both assumptions will be satisfied.

In practice the best way of attempting to satisfy these assumptions is to measure the recoil loopsto Hi = 0 as quickly as possible as is done here. This requires feedback control of the applied field,to ensure that Hi = 0 is reached but not overshot. Doing this at high field ramp rates requires ahigh sampling rate of the applied field and sample magnetisation.

39


Figure 2.2: An example of how the irreversible magnetisation is characterised by IRM and DCD mea-surements. The measurements shown here correspond to the same field strength

Making IRM/DCD measurements immediately after measurement of magnetic viscosity is also ofadvantage. Doing so reduces (but never entirely eliminates) the amount of magnetic viscosity thatthen occurs along the hysteresis loop. This is because recoil loops are measured along trajectoriesto fields lower than that which has caused the magnetisation or demagnetisation of the materialand so they predominantly probe lower energy states than those that have already been activated.By allowing most of the thermal activation to occur at the highest energy state (during the viscositymeasurement) then the subsequent thermal activation (and so time dependence) along the recoilloop is minimised. This is a consequence of time dependence depending both on field and themagnetic state of the material (and of course temperature).

The consequence of some irreversible processes inevitably occurring along recoil loops is that the as-sumptions of the DCD method are not entirely satisfied and so values of Mrev and Mirr determinedand ascribed to a particular field will be uncertain.

Emura et al. have suggested that the extent to which irreversible magnetisation changes occuralong recoil curves may be assessed by examination of the area of the recoil loops [108]. Figure 2.3shows the area of the DCD recoil loops in the sintered NdFeB materials studied in this work as afraction of the area of the major hysteresis loop. Similar plots, with the same or lesser magnitudeswere obtained for the other materials studied in this work.

The origin of the non-closed loops is unimportant in terms of assessing the validity of the IRM/DCDmethod. Their significance is that they indicate that irreversible changes in magnetisation areoccurring since hysteresis always indicates the presence of irreversible processes [108]. It is clear that

40


Figure 2.3: The area of the DCD recoil loops as a fraction of the area of the major hysteresis loop in thesintered NdFeB iron magnets.

the loops enclose very small fractions of the major loop, suggesting that the change in irreversiblemagnetisation along the recoil loops is small and so use of the DCD method is justified.

Uncertainty in Experimentally Determined Values of Mrev and Mirr

The magnitude of the uncertainty in values of Mirr and Mrev determined by the IRM/DCD methodmay be assessed by considering the magnitude of irreversible changes in magnetisation that occuralong the recoil loops. An estimate of this is the maximum difference in magnetisation at cor-responding fields between the upper and lower branches of the recoil loop. This is analogous tothe difference in the positive and negative remanences being a measure of the irreversible changesin magnetisation occurring around the major hysteresis loop. Thus if the maximum difference inmagnetisation between the upper and lower branches of a recoil loop measured from a field H isδMrecoil(H) then

Mirr(H) = MDCD ± δMrecoil(H), (2.3)

where the intrinsic uncertainty in the measured value of MDCD (±0.9%) has not been includedand so must be added to obtain the total uncertainty. Similarly

Mrev(H) = Mtot(H)−MDCD ± δMrecoil, (2.4)

41


Figure 2.4: The maximum separation of the lower and upper branches of DCD recoil loops in NdFeBmaterials.

where again the intrinsic uncertainties in the measured quantities must be added to give the totaluncertainty. During this work a C++ computer program was written to analyse recoil loops anddetermine (among other parameters) the value of δMrecoil for each loop. Figure 2.4 shows a plotof such data for DCD measurements in sintered and melt quenched NdFeB. The uncertainty in thevalues of Mirr and Mrev are, then, these values plus the 0.9% of the intrinsic uncertainty in themeasured value of MDCD.

Typically values of Mirr were determined as being ± 2%, with the proportionately greatest uncer-tainties being those measured around the remanent coercivity.

2.3.2 Determination of the Magnetic Viscosity Parameters from Con-

ventional Viscosity Measurements

The data obtained from conventional viscosity measurements are discrete samples of M(t) forparticular values of Ha. Correction for demagnetisation effects by

Hi = Ha −DM (2.5)

allows the data to be expressed as [M(t),Hi(t)], where the values of Hi(t) are time dependentbecause of the time dependence of the magnetisation. The magnetisation may then be resolvedinto it’s irreversible component, and so the data expressed as [Mirr(t),Hi(t)], by subtraction of

42


values of Mrev(Hi(t)), as determined by DCD/IRM measurements. This data may be used todetermine the parameters associated with magnetic viscosity discussed in 1.3.1.

During this work subtraction of Mrev(Hi(t)) from the data of total magnetisation was performedby fitting the discrete data of Mrev(Hi) obtained from IRM/DCD measurements with a movingquadratic fit in Mathematica. This allowed for accurate interpolation of the data and for easyaccess to the uncertainties in the interpolated values.

The discrete data of Mirr(t) and Hi(t) were then fitted, in Mathematica, to

f(t) = A+2∑

n=0

Sn(∆ ln(t))n+1 (2.6)

which is the lower order terms of equation 1.25, derived by El-Hilo [40]. In the case of the fit tothe Mirr(t) the values of the two higher order coefficients of these fits, S1 and S2 are related to thewidth of the energy barrier distribution, as discussed in section 1.3.1. The proportionately greaterthese values are the narrower the distribution is.

The families of curves ofMirr(t) andHi(t) may be used to determine the fluctuation fieldHf (H,Mirr)in a variety of ways. It may be determined by equation 1.12, where S and χiirr are found experi-mentally as [35]

S =∆Mirr

∆ ln(t)

∣∣∣∣ H (2.7)

and

χiirr =∆Mirr

∆H

∣∣∣∣ t (2.8)

In this case the values of S and χiirr must be carefully matched as being those for the same valueof irreversible magnetisation [137].

Hf may also be determined by equation 1.19 under the assumption that ∂Si/∂Hi|Mirr→ 0 and

so Hf = Λ. Experimentally this is achieved by

Λ =H2 −H1

ln(Mirr2/Mirr1)

∣∣∣∣ Mirr

(2.9)

where H1 and H2 are the values of field sampled from the Hi(t) curves at the same value of Mirr

and Mirr1 Mirr2 are the values of the time derivative of the same Mirr(t) curves sampled at thesame values of Mirr (and so at the times corresponding to H1 and H2). In order to approximate theexact differential of equation 1.19 it is necessary that the differences in field be small and this, in

43


turn implies that the values of Mirr will be very similar. However since these values are derivativesof experimentally sampled data then their uncertainty is large, frequently to the extent that theuncertainty in the values is greater than the difference between values itself, meaning Λ can not bedetermined.

In this work Hf was determined using the waiting time method [126, 35]. This involves determi-nation of Hf by equation 1.24. Experimentally this is achieved by

Hf =H1 −H2

ln(t2/t1)

∣∣∣∣ Mirr

(2.10)

where t1 and t2 are the times on neighboring curves of Mirr(t) corresponding to the same particularvalue of Mirr and H1 and H2 are the values of field sampled from the corresponding Hi(t) curvesat those times. Since the uncertainty in the values of t1 and t2 are much less than those in Mirr

then this method results in far less uncertainty in the values of Hf than the Λ method. O’Gradyet al. have illustrated this point graphically in [126].

The computation of equation 2.10 may be done using either data corrected for demagnetisationeffects, as was done in this work, or data against applied field. In this latter case the values of Hf

may be corrected following the method of Lyberatos et al. [35] as

Hif = Ha

f

(1−D ∂Mrev

∂Ha

∣∣∣∣ Mirr

)(2.11)

El-Hilo has also noted [39] that when field sweep rate effects are present then Hf is actually givenby

Hf =∆H

∆ ln(S/t)

∣∣∣∣ Mirr

. (2.12)

In all cases during this work computation via this expression yielded values of negligible differenceto the waiting time method. This is to be expected since ∆ ln(S/t) = ∆ ln(S) − ∆ ln(t) and∆ ln(S) << ∆ ln(t).

Uncertainty in Experimentally Determined Values of Magnetic Viscosity Parameters

The uncertainty in the values of the parameters of the fits to the experimental data of Mirr(t) andHi(t) were obtained directly from Mathematica by regression analysis. The initial uncertaintiesin the values of Mirr, obtained from IRM/DCD measurements were not taken to include that ofthe separation of the recoil loops (see section 2.3.1), since this represents a systematic error, asdemonstrated by figure 2.4 and so will effect neighbouring recoil loops similarly. The uncertaintyin these values was, then, assumed to be ±2%. The resulting uncertainty in the fit parameters are

44


small since viscosity measurements typically involved the sampling to 2000 points.

The uncertainty in the values of Hf arise from the uncertainty in the times corresponding toparticular values of Mirr and the values of Hi corresponding to those times. The initial uncertaintyin Mirr imposes an uncertainty on the corresponding values of t on each viscosity curve so care wastaken to choose values of Mirr corresponding to steep sections of the Mirr(t) curves (i.e. valuesnear the start of the overlapping curves - typically at t = 1s). These values of Mirr are also thoseat which the greatest number of recoil curves overlap in Mirr. Uncertainties in the values werepropagated through the analysis in Mathematica (principally equation 2.10) and resulted in valuesof typically ±15%. The uncertainty in Hf is least near the remanent coercivity, where the viscosityis greater, both because of the greater number of recoil curves sharing overlap in Mirr and thelesser uncertainty in t. The uncertainties are greater at extreme values of Mirr where there is littleviscosity.

2.3.3 The Measurement and Calculation of η

A method for the measurement of η, as defined by equation 1.31, was described by Cammarano etal. [59]. This method employs the data from a DCD/IRM test to determine η as a function of Hand Mirr . Since the value of Mirr is assumed to be constant along a given recoil loop then thechanges of magnetisation with field along the loop are due to reversible processes and correspondto χrev as a function of H at constant Mirr, as defined by equation 1.30. This may be integratedto give Mrev as a function of H at a constant value of Mirr. The determined form of Mrev(H)on different recoil loops correspond to different constant values of Mirr so for particular constantvalues of field, H, Mrev as a function of Mirr at constant H may be determined. Differentiation ofthis function yields η, as defined by equation 1.31.

Crew et al. refined this method by pointing out that it is not necessary to determine Mrev(H) atconstant Mirr by integration of χrev(H) on each recoil curve, but rather Mrev(H) may be founddirectly as [103]

Mrev(H) = Mtotal(H)−Mirr (2.13)

where Mirr is the constant value of the irreversible magnetisation for the particular recoil loop, asmeasured using the DCD/IRM method. This is illustrated in figure 2.5.

In figure 2.5 the constant value of Mirr for recoil loop 1 is MAirr and for loop 2 MB

irr. The value ofthe reversible magnetisation on each recoil curve may also be determined at any value of field bythe subtraction of equation 2.13, as illustrated in the figure. Thus by measuring many recoil loopsdata of Mrev vs Mirr at constant values of H may be built up, with each value of Mirr comingfrom a different recoil loop. Differentiation of these data gives η(Mirr) for constant values of H.

Crew et al. also proposed the convention of plotting η as a function of field by plotting the valueof η, at a particular H, corresponding to the largest possible value of Mirr. Since at a particularfield the largest possible value of Mirr is that corresponding to a point on the major hysteresis loopthen this was referred to as plotting η ‘along the hysteresis loop’.

45


Figure 2.5: The definitions of Mrev and Mirr used to determine η from IRM and (as is the case here)DCD measurements.

The implicit assumptions in this method of determining η are:

1. That no irreversible processes occur on the recoil loops.

2. That the reversible changes of magnetisation along the recoil loops are not time dependentbut depend only on the field.

These assumptions are the same as those of the DCD/IRM method and their implications have beendiscussed in section 2.3.1. In most materials the assumptions will never be exactly satisfied and souncertainty is introduced into the values of η obatined. Nevertheless the method of determining ηis justified whenever use of the DCD/IRM method is suitable. The χrev method is unsuitable forthe determination of η as one of the assumptions implicit in the use of equation 2.1 is that Mrev

is a function of field only.

During this study a different method to that suggested by Crew was used to differentiate the dataof Mrev vs Mirr at particular values of H [61]. Crew suggested that at each particular field, thatdifferentiation should be performed by means of a high order polynomial fit over the entire domainof the data. Instead a low order (linear or quadratic) fit over a small fraction of the domain (thatmost local to the hysteresis loop) was used. The reason for this may be seen by considering datafor Mrev vs Mirr measured by recoil loops from the initial magnetisation curve for a material suchas sintered Nd2Fe14B. Figure 2.6 shows that part of the curve of Mrev vs Mirr for Hi = 1.27 ×105Am−1 which is closest to the hysteresis loop.

It is clear that Mrev vs Mirr slopes upwards over a very small range of Mirr near the hysteresis

46


Figure 2.6: The part of the Mrev vs Mirr data that is closest to the initial magnetisation curve (whereHi = 1.27 × 105Am−1). The material is SN2 and measurements were performed from an initially DCEdemagnetised state.

loop (the leftmost portion of the curve), this is particularly observed at lower fields. This meansthat at this field η is first positive and then negative. Hence if η is plotted along the hysteresisloop then it should be positive. If the differentiation is performed by means of a polynomial fit ofthe entire Mrev vs Mirr curve, however, then η appears negative as the contribution of points fromrecoil loops furtherest from the point on the hysteresis loop dominate the fit. If the differentiationof the Mrev vs Mirr data is performed by fitting a low order polynomial over just those points thatare most local to the hysteresis loop, however, then the correct sign of η along the hysteresis loopmay be found.

The phenomena of η changing sign over a small range of Mirr close to the hysteresis loop wasobserved in all of the materials studied in this work and was seen to occur on both the initial mag-netisation curve and demagnetisation curve. Figure 2.7 shows the phenomena for data measuredby recoil loops from the demagnetisation curve in melt quenched NdFeB.

It appears, then, necessary to determine η along the hysteresis loop by differentiation of Mrev vsMirr data that is most local to the hysteresis loop. During this work the differentiation of the Mrev

vs Mirr data was performed by differentiating either a linear or quadratic fit to the ten pointsmost adjacent to the hysteresis loop. This means that the value of η on the hysteresis loop for apoint corresponding to H = Hn is determined from the first ten recoil loops measured from fieldsof magnitude greater than or equal to Hn.

The approach is further justified by consideration of the irreversible state of the material on the

47


Figure 2.7: The part of the Mrev vs Mirr data that is closest to the demagnetisation curve. The materialis MQ1.

recoil curves from which η is ultimately determined. The irreversible state of the material, ascharacterised by Mirr, is different for each recoil loop. This means that when determining η onthe hysteresis loop, at a field of H = Hn, it is only appropriate to consider the points of Mrev vsMirr that come from recoil loops with similar values of Mirr to that on the hysteresis loop at H= Hn. Recoil loops corresponding to values of Mirr that are significantly different to Mirr (Hn)on the hysteresis loop represent completely different irreversible states and so are of no relevanceto determining η on the hysteresis loop at H = Hn. Thus it is necessary to determine η byconsideration of only those values of Mrev vs Mirr that are most local to the hysteresis loop.

Uncertainty in Experimentally Determined Values of η

Crew estimated the error in the values of η determined using this method, but fitting high orderpolynomials over the entire Mrev vs Mirr data, to be of between 10 and 40%, with larger valuesof η being proportionately more accurate [61]. The method employed here of fitting data fromjust 10 recoil loops corresponding to values of Mirr most similar to that on the hysteresis loopincreases the uncertainty in the fits because the number of points being fitted to is less. Converselythe reduction in the order of the fit (and so the number of parameters - 2 or 3 as opposed to 10)increases the certainty in the fit and the derivative near the hysteresis loop.

Uncertainties in the values of η were computed using regression analysis in Mathematica. As withthe time dependence measurements the initial uncertainties in the values of Mrev and Mirr wereestimated as ±2%. Most of the actual uncertainty in the values of Mrev and Mirr arise from

48


systematic error caused by the assumption that no irreversible processes occur along the recoilloops to zero field. This error will be similar for adjacent recoil loops and so will not significantlyeffect the computations of the value of η since it is a measure of the derivative of Mrev with respectto Mirr and so is less dependent on the absolute values. Typically the uncertainty in the values ofη were found to be between ±15% and ±40%.

2.3.4 The Measurement and Calculation of Rotational Hysteresis Prop-

erties

Measurements of rotational hysteresis loss are best performed on samples that are rotationallysymmetric about the axis about which the measurement is performed. Where the sample is notrotationally symmetric then the bulk shape anisotropy of the sample will effect the measurements,possibly obscuring the effects of magnetocrystalline anisotropy. All of the materials studied in thiswork were magnetically isotropic and were prepared as rotationally symmetric samples, so thatshape anisotropy did not effect the measurements.

Measurement of Rotational Hysteresis Loss

During this study all measurements of rotational hysteresis loss were made either directly usingthe DABAAM (see chapter 5) or indirectly using the RSCM (see chapter 6). These devices weredesigned and constructed specifically for this work.

The determination of Wr(H), allows the effective anisotropy field to be found, according to themethod described in section 1.3.4, and the dimensionless rotational hysteresis integral, given byequation 1.34, to be calculated.

Calculation of the Rotational Hysteresis Integral

Determination of the rotational hysteresis integral involves the integration of Wr(H) against 1/Haccording to equation 1.34. In this work the integration of the discrete experimentally determineddata was performed by Riemann summation. Thus when applied to discrete experimental data

RH =1

2µ0Ms

m−1∑n=1

(Wnr +Wn+1

r )(1/Hn − 1/Hn+1) (2.14)

where Wnr and Hn are the values of rotational hysteresis loss and field respectively for the nth

measurement of data consisting of m such measurements, sorted by values of H.

The rotational hysteresis measurements of high coercivity materials, such as NdFeB, are frequentlylimited by the maximum field that may be applied using the experimental apparatus. This isbecause apparatus to measure rotational hysteresis is often limited by its geometry to use between

49


the poles of an electromagnet, thereby setting an upper limit for the applied field of about 2.1 × 106

Am−1. In such cases it is often not possible to measure the whole of the Wr(Ha) curve of a materialand so the value of RH determined by equation 2.14 will be too low. In such cases the value ofthe rotational hysteresis integral must be increased by additional integration of an extropolationof the incomplete data to Wr → 0 and 1/H → 0. Figure 2.8 shows such an extrapolation of themeasured data of Wr plotted against 1/Ha for sintered NdFeB. The data is incomplete because ofthe maximum available applied field of Ha =2.3 × 106 Am−1. Implicit in the extrapolation of thedata are the assumptions:

1. That the maximum field at which measurements are performed, Hm, is greater than the fieldat which Wr(H) peaks, Hp.

2. That Wr is monotonically decreasing for Ha > Hp.

The second assumption is likely to be true for all single phase materials or materials where phasesare exchanged coupled and so behave as a single phase[135]. This is apparent in the form of Wr(H)predicted by all of the models described and by the overwhelming evidence in the literature (seefor example [114, 118]). The second assumption is equivalent to saying that Wr(H) has only onepeak. This is also true of practically all data described in the literature.

Uncertainty in the Value of the Rotational Hysteresis Integral

Both the Riemann summation of the discrete experimental data and, in the case of incompletedata, the extrapolation of the Wr vs 1/H data introduce uncertainty into the determined valuesof RH . Such uncertainty is in addition to that already present due to the intrinsic uncertainties ineach of the measurements of Wr and H.

The uncertainty introduced by the Riemann summation is

12µ0Ms

m−1∑n=1

(Wnr −Wn+1

r )(1/Hn − 1/Hn+1) (2.15)

which is zero in the limit that ∆H → 0. Equation 2.15 differs from equation 2.14 by the subtrac-tion of the Wr values and represents the summation of just the triangular areas formed by thetrapezoidal area involved in each component of the original summation.

The uncertainty introduced by the extrapolation of incomplete data may be determined by ex-amination of figure 2.8, which shows the measured form of Wr plotted against 1/Ha for sinteredNdFeB. The two extreme cases of the linear extrapolations of the Wr vs 1/Ha curve, subject to theabove assumption are also shown in figure 2.8. The linear extrapolation directly to zero will almostcertainly provide an overestimate of the value of RH since Wr(Ha) vs 1/Ha is concave upwardsat high fields (and small values of 1/Ha) in practically all materials. Similarly the extrapolationrepresenting truncation of the data at the maximum field of measurement will provide an under-

50


Figure 2.8: Incomplete rotational hysteresis data for sintered NdFeB. The extrapolation to obtain a valuefor the rotational hysteresis integral is shown, together with the bounds on the uncertainty.

estimate of RH . The uncertainty from the extrapolation procedure, then, is approximately ± halfthe area of the triangle formed by the extreme extrapolations. That is

Wmr

4µ0MsHm(2.16)

where Wmr is the value of Wr measured at the maximum field of measurement, Hm.

Both such estimates of the uncertainty include the extreme cases. In general the value of RHdetermined by equation 2.14 (including the extrapolation of incomplete data) will be much closerto the true value than the uncertainty estimate suggests.

The value of RH calculated from the data in figure 2.8 (for Wr in SN2) is RH = 1.0 ± 0.2. Theuncertainty is made up of 0.01 due to the Riemann summation, 0.14 due to the extrapolation ofthe incomplete data and 0.05 due to the intrinsic uncertainties in the measured values of Wr andH. The data shows that at the maximum applied field the rotational hysteresis is still over 50%of its’ peak value and looks likely to vanish only in a field of greater than twice the maximumapplied value. Nevertheless, the relatively small value of the uncertainty (20%) shows that evenfrom measurements limited by the available applied field strength RH may be determined to areasonable degree of accuracy.

51


The Time Dependence of Rotational Hysteresis

Measurements of rotational hysteresis using torque or search coil measurement of the magnetisationvector are usually performed with the sample being rotated between individual measurements (incontrast to the RSCM). Direct measurements of the rotational hysteresis loss[138, 139], on theother hand, require that the sample is rotating during the measurement. The time dependence ofthe magnetisation means that it is likely that the results of the two kind of measurements maydiffer. In the case of the static measurements viscosity may serve to reduce the angle between M

and H and so reduce the rotational hysteresis torque. Dynamic measurements, then, will measurethe upper bound on the rotational hysteresis loss, although care must be taken to ensure this doesnot include eddy current losses. This time dependence of rotational hysteresis has been largelyneglected in the literature. The reported agreement in the measurement of static and dynamicmethods[140, 141] suggests that in the vast majority of cases the effect is negligible over the timescales involved and so may be ignored. It is likely that as new and more sensitive measurementtechniques become available then measurement of this time dependence will provide further insightinto the nature of thermal activation of magnetic materials.

Rotational Hysteresis Measurement Regimes

Measurement regimes similar to the SSL and MSL regimes of alternating hysteresis can also beapplied to rotational hysteresis. The individual measurements of either the rotational hysteresisloss or of the magnetisation vector can be performed either successively, with the field strengthincremented after each measurement (analogous to MSL), or each measurement can be performedfrom an initially demagnetised state (analogous to an SSL IRM measurement). During this studyall measurements of rotational hysteresis were made with the field strength being progressively in-cremented between measurement points (MSL). All of the samples were rotationally demagnetised,immediately prior to the commencements of the measurements. Where additional measurementswere required to fill in gaps in the data then the sample was re-demagnetised before the additionalmeasurements were performed. In all cases these extra points were found to be in excellent agree-ment with the previously measured data, suggesting that there is negligible difference in the resultsof the SSL and MSL regimes for rotational hysteresis measurements.

Although it was not studied in detail it was noticed during the course of experimentation that thevalue of the rotational hysteresis loss, Wr(H), seems to be remarkably independent of the previousmagnetic history of the material. In the materials studied during this project it appeared thatthe measured values of Wr at a particular value of H were the same regardless of whether H wasincreased or decreased prior to the measurement. This is in stark contrast to alternating hysteresismeasurements, but would explain the apparent equivalence of the SSL and MSL regimes in therotational hysteresis case.

52

3. The Samples Studied

THIS chapter provides details of the samples on which measurements were performed duringthis work. The materials from which the samples were made are described and the methods

by which they were manufactured are outlined. Particular emphasis is given to consideration ofthe micro-structure of the materials, which plays a large part in determining which mechanismsof magnetisation change are possible in each material. When preparing a sample for magneticmeasurements care must be taken to ensure that the sample is such that its’ apparent magneticproperties are representative of the bulk of the sample. This requires that the geometric demag-netisation properties of a sample are well known and that the surface of the sample is smooth, sothat local deviations in magnetic flux due to roughness are avoided. The techniques employed toprepare the samples so that these criteria are met are discussed. Finally the various methods bywhich the samples were demagnetised are described.

3.1 The Materials

All but one of the materials studied during this work are commercially available and were obtaineddirectly from the manufacturer. The one sample that was not commercially available was manu-factured at the University of Western Australia for research purposes. The precise details of themanufacturing process for all of these materials have not been divulged by their manufacturersdue to their commercial value. The general processes by which they were produced, though, areas described here.

3.1.1 Rare Earth-Iron Permanent Magnets

Four materials of rare earth-iron composition (in this case NdFeB) were studied. These materialswere produced by two distinct methods: sintering and melt-quenching.

53


Sintered Materials

Two of the NdFeB materials studied were produced by the sintering process. This method ofproducing rare earth-iron permanent magnets was developed by Sumitomo Special Metals duringthe mid 1980s and is responsible for the highest energy product permanent magnets. The processinvolves the following steps:

1. Production of ingots of the desired composition, usually by induction melting of the rawconstituents.

2. Pulverization of the ingots followed by mechanical milling to produce a fine powder of thematerial.

3. Pressing of the powder into ‘green’ compacts. This may be performed in a magnetic field toproduce anisotropic materials.

4. Sintering (or heating) in an inert atmosphere or vacuum. Typically samples may be heatedto temperatures of ∼ 1350K for several hours during the sintering.

5. Post sintering heat treatments, involving heating at temperatures lower that the sinteringtemperature followed by rapid cooling are also common. The purpose of these treatments isto optimize the grain structure of the material (size and shape).

Sintered rare earth-iron permanent magnets usually consist of large spherical grains (typically 5- 40 µm in diameter) of the primary magnetic phase, often surrounded by thin rare earth richphases, a few nm thick[25]. These boundary phases have been found to increase the coercivityof the materials and so are desirable. Thus ingots produced at step 1 of the process are usuallymade rare earth rich in order to encourage the formation of this non-magnetic layer. In mostcases the primary magnetic phase is either Nd2Fe14B or Pr2Fe14B, which form hexagonal crystalstructures with uniaxial anisotropy along their major axis (the magnetocrystalline anisotropy inthe hexagonal plane being negligible). The Nd materials are of the greater commercial importancethan the Pr materials, despite having lower coercivities. This is principally due to the high expenseof Pr, although the Nd materials also have slightly higher Curie temperatures.

The domain wall width in NdFeB is around 20 nm[142, 143]. Allowing for the presence of inter-actions, Gronnefeld et al. have calculated that in a material of NdFeB then the critical diameterbelow which grains are single domain is 190 ± 20 nm[144]. Thus all of the grains in sintered rareearth-iron materials contain multiple domains and domain walls. The mechanisms of magnetisa-tion change in these materials, then, are determined by the behaviour of the domain walls withinthe grains.

One of the sintered rare earth-iron materials studied (SN1) was supplied by Sumitomo SpecialMetals Company Ltd., Japan. The other, designated SN2, was obtained from the Electron EnergyCorporation, USA. The hysteresis loops of these materials are shown in figure 3.1. Both SN1 andSN2 have a nominal composition of Nd15Fe77B8 and a primary magnetic phase of Nd2Fe14B. Theyhave a very low degree of alignment (if any) and may be considered to be isotropic.

54

The Samples Studied

Figure 3.1:The hysteresis loops of the sintered NdFeB materials.

Melt-Quenched Materials

Production of materials by melt-quenching involves the following steps:

1. Melting (usually induction melting) of the constituents in either a vacuum or an inert atmo-sphere.

2. Spraying of the liquid material onto the edge of a rapidly spinning wheel, of high thermalconductivity. The purpose of the wheel is to rapidly remove the heat from the liquid andso quench the material. Using this method very high quench rates are possible, typically ofthe order of 106 − 107Ks−1. If the liquid material is sprayed continuously onto the wheelthen ribbons of material are formed. The micro-structure of these ribbons may be alteredby varying the wheel speed and so the quench rate of the material. Using high wheel speedsamorphous ribbons may be produced or the size of crystallites in the ribbon controlled.

3. The ribbons are ground into a powder and then either

• mixed with a resin and pressed or

• hot pressed, in a temperature of typically 1000K.

The heat treatment implicit in the hot pressing approach produces grain growth in thematerial. In this way materials with particular grain sizes may be grown from amorphousribbons or ribbons composed of small crystallites.

4. If materials are hot pressed then they may be rendered anisotropic by forcing them to deformperpendicular to the pressing direction. This is known as die upsetting.

55


Figure 3.2:The hysteresis loops of the melt quenched NdFeB materials.

Steps 1 and 2 are often referred to as ‘melt spinning’ and so the materials produced are oftendescribed as ‘melt spun’, rather than melt quenched.

The hysteresis loops of the melt-quenched materials are shown in figure 3.2

The materials MQ1 and MQ2 were produced by Delco Remy (General Motors), USA and aremarketed under the name Magnequench. MQ1 was produced by melt spinning followed by resinbonding, whereas MQ2 was made by melt spinning followed by hot pressing. The precursor toeach materials had nominal composition of Nd14Fe81B5 resulting in primary magnetic phase ofNd2Fe14B with small amounts of a Nd rich phase. Both materials are isotropic.

The grain size of the melt quenched materials in generally less than that required to supportmultiple domains (estimated at 190 ± 20 nm[144]). This size is not absolute, but rather reflectswhat applies to a typical grain in such a material, where the state of the grain is that whichis most energetically favorable. Nevertheless this size implies that melt quenched materials arepredominately composed of single domain grains. Thus the mechanisms of magnetisation changein melt quenched materials are less influenced by domain wall behaviour and more controlled byrotation mechanisms (see section 1.2). In the case of MQ1 the typical grain size of 30 nm is closeto the domain wall width for Nd2Fe14B of 20 nm and so it is likely that practically all of the grainsare single domain. Indeed Folks et al. have, on the basis of experiment, estimated that 94% ofthe grains in MQ1 are single domain[90]. The larger grain sizes produced by heat treatment of themelt spun material, as in the case of MQ2 (grains of ∼ 100 nm), means that a larger proportion ofgrains may contain domain walls. Folks et al. have estimated that 56% of the grains in MQ2 aresingle domain[90].

56

The Samples Studied

Figure 3.3:The hysteresis loop of AlNiCo 7.

3.1.2 AlNiCo

The AlNiCo sample studied was obtained from Oriel Pty. Ltd., NSW Australia and is composedof isotropic AlNiCo 7. Despite its name the principal element in this material is iron, accountingfor approximately 45 % of the mass. The remaining mass is composed of aluminium (∼ 8 %),nickel (∼ 15 %), cobalt (∼ 27 %) and lesser amounts of other metals such as copper and titanium.AlNiCo 7 is not commonly commercially available as it has an inferior remanence to most othertypes of AlNiCo. Like many of the other types of AlNiCo it is produced by casting followed bya very high temperature (∼ 2000K) heat treatment. The resulting material is made up of largegrains able to support multiple domain walls.

The hysteresis loop of the AlNiCo sample studied is shown in figure 3.3.

3.1.3 MnAlC

The MnAlC sample studied was produced by D. Crew, of The University of Western Australia, bymechanical alloying. A powder consisting of 72% manganese, 27% aluminium and 1% carbon wasmilled for 16 hours in an argon atmosphere, using a hardened steel mill with ten 12 mm steel balls.The resulting powder was then pressed into cylindrical form and heated to 1323 K for ten minutesbefore being quenched into water. The material was given a final heat treatment at 873 K for tenminutes. The resulting material has grain sizes of ∼ 300− 500nm [145] and is isotropic. Since thedomain wall width in MnAlC is ∼ 15nm [146], then this means that practically all of the grains in

57


Figure 3.4:The hysteresis loop of the MnAlC material.

the material are able to support multiple domains.

The hysteresis loop of the MnAlC sample studied is shown in figure 3.4.

3.1.4 Sample Geometric Demagnetisation Effects

Sample geometry is of importance in magnetisation measurements both because of shape anisotropyand because of demagnetisation effects. Along a particular axis in a sample the internal field, Hi,experienced by a sample may be calculated as

Hi = Ha −D ·M (3.1)

where D is the demagnetisation factor for the particular direction of measurement in the sampleand Ha is the applied field. In general D is only well known for ellipsoids of revolution [147], infinitesheets and needles.

Since consideration of demagnetisation effects are of particular importance in high moment mate-rials, such as permanent magnets, then it is important that bulk samples are prepared with suchgeometries. In particular, spherical samples have the advantage of having a precisely known demag-netisation factor (D = 1

3 ) and no bulk shape anisotropy. The other commonly studied geometryis that of the needle. In this case measurements are made along the axis of the needle, which hasnegligible demagnetisation and very high shape anisotropy, with the easy direction being along theaxis.

58

The Samples Studied

Surface roughness may also contribute significantly to the apparent sample properties by alteringthe uniformity of the magnetic flux. Where the bulk properties are of interest, then, it is importantthat samples are prepared with a high polish. In the case of an ideal spherical sample (i.e. a sampleof uniform density and composition), a smooth surface ensures absolute uniformity of the flux andso ensures that the apparent bulk magnetic properties of the sphere are representative of theintrinsic properties of the material, after correction for demagnetisation effects by equation 3.1.

3.2 Sample Preparation

All of the samples, other than MnAlC, were prepared as highly polished spheres. This ensures thatthe bulk samples have no shape anisotropy, their demagnetisation factor is D = 1

3 and that theeffects of the sample surface on the magnetic properties are minimised. For the VSM measurementsthe spheres had a radius of approximately 5 mm. For some of the samples it was necessary to reducetheir diameter so that rotational hysteresis measurements could be successfully performed, as thetorque acting on the samples at their initial diameter was too great for measurement. In thesecases the samples were ground down to the smaller size, repolished, and then remounted. Thisensured that both the VSM and rotational hysteresis measurements were performed on exactly thesame material.

The MnAlC sample was measured in the form in which it was pressed, namely as a cylinder ofheight 1.82 mm and radius 2.53 mm. This sample was too small to be ground into a sphere by themethod employed for the other samples. The surface of the cylinder had a smooth appearance andwas not further polished. It is likely, though, that the effects of the surface on the measurementsfor this sample are greater than for the other samples.

The VSM measurements of this sample were performed along the axis of the cylinder and theDABAAM measurements were performed at right angles to the axis. In the case of the VSMmeasurements the demagnetisation factor along the cylinder axis was estimated according to thecalculations of Chen et al. as 0.56[147]. A rough estimate of the demagnetisation factor in theDABAAM case may be made by approximating the cylinder as equivalent to an ellipsoid of rev-olution and then making use of the fact that the sum of the demagnetisation factors along anythree orthogonal axes must equal 1. This gives a crude estimate for the demagnetisation factorin the DABAAM measurements as D = (1 − 0.56)/2 = 0.22. It is likely that the correction ofmeasurements for demagnetisation effects in this sample is inferior to those of the other samples,for which the demagnetisation factor was well known.

For the spherical samples the uncertainty in the demagnetisation factor is related to the uncertaintyin the measurements of the sphericity of the samples, which involves multiple measurements acrossdifferent diameters. For the sphericity of the samples prepared Folks has estimated that theresultant uncertainty in the value of D is ∼ ±0.5% [148]. In the MnAlC sample the uncertainty inthe value of D along the axis of the cylinder is ±5% and across the axis of the cylinder ±10%.

59


Figure 3.5:A cross section of the type of sample buckets used throughout the work.

3.2.1 Preparation of Sample Spheres

The sample spheres were prepared with diameters of between 1.8 and 6.0 mm, according to themethod described by Folks et al. [148]. This method involves the grinding of rough spheres byattrition milling and then polishing them in a specially designed machine, using successively finergrades of diamond paste. The resulting spheres have a sphericity of ±5µm and have a mirror polish,with scratches of depth and width of less than 250 nm, as verified by microscopy measurements.The diameter of each sphere was determined by averaging micrometer measurements across 10different points on the sphere. This allowed the determination of sample volume to within ±0.6%.

3.2.2 Mounting the Sample Spheres

The sample spheres were mounted in polycarbonate buckets machined to be of outside diameter6.1 mm and length 10.5 mm. The depth and diameter of the bore of the buckets was customisedfor each sample, to ensure that when a sample sat in the bucket its’ centre was coincident withthe centre of the bucket - both along the bucket length and radially, as shown in figure 3.5. Thiswas of particular importance for RSCM measurements as it ensured that the sample spheres wererotated about their centre during the measurement.

The samples were held in place in the buckets using either wax or resin. For rotational hysteresismeasurements, where the torques acting on the samples could be quite large, then the resin wasrequired and in some cases (see chapter 6) proved to be inadequate. For VSM measurements,where the forces on the samples are small, wax was sufficient and more convenient as it allowed thesample to be easily removed for thermal demagnetisation and/or repolishing. Prior to mountingthe samples the buckets were cleaned in a hot detergent solution in an ultrasonic bath to ensurethat any impurities (especially metallic specks from the machining) were removed. VSM mea-

60

The Samples Studied

surements verified that the buckets, wax and resin made negligible contributions to the magneticmeasurements, being diamagnetic.

3.3 Demagnetisation of the Samples

During this work measurements were performed from four different zero net magnetisation states:the virgin state, the thermally demagnetised state, the DC cyclic erasure (DCE) state and therotationally demagnetised state. A material exists in a virgin state when it has never previouslybeen magnetised. This is the state of all isotropic and many anisotropic materials immediatelyafter manufacture. The thermally demagnetised state is obtained by heating a material to aboveits Curie temperature, making it temporarily paramagnetic, and then cooling it in zero field.Demagnetisation by DC cyclic erasure (DCE)[93] involves cycling the material through successivelysmaller hysteresis loops, by periodically reducing the magnitude of the applied oscillating field. Therotationally demagnetised state is produced by rotating a sample in a field which diminishes fromthat required to saturate the sample to zero as the sample rotates.

Ideally the virgin state and the thermally demagnetised state are the same; typically characterisedby the grains in the material containing multiple domains, should the micro-structure allow it. Insome materials, such as AlNiCo, thermal demagnetisation may alter the structure or phase of thematerial by acting as a heat treatment, similar to those commonly used during magnet manufacture.In such materials the thermal demagnetisation changes the nature of the material itself and so isinappropriate. Similarly in many resin bonded materials the material Curie temperature is abovethe melting point of the resin holding the material together and so thermal demagnetisation is notpossible. Where thermal demagnetisation is not possible then measurements must be performedfrom either a virgin state, a DCE demagnetised state or a rotationally demagnetised state.

Like the virgin and thermally demagnetised states the AC demagnetised and rotationally demag-netised states are also similar. The states are generally characterised by few domain walls withingrains, with saturated grains oriented so as to produce zero net magnetisation. In the case of ACdemagnetisation the moments of the grains are loosely oriented about a single axis in the material,which imparts uniaxial anisotropic behaviour to the material. This has been previously discussedin chapter 1. In rotationally demagnetised states the moments of the grains are uniformly orientedwithin a plane in the material, which imparts planar anisotropy to the material. Thus, unlike thevirgin and thermally demagnetised states, the properties of a material in an AC or rotationallydemagnetised state depends on the previous magnetic history of the material, immediately prior todemagnetisation. AC and rotational demagnetisation retain some memory of the demagnetisationprocedure.

Ideal demagnetisation of a sample involves not only reducing the magnetisation to zero but alsoleaving the material in it lowest energy state. Both McMichael et al. [49, 50] and El-Hilo et al.[93] have studied the effect of the demagnetisation process on ∆M plots and have concluded thatinitially AC demagnetised states are necessary for such plots to be meaningful. These studies havebeen previously discussed in section 1.3.2. El-Hilo et al. has also studied the conditions necessary

61


to achieve quasi AC demagnetisation by DCE demagnetisation in CoNiCr thin films [94]. In thatstudy it was found that a field decrement between DCE cycles of at most 5% of the remanentcoercivity was necessary to approximate AC demagnetisation. Furthermore reference [93] showsthat when the field decrement in the DCE process is too large then ∆M plots become positive, evenin the presence of demagnetising interactions. This is caused by the large field steps leaving higherenergy non-equilibrium magnetic configurations within the material making the sample easier tomagnetise.

In general it is difficult to AC demagnetise permanent magnets because it is difficult to cycle thenecessarily large fields at the high rates required for AC demagnetisation (typically 50 Hz). It isnecessary, then, to either simulate AC demagnetisation by very small field step DCE, along thelines described by El-Hilo et al. or to rotationally demagnetise samples. Details of the specificprocesses used to achieve thermal, DCE and rotational demagnetisation are given below.

3.3.1 Thermal Demagnetisation

In cases where the sample was suited to thermal demagnetisation, it was performed by heatingthe samples, over a period of about 30 minutes, to a temperature 10 - 15C above their Curietemperature. Once at the maximum temperature the samples were then slowly cooled back toroom temperature over a period of 12 hours. In all cases the heating was performed in a vacuum ofpressure less than 6 × 10−3 Pa, so as to avoid potential oxidation of the samples. The slow coolingwas to minimise the possibility of changes in crystal structure associated with rapid cooling rates.

3.3.2 Rotational Demagnetisation

While it is difficult to cycle large fields at rates suitable for AC demagnetisation it is easy to rotatesamples at high rates within them. Materials may be rotationally demagnetised by rotating them ina field sufficient to cause magnetic saturation and then slowly decreasing the field strength to zero,whilst continuing to rotate the sample at the same rate. This method has been previously describedby Flanders[149] and is analogous to AC demagnetisation except that the demagnetisation occurswithin a plane, rather than along an axis. Studies comparing the rotationally demagnetised andAC demagnetised states have not been reported but it is expected that they will be equivalent.

Samples were rotationally demagnetised with a rotational frequency of 10 Hz (i.e. ω = 20πHz)and a constant field ramp rate of -4500 Am−1 s−1. A slow field ramp rate is necessary to ensurethat regions requiring similar fields for activation are left with a wide distribution of orientations.The final magnetic moments of the samples were examined using a VSM to measure the threeorthogonal components. Rotational demagnetisation was found, in all cases, to produce a netmagnetisation of less than 0.5% of the maximum remanent magnetisation of any of the materialsexamined.

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The Samples Studied

Figure 3.6: The magnetic trajectory of DCE demagnetisation of SN1, using the procedure described inthe text. The final remanent moment of the sample was 0.5% of the initial (maximum) remanent moment.

3.3.3 DCE Demagnetisation

The procedure used for the DCE demagnetisation of materials was to first apply a field sufficientto saturate the material and then to oscillate the field back and forth along the axis of the mag-netisation with decreasing amplitude. Throughout this work the amplitude of the field for eachsuccessive oscillation was that required to produce a magnitude of magnetic moment equal to90% of that achieved on the previous oscillation. The demagnetisation process was halted oncethe remanent moment of the sample fell below 0.5% of the maximum remanent moment of thematerial. This procedure was made possible by the feedback control of the software describedin section 2.1.1, giving the ability to measure to a specified magnetic moment. Figure 3.6 showsthe trajectory described by this procedure during the DCE demagnetisation of SN1. In all of thesamples studied this procedure produced a net magnetisation of less than 0.5% of the maximumremanent magnetisation of the individual material.

This procedure is different from that used by El-Hilo et al., in that the field decrement varies foreach cycle. Initially the field decrements will be larger than the maximum of 5% of Hr foundby El-Hilo et al. to be necessary to achieve quasi AC demagnetisation in thin films. As themoment becomes small the field decrements will be less than this maximum. Examination of IRMdata shown in figure 3.7 measured from both a rotationally and DCE demagnetised state in SN1shows the effect of these initially large field decrements. The data measured from the rotationallydemagnetised state is smooth, whereas the data from the DCE demagnetised state contains smallpeaks. The field strength at which each of these peaks occurs corresponds almost exactly with the

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Figure 3.7: The derivative of the IRM curve in SN1. The measurements are performed from both aDCE and rotationally demagnetised state. Uncertainties have not been shown so as to not obscure thedifferences between the curves.

field turning points of the DCE demagnetisation process.

It is observed that despite the small peaks in the data measured from the DCE demagnetisedstate the data are still in good agreement. Clearly no significance can be attributed to the smallpeaks caused by the DCE demagnetisation process and when these peaks are discounted then thegeneral form of the curves is qualitatively similar. In all of the materials studied within this workIRM measurements and ∆M plots produced from rotationally demagnetised (quasi AC) and DCEdemagnetised states were seen to produce plots in good qualitative agreement. In particular noneof the ∆M plots were seen to be made positive by magnetisation inhomegeneites caused by thelarger field steps of the DCE demagnetisation. It is possible that the 5% of Hr rule found to betrue for CoNiCr thin films is unnecessarily strict for bulk permanent magnets

Nevertheless remanence data measured from a rotationally demagnetised state is to be preferredto that measured from a DCE demagnetised state, since it inevitably produces a lower energystate. All rotational hysteresis measurements and most VSM measurements were performed fromrotationally demagnetised states. The actual initially demagnetised state for each measurement isdescribed within the captions of the plots of data presented within this work.

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4. Investigation of the Mechanisms of

Magnetisation Change in Linearly

Alternating Magnetic Fields

THE mechanisms of magnetisation change along the initial magnetisation curve and the de-magnetisation curve of the materials described in chapter 3 are investigated by measurements

of remanent magnetisations, viscosity and the dependence of the reversible magnetisation on theirreversible magnetisation. A combination of such measurements allows the magnetisation mech-anisms occurring within the materials to be well characterised. It is found that magnetisationchange in the sintered NdFeB materials is controlled by a combination of domain nucleation anddomain wall pinning within the grain boundaries, such that the field required for nucleation, Hn

is greater than that required for domain wall unpinning, Hp. The same mechanism controls mag-netisation change in the larger grains of the melt quenched materials with the majority of theirgrains changing magnetisation by incoherent rotation. In MnAlC and AlNiCo the magnetisationchange is found to be controlled by domain wall pinning (Hp > Hn) within the grain interiors.

4.1 Characterisation of the Mechanisms of Magnetisation

Change

The theory behind interpreting measurements of the hysteresis and time dependence of magneti-sation has been discussed in detail in chapter 1. Here the results of measurements suitable forperforming such characterisation are presented. The samples studied are those described in chapter3 and are the same as those used for the subsequent rotational hysteresis measurements, presentedin chapter 8. The specific measurement methods have been described in chapter 2.

The assumption that irreversible magnetisation changes on recoil loops are negligible is an impor-tant assumption of the IRM/DCD method of determining irreversible magnetisation. The validityof this assumption for each material was investigated by measuring the work done during cyclesof the recoil loops, in a manner similar to that described by Emura et al. [108], and by examiningthe maximum separation of the lower and upper branches of the recoil loops. Examples of such

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measurements have been presented in chapter 2. The assumption that few irreversible processesoccur on the recoil loops is satisfied in all of the materials. The slight openness of the loops impliesuncertainty in the values of irreversible and reversible magnetisation and these values are statedin the captions of the data presented. Error bars are not shown on the plots because of the highdensity of data.

4.2 Results and Discussion

4.2.1 SN1 and SN2

The two sintered NdFeB materials studied consist of large spherical grains that are easily ableto accommodate domain walls. It is therefore expected that the magnetisation reversal in thesematerials will involve domain wall processes.

Figures 4.1 and 4.2 show the remanence magnetisation measurements for SN1 and SN2 respectively.It is apparent that the nature of the magnetisation along the initial magnetisation curve is verydifferent when measured from an DCE or rotationally demagnetised state to that when measuredfrom a thermally demagnetised state. Magnetisation being easier from the thermally demagnetisedstate is characteristic of materials in which the reversal is controlled by nucleation (i.e Hn > Hp).In such materials domain walls already exist in grains in the thermally demagnetised state but notin the DCE (or rotationally) demagnetised state. Thus from the thermally demagnetised state thedomain walls are able to sweep through the grains, increasing the magnetisation at lower fields.In the DCE and rotationally demagnetised states magnetisation change can not take place untildomains have been nucleated, which occurs at higher fields.

Differentiation of these curves may allow visualisation of the critical field distributions correspond-ing to the magnetisation mechanisms. These distributions are presented in figures 4.3 and 4.4.

The similarity of the large DCD peak to those of the IRM data measured from an DCE or rota-tionally demagnetised state suggests that the same mechanism is controlling these cases. A largermagnitude in the DCD peak is to be expected, since the curves have not been normalised. Itis likely that this peak represents the nucleation field distribution. The lower, broader peak inthe data measured from a thermally demagnetised state most likely represents the pinning fielddistribution. The peak at very low fields is likely to represent a distribution of very weakly pinnedor metastable domains in the thermally demagnetised state. Such domains require very low fieldsto be rapidly swept into the grain boundaries.

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Investigation of the Mechanisms of Magnetisation Change in Linearly Alternating Magnetic Fields

Figure 4.1: The irreversible magnetisation of SN1 vs internal field. The uncertainties in the values ofMirr obtained from IRM measurements from a thermally demagnetised state are approximately ±5%. Inthe other cases they are approximately ±2%.

Figure 4.2: The irreversible magnetisation of SN2 vs internal field. The uncertainties in the values ofMirr are approximately ±2%.

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Figure 4.3: The critical field distributions of SN1 vs internal field. The uncertainties in the values of thederivatives are of the order of ±20% for IRM measurements from a thermally demagnetised state and ±8%for the other measurements.

Figure 4.4: The critical field distributions of SN2 vs internal field. The uncertainties in the values of thederivatives are of the order of ±8%

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The difference in the mechanisms on the demagnetisation and initial magnetisation curves measuredfrom the thermally demagnetised state are also reflected in the ∆M plots, as shown in figures 4.5and 4.6.

When the IRM measurements are performed from a thermally demagnetised state the plots arepredominately positive, reaching maximum values of about 1 in the case of SN1 and 0.8 for SN2.This form of plot is that predicted by Thomson et al. for systems in which nucleation is the control-ling mechanism on the demagnetisation curve, with pinning controlling the initial magnetisationcurve [54]. The reasonably high value of ∆M ∼ 1 is consistent with the significant overlap of thepinning and nucleation field distributions, suggested by figures 4.3 and 4.4.

In contrast, the ∆M plots for the IRM measurements performed from the DCE and rotationallydemagnetised states are zero for much of the field range before a broad negative peak. This is furtherevidence that the mechanism controlling the magnetisation change is the same on both the initialand demagnetising curve for these cases. The negative values of ∆M here are probably indicativeof strong negative dipolar interactions in these materials. It appears that these interactions alsoeffect the ∆M plots measured from the thermally demagnetised state, although in this case thetransition to negative values is delayed by the competing effect of the different mechanism keepingthe plot positive.

Figure 4.5: ∆M for SN1 with the IRM measurements performed from different states. The uncertainty

in the value of ∆M measured from the thermally demagnetised state is of the order of ±0.13. In the other

cases it is approximately ±0.08.

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Figure 4.6: ∆M for SN2 with the IRM measurements performed from different states. The uncertaintyin the value of ∆M measured is of the order of ±0.08.

Figure 4.7: Mrev vs Mirr in SN1 measured from an initially thermally demagnetised state. The uncer-

tainty in the magnetisation values is of the order of ±5%.

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Measurements of the dependence of the reversible magnetisation on the irreversible magnetisationmay offer further insight into the mechanisms of magnetisation change occurring within thesematerials. Figure 4.7 shows the form of Mrev vs Mirr for IRM measurements performed fromthe thermally demagnetised state in SN1. By taking the gradient of these curves where theymeet the initial magnetisation curve (the left hand of the curves in figure 4.7), then η during themagnetisation process may be determined (see section 2.3.3).

Data for η along the initial magnetisation curve of each material are shown in figures 4.8 and 4.9.

In lower fields the form of η vs Hi is similar to that predicted by Crew for the reversible mag-netisation being produced by domain wall bowing (see figure 1.5). The initially positive values ofη indicate that each irreversible process leads to increases in the reversible magnetisation. Thisis most likely a result of the total area of domain walls within each grain increasing with eachirreversible step, meaning that there is a greater area of wall available for reversible bowing. Oncethe area of domain walls has reached an equilibrium then the value of η is seen in this case todecrease to a constant value of -1.

For a magnetisation change mechanism to produce a value of η = -1 it is necessary that eachirreversible change in magnetisation results in an equal but opposite change in reversible mag-netisation. This is equivalent to saying that irreversible changes are the conversion of reversiblemagnetisation into irreversible. It also implies that the total magnetisation remains unchanged atconstant field, which is consistent with the observation that there was negligible magnetic viscosityalong the initial magnetisation curve of these materials.

In terms of domain wall pinning a value of η = -1 is consistent with, but not necessarily indicativeof, the Friedel condition being satisfied (see section 1.2.3). Thus η = -1 is consistent with themagnetisation change being controlled by strong domain wall pinning over a statistically uniformdistribution of pinning sites within the grain interior. In this case the domain walls bow outenclosing a volume equivalent to that containing a another pinning site before breaking free of apinning site and being irreversibly pinned in the final bowed position.

Crew has proposed an alternative mechanism of magnetisation change for materials in which η =-1 [61]. He suggests that if the interior of the grains are defect free then a mechanism by whichthe domain walls sweep rapidly through the domain before being eventually annihilated in thegrain boundaries may explain the value of η = -1. In this model the domain walls are pinned inthe grain boundaries and bow reversibly throughout the defect free interiors of the grains. Afterbowing through the interiors of the grains they are either irreversibly pinned in the boundaryor annihilated and so the decrease in the reversible magnetisation is equal to the increase in theirreversible magnetisation.

Further information in discriminating between these two possible mechanisms may be had byconsidering the forms of η along the initial magnetisation curves of the sintered materials measuredfrom DCE and rotationally demagnetised states, which are also shown in figure 4.8 and 4.9. Theform of η vs Hi in these cases is similar to that of the thermally demagnetised state, except thatthe approximately constant value reached by η is now about -0.5.

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Figure 4.8: η along the initial magnetisation curve in SN1, with measurements performed from a variety

of demagnetised states. The uncertainties in the values of η are of the order of ±0.3 in the thermally

demagnetised case and ±0.2 in the other cases.

Figure 4.9: η along the initial magnetisation curve in SN2, with measurements performed from a variety

of demagnetised states. The uncertainties in the values of η are of the order of ±0.2.

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During DCE, AC or rotational demagnetisation the domain walls will be driven into the grainboundaries where they will be pinned or annihilated. In a demagnetised state half of the grainswill be magnetised along one direction and half along the other. During the magnetisation thegrains that are initially magnetised in the direction of the field will contribute a value of η = 0,since the irreversible processes will result in no change in the reversible magnetisation but merelyannihilation of the domain walls already pinned within the boundary regions of the grains. Bycontrast, in those grains that are aligned opposed to the field the domain walls will bow reversiblythrough the grain before being annihilated in the grain boundary giving a value of η = −1. Thenet result is η = −0.5.

Thus the dependence of the reversible magnetisation on the irreversible magnetisation during themagnetisation process in the sintered material is consistent with the model of the magnetisationchange being controlled by pinning of the domain walls within the grain boundaries. The domainwalls are able to reversibly bow through the grain interiors unimpeded. This is consistent with theobservation of Fidler that the interior of the grains in sintered NdFeB are largely defect free [150].The characterisation is also consistent with that of Liu et al. [151] and Tomka et al. [152].

Figures 4.10 and 4.11 show the form of the fluctuation plotted against irreversible magnetisationin SN1 and SN2. The determination of Hf was made using the waiting time technique (see section2.3.2).

Figure 4.10: Hf vs Mirr along the demagnetisation curve in SN1. Note that the evolution is from positive

values of Mirr to negative (right to left).

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Figure 4.11: Hf vs Mirr along the demagnetisation curve in SN2. Note that the evolution is from positive

values of Mirr to negative (right to left).

It is clear that Hf slowly decreases in magnitude during the magnetisation reversal in both ma-terials. This is similar to that observed by El-Hilo et al. in a system that has distributions ofparticle volumes and anisotropy [153]. The materials here are likely to also have such sources ofenergy barrier distribution and the slight overlap of the pinning and nucleation field distributionssuggested by figures 4.3 and 4.4 indicates that more than one mechanism may control magneti-sation reversal in these materials. Thus the factors contributing to the variation of Hf in thesematerials are complex. Previous reports of the variation of Hf in sintered NdFeB have reportedit increasing [81], decreasing [128] or remaining constant with field [62]. More recently El-Hilo etal. have studied the form of the fluctuation field in a simulated model of granular materials [39].Their work is significant in that it considers the variation of Hf during reversal in a system inwhich the distribution of energy barriers is a composite of model distributions. In their model Hf

is seen to change during reversal, either increasing, decreasing or remaining constant accordingto the relative standard deviations of the anisotropy field and volume distributions. Similar suchdistributions within these materials will control the variation of Hf .

The magnitudes of the fluctuation field observed are consistent with those reported previously insintered NdFeB (see for example [132]). By equation 1.16 these values imply activation volumes ofdiameter 10 - 20 nm, which is similar to that of the domain wall width in NdFeB and so consistentwith a nucleation mechanism controlling the demagnetisation. The extensive studies of Givord etal., over a range of temperatures, have also found the activation volumes in these materials to beproportional to the domain wall width [79].

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4.2.2 MQ1 and MQ2

The melt quenched materials have smaller grain sizes than the sintered materials, with manygrains too small to support domain walls. Therefore it is to be expected that rotation mechanismswill play a significant role in their magnetisation change. Thermal demagnetisation is difficult inthese materials because of the risks of grain growth and melting the resin bonding the materials.Therefore only results for measurements made from DCE and rotationally demagnetised states arepresented here.

Figures 4.12 and 4.13 show the irreversible magnetisation in the melt quenched samples, as deter-mined by IRM and DCD measurements.

Since the IRM measurements are made from a rotationally demagnetised state in the case of MQ1and a DCE demagnetised state in the case of MQ2 then it may reasonably be expected thatthe magnetisation processes are the same for both the magnetisation and demagnetisation. Theagreement in the peaks of the critical fields in the magnetising and demagnetising cases, shown infigures 4.14 and 4.15 is consistent with this. Thus in these cases ∆M plots may give informationconcerning the interactions within the materials. These are presented in figures 4.16 and 4.17.

Figure 4.12: The irreversible magnetisation of MQ1 vs internal field. The IRM measurement is madefrom a rotationally demagnetised state. The uncertainty in the values of Mirr are of the order of ±2%.

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Figure 4.13: The irreversible magnetisation of MQ2 vs internal field. The uncertainty in the values of

Mirr are ±3%. Note that the fluctuations in the data are a result of the DCE demagnetisation process

rather than real variations in the critical fields.

Figure 4.14: The critical field distributions of MQ1 vs internal field. The IRM measurements were made

from a rotationally demagnetised state. The uncertainty in the values of the derivative are ±8%.

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Figure 4.15: The critical field distributions of MQ2 vs internal field. The IRM measurements were made

from a DCE demagnetised state. The uncertainty in the values of the derivative are of the order of ±12%.

Figure 4.16:∆M for MQ1. The uncertainty in the values of ∆M are of the order of ±0.8

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Figure 4.17: ∆M for MQ2 with the IRM measurement performed from a DCE demagnetised state. The

uncertainty in the values of ∆M are of the order of ±0.11

The ∆M plots suggest that magnetising interactions occur within MQ1 and demagnetising inter-actions within MQ2. The most likely source of the magnetising interactions in MQ1 are exchangeinteractions. In MQ2 the demagnetising interactions are probably caused by dipolar coupling.

Insight into what mechanisms are occurring in both cases may be gained from looking at the formof η along the demagnetisation curves, as presented in figures 4.18 and 4.19.

The positive values of η in the lower fields are similar to that predicted by Crew for reversiblemagnetisation mediated by domain wall bowing (see figure 1.5). This is suggestive of domain wallprocesses, presumably similar in nature to those that occur in the sintered NdFeB materials. Asdomains are nucleated then a greater area of walls is available for reversible bowing and so η in-creases. The walls rapidly sweep through the grains where they are either immediately annihilatedor pinned (because of the overlap in the pinning and nucleation field distributions) prior to anni-hilation - at which point η decreases. At higher fields, however, the form of η resembles that offigure 1.4, which indicates rotation of the magnetisation vector away from easy axes. This impliesthat a rotation mechanism is controlling the magnetisation change. Since the grain sizes in thesematerials are of about the size required to support domains then the rotation will undoubtedly beincoherent.

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Figure 4.18: η along the demagnetisation curve of MQ1. The uncertainties in the values of η are of the

order of ±0.1.

Figure 4.19: η along the demagnetisation curve of MQ2. The uncertainties in the values of η are of the

order of ±0.15.

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In MQ1 the lower field domain wall processes occur over a field range of about 1.2 × 106 Am−1 andin MQ2 a range of about 1.2 × 106 Am−1. Examination of figures 4.12 and 4.13 show that thesefield ranges correspond to only a few percent of the total irreversible magnetisation changes in MQ1and about 40% of the irreversible magnetisation changes that occur during the demagnetisationprocess in MQ2. This suggests that just a few percent of the grains are multi domain in MQ1 andabout 40% in MQ2. These estimates are in excellent agreement with the estimate of Folks et al.that 6% of the grains in MQ1 are multi-domain and 44% of the grains in MQ2 [90].

Folks et al. have also reported ∆M measurements in MQ1 and MQ2 where the IRM measurementsare made from a virgin state [90]. In both materials ∆M is seen to be positive. This suggeststhat in MQ2 the magnetisation from the virgin state in the larger grains is controlled by domainwall pinning, just as it is in the sintered materials. In MQ1 the positive form of ∆M from boththe virgin and rotationally demagnetised states is consistent with incoherent rotation controllingmagnetisation from both states and the interactions being positive.

The evident distribution in the grain sizes in these materials, the combined mechanisms of incoher-ent rotation and domain wall nucleation and the different forms of interactions in these materialsmeans prediction of the evolution of Hf during their reversals is difficult. The measured forms arepresented in figures 4.20 and 4.21.

In MQ2 the variation of Hf is similar to that in the sintered NdFeB materials, but in MQ1 itincreases during the magnetisation reversal. The form of Hf in MQ1 is consistent with the mea-surements of Crew et al. in this material [62] using the technique of Estrin et al. and demonstrates

Figure 4.20: Hf vs Mirr along the demagnetisation curve in MQ1. Note that the evolution is frompositive values of Mirr to negative (right to left).

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Figure 4.21: Hf vs Mirr along the demagnetisation curve in MQ2. Note that the evolution is frompositive values of Mirr to negative (right to left).

the equivalence of the waiting time technique used here (although it allows for less uncertainty inthe values of Hf - see section 2.3.2). The similar evolution of Hf in MQ2 and the sintered materialsmay be evidence that nucleation controls the magnetisation reversal in the larger grains of MQ2.Hf increasing in MQ1 suggests that a different mechanism controls the magnetisation change inpractically all grains in this material (i.e. incoherent rotation) although the effect that the positiveinteractions in this material will have is unclear. It is certainly the case in both MQ1 and MQ2(and more generally) that the variation of Hf during reversal will be influenced by the distribu-tions of sample parameters and their effects on the distribution of energy barriers to reversal withchanging field.

4.2.3 MnAlC and AlNiCo

The MnAlC and AlNiCo 7 samples studied both have large grains able to support multiple domains.It is expected that the magnetisation reversal in these materials will involve domain wall processes.It is not possible to thermally demagnetise AlNiCo without changing its magnetic properties sodata for this material is only presented from DCE and rotationally demagnetised states.

The approximate critical field distributions, obtained from IRM and DCD measurements in AlNiCoand MnAlC are presented in figures 4.22 and 4.23.

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Figure 4.22: The critical field distributions of AlNiCo vs internal field. The IRM measurement is made

from a rotationally demagnetised state. The uncertainty in values of the derivative is ±28% for the DCD

measurements and ±10% for the IRM measurements.

Figure 4.23: The critical field distributions of MnAlC vs internal field. The uncertainty in values of the

derivative is ±18% for the DCD measurements and ±8% for the IRM measurements.

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In the case of MnAlC the similarity in critical fields in the demagnetisation and the magnetisationcases from both the thermal and DCE demagnetised states suggests that similar mechanismscontrol the magnetisation change in each case. This is in contrast to the very different switchingfield distributions observed for measurements from the DCE and thermally demagnetised statesin the sintered rare-earth iron materials. It suggets that domain wall pinning may control themagnetisation in MnAlC. In AlNiCo the similarity in the distributions measured in demagnetisingand magnetising case from a rotationally demagnetised state also suggests that the same mechanismcontrols both magnetisation and demagnetisation. ∆M plots, then, may give information aboutthe interactions in these materials when measured from the DCE and rotationally demagnetisedstates. In the case of MnAlC the ∆M plot measured from the thermally demagnetised state mayalso give information concerning the mechanism controlling the magnetisation change. ∆M plotsfor AlNiCo 7 and MnAlC are presented in figures 4.24 and 4.25.

There is some small overlap between the nucleation and pinning field distributions in MnAlC,as implied by the small positive values of ∆M when measured from the thermally demagnetisedstate, but in general Hn < Hp (compare figure 4.25 with figure 1.3). This is the opposite to theassessment for the sintered rare-earth iron materials. In both materials the broad negative peaks in∆M when measured from a DCE or rotationally demagnetised state is evidence of demagnetisinginteractions. These are most likely the result of dipolar interactions.

Figure 4.24: ∆M for AlNiCo with the IRM measurements performed from different demagnetised states.

The uncertainty in the values of ∆M are ±0.14.

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Figure 4.25: ∆M for MnAlC with the IRM measurements performed from different demagnetised states.

The uncertainty in the values of ∆M are ±0.11.

The form of η along the magnetisation curve in both materials is shown in figures 4.26 and 4.27.

Examination of the figures reveals in each case a form similar to that of figure 1.5 which is consistentwith a domain wall pinning mechanism. This is consistent with previous characterisations of thesematerials [145, 26] and reports of microscopy [154]. In the case of MnAlC η approaches a constantvalue of about -0.05 which means that each change inMrev (via domain wall bowing) is accompaniedby about 20 times the change in Mirr (by domain wall unpinning, propagation and repinning).This suggests a reasonably low density of pinning sites within the grains. In AlNiCo the constantvalue for η is about -0.3, suggesting a rather higher density of pinning sites, such that with eachunpinning event the domain walls sweep out regions of about three times the area enclosed by thebowing prior to the unpinning.

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Figure 4.26: η along the initial magnetisation curve of AlNiCo. The uncertainties in the values of η are

approximately ±0.08.

Figure 4.27: η along the initial magnetisation curve of MnAlC. The uncertainties in the value of η are

approximatetly ±0.05.

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4.3 Conclusions

Individually none of the methods employed here is able to provide sufficient information for thecharacterisation of the magnetisation mechanisms occurring in the materials studied. Used inconjunction, however, then considerable confidence may be had in their characterisation. Con-sideration of remanent magnetisations and the dependence of the reversible magnetisation on theirreversible magnetisation have proved particularly instructive.

In all of the materials the critical field distributions measured from a DCE or rotationally de-magnetised state were similar suggesting that similar mechanisms control the magnetisation anddemagnetisation, when the magnetisation is measured from a DCE, rotationally or presumably anAC demagnetised state. In these cases ∆M curves may provide insight into the interactions withinthe materials. In all cases other than MQ1 the interactions appear to be negative dipolar coupling.In MQ1 there is evidence of positive exchange interactions.

The critical field distribution measured from the thermally demagnetised state differed from thaton the demagnetisation curve in the sintered NdFeB materials. This is evidence that differentmechanisms control the demagnetisation and magnetisation from a thermally demagnetised statein these materials. In contrast, in MnAlC the evidence is that the same mechanism controls bothcases. It may therefore be concluded that measurements of IRM data from both thermally andAC or rotationally demagnetised states may provide greater insight into the mechanisms occurringwithin permanent magnets than measurements from just one of the states.

The interpretation of ∆M data in terms of the two coercivity model of Thompson et al. [51, 54]has been found to be useful in the study of these permanent magnets. It is noted that the methodof interpretation of Thompson et al. does not require the specific mechanisms of nucleation anddomain wall pinning within grain interiors but rather requires that one of the two mechanisms mustoccur prior to the other being possible on the demagnetisation curve. Here such interpretation wasapplicable to the sintered NdFeB materials, MnAlC, AlNiCo and the larger grains in MQ2.

Measurements of the dependence of the reversible magnetisation on the irreversible magnetisationwere particularly useful in distinguishing between the mechanisms occurring in these materials. Thedifferent forms of η(H) predicted for rotation and domain wall processes provided clear evidenceof rotation mechanisms in MQ1 and MQ2 and were consistent with the fraction of grains thoughtto be multi-domain in these materials. The constant value of η = −1 achieved in SN1 and SN2is particularly striking and the similarly constant value of η = −1/2 measured from DCE androtationally demagnetised states provides very strong evidence of the characterisation of thesematerials.

Interpretation of the significance of the evolution of the fluctuation field during magnetisationreversal in these materials is complex. El-Hilo et al. have demonstrated that the variation ofHf during reversal is influenced by the distribution of activation energies, which is in turn acomposite of the distributions of various material parameters [39]. Variation in the width of thosedistributions may cause Hf to either increase, decrease or remain constant during reversal. Most

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bulk permanent magnet materials, such as those studied here, will have considerable distributions inmaterial parameters, as is evident in the range in particle volumes in the melt quenched materials.Further studies into the effect on Hf of interactions and material distributions in bulk materials,as recommended by Tomka et al. [152] and along the lines of El-Hilo et al. [39] are necessary toclarify the significance of the variation of Hf during reversal in these materials.

The magnitudes of Hf measured in the NdFeB materials are similar to those reported by otherauthors and consistent with activation volumes of dimensions similar to the domain wall widthand so in the case of the the multi-domain grains provide evidence of domain wall processes [79].This is consistent with the conclusion that reversal in SN1 and SN2 and the larger grains of MQ2is controlled by nucleation since it requires creation of a domain wall. In the smaller grains andin MQ1 where most of the grains are too small to contain domain walls, however, the significanceof the activation volume being similar to the domain wall width is less clear. In these cases theactivation volume will be governed by the activation energy requirements of incoherent rotationprocesses. It is not unreasonable to expect this to be similar to the domain wall width, particularlysince the grain sizes are close to being large enough to support domains and the domain wall widthis ultimately a consequence of coupling energy considerations of neighboring spins.

As a result of the measurements performed, the magnetisation change in SN1 and SN2 was deter-mined to be controlled by a combination of domain nucleation and domain pinning within the grainboundaries. There is overlap in the nucleation and pinning fields in these materials but on averageHn > Hp. This implies that when measured from the thermally demagnetised state the magneti-sation is controlled by domain wall pinning. The data here are consistent with this occurring inthe grain boundaries, with the domain walls free to bow reversibly unimpeded through the defectfree grain interiors. The demagnetisation of these materials and the magnetisation when measuredfrom an AC or demagnetised state will be predominately controlled by domain nucleation. This isconsistent with previous characterisations of these materials [152].

The data for the melt quenched materials are consistent with the magnetisation change in thesematerials being predominately controlled by an incoherent rotation mechanism. The data are alsoconsistent with a fraction of the grains in these materials containing domain walls. In MQ1 it isestimated that a few per cent of of the volume of grains are multi domain and in MQ2 it is estimatedthat the proportion is 40%. A mechanism similar to that proposed for the sintered materials willcontrol the magnetisation change in these larger grains. Such findings are also consistent withprevious characterisations of these materials [103].

In MnAlC and AlNiCo the data are consistent with the magnetisation being controlled almostentirely by domain wall pinning, such that Hp > Hn. In MnAlC there is evidence of only a smalloverlap in the nucleation and pinning field distributions. The measurements here suggest that thedensity of pinning sites within the interior of the grains in MnAlC is lower than that in AlNiCo.

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88

5. Measurement of Rotational Hysteresis

and Resistivity by Angular Acceleration

Based Methods

A SIMPLE method of measuring both magnetic rotational hysteresis losses and electrical re-sistivity in isotropic media is described. The method involves measuring the angular deceler-

ation of a sample set rotating in a magnetic field. By using air bearings to minimise friction, andallowing the moment of inertia of the rotating system to be varied, a maximum sensitivity of 3 ×10−7 J cycle−1 with a precision of 8 × 10−8 J cycle−1 has been achieved for rotational hysteresisloss measurements. The method has been used to measure rotational hysteresis losses as a functionof applied field strength in isotropic samples of γ-Fe2O3, AlNiCo and MQ1. Measurements of theresistivity of Cu, Ni and AlNiCo, together with a rough measurement of the saturation magnetisa-tion of Ni, are reported. The data from these measurements was found to be consistent with thatobtained by established methods of measurement.

5.1 Introduction

Measurements of rotational hysteresis losses in isotropic materials may be used to investigate themagnetic reversal mechanisms occurring within isotropic media (see section 1.3.4) [118, 116]. Themajority of such measurements are performed using the static measurement technique of torquemagnetometry (see for example [89, 121]) although dynamic methods involving search coil tech-niques are also sometimes employed [155, 156]. Also reported in the literature are techniques ofmeasuring rotational hysteresis losses in isotropic media using angular acceleration measurements[157, 158], although these techniques have suffered from an inability to properly distinguish damp-ing caused by rotational hysteresis from that caused by eddy currents. Here such measurementtechniques are refined, so that these two effects are explicitly separated.

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5.1.1 Measurement of Rotational Hysteresis Losses and Resistivity by

Angular Acceleration Based Methods

In 1957 Kelly [157] proposed a method of measuring rotational hysteresis in ferromagnetic materialsby mounting disks of the materials on a spinning top and measuring the increasing time it tookthe top to complete successive cycles when spun in a field. Kelly’s top consisted of a Lucite rodwith sapphire bearings on each end and a disk of the sample mounted about the rod. The timetaken to complete each cycle was measured by means of a photocell picking up a single pulseof a light beam reflected from a mirror mounted on the top. Using this technique Kelly, andlater Cecchetti [138], were able to measure rotational hysteresis torques and to make empiricaleddy current measurements in saturated samples. The restrictions on sample geometry, however,meant that this technique was unable to properly distinguish between rotational hysteresis andeddy current damping when they occurred concurrently. This meant that in conductive samplesthis device was limited to measurements of “rotational power losses”, as distinct from rotationalhysteresis losses.

The idea of determining rotational hysteresis by measuring the damping of a rotating system wasused by Stephenson [140] in 1980 and then by Flanders [141, 158] in 1982, although Flanders makesno reference to Stephenson’s work and so appears to have developed the same ideas independently.Both authors employed a gas powered turbine located between the poles of an electromagnet onwhich a sample is mounted. The turbine is driven with a constant torque and it’s equilibriumfrequency in zero field, f0, measured. In the presence of a field the additional damping torquesof rotational hysteresis and eddy currents may be present and serve to reduce the equilibriumfrequency of the turbine by a factor ∆f . This factor is directly proportional to the additionaldamping torque on the system and so may be related to rotational hysteresis loss and eddy currentdamping. Using this method Stephenson achieved a torque sensitivity of 1 × 10−7 Nm and Flanderswas able to measure resistivity accurate to within a few percent in the majority of cases and towithin 25% in the worst case. The advantage of this method is that in non conducting samplesit allows the measurement of rotational hysteresis as a function of rotational frequency. Thedisadvantage is that it is unable to differentiate between torques due to eddy currents and torquesdue to rotational hysteresis when they occur simultaneously, meaning it is only able to measurerotational hysteresis in non-conducting samples.

5.2 The Dual Air Bearing Angular Acceleration Magne-

tometer

The method of Kelly offers the advantage that as the rotational velocity of the sample changesduring measurement then it should be possible to differentiate between rotational hysteresis lossesand eddy current losses, in a single measurement, by examining the velocity dependence of thedamping. With this in mind a device was constructed to allow similar measurements. Care wastaken with the design to allow different sample geometries to be used and to minimise the friction,

90

Measurement of Rotational Hysteresis and Resistivity by Angular Acceleration Based Methods

Figure 5.1: Diagram of the Dual Air Bearing Angular Acceleration Magnetometer (DABAAM). Theparts represented in grey are supported by the air bearings and rotate during measurement. The otherparts remain stationary.

which limits the sensitivity of the device. The resulting device for measuring rotational hysteresis inisotropic media is known as the Dual Air-Bearing Angular Acceleration Magnetometer (DABAAM)and is represented by Fig. 5.1 and shown in figure 5.2.

The device consists of a vertically orientated 360 mm long, 5 mm diameter carbon composite shaft,the bottom end of which is attached to a polycarbonate sample holder and is free to rotate betweenthe poles of an electromagnet, capable of generating a maximum flux density of 2.8 Tesla. Thematerial of the shaft was chosen for its’ high rigidity and low conductivity. To the top end of theshaft is attached an aluminium disk, to which further brass disks may be added to alter the momentof inertia of the device. The shaft is supported by two contactless air bearings, both of which consistof an nitrogen gas inlet into a chamber, in which the gas pressure becomes homogeneous beforeescaping through 0.75 mm diameter holes, orientated so as to provide gas jets for supporting theshaft.

The upper air bearing consists of 9 upwardly directed jets, equally spaced about a circle of diameter60 mm, and 9 radially outward orientated jets. The jets act on the inside of a hollow aluminiumcylinder, of diameter 74.7 mm and height 20 mm, which is attached to the shaft through its’ closedtop end. The upwardly directed jets support the weight of the device and the radially directed jetsprevent radial motion. The air spacing in this bearing is 0.34 mm.

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Figure 5.2:The Dual Air Bearing Angular Acceleration Magnetometer (DABAAM)

The lower air bearing consists of two rows of six equally spaced radially inwards directed jets,which act on a solid polythene cylinder of length 21 mm and diameter 31.88 mm which is attachedto the shaft two thirds of the way down. The air spacing of 0.06mm in this cylinder is significantlyless than for the upper bearing. This ensures that the bearing operates at a higher pressure andso is able to withstand any sideways magnetic forces that may be exerted on a sample due to anyslight inhomegeneity in the field.

Attached to the shaft between the upper air bearing and the disk at the top is the angular positiontransducer. This consists of a thin disk, coded with 500 equally angularly spaced slots, which isfree to rotate between a diode laser and a photodiode, such that the photodiode only senses thelaser when a slot is directly above it. These light pulses are timed, to within 1µs by a computer,with a digital data acquisition board.

The sample holder consists of a polycarbonate cylinder, capable of holding cylindrical samples ofa fixed diameter by a tight friction fit. Spherical samples are first set in resin in polycarbonatebuckets of the correct diameter to make them suitable for measurement. In samples with very largerotational hysteresis, such are NdFeB, then it is necessary to machine a slot in the top of thesebuckets so that a square section on the end of the shaft may prevent the sample from rotating, dueto the insufficient friction fit of the holder.

5.2.1 The Mechanics of the Measurement

Samples in the form of cylinders or spheres are attached to the end of the shaft which is free torotate around a vertical axis between the poles of an electromagnet. The torque, τ , acting on therotating system is given by

τ = Iα (5.1)

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where I is the moment of inertia of the device and α is the angular acceleration. I may be altered byattaching a disk with an appropriate calibrated moment of inertia to the top of the shaft, above theupper air bearing. When the magnetic field provided by the electromagnet is such that rotationalhysteresis occurs in the material then τ consists of a magnetic (rotational hysteresis ) component,a frictional component and, in the case of conductive samples, an eddy current component. Themechanical frictional torque acting on the device is minimised by the two air bearings. These resultin a frictional torque, τf that depends on the gas pressure in the bearings. This pressure remainsconstant during a measurement. τf consists of a constant kinetic component, τ0, and a changingviscous component directly proportional to the angular velocity of the shaft. The torque due tothe induced eddy currents, τe is also directly proportional to the angular velocity. Hence:

τf + τe = τ0 + kfω + keω = τ0 + kω (5.2)

For the range of gas pressures used, τ0 was found to range from 1.4 × 10−6 Nm to 4.2 × 10−5

Nm. kf , is independent of the sample but depends on the bearing pressure. ke depends on thegeometry and conductivity of the sample. k is the sum of the two. In isotropic materials themagnetic (rotational hysteresis ) component of the torque, τm, is constant throughout rotation inan unchanging magnetic field. Hence 5.1 may be rewritten as

τ = τf + τe + τm = τ0 + kω + τm = Iα (5.3)

The solution to this equation of motion is given by

θ(t) =(e[ ktI ] − 1)(ω(0) + τ0+τm

k )I − (τ0 + τm)tk

+ θ(0) (5.4)

where θ(t) is the angular position of the shaft as a function of time, t, and ω(0) is its initial angularvelocity. The initial angular acceleration, α(0), is

α(0) =kω(0) + τ0 + τm

I(5.5)

Since kt/I is always less than 0.04, even at high fields and over the longest measurement times ( 10s), then the exponential in 5.4 may be expanded as a series and the result rewritten in terms ofthe initial angular acceleration, α(0):

θ(t) = θ(0) + ω(0)t+ α(0)(t2

2+kt3

6I+k2t4

24I2

)+O[t]5 (5.6)

The terms O[t]3 and greater are diminishing (for kt/I 1 as is the case here) and contributeless than 1% of the total. Hence θ(t) may be approximated by a quadratic with the O[t]3 termsand greater being discarded. In all cases this approximation has been found to fit the measureddata with a very high degree of accuracy (R ≈ 0.9999). The use of the approximation is preferredbecause it is computationally more efficient.

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5.2.2 Measurement of Rotational Hysteresis Losses

To make a measurement of rotational hysteresis losses at a particular field the shaft is set rotatingand θ(t) digitally recorded by an angular position transducer from which ω(0) and α(0) are deter-mined by fitting equation 5.6 to the data. By plotting α(0) vs ω(0) for several measurements atthe same field a linear relation given by 5.5 is found. τm is separated from τ0 by subtraction ofmeasurements made without a sample. Hence the rotational hysteresis loss per cycle, Wr , may befound by

Wr =∫ 2π

0

τmdφ = 2πτm (5.7)

5.2.3 Measurement of Resistivity

In the absence of unknown magnetisation or magnetoresistance, or when a sample is magneticallysaturated and so has no domains, the torque induced by eddy currents may be determined forparticular sample geometries. In the case of a sphere it may be determined by simple integrationof Faraday’s law, giving the emf induced about a loop in the presence of a time varying appliedmagnetic flux density (see appendix A.1). Such an approach assumes the absence of screening orskin depth effects that may be present at high rotational velocities. The integration may be usedto derive the power dissipated by eddy currents in a sphere made to rotate in a field:

Pe =2πω2B2r5

15ρ(5.8)

and since τe = keω = Peω , here, then for the particular case of a spherical sample of radius r and

resistivity ρ, ke of equation 5.2 is given by

ke =2πB2r5

15ρ(5.9)

Since the value of ke, as a function of B2 may be found experimentally then this equation may beused to determine the resistivity, ρ, of a sample.

The analysis is based upon the same form of measurements as those used to determine rotationalhysteresis losses. To measure resistivity a linear plot of α(0) vs ω(0) of the form of equation 5.5 is,again, found at each particular field but in this case the gradient, rather than the intercept is ofinterest. By expanding k then it can be seen than this gradient, me is given by

me =kfI

+keI

(5.10)

kf is independent of B but as 5.9 shows ke depends on B2. Hence by plotting these experimentallydetermined values of me vs the square of the applied magnetic flux density, B2 we obtain a linewith an abscissa axis intercept of kf

I and a gradient of 2πr5

15Iρ , from which the resistivity, ρ, isfound. It is emphasised that this analysis is only valid for a spherical samples with negligiblemagnetoresistance. The analysis also relies on B being uniform throughout the sample, which will

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not be the case when magnetic domains are present.

5.2.4 Measurement of Magnetisation

The presence of an unknown magnetic moment in a sample, which contributes an uncertain amountto the magnetic flux density, prevents the determination of the sample’s resistivity. In certain spe-cial cases when the resistivity is known a priori, however, and there is negligible magnetoresistancethen the eddy current parameter, ke, may be used to determine the magnetisation of the sam-ple. These special cases are those where the flux density in the sample is sufficiently uniform forthe eddy current calculations of section 5.2.3 to be valid. This will be the case when a materialis uniformly magnetised throughout its volume. Uniform magnetisation occurs in paramagneticand diamagnetic materials at all fields and in ferromagnetic materials only when the applied fieldis sufficiently high to magnetically saturate the material. In such cases there is zero rotationalhysteresis and so the direction of magnetisation is aligned with the applied field.

The parameter ke may be measured as a function of field by rearranging equation 5.10 as

ke(Ha) = me(Ha)I − kf (5.11)

where kf has been previously determined from measurements in zero field and me(Ha) is thegradient of the line of best fit for α(0) vs ω(0) at the particular applied field, as before. Themeasured form of ke(Ha) may then be used to determine the value of B, using equation 5.9.Combination of this equation with the scalar form of the constitutive equation (as is appropriatehere)

B = µ0(Ha + (1−D)M) (5.12)

gives

M(Ha) =32

√

15ke(Ha)ρ2πr5

µ0−Ha

(5.13)

where M(Ha) is the value of the magnetisation as a function of applied field and ke(Ha) is theexperimentally determined parameter. As the equation only holds for spherical samples the sub-stitution D = 1

3 has been made.

Such analysis to obtain the magnitude of the magnetisation of the sample is only valid when thereis negligible magnetoresistance (ρ doesn’t vary with M) and negligible screening effects are present.The effect of magnetoresistance is to decrease the resistivity with field, which would serve to in-crease the value of ke and lead to artificially large values for the calculated magnetisations. If highrotational velocities were used for the measurements then screening effects may become appar-ent. This would serve to decrease the eddy currents and so decrease ke and the correspondinglycalculated magnetisation.

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Figure 5.3: A calibration plot for the DABAAM. The moment of inertia of the device may be determinedfrom such plots.

5.2.5 Calibration

Calibration of the device requires that its moment of inertia, I0, is determined. The frictionparameters, τ0 and kf , are determined with each measurement and so don’t need to be examinedseparately. The moment of inertia is determined from repeated measurements with the same,constant, rotational hysteresis torque. The moment of inertia used in each measurement is variedby attaching to the top of the shaft one of seven brass disks, each with a moment of inertia knownto within 1%. From the measurements performed for each disk a plot of α(0) vs ω(0) is produced,using the method described in section 5.2.1. This again gives a linear relation described by

α(0) =kω(0) + τ0 + τm

I0 + Idisk(5.14)

As shown in figure 5.3, plotting the reciprocal of the abscissa axis intercept of this line, I0+Idiskτ0+τm

,against the moment of inertia of the disk for which each fit is performed, Idisk, yields a linear graphwith gradient of 1/(τ0 + τm) and abscissa axis intercept of I0/(τ0 + τm). Division of the interceptby the gradient, then, yields the moment of inertia of the device, I0.

Equation 5.14 suggests that the calibration of the device could be performed in zero field, whereτm = 0. In practice it was found that better results were obtained when the calibration measure-ments were performed with a rotational hysteresis torque present. This is because of the slightvariation of τ0 with the different weight of the disks used to alter the moment of inertia. A greaterweight reduces the spacing in the upper air bearing causing more gas to flow through the lower

96


bearing which, due to the fine spacing of the lower bearing, leads to a slight change in the friction.The rotational hysteresis torque, τm, is unchanged with varying weight, however, and so if τm ischosen such that it is of much greater magnitude than τ0 then the variations in τ0 may be renderedinsignificant.

I0 was determined as being (3.1 ± 0.2) × 10−5 kg m2. The uncertainty of I0 contributes relativelylittle to the uncertainty of most rotational hysteresis measurements. The large torques involved inmost such measurements require that the measurements are performed with the overall moment ofinertia increased by attaching the disks of known moment of inertia to the top of the device. Thesedisks have significantly larger moments of inertia than the device, with lower relative uncertainties,so the overall relative uncertainty contributed by the uncertainty in I0 is diminished. In the caseof measurements of resistivity, magnetisation or very small hysteresis, however, the uncertaintyin I0 contributes significantly to the overall uncertainty of the measurement. In these cases thetorques are small so the measurements are best performed using just the moment of inertia of thedevice. Thus the relative uncertainty in I0 contributes directly to the uncertainty of resistivity andmagnetisation measurements.

5.2.6 Other Effects of Eddy Currents

Eddy currents induced in conductive samples may provide information about the resistivity ormagnetisation of the sample, as described in sections 5.2.3 and 5.2.4. It is emphasised that thevalidity of such analysis must be assessed, in each case, after consideration of the possible effects ofmagnetic domains or magnetoresistance altering the resistivity of the sample. These effects aside,however, there is another way in which eddy currents may compromise the measurements. Whena sample is rotating with some component of it’s moment remanent (ie: rotating with the sample,unperturbed by the field) then this moment produces a rotating field which may induce eddycurrents in the iron pole faces of the electromagnet, removing energy from the rotating system.

The significance of this image effect was examined by measurement of a sphere with a larger rema-nent moment than was present for any of the other measurements. Measurements were performedin zero field using a sphere of NdFeB with high remanent magnetisation (Mr = 7.8 × 105 Am−1),giving a remanent moment of 5.0 × 10−2A m2. The field produced by such a moment at the polefaces is of the order of 7 × 103 Am−1 so it is to be expected that the eddy currents induced in thepole faces are very small and this was found to be the case. No increased damping was measurable,suggesting that the effect is less than the sensitivity of the instrument.

5.2.7 Heating of the Samples

As the samples are rotated many times during measurement heat generated by hysteresis may besignificant. Consideration of the possible rise in the equilibrium temperature of the samples showedthat even with continuous rotation at a high rate the maximum temperature rise that could beexpected was of the order of 6C. This is because the thermal conductivity of the sample holder and

97


rod is sufficient to remove heat from the sample. During the course of actual measurements the shaftrotates at lower velocities than for this worst case and spends some time motionless, therefore it isexpected that any temperature rise would be significantly less than 6C. No discernible temperaturerise has been noticed and so it is expected that changes in the magnetic, or conductive propertiesof the samples due to heating are insignificant.

5.2.8 Sensitivity and Precision

Theoretical sensitivity

Ignoring the uncertainty in friction and the sample parameters, the maximum achievable sensitivityof the device is limited by the resolution of the angular position transducer and by the momentof inertia of the device. The angular position transducer consists of a disk with 500 identical slotscut in it, equally spaced 2π

500 apart, through which a stationary laser is able to shine. As the diskrotates a light sensitive diode located on the other side of the disk to the laser detects these laserpulses with the time of each pulse being measured by a computer. It is this time resolution, then,that limits the sensitivity of the determination of angular velocity using the device.

From equation 5.6 it can be deduced that for the same value of ω(0) a difference in time of δt overthe 15 revolutions of a measurement corresponds to a difference in α(0) of

δα(0) =2(30π − ω(0)(t+ δt))

(t+ δt)2− 2(30π − ω(0)t)

t2(5.15)

Substitution of this δα(0) into equation 5.5 reveals that

δα(0) =δkeω(0)

I+δτmI

(5.16)

In general, then, the intrinsic uncertainties in the measurement of rotational hysteresis loss or eddycurrent effects (resistivity or magnetisation) are linked. The maximum sensitivity of a measurementof either type is achieved when the other effect is not present.

For a measurement of rotational hysteresis losses, then, the maximum sensitivity is

δWr = 2πIδα (5.17)

The timing computer is able to measure the time to a resolution of 1µs and the fastest measurements(which have the least sensitivity) correspond to total measurement times of about 2.5 s and ω(0) ≈35 rad s−1. Substitution of these values into equation 5.15 and then for δα in to equation 5.17gives a theoretical sensitivity of 2 × 10−10 J cycle−1.

This sensitivity is calculated considering just the moment of inertia of the device. Equation 5.17shows that this is when the device is at its most sensitive. When I is increased, to make possible

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the measurement of samples with greater rotational hysteresis, then the sensitivity will worsen.For measurements of larger torques, made using the disk with the greatest moment of inertia thenthe theoretical sensitivity is 4 × 10−8 J cycle−1.

The worst case of a determination of resistivity involves only two measurements. In this case thenthe uncertainty in the gradient of the line of best fit me vs B2 is 2δα

ω(0)∆B2 , where ∆B is differencein the flux density at which the two measurements were performed. In this case the maximumsensitivity of the measurement of ρ is

δρ =15Iδαρ2

πr5ω(0)∆B2(5.18)

For the parameters used above then the maximum sensitivity for a measurement of the resistivityof Cu (ρ =1.7 × 10−8 Ωm), for example, is 3 × 10−12 Ωm. It is noted also that as the sensitivitydepends on r5 then any small uncertainty in the value of r will lead to a large intrinsic uncertaintyin the value of ρ.

Substitution of equation 5.16 into equation 5.13 gives the maximum sensitivity of a magnetisationmeasurement as

δM =3√

15Iρδα2πr5ω(0)

2µ0(5.19)

For a sample with ρ = 5 × 10−6 Ωm and all other parameters as before, the maximum sensitivityof a magnetisation measurement by the DABAAM is 4.5 × 103 Am−1 . This is much worse thanthe actual sensitivity of more common methods of measuring magnetisation and suggests that theDABAAM is not useful for measuring magnetisation.

Actual Sensitivity

Achieving the theoretical sensitivity of a device is rarely, if ever, possible. In the case of theDABAAM the actual sensitivity achieved may be estimated from a measurement of Wr vs Ha

for a very small magnetic sample. Figure 5.4 shows such data measured for a single layer 8 mmdiameter punched disk of commercially available BASF video tape. Here each point represents asingle (unaveraged) measurement of Wr, with each measurement consisting of recording θ(t) for10 different values of ω(0) (to which equation 5.6 was fit). The mass of the tape sample was 3.4mg, suggesting that the mass of magnetic material was ∼ 0.4 mg[27]. It can be seen that despitethe small amount of magnetic material such measurement is still within the achievable sensitivityof the device, with the quality of the data being effected by the precision.

Whilst averaging may be used to reduce the impact of the precision, for practical purposes thesensitivity is still limited by it. In this case the actual sensitivity of the device is estimated to beapproximately a third of the peak in Wr, or about 3 × 10−7 J cycle−1.

The actual sensitivity of measurements of magnetisation and resisitivity have not been determined.

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Figure 5.4: The Rotational Hysteresis Loss vs Applied Field in a single layer 8 mm diameter punched diskof BASF video tape, measured using the DABAAM. The uncertainties in the values ofWr are approximately±10−7 J cycle−1

Precision

The actual precision of the device was estimated by repeated measurements of zero torque. Zerotorque was chosen since it represents the most difficult test for the device, since the variations in τfwith any slight wobble or vibrations of the device are relatively most significant. This is in contrastto the calibration measurements, which were performed at a high (constant) value of torque, so asto reduce the relative uncertainty contributed by slight variations in τf .

Twenty five determinations of zero torque, each consisting of ten measurements of θ(t) (250 mea-surements in all), were made. The mean of the determined torque values was -2 × 10−8 J cycle−1,the range 3 × 10−7 J cycle−1 and the standard deviation 8 × 10−8 J cycle−1, suggesting a precisionof 8 × 10−8 J cycle−1.

As the data presented in figure 5.4 is measured without averaging then it is to be expected thatthe magnitude of the scatter of the data about the true form of Wr(Ha) for this material will be ofabout twice this order (since for random Gaussian scatter 95% of the measurements are expectedto lie within 2 standard deviations of the mean). The arrow in figure 5.4 represents four times theestimate of the precision (±2στ ). It is apparent in the figure that the scatter is contained withinthis range and so is consistent with the value of 8 × 10−8 J cycle−1 as the precision of the device.

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5.2.9 Uncertainties in the Measured Quantities

The uncertainties in the quantities measured by the DABAAM were estimated by linear regressionanalysis of the fits to the raw data. This analysis was performed using Mathematica. The fits of5.6 to the measured data of θ(t) almost universally produced very high correlation coefficients (R≈ 0.9999), indicating the mechanics of the motion are well characterised by equation 5.6. Linearregression analysis of these fits typically revealed uncertainties in the value of ω(0) of about 0.1%and in α(0) of about 2%. The relative (fractional) uncertainties were observed to be independentof the torque acting during the measurements, for each sample measured. The value of θ(0) isnot used in further calculations but is expected to be zero. Typically this was found to have amagnitude of about 0.02 radians with an uncertainty of ±0.002 radians.

Such small uncertainties in the values of α(0) and ω(0) are to be expected as each measurement ofθ(t) consists of 3850 points and so allows for a high degree of confidence in the fits. Similarly smalluncertainties in these quantities were found across the entire range of measurements made usingthe device, from the extreme cases of very small torques, such as in measurement of the singlelayer of video tape presented in figure 5.4, to the large torques in the measurement of rotationalhysteresis in a NdFeB magnet, such as that presented in figure 5.7.

Linear regression analysis of the fits of equation 5.5 to the determined values of α(0) and ω(0)revealed rather larger uncertainties in the parameters. This is to be expected as each fit is typicallyperformed to a set of just 10 (ω(0), α(0)) pairs. Uncertainties in the values of the parameter(τ0 + τm)/I (the abscissa axis intercept of the fit), from which the rotational hysteresis loss isdetermined, varied with the torques and the samples but were generally less than 5%. It is notedthat this uncertainty is due to that in the torques and not in I, since that remains constant duringthe measurements. Therefore the uncertainties in the value of the rotational hysteresis loss arelarger than this and are indicated by error bars on the plots of the data. Uncertainties in thevalues of the gradient of the fit, from which M or ρ may be determined, were typically around16%.

The uncertainties in the values of the rotational hysteresis integrals calculated take into accountthe uncertainty in the individual measurements of Wr, the uncertainty introduced by the Reimannintegration of the discrete data and, in the case of incomplete data, the extrapolation of the datato 1

H → 0, as described in section 2.3.4.


5.3.1 Rotational Hysteresis Loss Measurements

The device has been used to measure rotational hysteresis losses in a wide range of isotropicmaterials. Rotational hysteresis losses as a function of applied field for samples of Co modifiedγ-Fe2O3 , AlNiCo 7 and MQ1 are presented in Figs. 5.5 - 5.7. The properties of AlNiCo 7 and

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Figure 5.5: The Rotational Hysteresis Loss vs Applied Field in Co-γ-Fe2O3, measured using theDABAAM.

MQ1 are described in chapter 3 and those of the Co modified γ-Fe2O3 sample are described inappendix B. These materials were chosen as they span three orders of magnitude in both coercivityand Wr and so represent a good test of the device.

Co modified γ-Fe2O3

In cobalt modified γ-Fe2O3, Wr peaks at a value of about 2.2 Jm−3cycle−1 (5.3 Jkg−1cycle−1) inan applied field of 5.2 × 104 Am−1. This is in reasonable agreement with previous results [159]which show a similarly shaped curve for pure γ-Fe2O3 of particle size ∼250 nm with Wr peakingat ∼2.2 Jkg−1cycle−1 in a field of 5 × 104Am−1. The difference in the value of Wr may be due tothe cobalt modification of the material measured here, which also serves to increase the coercivity.

The form of Wr vs Ha may be used to calculate the rotational hysteresis integral (see sections 1.3.4and 2.3.4). The value of this integral is often used as a means to probe the magnetisation reversalmechanism occurring within a material by comparison with calculated values, such as those intable 1.1.

For the data presented in figure 5.5 a value of RH = 1.24 ± 0.11 was calculated. Ignoring theeffects of interactions, this would suggest that the magnetisation in the Co modified γ-Fe2O3

reverses in an incoherent manner. Such a conclusion is in agreement with previous studies ofγ-Fe2O3 which showed that for particles of greater than 100 nm then the magnetisation reversalmechanism is incoherent [159]. This is to be expected since the particles here are at about the

102


critical size necessary to contain domain walls. Thus it is likely that some of the larger particleschange magnetisation through domain wall movement. The remaining particles are likely to reversethrough an incoherent rotation mechanism.

AlNiCo

Two sets of data are presented in figure 5.6, one set measured using the DABAAM and anothermeasured using a Rotating Sample and Coil Magnetometer (see chapter 6), which is a techniqueemploying search coils. The close agreement of the data obtained using two independently cali-brated and altogether different techniques suggests that the accuracy of the data is good.

The value ofRH = 0.57±0.05 suggests that the magnetisation reverses by an incoherent mechanism.The value of the rotational hysteresis integral for sintered AlNiCo (particularly in this case) is verylow for a material consisting of large multi-domain grains. It is to be expected that AlNiCo 7changes it’s magnetisation during rotational hysteresis by mechanisms involving domain walls andso a larger value of RH (something closer to π) would be expected. The low value may be due tothe unusually high saturation magnetisation of AlNiCo. This point is discussed further in chapter8

Nevertheless the conclusion that AlNiCo changes it’s magnetisation by an incoherent mechanismsis consistent with the conclusions of previous studies of AlNiCo [160, 119]. The value of RH =0.57±0.05 for the sample studied here also falls in the range of RH = 0.43 - 1.40 found by McRobbie[160] for measurements performed on textured AlNiCo at various angles to the axis of alignment(0 - 90). It is reasonable to expect that the isotropic sample measured here would fall somewherein this range.

MQ1

Measurements of rotational hysteresis losses in melt quenched NdFeB magnets have been reported[121]. After correction of these results [161] the peak value of Wr is found to be between 2.3 ×106 Jm−3cycle−1 and 4.9 × 106 Jm−3cycle−1 (depending on the quench rate) at Ha = 1.75 × 106

Am−1. The measurements for MQ1 shown in figure 5.7 are in agreement, showing Wr with a peakvalue of about 410 Jkg−1cycle−1 (or 2.5 × 106 Jm−3cycle−1), in an applied field of 1.6 × 106Am−1.

Despite the incompleteness of the data presented for MQ1 a reasonable estimate of the rotationalhysteresis integral, RH , may be made, using the method described in section 2.3.4. The value forRH estimated for MQ1, then, is RH = 1.55 ± 0.33. This falls into the range of the Shtrikman- Treves theory, suggesting an incoherent rotation mechanism for magnetisation change. SinceMQ1 consists of single domain grains of dimensions of about the domain wall width then sucha mechanism would be expected. The grains are too small to support domain mechanisms butsufficiently large to allow non homogeneous orientation of the constituent spins.

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Figure 5.6: Comparison of the Rotational Hysteresis Loss vs Applied Field in AlNiCo, measured using

the DABAAM and the RSCM

Figure 5.7:The Rotational Hysteresis Loss vs Applied Field in MQ1, measured using the DABAAM.

104


Figure 5.8: α(0) vs ω(0) for DABAAM measurements of a sphere of copper. The applied applied fluxdensity was 2.01 Tesla.

5.3.2 Resistivity Measurements

Measurements of the resistivity of a sphere of copper was performed. Since copper has no magneti-sation the resistivity may be determined using measurements over the entire flux density range of 0- 2.7 Tesla. The resistivities of nickel and AlNiCo were also measured. Both of these materials areferromagnetic and so the resistivity measurements were performed only in applied fields sufficientto magnetically saturate the samples. The magnetoresistance in these samples is of the order of0.5%, which is insufficient to significantly effect the measurements.

Copper

Figure 5.8 shows a typical plot of α(0) vs ω(0) obtained from measurements of a 5mm diameter99.99% pure Cu sphere. In this particular case the applied flux density was 2.01 Tesla. The linearform of this plot, even at the higher rotational velocities, indicates that the effect of screening inthis sample is negligible. Screening effects would produce a plot that curved upwards at highervelocities.

Figure 5.9 shows a plot of the gradient, me, of these lines α(0) vs ω(0) at each particular flux densityversus the square of the applied flux density, B2, as described in section 5.2.3. The gradient ofthe line of best fit is (1.4 ± 0.2) × 10−3, giving a value for the resistivity of (1.7 ± 0.3) × 10−8

Ωm. This value is in excellent agreement with the accepted value of 1.707 × 10−8 Ωm for copperat 25C.

105


Figure 5.9: me vs B2 for DABAAM measurements of Cu. The resistivity may be determined from thegradient of these data.

Nickel

A plot of me vs B2 for a 5.11 mm diameter sphere of 99.9% pure Ni is presented in figure 5.10. Theplot is linear as the fields used are all sufficient to saturate the sample and so no domain walls arepresent. The effect of this magnetisation, Ms, on the magnetic flux density, B, may be ignored asit serves only to change the abscissa axis intercept, leaving the gradient unchanged. The gradientof the line of best fit is (3.7 ± 0.3) × 10−3, giving a resistivity of ρ = (6.6 ± 0.9) × 10−8 Ωm,compared with the accepted value of 6.87 × 10−8 Ωm for nickel at 25C.

AlNiCo

The plot of me vs B2 for a 5.33 mm diameter sphere of isotropic AlNiCo is shown in figure 5.11.Measurements were only performed at fields sufficient to saturate the sample. The scatter aboutthe linear form in this data is because the eddy current damping is much less than in copper ornickel. This is because the resistivity of AlNiCo, is high compared with the pure metals previouslyexamined. The line of best fit has a gradient of (1 ± 0.3) × 10−3, giving a resistivity of ρ = (5 ±2) × 10−7 Ωm.

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Figure 5.10: me vs B2 for DABAAM measurements of a nickel sphere in fields sufficient to saturate thesample.

Figure 5.11: me vs B2 for DABAAM measurements of the sample of AlNiCo 7 in fields sufficient to

saturate the sample.

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5.3.3 The Saturation Magnetisation of Nickel

The data collected for nickel may be alternatively analysed to obtain the value for the saturationmagnetisation of the sample. The analysis described in section 5.2.4 was applied to data obtainedin fields sufficient to saturate the sample, using the accepted value for the resistivity of Ni, 6.87× 10−8 Ωm. Figure 5.12 shows the measured form of M(Ha) for the sphere of Ni, as measuredusing the DABAAM (the diamonds) and the VSM (the continuous line). In fields of greater than3 × 105 Am−1 the sample is saturated and so the magnetisation is constant, as shown by bothmethods. In a ferromagnetic sample this is the only value of the magnetisation that may be foundby the DABAAM as it is the only time the magnetisation is uniform throughout the sample.

The value of Ms determined by the DABAAM in this case (∼ (6 ± 1) × 105 Am−1 ) is too high(by about 25%). This is most likely due to inaccuracy in the actual value of the resistivity ofthe sample, which will effect both the absolute values and the slope of the line of best fit. It islikely that if the actual value of ρ could be determined by an independent means then a betterestimate of the saturation magnetisation could be made from such DABAAM measurements. Thepoor sensitivity of magnetisation measurements with the DABAAM, together with the limitationof only being able to measure saturation magnetisation means it is unlikely to be of interest formagnetisation measurements.

Figure 5.12: The magnetisation of Ni as a function of applied field, as determined by DABAAM (dia-

monds) and VSM measurements (continuous line).

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5.4 Conclusions

Earlier methods of measuring rotational hysteresis and resistivity in isotropic media using angularacceleration based methods have been refined. The method presented here distinguishes betweenlosses due to rotational hysteresis and losses due to eddy currents by examining the velocity de-pendence of the damping torque. The advantage of this approach is that it does not requireassumptions about the flux density in the sample. The data may be used to determine both therotational hysteresis losses and the resistivity of spherical samples. In particular cases, where theresistivity of a sample is known a priori, then the data may be used to calculate the saturationmagnetisation of the sample.

A device designated as the “Dual Air Bearing Angular Acceleration Magnetometer” (DABAAM)was built to perform these measurements. The DABAAM utilises two air bearings in order tominimise the effects of friction on its measurement of angular acceleration. The frictional propertiesof the bearings were characterised and the intrinsic maximum sensitivities of the device calculated.

The measured forms of rotational hysteresis loss, Wr, as a function of applied field strength, Ha,were found to be consistent with measurements performed using other methods. The data wasalso used to calculate the rotational hysteresis integral, RH , which may be used to investigate themechanism by which the magnetisation changes within the samples. Each of the samples was foundto change magnetisation by incoherent mechanisms. These conclusions are consistent with thoseof previous studies.

Measurements of the resistivity of copper and nickel were performed and found to be in excellentagreement with the accepted values. The resistivity of AlNiCo was also determined from measure-ments performed in fields sufficient to saturate the sample. The eddy current measurements alsoallowed the saturation magnetisation of nickel to be measured, which was found to be in roughagreement with that determined by VSM measurements. The DABAAM was, then, found to beuseful for resistivity measurements but not for magnetisation measurements.

The consistent agreement of the above results with those obtained using established methods ofmeasurement suggests that the DABAAM may be used for reliable measurements of rotationalhysteresis and resistivity in isotropic media. Sample magnetisation may also be determined inspecial cases. The measurements have indicated that the rotational hysteresis measurements havea sensitivity of 3 × 10−7 J cycle−1 with a precision of 8 × 10−8 J cycle−1.

109


110

6. Measurement of the Magnetisation

Vector During Rotational Hysteresis in

Isotropic Materials

A METHOD of measuring rotational hysteresis in isotropic media where the magnetisationvector is explicitly determined is described. The method employs a rotating former posi-

tioned between the pole pieces of an electromagnet. Fixed inside the former are two sets of searchcoils connected in series and held mutually orthogonal. Each set of coils consists of a sense coil,surrounding the sample space, connected to a reference coil which is wound in opposition andremoved from the sample so as to cancel out the voltages generated in the coils by the field as theformer is rotated. The voltage signals from the series connected coils are passed through slip ringsmounted at the top of the former and measured using a lock-in amplifier, using the signal from oneof the reference coils as the phase and frequency reference. The phase, φ, is a direct measure of theangle between the magnetic moment of the sample and the applied field and, after calibration, theamplitude is a measure of the magnitude of the moment, µ. This information is not available fromconventional methods of measuring rotational hysteresis, where only the torque or energy loss ismeasured. It allows rotational hysteresis data to be presented as a function of internal field and therotational hysteresis integral corrected for demagnetisation effects to be computed, which is alsonot conventionally possible. Results are presented for measurements of isotropic samples of Al-NiCo, Sr-ferrite and an exchange-spring magnet. Similarity in the form of M(Hi) in the rotationaland alternating hysteresis cases was interpreted as evidence that the mechanisms of magnetisationchange are the same in both two cases. The (calculated) loss data from these measurements wasfound to be consistent with that obtained by established methods of measurement.

6.1 Introduction

Methods of directly measuring the magnetisation vector during rotational hysteresis have beenreported[155, 162]. Flanders describes a ‘Rotating Sample Magnetometer’ (RSM) in which samplesare rotated, within a field, about an axis that is offset from the sample, as shown in figure 6.1 [149].Search coils are mounted close to the space in which the sample rotates and oriented so as to bemutually orthogonal to both the field and axis of rotation. The oscillatory voltage induced in them

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Figure 6.1:The Rotating Sample Magnetometer of Flanders.

by the changing magnetic flux (because of the periodically rotating sample) is measured using lock-in techniques, similar to a VSM. Using this method it is possible to measure the total magneticmoment of a sample during rotational hysteresis. The disadvantage of Flanders’ approach is thatrelatively high rotational frequencies are needed (Flanders required 80 Hz), making eddy currentsin conductive samples significant. These eddy currents induce an additional magnetic moment inthe sample which both perturbs the voltage signal in the sense coils and alters the magnetic stateof the material itself. The use of high frequencies for measurement means that the method is onlysuitable for use on non conductive materials, which was recognised by Flanders [149].

A more recent variation of the approach of Flanders is to employ a VSM with two sets of sense coilsarranged so as to be orthogonal to both the field and to each other. The vibration unit is orientedsuch that the vibrations of the sample are transverse to the applied magnetic field, produced byan electromagnet. A motor is mounted on the end of the vibrator such that the sample (on theend of a non-magnetic shaft so as to keep the motor away from the magnetic field) is able to berotated within the field between the coils. Rotating the sample with the vibrator on enables theorthogonal components of the magnetic moment to be measured during the rotational hysteresisby traditional VSM lock-in techniques applied to both sets of coils.

Another type of measurement, which allows the determination of the magnetisation vector duringrotational hysteresis, has been utilised in the study of electrical steel sheets, particularly those usedin electric motor cores (e.g. Fe-Ni, Fe-Si and Fe-Si-Al) [163, 164]. This approach, pioneered byBrix[165] and improved by Zhu et al. [162] and Enokizono et al. [166], makes use of a four pole(two phase) electromagnet to generate a rotating field in which a square specimen is located. Theelectromagnet has a laminated or ferrite core and so is suitable for use at relatively high frequencies(up to a few kHz). Pairs of orthogonal B and H sense coils measure the magnetic response and

112

Measurement of the Magnetisation Vector During Rotational Hysteresis in Isotropic Materials

a feedback system allows either B or H to be controlled, so as to maintain circular or ellipticalrotational forms. Knowledge of the vector forms of B and H allow M to be determined throughthe constitutive equation. This approach is eminently suitable for the study of materials in similarconditions to those found in electric motor cores as B may be precisely controlled. The approach isnot as suitable for the precise study of permanent magnet materials, however, as the field strengthsavailable from such electromagnets are insufficient (generally less than 5 × 10−4Am−1) and theuniformity relatively poor.

The above methods of measuring the magnetisation vector during rotational hysteresis all employstationary search coils (in the frame of reference of the laboratory) and rotate either the sample(Flanders’ RSM) or produce a rotating field in which the sample is held stationary (the methods ofZhu et al. and Enokizono et al.). In the study of permanent magnet materials, uniform magneticfields of high intensity are required so approaches in which the field is kept stationary in the frameof reference of the laboratory are preferable. As such, the use of a VSM with sample rotationfacilities or an RSM are to be preferred of the methods above.

An alternative approach to the RSM or VSM, in which the field is held stationary in the frameof reference of the laboratory, is also possible: one in which both the sample and the sense coilsare rotated in a stationary field. The simplest way to implement such approach is to rotate thesample and coils together. In this case the search coils will measure only that component of themagnetisation that rotates within the sample (in the reference frame of the sample). This is incontrast to both Flanders’ RSM and to VSMs, which measure the total magnetisation. Only thatcomponent of the magnetisation that rotates in the frame of the reference of the sample contributesto the rotational hysteresis loss. Hence the rotating coil method measures only that component ofthe magnetisation that contributes to the rotational hysteresis.

Measurement of just this component of magnetisation may be advantageous where measurementsare being performed on a sample from a previously magnetised remanent state. In zero field theremanent moment of the rotating sample merely rotates with the sample (and so is stationary inthe reference frame of the sample). As the strength of the field is successively increased, however,then eventually some component of the magnetisation will become activated by the field and willassume a stationary orientation at some angle to it (and so will be rotating in the reference frameof the sample). This is the onset of rotational hysteresis. Initially the activated component ofthe magnetisation will be much less than the unactivated remanent component and so if the totalmoment is measured then the signal from the activated component (which is of interest) may beswamped by the much larger signal from the unactivated remanent component. Thus if a sample hasa remanent moment then techniques that measure the total magnetisation will have poor sensitivityto the component of magnetisation that is actually contributing to the rotational hysteresis, becauseinstrumentation amplification levels will be determined by the remanent moment. A methodthat employs rotating coils, however, measures only that component of the magnetisation thatcontributes to the rotational hysteresis and so its sensitivity is independent of the initial remanentstate of the sample. Such a method is described here.

113


Figure 6.2:The essential features of the RSCM.

6.2 The Rotating Sample and Coil Magnetometer

6.2.1 Physical Description

The essential physical features of the Rotating Sample and Coil Magnetometer (RSCM) are rep-resented by figures 6.2, 6.3 and 6.4.

The device consists of a hollow Tufnol (linen in resin) tube held in place by a single bearing sothat it is free to rotate about its central axis between the poles of a 1.7 Tesla electromagnet. Thelength of the tube is 600 mm, it has an inner radius of 11 mm and an outer radius of 16 mm. Thetube is rotated, via a cog and chain drive, by a stepper motor with a 0.9 step size. The axis ofrotation of the tube passes through the centre of the field space of the electromagnet and is fixed

114


Figure 6.3:How the components of the RSCM interact.

orthogonal to the field direction. The stepper motor is capable of rotating the tube at up to 10 Hz(≈ 63 rad s−1), with the rotation speed being controlled by a digital controller which is triggeredby TTL pulses from a signal generator.

Fixed inside the Tufnol tube are two pairs of orthogonal coils, the sense coils and the compensationcoils. Both pairs are positioned so that they rotate within the space of field homogeneity betweenthe electromagnet pole pieces. During rotation in a field periodic voltage signals are generatedin these coils, according to Faraday’s Law. These signals are passed through slip rings, with anoise rating of 2mV per root Hertz, mounted in the top of the Tufnol tube. The sinusoidal signalsare transmitted by coaxial cable to a digital lock-in amplifier which determines the amplitude andphase of the signals. The lock-in has a time constant of 1 s and obtains its reference frequencyand phase from another signal generator, which shares its clock with the one triggering the steppermotor controller, as shown in figure 6.3. The data from the lock-in amplifier is logged by a PC forfurther analysis.

The sense coils consist of an inner coil of 350 turns surrounded by another coil, arranged orthogonalto it, of 520 turns. The axes of both coils are aligned orthogonally to the axis of rotation so that theaxis of each coil aligns with the field twice during each revolution. For each of the sense coils thereis a corresponding compensation coil, wound so as to have a matching inductance. Both of thecompensation coils are aligned with their corresponding sense coil and connected, via twisted pairs,in series opposition. They are sufficiently removed from the sense coils so that they experience anegligible component of any flux produced by a magnetic sample held within the sense coils butstill experience the same applied field. The purpose of the compensation coils is to reject that partof the voltage signal which is generated in each coil pair due to rotation in the applied field of theelectromagnet.

115


When measurements are performed on a magnetic sample then it is mounted in the space sur-rounded by the sense coils. As the sample and the coils are rotated in the field then any activatedcomponent of magnetisation also rotates and so alters the magnetic flux passing through the sensecoils. If the sense and compensation coils were perfectly matched then the signal from the seriesconnected pairs would be solely due to the changing flux associated with changes in the activatedcomponent of magnetisation of the sample. In practice the matching of the sense and compensationcoils is not quite perfect and so data for measurements with no sample must be subtracted fromthe data for measurements with a sample in order to completely remove the field induced voltagesignals. The procedure for doing this is described in section 6.2.3

The sense and compensation coils are wound onto polycarbonate formers, as shown in figure 6.4.The formers of the compensation coils are identical to those of the sense coils. They are designedso that one fits within the other. Together they hold a sample bucket (in the sense coils), whichcontains a sample, such that the centre of the bucket is held at the centre of the coils and is unableto move relative to the coils. This is achieved in the manner shown in figure 6.4. The samplebucket is held snugly in the hole in the inner former such that it protrudes out equally either endof the hole. This fit is sufficiently snug so that the friction between the bucket and the former issufficient to prevent rotation of the sample bucket within the former. During rotational hysteresisthe torques that could produce rotation about this axis are negligible and so a sufficient degree offriction is not difficult to obtain. The inner former, with the sample held in it, is then inserted intothe hole of the outer former, so that the walls of the hole in the outer former touch the protrudingends of the sample bucket and prevent it from moving along it’s axis within the inner coil, asindicated in figure 6.4. This outer former then fits snugly into the bore of the Tufnol tube. Thisfit ensures that both the coils and sample are rotated about their centre and prevents any rotationof the inner former within the hole of the outer former. Once together the formers make up thesample holder.

After insertion into the Tufnol tube the sample holder is fixed into place by a non-magnetic brassscrew which passes through both the Tufnol tube and the top of the sample holder. Together theTufnol tube and the sample holder fulfill five important criteria:

1. The polycarbonate formers maintain the shape of each of the sense coils.

2. They maintain at all times the axes of the coils mutually orthogonal to each other and to thecentral axis of the Tufnol tube (the axis of rotation).

3. The sense and compensation coils are kept aligned.

4. Any sideways or rotational motion of samples is prevented.

5. The samples are held at the centre of the sense coils and the space between the sense coilsand the samples is minimised, so that the coupling between the coils and the samples ismaximised.

116


Figure 6.4: The sense coils of the RSCM. The sample bucket fits into the inner coils, which then fits intothe outer coils. This then fits into the main rotating shaft of the RSCM.

6.2.2 The Form of the Signals Induced in the Coils

The Signal due to the Magnetic Field

When the RSCM is rotated in a magnetic field then voltage signals are induced in the coils asthe area they present to the flux changes. This area varies sinusoidally so the induced voltagesignals are also sinusoidal. In the absence of a sample the net voltage signal from each of thesense/compensation coil pairs is small and sinusoidal and results from the imperfect matching ofthe effective cross sections of the opposing coils. This signal may be expressed as the sum of thesignal from each of the coils as

VH(t) = ωNcAcµ0Ha sin(ωt+ φ0) + ωNsAsµ0Ha sin(ωt+ φ0 + φcs) (6.1)

where ω is the rotational frequency of the RSCM; Nc and Ac are the number of turns and theeffective area of those turns in the compensation coil; Ns and As are the same quantities for thesense coil; Ha is the magnitude of the applied field; ωt+φ0 is the angle between the compensationcoil axis and the applied field, with φ0 being the angle at t = 0; and φcs is the phase differencebetween voltages induced in the compensation coil and those induced in the sense coil. Ideally φcsshould be π but in reality the imperfect winding of the coils means that the coils are not orientedperfectly opposed.

Equation 6.1 may be represented vectorially as in figure 6.5, which shows the magnitudes and

117


Figure 6.5:A vectorial representation of the field signal in the RSCM.

phases of the sinusoidal signals. Hence the total sinusoidal voltage signal due to the field may bemore simply expressed in terms of [magnitude, phase] coordinates, V ωH as

V ωH =[xµ0Haω , arcsin

(NsAs sinφcs

x

)+ φ0

](6.2)

where the superscript in V ωH is taken to mean that the signal has an angular frequency of ω, suchthat if V ωH = [V ωH , φ] then VH(t) = V ωH sin(ωt+ φ). x is a constant given by

x =√

(NcAc)2 + (NsAs)2 + 2NcNsAcAs cosφcs. (6.3)

If the sense and compensation coils were perfectly matched and aligned then x would be zero andthere would be no signal due to the field.

The Signal due to Magnetisation in an Isotropic Sample

When the RSCM is rotated in a field with a magnetic sample in the sample holder, then the signalfrom each of the sense/compensation coil pairs may differ from the no sample case. In isotropic ma-terials this will occur when some component of the magnetisation of the sample becomes activatedand remains at a constant angle to the field. This occurs at the onset of rotational hysteresis. Thesense coils see the activated magnetisation of the sample as a moment which adds to the flux theyexperience. The additional flux due to the moment of the sample induces an additional voltage inthe sense coils given by

VM (t) = yωNsAsµ0µact sin(ωt+ φ0 + φcs + φM ) (6.4)

where µact is the magnitude of the activated component of the sample moment; φM is the constantangle between the applied field and µact; and y is a constant that takes into account the couplingbetween the sense coil and the flux of the sample. The other parameters are defined as in equation6.1.

This signal may also be expressed more simply in [magnitude, phase] coordinates, in the manner

118


of equation 6.2, asV ωM = [zµactω , φ0 + φcs + φM ] (6.5)

where z is a calibration constant such that

z = yµ0NsAs. (6.6)

The Signal due to Magnetisation in an Anisotropic Sample

In anisotropic materials the behaviour of the magnetisation is more complex. In samples withuniaxial anisotropy the activated component of the magnetisation vector will oscillate at twice therotational frequency about some mean angle to the field direction. In samples with cubic anisotropythe oscillation may be at 3 or 4 times the rotation frequency, depending on the alignment of the bulkanisotropy axes with the axis of rotation. The degree of oscillation will depend on the magnitudeof the sample magnetisation and the strength of the anisotropy. The mean phase, however, willbe independent of the degree of sample anisotropy and it is this phase that is of importance whenconsidering the rotational hysteresis loss.

In the case of measurements without a sample, or measurements with an isotropic sample, then theentire voltage signal occurs at the frequency of the rotation of the RSCM. In anisotropic materials,however, the signal also contains higher harmonics. The voltage signal of these higher frequencyharmonics may be expressed as

VK(t) =nmax∑n=2

V nωK (6.7)

whereV nωK = [zJnµactω , φ0 + φcs + φM ] . (6.8)

Jn are constants that are proportional to the magnetocrystalline anisotropy constants and nmax isthe maximum value of n for which Jn is not zero. Note that although the magnitude of the higherharmonics depends on the rotational frequency the signals themselves are (by definition) at higherfrequencies, as indicated by the superscript. The total voltage signal due to the activated compo-nent of magnetisation in an anisotropic material is, then, the sum of the fundamental component,V ωM , given by equation 6.5, and these higher frequency components,V nωK .

Only that part of the voltage signal at the rotation frequency, V ωM , contains information about therotational hysteresis loss per cycle. The higher frequency oscillations result in no net work duringa complete revolution of the sample. This is because the work done as the magnetisation oscillatesaway from the field is recovered as it oscillates back towards the field. This is analogous to torquemeasurements of anisotropic materials, where the net work done in a single revolution dependsonly on the mean offset of the torque curve from the zero torque position, rather than on the shapeof the curve itself. Thus the RSCM may be used to study rotational hysteresis in both anisotropicand isotropic samples, through measurement of only that component of the total induced voltagesignal that is at the rotational frequency of the RSCM.

119


Figure 6.6:A vectorial representation of the makeup of Brot during rotational hysteresis in the RSCM.

The Signal due to Eddy Currents Induced in the Sample

When the RSCM is used to measure a sample which is conductive then eddy currents are inducedin the sample as it is rotated in the magnetic field, according to Faraday’s Law. These eddycurrents are induced in the sample both by changes in the flux density contributed by the rotatingapplied field (in the frame of reference of the sample) and by changes in the flux density due toany activated component of magnetisation in the sample. In the frame of reference of the sampleboth of these components of the flux density are constant in magnitude but rotate in directionwith time. The eddy currents produced circulate about an axis orthogonal to the axis of rotationof the sample. If the sample is spherical then the magnetic moment of these eddy currents is givenby

µe =2πωBrotr5

15ρ(6.9)

where r is the radius of the spherical sample, ρ is its bulk resistivity and Brot is the magnitude ofthe rotating magnetic flux density. Equation 6.9 is derived in Appendix A.2.

Brot is given by the constitutive equation

Brot = µ0(Ha + (1−D)Mact), (6.10)

where D is the demagnetisation factor of the sample, which in the case of a sphere is 13 . This may

be represented by the vector diagram shown in figure 6.6. From the figure it is apparent that themagnitude, Brot, and phase, φB , of this rotating flux density are given by

Brot =

√(µ0Ha)2 +

(23µ0

µact43πr

3

)2

− 43µ2

0Haµact43πr

3cos(π − φM ) (6.11)

andφB = arcsin

(2µ0µact

3Brot 43πr

3sin(π − φM )

). (6.12)

where 43πr

3 is the volume of the sample. In the case where the sample has no activated magneti-sation then Brot is simply µ0Ha and φB = 0.

120


The moment due to the eddy currents, µe, is distributed over the sample volume and so it inducesa voltage in the sense coils but not in the compensation coils. Thus an expression for the voltage,V ωe , induced in the sense coil by the eddy currents may be obtained by substitution of µe in placeof µact and φe in place of φM in equation 6.5. This leads to the expression

V ωe =[z

2πBrotr5

15ρω2 , φ0 + φcs + φe

](6.13)

where φe is the phase of the eddy current moment, given by

φe = φB + π (6.14)

since µe opposes Brot. Unlike the voltages induced by the field and the activated componentof magnetisation the magnitude of this voltage signal depends on the square of the frequency ofrotation.

The significance of the signal induced by the eddy currents may be examined by considering theratio V ωe

V ωM

over the range of values of Ha and Mact involved in a typical measurement. Figure 6.7shows a plot of this ratio as a function of Ha and Mrot for a sphere of 5 mm diameter and resistivityρ = 5 × 10−7 Ωm being rotated at 9 Hz. Such a measurement may be considered as a worst case,in terms of the perturbation of B by eddy currents, because of the large diameter of the sample,the low resistivity (lower than most magnetic materials) and high rotational frequency (9 Hz is themaximum for the RSCM). Even in this case it is apparent that the contribution to the measuredsignal from eddy currents is insignificant in all cases other than when Mact is very small. WhenMact is sufficiently small for the eddy currents to contribute a significant fraction of the signal fromthe sample then the rotational hysteresis itself is vanishing and so the measurement is of limitedinterest. Thus it is generally not necessary to consider the effects of eddy currents when performinga measurement on a magnetic material in the RSCM.

The Total Signal With a Sample

The total voltage signal when a measurement is performed with a sample in the sample holder isthe sum of the individual components described, that is

V (t) = V ωH + V ωM + V ωe +nmax∑n=2

V nωK . (6.15)

If the higher frequency components of the signal are filtered out, or if the sample is isotropic, thenthe total measured voltage is

V ωmeas = V ωH + V ωM + V ωe . (6.16)

In most cases the last term can be neglected because for most measurements the eddy currentcontribution to the signal is negligible. Regardless of the relative significance of each term this isa simple sinusoidal signal at an angular frequency of ω.

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Figure 6.7: The ratio of the magnitude of the signal due to eddy currents, V ωe , to the magnitude of thesignal due to the activated component of magnetisation, V ωact, plotted as a function of both the magnitudeof the applied field, Ha, and of the magnitude of the activated component of magnetisation Mact. Theactivated component of magnetisation and the applied field are assumed to be aligned, which is the casethat produces the greatest eddy currents. The plot is for a rotational frequency of 9 Hz and a sampleradius of 5 mm diameter with ρ = 5 × 10−7Ωm. It can be seen that the relative contribution of eddycurrents is insignificant at all but very small amounts of activated magnetisation.

6.2.3 The Method of Measurement

The sinusoidal nature of the signal suggests that measurement with a lock-in amplifier is appro-priate. Indeed, since it is the amplitude and the phase of the signal that are of interest then alock-in amplifier is ideal as it provides a direct measurement of these parameters. Using a lock-inamplifier also provides a convenient means to filter the higher harmonics of the rotational frequency(possibly due to sample anisotropy), to ensure that only the component of the signal that reflectsthe rotational hysteresis is measured.

The lock-in amplifier was set up so that it measured only that component of the total voltage signalthat occurred at the rotation frequency of the shaft, with a time constant of 1 s. As shown in figure6.3 the reference frequency to the lock-in amplifier is supplied by a signal generator that sharesits clock with the signal generator triggering the stepper motor controller. It was not possibleto use the single frequency source as both the frequency reference and the motor trigger as thestepper motor controller requires a signal at 400 times the desired rotation frequency of the motor(since there are 400 steps per revolution). In the absence of miss-stepping of the motor, then, thepresent setup ensures that the reference frequency provided to the lock-in amplifier is the same asthe actual rotation frequency of the RSCM shaft. Lock-in amplifier measurements were made with

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maximum dynamic reserve and a 50 Hz low pass filter of fall off 24 dB/oct was employed.

Since it is necessary that the phase of the signals are measured relative to the field direction, then itis necessary to have a repeatable way in which to assign the field direction zero phase. This requiressetting φ0, of equations 6.1 - 6.13, to zero. This is achieved by measuring the voltage signal acrossthe relevant compensation coil, according to which compensation/sense coil pair is being used inthe measurement. This signal is induced solely by the field, and so may be designated as havingzero phase. All other phases are then measured by the lock-in amplifier relative to this phase. Thiszero phase assignment is usually performed in the maximum field possible (i.e. 1.4 × 106 Am−1)so that the signal across the compensation coil is at its greatest. If a sample is present then itcan then be rotationally demagnetised before commencement of the measurements. In the case ofmeasurements performed from a thermally demagnetised initial state the zero phase assignment isperformed in a small applied field, typically 4 × 104 Am−1. Care must be taken to ensure that theRSCM is rotated continuously at the same frequency from the time of the assignment of the zerophase till the conclusion of the measurements, so that the zero phase direction remains alignedwith the actual field. A check that this is still the case can be performed at the end of a series ofmeasurements on a sample (i.e. measurements over a range of field values), by again measuringthe phase of the compensation coil signal to verify that it is still zero.

Under the current measurement configuration the lock-in amplifier is controlled by a PC runningcustom software, written specifically for the RSCM. The software sets the appropriate gain anddynamic range on the lock-in amplifier for each individual measurement. The data from theindividual measurements consists of the mean signal amplitude and phase, measured at a constantfield strength over a period of 30 s at a sampling rate of 16 Hz. The sampling for each measurementstarts 40 s after a change of field strength, so as to allow the electronics time to equilibriate becauseof the long time constant used. For each measurement the software also measures the signal from aHall probe which is located between the poles of the electromagnet, so as to accurately determinethe field strength at which the measurement is performed. Statistical analysis of the sampled datais performed by the computer in order to estimate the minimum experimental uncertainty in thevalues obtained.

The electromagnet is powered by a power supply, which provides precise field ramp rates andconstant field stability of better than 1 part in 10000. This power supply is equipped for remotecomputer control, although this is yet to be implemented in the RSCM control software. At presentthe software alerts the user when a change of field is required and the change must be performedmanually.

Before measurement of a sample can be performed it is necessary that the signal due to the fieldbe well characterised at each field strength of interest. This is done be performing measurementswithout a sample. This is necessary so that the field signal, V ωH (Ha), may be vectorially subtractedfrom the total signal, V ωmeas(Ha), obtained from a measurement of a sample. The remaining signalis then that which is due to the presence of the sample. Once the field signal, V ωH (Ha) has beencharacterised over the range of Ha values used for measurement then this subtraction is performedautomatically by the software as measurements are performed on a sample. It is necessary that

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characterisation of the field signal be repeated at regular intervals so that any slight changes inthe measurement circuit characteristics (due to corrosion, strain, wear etc.) do not effect the finalsignal which is attributed to the sample.

The Measurement Procedure

The general measurement procedure, then, is:

1. Measurement of the signal without a sample, V ωH (Ha), as a function of field strength.

2. Measurement of the signal with a sample in the sample holder, V ωmeas(Ha), at the same fieldvalues as those used in step 1.

3. Subtraction of the field signal, V ωH (Ha), from the total signal, V ωmeas(Ha), to give V ωsamp(Ha) =[V ωsamp , φsamp], where

V ωsamp =√

(V ωmeas)2 + (V ωH )2 − 2V ωmeasV ωH cos(φmeas − φH) (6.17)

andφsamp = arcsin

(V ωHV ωsamp

sin(φmeas − φH))

+ φmeas − φH . (6.18)

V ωsamp(Ha) is the voltage signal that can be attributed to the presence of the sample.

If measurements are being made of a large conductive sample with a very small activated componentof magnetisation then it may be necessary to remove the eddy current component of V ωsamp. Inorder to separate V ωM (Ha) from V ωe (Ha) measurements of the sample at more than one angularfrequency, ω, must be made and then advantage taken of the fact that the magnitude of V ωe (Ha)depends on ω2 whereas, in the absence of rotational rate dependence of the magnetisation, themagnitude of V ωM (Ha) is linear in ω. The phase of both components of the signal are independentof ω and hence

V ω2M =

ω2

ω1V ω1M (6.19)

and

V ω2e =

(ω2

ω1

)2

V ω1e (6.20)

where ω1 and ω2 are two different frequencies at which the measurements are performed. Hencethe separation of the signal components may be achieved by

V ω1M =

V ω1samp −

(ω1ω2

)2

V ω2samp

(1− ω1ω2

)(6.21)

and

V ω1e =

V ω1samp − ω1

ω2V ω2samp

(1− ω2ω1

). (6.22)

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Since these are vector equations then the subtractions must be performed in a manner similar tothat achieved by equations 6.17 and 6.18.

Although in the majority of cases it is not necessary to perform such a separation of the signalcomponents, this procedure can be applied to most samples, regardless of whether or not they areconductive. In most samples this procedure will make a negligible difference to the signal attributedto the activated component of magnetisation of the sample, because the eddy currents will berelatively insignificant. This method of separation of signal components is unsuitable, however, insamples where the activated component of magnetisation is field rotation rate dependent.

Interpretation of the Signal Due to the Sample

Once the field signal has been removed from the total signal then the remaining signal, V ωM (Ha)is that due to the activated component of magnetisation of the sample. From equation 6.5, it isapparent that

µact(Ha) =V ωM (Ha)zω

(6.23)

andφM (Ha) = φsamp(Ha)− φcs − φ0(Ha) (6.24)

where φsamp is the phase of V ωsamp. The reason φ0 is written as a function of Ha is explained insection 6.2.6.

From equation 6.23 the magnitude of the activated component of magnetisation, Mact(Ha), issimply

Mact(Ha) =V ωM (Ha)zωVolsamp

, (6.25)

where Volsamp is the volume of the sample.

The phase of Mact(Ha) is simply φM (Ha).

6.2.4 Expressing the Data in Terms of Internal Field

Data of Mact(Ha) during rotational hysteresis allows Wr to be found as a function of either appliedfield, Ha (as is conventional), or of internal (intrinsic) field, Hi. This may be achieved using thetransformations

Hi =√

(DMact)2 +H2a − 2DMactHa cos |φM (Ha)| (6.26)

andφiM (Hi) = arcsin

(DMact sinφM (Ha)

Hi

)+ φM (Ha) (6.27)

which are derived from figure 6.8. The superscript ‘i’ in φiM is used to indicate that the phase isnow between Mact and Hi, rather than between Mact and Ha.

It is not possible to find Wr as a function of Hi from either torque or direct measurements of rota-

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Figure 6.8: A vector diagram showing the orientations of the activated component of the magnetisationrelative to the applied and internal fields.

tional hysteresis losses alone as they do not provide sufficient information about the magnetisationvector. Expression of the data as a function of Hi may be useful for comparison with alternatinghysteresis data, which is more commonly corrected for demagnetisation effects.

6.2.5 Calibration

Calibration of the device requires that the parameter z and the angle φcs be determined. z is afactor which depends on the geometry of the sample, its position within the sense coil, the numberof turns in the coil, the area enclosed by the coil and the shape of the coil. If all measurements areperformed using the same sense coil then the dependencies on the coil may be ignored. Likewise ifall measurements are performed on spherical samples which are centred within the sense coil thenthe dependence of z on the sample geometry and position may be ignored. φcs is the angle by whichthe orientation of the compensation and the sense coil differ. If the same sense/compensation coilpair is used for each measurement then φcs does not change.

In order to determine z and φcs measurements must be performed on samples for which µact andφM are well known. When a sample is rotated in a field that is sufficient to magnetically saturateit then µact is the entire sample saturation moment and φM is zero, since the moment aligns withthe field at saturation. Thus measurements of any sample for which the saturation moment hasbeen previously determined by an independent means, such as vibrating sample magnetometry,may be used to calibrate the RSCM.

If measurements are performed in a field sufficient to saturate the sample then the determinedV ωM (Ha) is constant with field. Hence z and φcs may be determined by

z =V ωsampµactω

(6.28)

andφcs = φsamp(Ha)− φ0(Ha), (6.29)

since φM = 0 at saturation.

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6.2.6 Characteristics of the Electromagnet

Although φ0, the phase of the field relative to the compensation coil, can be set to zero by followingthe procedure described in section 6.2.5 this assignment only takes place at a single value of field,typically the maximum that the electromagnet can produce. As the field magnitude is alteredduring the measurements then it is possible that the electromagnet pole faces may not uniformlychange polarisation. The electromagnet used with the RSCM is of high commercial quality andso it is likely that these effects are very small. If this does occur, however, then the apparentphase, φ0, of the field relative to the compensation coil will change. This will occur because of veryslight changes in the field direction and changes in the field magnitude between the compensationand sense coils. For this reason φ0 is included as a function of Ha in equation 6.24, so thatthe electromagnet characteristics do not effect the measurements. The form of φ0(Ha) may bedetermined from measurements of V ωH (Ha) by means of equation 6.2, where the constant arcsinterm may be determined at the field at which φ0 is set to zero.

6.2.7 Sensitivity and Precision

At the frequencies used (1 - 10 Hz) the lock-in amplifier has an amplitude sensitivity of 1µV anda phase sensitivity of 0.01.

This limitation in the sensitivity of the voltage measurements imposes limitations on the sensitivityof the values of Mact and φM determined using the RSCM. The sensitivity of measurements ofMact in a non conducting sample is given by

δMact =δV

zωVolsamp(6.30)

and so the sensitivity improves as the sample volume is increased and measurements are performedat higher frequency. For a typical sample radius (r = 2.2 mm) and measurement frequency (3 Hz)the theoretical sensitivity of a measurement of Mact is 3 × 102 Am−1.

The sensitivity of measurements of φM is limited by the difference in the with sample and with-out sample measurements. Hence the sensitivity of the φM measurements is, in the best case,0.03. When measurements of φM are made in samples where eddy currents make a significantcontribution to the signal then the sensitivity is less.

Following calibration using a Ni sphere with Ms = 4.96 × 103 Am−1 (determined previously byVSM measurements) the RSCM was used to measure the saturation magnetisation of a sample ofAlNiCo. The saturation magnetisation determined by the RSCM was 6.49 × 105 Am−1, comparedwith 6.63 × 105 Am−1 as determined by VSM measurements. By repeated measurement theprecision of the measurement of the moment was found to be ± 2 × 103 Am−1. The precision ofthe phase, φM was found to be ±0.3 or ± 0.0005 radians.

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6.2.8 Uncertainties in the Measured Quantities

In practice the amplitude noise within the bandwidth of the measurement was found to be ofthe order of ±10µV and the phase noise ±0.1. It is likely that the main source of this noiseis the slip rings through which the signals pass, as these will produce noise with components atthe rotational frequency. The uncertainty in the voltage measurements of the RSCM is, then,ultimately determined by the slip rings and is about ten times that of the theoretical sensitivity.Since 480 samples were made for each measurement with the RSCM then the standard error inthe actual measured voltages is very low and can approach the theoretical sensitivity.

The uncertainty of RSCM measurements were estimated by propagation of the standard error in thevoltage measurements through the vector subtraction of equations 6.17 and 6.18. The propagationof this uncertainty through the vector subtraction was performed in Mathematica. Since vectorsare involved the uncertainty in the voltage and the phase are related and so the final uncertaintyis amplified during the subtraction. The ultimate uncertainty in the magnetisation is given by:

δMact = δV ωM + δz + δV olsamp (6.31)

where the quantities in equation 6.31 are relative values.

The uncertainties in the values of Wr computed from the RSCM data were computed taking intoaccount the uncertainty in both Mact and φM .


6.3.1 Calibration

The RSCM was calibrated by the procedure described in section 6.2.5 using measurements of aNickel sphere of diameter 5.11 mm, which had been previously characterised by VSM measurementsas having a saturation moment of 3.27 × 10−2 Am2. Measurements were performed at 1 Hz (i.e.ω = 2π Hz), 3 Hz and 9 Hz.

After calibration Mact(Ha) and φM (Ha) were measured for the Ni sample. These data are shownin figure 6.9. As expected the phase of the magnetisation relative to the field is close to zero atall field strengths, particularly at those sufficient to saturate the sample. Also plotted in figure 6.9is the initial magnetisation curve of the Ni sample, as measured from a rotationally demagnetisedstate in a VSM. The excellent agreement between the initial magnetisation curve and the form ofMact(Ha) determined by the RSCM suggests that the calibration is valid over the entire range ofmagnetic moments exhibited by the Ni sample.

The calibration constant z was determined as (1.69 ± 0.01) × 10−2 and φcs 1.58±0.02 radians.

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Figure 6.9: The magnitude (crosses) and phase (diamonds) of the magnetisation of Ni as measured bythe RSCM. The uncertainty in the phase values is not shown and is ±0.02. Also shown in the plot is themagnetisation measured by the VSM (solid line).

6.3.2 Rotational Hysteresis Measurements

AlNiCo

Mact(Ha) and φM (Ha) for the AlNiCo 7 sample (see chapter 3), measured by the RSCM at arotational frequency of 3 Hz, are shown in figure 6.10, along with the initial magnetisation curve,measured from a rotationally demagnetised state using a VSM. It is noted that when plottedagainst applied field the two magnetisation curves are in close agreement. The angle between Mact

and Ha increases to its greatest value of 0.17±0.005 rad (≈ 10) at an applied field strength ofapproximately 2.2 × 105Am−1 and then decreases back to zero as the sample saturates.

From the data of Mact(Ha) and φM (Ha) it is possible to calculate the rotational hysteresis lossper cycle, Wr(Ha), by

Wr(Ha) = 2πµ0Mact(Ha)Ha sinφM (Ha) (6.32)

This data is presented in figure 6.11 along with the form of Wr(Ha) determined for the same AlNiCosample by DABAAM measurmements[139] (see chapter 5). The excellent agreement between thedata obtained by the RSCM and that obtained by the DABAAM suggests that the RSCM methodis reliable.

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Figure 6.10: The activated component of magnetisation vector, [Mact , φM ], during rotational hysteresis

at 3 Hz in AlNiCo, as measured using the RSCM. Also shown (as the solid curve) is the initial magnetisation

curve for AlNiCo, measured using a VSM, from an initially rotationally demagnetised state.

Figure 6.11: The rotational hysteresis loss for AlNiCo as measured by the RSCM (squares) and the

DABAAM (crosses).

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Wr(Hi) for the AlNiCo material measured is shown in figure 6.12. The transformation of Ha toHi was achieved using equation 6.26. The data of Mact and φM may be similarly expressed as afunction of Hi, using the transformations given by equations 6.26 and 6.27. These data are shownin figure 6.13.

It is apparent from figure 6.13 that the transformation to internal field slightly alters the agreementbetween the alternating field and rotational field magnetisation data. This is because the trans-formation of magnitude of the rotating field, given by equation 6.26, is different from that used tocorrect the alternating hysteresis measurements when φM 6= 0. When φM = 0 then equation 6.26reduces to the (scalar) equation of Hi = Ha−DM and alternating field and rotating field data arecorrected for demagnetisation effects by the same linear transformation.

Similarly the phase of the magnetisation is altered by transformation to internal field. This isbecause a different phase is expressed in figure 6.13 than in figure 6.10. The phase is now relativeto the internal field, rather than the applied field. This is expressed by the superscript i in the newnotation, φiM , for the phase of the magnetisation. φiM will always be larger than φM because thedemagnetising field causes the internal field to lead the applied field, which the magnetisation lags(see figure 6.8). In the case of the AlNiCo sample studied here the maximum magnitude of φiM is0.335±0.007 radians (≈ 19) compared to the maximum magnitude of φM of 0.17±0.005 radians. Itis likely that this relative difference will be larger in a sample such as AlNiCo than in most materialsbecause of the very large magnetisation in relatively small fields leading to correspondingly largedemagnetisation effects.

The value of the rotational hysteresis integral, RH , (see section 1.3.4) may be computed from thedata obtained by the RSCM. Since RSCM measurements allow Wr to be expressed as either afunction of Ha or Hi then it is possible to perform the integration with respect to either 1/Ha or1/Hi, this is generally not possible with torque magnetometer measurements. The two approachesyield different values for the rotational hysteresis integral. The reasons for this difference andits’ consequences are discussed further in chapter 8. Theoretical values of RH , such as those intable 1.1 are typically computed for models of magnetisation that do not take account sampledemagnetisation effects. As such, experimentally determined values of RH where the integration isperformed with respect to 1/Hi are more suitable for comparison with theoretical values than thoseobtained when the integration is performed with respect to 1/Ha. Here, values of RH obtained byintegration with respect to 1/Hi are denoted as RiH and those obtained by integration with respectto 1/Ha as RaH . It should be noted that the values of RH presented in chapter 5 and experimentalvalues (based on measurements of actual materials) presented in the literature are values of RaH .

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Figure 6.12: The rotational hysteresis loss as a function of internal (intrinsic) field for AlNiCo, as measured

by the RSCM.

Figure 6.13: The activated component of magnetisation vector, [Mact, φiM ], as a function of internal

(intrinsic) field during rotational hysteresis at 3 Hz in AlNiCo, as measured using the RSCM. Also shown

(as the solid curve) is the initial magnetisation curve for AlNiCo, measured using a VSM.

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The value of RH computed from the RSCM Wr(Hi) data is RiH = 1.02 ± 0.12. When computedusing Wr(Ha) data RaH = 0.58 ± 0.06. This value is very close to the value of RaH = 0.57 ± 0.05obtained from DABAAM measurements of the same sample. Given that the grains within theAlNiCo are large and able to contain domain walls then comparison with the results of modellingby Elk et al. for domain wall pinning[77] may be appropriate. Elk et al. suggest RH ≈ 3.2when domain wall pinning controls the rotational hysteresis in an isotropic material. The value ofRiH = 1.02± 0.12 obtained here, then, is surprisingly low. Indeed, an RH value such as 1.0 wouldbe traditionally interpreted as being indicative of an incoherent rotation magnetisation mechanism(see for example [120, 118]), since it lies in the 0.42 - π of the Shtrikman - Treves theory (see table1.1). Comparison with such a model, though, is likely to be inappropriate for this multi-domaingrain material.

Ex1

Figure 6.14 shows the form of Mact(Ha) and φM (Ha) as determined by the RSCM for Ex1, whichis described in appendix B. This measurement demonstrates a difficulty in using the RSCM tomeasure highly polished spherical samples of a material that exhibits a large rotational hysteresistorque. At an applied field of approximately 7.8 × 105Am−1 the apparent phase of the magnetisa-tion changes suddenly to lead rather than lag the field. This is not an actual occurrence but ratheris symptomatic of the rotational hysteresis torque on the sample becoming so great that the resinin which the sample is set is unable to prevent rotation of the sample within the sample bucket.

Figure 6.15 shows the magnitude of the rotational hysteresis torque τM (Ha) for the Ex1 sphere,as determined by DABAAM measurements. At a field of 7.8 × 105Am−1 the rotational hysteresistorque is τM = 5.4 × 10−3 Nm. This is the minimum (critical) torque that is required to rotatethe sample within the sample bucket. The torque remains greater than this critical value until thefield reaches about 1.05 × 106Am−1 and so the sphere rotates within the sample bucket in thefield range 7.8 × 105Am−1 to 1.05 × 106Am−1. This is consistent with the large changes in theapparent φM that occur in figure 6.14 in this field range. During this time the sample oscillatesin phase relative to the field between the (larger) angle at which M ×H is equal to the maximumstatic frictional torque on the sample and the (smaller) angle at which it is equal to the kineticfrictional torque. The torque acting on the sample during this cycle is shown in figure 6.16. Thespecific coordinates of the points A, B and C will vary slightly (and unpredictably) with orientationand time. The sample only rotates relative to the sample bucket during the step C → A, whichwill be a relatively quick slip. For most of the time (A → B → C) the sample remains stationaryin the frame of reference of the sample bucket. With each slip the value of φ0 is altered in anunpredictable way because the frictional torques on the sample are not necessarily uniform withorientation. Thus φM cannot be determined. The measured value of Mact, however, is reliableas the value of Mact is independent of φM and is unaffected by the higher frequency componentadded to the signal by the slipping of the sample.

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Figure 6.14: The activated component of magnetisation vector, [Mact , φM ], during rotational hysteresis

at 3 Hz in Ex1, as measured using the RSCM. Also shown (as the solid curve) is the initial magnetisation

curve, measured using a VSM. The sudden changes in φM are not physical, as explained in the text.

Figure 6.15: The rotational hysteresis torque acting on the Ex1 sample during rotational hysteresis, as

determined by DABAAM measurements.

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Figure 6.16:The frictional torque acting on a sample in the RSCM when it is slipping.

Data of Mact(Ha) for measurements that are subject to sample slipping may be combined with dataof Wr(Ha), obtained by DABAAM measurements, to obtain the form of φM (Ha), using equation6.32. Correction for demagnetisation effects may then be applied. Figure 6.17 shows the calculatedform of φM (Hi) for Ex1. The uncertainty in φM now arises from the uncertainty in both the RSCMand DABAAM measurements. Excellent agreement is seen between the calculated form of φM (Ha)and the measured form prior to the onset of sample slipping. This is indicative of the rotationalhysteresis being very similar during both the RSCM and DABAAM measurements.

The procedure of calculating Mact(Ha) assumes (a) that the DABAAM data is not also subject toslipping and (b) that the actual rotational hysteresis is the same in each case. The first assump-tion is readily satisfied as smaller samples are used for DABAAM measurements than for RSCMmeasurements so that the moment of inertia of the device is sufficient to allow enough revolutionsof the sample for measurement. Smaller samples have a higher surface area to volume ratio and soare more easily held by the resin. In the case of the RSCM larger samples are desirable so as toenhance the signal strength and so signal to noise ratio. The assumption that the actual rotationalhysteresis (per unit volume) is the same during both measurements is satisfied by using the samematerial and performing the measurements over similar time scales, so as to remove any differ-ences due to possible viscosity effects. Using absolutely identical time scales is not possible sincethe DABAAM measurements are performed over a range of frequencies. Nevertheless differencesin time dependent effects are likely to be negligible over the short time scales involved in bothmethods.

Figure 6.18 shows Wr(Hi) for Ex1. This plot represents a correction for demagnetisation effects ofthe Wr(Ha) data obtained by DABAAM measurements.

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Figure 6.17: The activated component of magnetisation vector, [Mact , φM ], during rotational hystere-sis at 3 Hz in Ex1, as measured using the RSCM. The form of Mact is obtained directly from RSCMmeasurements and φM from a combination of RSCM and DABAAM measurements.

Figure 6.18:The rotational hysteresis loss, Wr, as a function of internal field, Hi, in Ex1.

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Figure 6.19: The activated component of magnetisation vector, [Mact , φM ], during rotational hysteresisat 3 Hz in SrFe, as measured using the RSCM. Also shown (as the solid curve) is the initial magnetisa-tion curve, measured using a VSM, from an initially rotationally demagnetised state. The form of Mact

is obtained directly from RSCM measurements and φM from a combination of RSCM and DABAAMmeasurements.

The values of the rotational hysteresis integral for Ex1 are RiH = 2.44 ± 0.43 and RaH = 1.65 ±0.25. The relatively high uncertainties in the values is caused by the non-vanishing nature ofthe rotational hysteresis at the highest measurement field. The large differences between RiHand RaH are indicative of the large demagnetisation effects in this material, since it achieves highmagnetisation in relatively low fields. The value of RiH suggest that domain wall processes controlthe magnetisation in this material.

SrFe

The rotational hysteresis properties of a sample of commercial SrFe (described in appendix B) areshown in figures 6.19 and 6.20. As for Ex1, these plots were produced using a combination ofDABAAM and RSCM measurements. Once again there is excellent agreement between the formof Mact(H) and the initial magnetisation curve. This is particularly striking because of the kinkin both curves.

The value of the rotational hysteresis integral calculated from these data is RiH = 0.75 ± 0.07(RaH = 0.58 ± 0.05). Since the SrFe material consists of single domain grains (see section B.2)then comparison with the values of the various models of rotation in single domain particles isappropriate. The value of RiH = 0.75 ± 0.07 is within the range of 0.67 − π of Ishii and Sato’smodel of fanning in touching spheres[76] and the 0.42 − π of the Shtrikman and Treves theory of

137


Figure 6.20:The rotational hysteresis loss, Wr, as a function of internal field, Hi, for SrFe

curling. This suggests that the mechanism of magnetisation change is some form of incoherentrotation.

Interpretation of Mact and φM

In each of the materials studied, the measured form of Mact(Hi) during rotational hysteresis issimilar to, but always greater than, Mtotal(Hi) along the initial magnetisation curve, for measure-ments made in a linearly applied field. The time scales over which these two forms of measurementsare made are quite different and so the possibility of viscosity being responsible for this differenceshould be considered. The measurements of rotational hysteresis are made over short time intervals(since the field is constantly rotating at a reasonably high rate). In contrast the VSM measure-ments are made with a slow rate of change of field and so will be more effected by viscosity. Theeffect of viscosity on the initial magnetisation curve is to increase the magnetisation which wouldcause the initial magnetisation curve to be above the curve of Mact, which is the opposite to whatis observed here. The difference is not being caused by magnetic viscosity.

The larger magnetisation in the rotational hysteresis case may be a consequence of a greaterproportion of grains seeing a larger component of the field during some point in the sample’srotation and so becoming activated. This is a consequence of the field during rotational hysteresissampling a range of orientations, rather than a single orientation, as with alternating hysteresis. Ifit is assumed that all grains (or regions that are able to be individually activated) are of the samesize then interpretation of Mact and φM is made easier. The activated component of magnetisation,Mact, may interpreted as a measure of the fraction of grains activated by the field at any instant

138


during the rotational hysteresis (the actual fraction being ∼ Mact/MS). Wr is proportional tothe average effective activation energy of the activated grains and φM is simply the angle requiredbetween M and Ha such that Wr = 2πM ×Ha.

6.4 Conclusions

A device that measures the activated component of the magnetisation vector during rotationalhysteresis has been successfully implemented. The device may be useful for measuring the rota-tional hysteresis properties of spherical magnetic samples over the frequency range of 1 - 9 Hz.Measurement at different frequencies allows the determination (and removal) of eddy current ef-fects, although in most samples these are relatively insignificant. The rotational hysteresis losses ofAlNiCo determined by RSCM measurements were found to be in excellent agreement to those deter-mined by DABAAM measurements, suggesting that both these independently calibrated methodsare reliable.

Measurement of the magnetisation vector during rotational hysteresis allows the measurementsto be expressed as functions of the internal field strength, Hi. This may be useful in comparingsamples of different geometries. Expression of the data versus Hi is not possible for measurementsmade by torque magnetometry or by methods that directly measure the rotational hysteresis loss.

Knowledge of Wr(Hi) allows the rotational hysteresis integral to be computed by integration withrespect to 1/Hi. These values are greater than those computed by integration with respect to 1/Ha

and are more suitable for comparison with values obtained from modelled data where demagneti-sation effects are not considered. The values of the rotational hysteresis integral computed for thedata suggest that all of the materials examined change magnetisation incoherently.

In each material the measured form of Mact(Hi) during rotational hysteresis was similar to butgreater than, Mtotal(Hi) along the initial magnetisation curve, for measurements made in a linearlyapplied field. The larger magnetisation during rotational hysteresis is not a consequence of timedependence and was interpreted as being due to the field being able to activate grains with awider distribution of anisotropy orientations. The similar form of M in both the rotational andalternating hysteresis cases also suggests that similar mechanisms occur in the two cases.

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140

7. Modelling of Rotational Hysteresis

RROTATIONAL and alternating hysteresis measurements were simulated using a model ofmagnetisation change based on the Stoner-Wohlfarth model of coherent rotation, modified

to include different forms of anisotropy and interactions. Such a model is not intended to be rep-resentative of any of the materials studied within this work, rather it is studied as it may allowmore general trends in the dependence of rotational hysteresis on interactions and distributionsin anisotropy to be examined. It was found that both the form of Wr(H) and the value of therotational hysteresis integral depend on interactions, the geometry of the anisotropy and distribu-tions in the activation energies. Such predictions are consistent with experimental data reportedin the literature. The results suggest that interactions within a material must be considered wheninterpreting the value of the rotational hysteresis integral.

7.1 Introduction

Various models of magnetisation change through irreversible rotation have been developed. Irre-versible coherent rotation was first studied by Stoner and Wohlfarth [3] and has subsequently beenexamined by many authors, some of whom who have extended the Stoner and Wohlfarth model toinclude time dependence[84, 167] and interactions[5]. The model has been discussed previously inchapter 1.

Since interactions and distributions in anisotropy are significant in most permanent magnet mate-rials then their effect on rotational hysteresis (and the value of RH) needs to be examined. Timedependence may also significantly effect both the alternating and rotational hysteresis of real mate-rials and so its effect on the value of RH should also be considered. In this chapter the effects of themodel parameters on the rotational hysteresis loss of a randomly orientated interacting ensembleof particles, that change their magnetisation by coherent rotation, are examined. In particular,this systems allows the effects of interactions and distributions in anisotropy on the rotationalhysteresis integral to be examined. This model is not representative of any of the materials studiedin this work but rather allows more general trends about the effects of material parameters to bepredicted.

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7.2 Description of the Model

The model of magnetisation change by irreversible coherent rotation was implemented as a C++Monte-Carlo simulation. During this study the computer program was used to evolve 1000 ran-domly oriented regions (in three dimensional space). As implemented the model allows for a numberof modifications to the original Stoner-Wohlfarth model [3], specifically:

1. Regions may be modelled with either uniaxial, planar or cubic anisotropy. The programallows for each type of anisotropy to be present simultaneously, although during this studyonly ensembles of homogeneous anisotropy geometry were considered.

2. Distributions in the value of the anisotropy constants are allowed.

3. Interactions between the regions are modelled. Three interaction regimes were catered for:mean field interactions, true dipolar interactions and mean field interactions with local ran-dom perturbations. The application of periodic boundary conditions was also catered for.During this study only the case of mean field interactions was considered, since calculatingtrue dipolar interactions is O(n2) and so was found to be prohibitively slow.

For a measurement at an applied field the individual magnetisations of the regions were successivelyevolved by rotation of 1 towards their lowest energy orientation. For each region the direction ofthis orientation is determined by the combined effects of the region anisotropy, the applied fieldand the interaction field. The regions were evolved in a random order until at least 95% of themhad not changed magnetisation during their previous 5 evolution steps. At this point the systemwas considered to be stable and so the magnetisation recorded and then the applied field changedfor the commencement of the next measurement.

Interactions between regions were simulated using the mean field approximation proposed by Ather-ton and Beattie[5]. In this approach the effect of interactions are simulated as being equivalent toa mean field which is proportional to the net magnetisation of the ensemble, that is

Hint = γM (7.1)

where γ is constant. During this study a range of values for γ of between -1 and 1 were used. Thisapproach provides an imperfect but reasonable representation of interaction effects[5] and was usedbecause of the high computation speeds it allows.

Distribution in the anisotropy constants were simulated as being approximately log-normal, withthe mean and standard deviation of the constants being user defined. For uniaxial anisotropy onlypositive constants were used during this study. For cubic anisotropy the constants were either allpositive or all negative. In the cubic case positive anisotropy constants result in the easy axesbeing normal to the faces of the cube (three axes) and a negative constant results in the easy axesbeing through the corners of the cube (four axes). Where the standard deviation of a distributionwas such that the distribution contained both positive and negative constants then the distributionwas truncated at zero and renormalised. This alters the form of the distribution slightly, although

142

Modelling of Rotational Hysteresis

it is believed that for the range of mean values used during this study the renormalised forms wereall still physically reasonable. During this study K1 = ±0.5 and σK1 ranged from 0 to 0.5. Thesecond order anisotropy coefficient was set to zero in all cases.

The pseudo-random number generator used during the simulation was the KISS generator whichwas developed by the Department of Mathematics at the University of Western Australia. Itwas initially seeded with the computer system time. This generator has been shown to generatenumbers of a high degree of randomness and has a very large period of 1032, which is much morethan the number of random samples required during a simulation.

7.3 Experimental Procedure

The program contained routines to simulate the measurement of hysteresis loops, IRM/DCD testsand rotational hysteresis. For each type of measurement the procedure was analogous to theprocedure used for the measurement in a real material. The only significant difference was thenecessarily discrete nature of the field steps used in the modelled measurements. For most mea-surements the same field path as used in real measurements was followed, even to the extent ofperforming measurements in an SSL regime (see section 2.2.1).

7.4 Uncertainty

The uncertainty in the data collected from the simulation is considered to be negligible because ofthe reasonably large number of regions that are evolved and the exhaustive nature of the evolutiondescribed in section 7.2. Repeated measurements made using the model confirmed that this wasthe case, showing excellent reproducibility.


7.5.1 Verification of the Implementation of the Model

In the simple case of a single value ofHK and no interactions (i.e. the Stoner-Wohlfarth model) thenthe model was verified to produce both alternating and rotational hysteresis data corresponding tothat in the literature, as may be seen by comparing figures 7.1 and 7.2 with [3]. Furthermore theeffect of the mean field interactions of the hysteresis loops was seen to be in agreement with theresults of Atherton et al., from whom the method is derived [5]. These results are shown in figure7.3, which shows loops for different values of the interaction parameter. Such agreement allows fora high degree of confidence in the output of the implementation of the model.

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Figure 7.1:The alternating hysteresis of the model with no interactions and a single value of HK

Figure 7.2:The rotational hysteresis loss of the model with no interactions and a single value of HK .

144


Figure 7.3: The alternating hysteresis of the model in the case of a single value of HK and different values

of interactions, as determined by γ.

7.5.2 The Effect of Interactions

Figure 7.4 shows the rotational hysteresis loss for different values of the mean field interactionparameter, γ.

It is apparent that as the interactions become more negative (demagnetising) then the rotationalhysteresis losses occur at greater fields. Similarly as the interactions become more positive then therotational hysteresis occurs at lower fields. The behaviour of the fields at which rotational hysteresislosses occur, increasing as interactions become more negative (demagnetising) and decreasing asthey become more positive (magnetising), is the opposite to that of the coercivity, for alternatinghysteresis measurements. This may be seen by comparison with figure 7.3, which shows thatthe coercivity of the major alternating hysteresis loop decreases as the interactions become morenegative (demagnetising) and increases as they become more positive (magnetising).

The effect of interactions is not just to change the field at which the peak in the rotational hysteresislosses occur but to shift (and change the shape of) the entire Wr curve. The cause of this maybe understood by examining the magnetisation during the rotational hysteresis, as shown in figure7.5. Demagnetising interactions (γ < 0) serve to miss-align the particle moments and so reducethe overall magnetisation. IRM measurements in this system reveal that irreversible magnetisationchanges do not occur until M = 0.42, when the reversible magnetisation starts to be convertedinto irreversible magnetisation. Since the overall magnetisation is reduced by negative interactions

145


Figure 7.4: The rotational hysteresis loss vs field in the model of coherent rotation. The different curvescorrespond to different values of the interaction parameter, γ. HK = 1 for each curve.

then the irreversible magnetisation processes do not occur until higher fields are reached. Onlythe irreversible component of the net magnetisation can be at an angle to the internal field (on amacroscopic scale) and so only the irreversible magnetisation can contribute to rotational hysteresis.Since this does not occur until higher fields then the whole Wr curve is shifted to higher fields bythe negative interactions. The opposite occurs when there are magnetising interactions (γ > 0),which cause irreversible magnetisation processes to occur in smaller fields and so shifts the Wr

curve to lower fields. In these cases the maximum magnitude of Wr is also decreased. This isbecause

Wr = 2πMHsinφM , (7.2)

and the field at which the maximum occurs is lower but M and φM are the same as before.

The argument that demagnetising interactions shift the irreversible magnetisation changes alongthe magnetisation curve to higher fields is quite general and does not just apply to the modelstudied here. Thus it is likely that similar trends in the evolution of the form of the rotationalhysteresis loss with interactions will be observed in real materials.

This dependence of the field at which the rotational hysteresis peaks on interactions is the inverseof the dependence of the (alternating hysteresis) coercivity on the interactions. The coercivityof the system increases slightly (relative to the no interactions case) as the interactions becomemore positive and decreases as they become more negative. Again this is more generally the casein real materials. Thus in the presence of interactions the coercivity and the field at which the

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Figure 7.5: The active component of magnetisation vs field during rotational hysteresis in the modelof coherent rotation. The different curves correspond to different values of the interaction parameter, γ.HK = 1 for each curve.

rotational hysteresis peaks are necessarily different. For systems with negative interactions (suchas magnetostatic interactions) the peak in the rotational hysteresis will occur at fields greater thanthe coercivity. This is observed in the vast majority of real materials (see for example chapter 5).

The rotational hysteresis integral, RH (equation 1.34), may be calculated from the data shown infigure 7.4 and expressed as a function of the interaction parameter, as shown in figure 7.6.

Clearly in this system RH is strongly effected by the presence of interactions. The value of therotational hysteresis integral increases as the strength of (negative) demagnetising interactions in-creases and decreases as the strength of (positive) magnetising interactions increases. It is believedthat, like the behaviour of HMax(Wr), this will also be more generally the case, for real materi-als and other mechanisms of magnetisation change. Demagnetising interactions mean that largerfields are needed to produce magnetisation than in non interacting cases. This, in turn, means thatrotational hysteresis losses occur at higher fields and so the values of RH are lower than in the nointeractions case. Similarly magnetising interactions produce greater magnetisation in lesser fieldsand so greater rotational hysteresis loss in low fields, which leads to greater values of RH .

The fact that RH depends on interactions is in stark contrast with the early claims of Jacobs [9],that the value of the rotational hysteresis integral is independent of interactions. Jacobs’ originalclaim that RH does not depend on interactions is not supported in his paper by way of any furtherexplanation or evidence of why this should be the case. No such subsequent explanation has beenfound in the literature either, despite repeated citation of and reliance on Jacob’s claims (see, for

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Figure 7.6: The rotational hysteresis integral vs the interaction parameter in the model of coherentrotation. A single value of the anisotropy field, HFK = 1, is used.

example [72, 121]). The effect of interactions on the value of RH has been noted by other authors(see for example [75]) but has not been widely acknowledged. Shtrikman and Wohlfarth haveshown, on the basis of theoretical arguments, that the value of RH is only strictly indicative of themechanisms of magnetisation change where interactions can be neglected [74]. Thus the dependenceof the value of RH on interactions as observed here extends to other theoretical models.

Experimental evidence that RH depends on interactions in real materials is also to be found in theliterature. Bottoni has measured the value of the rotational hysteresis integral at different packingdensities in 15 different samples of magnetic recording particles [72, 117]. In [72] interactions werecharacterised by ∆M and Interaction Field Factor (IFF) measurements and it was shown thatas the magnitude of the (negative, demagnetising) interactions increased then the value of RHdecreased. Similarly in [117] it was shown that as packing density increased (and so the negativemagnetostatic interactions increased) then the value of the rotational hysteresis integral decreased.These findings are consistent with the results here. Bottoni states that the rotational hysteresisintegral is ‘independent of packing’ [72] and so interprets this decrease as being indicative of themechanisms of magnetisation change becoming more coherent with increasing interactions andparticle aggregation. The result here, though, suggests that the decrease in RH as the interactionsbecome more negative may be caused by the interactions themselves. It is likely that both effectscontribute to the decrease in RH with increasing packing density.

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Figure 7.7: The rotational hysteresis loss, Wr(H), of the model of coherent rotation for different standarddeviations of the anisotropy field distribution.

7.5.3 The Effect of a Distribution of Anisotropy

Figure 7.7 shows the rotational hysteresis loss for non-interacting isotropic ensembles of uniaxialparticles with approximate log-normal probability distribution functions of anisotropy field. Eachof the ensembles has approximately the same mean anisotropy field (HK ≈ 1) but different standarddeviations, as indicated in the figure. The small differences in the means is a consequence of thetruncation and renormalisation of the PDF to avoid negative values, as discussed in section 7.2.

It is apparent that as the width of the anisotropy field distribution increases then the field at whichthe peak in Wr occurs also increases. The position of the peak is well represented by the empiricalequation

HMax(Wr) =HK

2+ σHK . (7.3)

Bottoni has studied the effect of the strength of anisotropy on rotational hysteresis in magneticrecording particles [114] and has obtained experimental data that is consistent with the form ofthis equation. The shape of the Wr(H) curves obtained by Bottoni in [114] suggests that the widthof the anisotropy field in the materials studied was reasonably small.

Figure 7.8 shows the alternating hysteresis of this system of uniaxial particles with different widthsin the anisotropy field distribution. It is apparent that the coercivity decreases with increasingwidth of the anisotropy field distribution. This is consistent with the Gerlach principle, which

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Figure 7.8: The major alternating hysteresis loop of the model of coherent rotation for different valuesof the standard distribution of the anisotropy field distribution and no interactions.

states that when a magnetic material has a distribution of coercivities then the lower coercivitycomponents dominate the magnetic properties [99]. In contrast, the field at which the peak in therotational hysteresis losses occurs increases with increasing width of the anisotropy field distribu-tion. For systems with a distribution of anisotropy fields then the components with the higheranisotropy fields dominate the rotational hysteresis loss.

Figure 7.9 shows the value of the rotational hysteresis integral, RH , as a function of the interactionparameter for different values of standard deviation of the anisotropy field distribution, σHK . It isapparent that the nature of the dependence of RH on the width of the anisotropy field distributiondepends on the interactions. When there are no interactions then RH is independent of the widthof the distribution, despite the dramatically different form of Wr(H) for different widths. Similarlywhen interactions are negative then the value of RH is independent of the width of the anisotropyfield distribution. When interactions are positive, however, then the value of RH increases as thewidth of the anisotropy field distribution is increased.

7.5.4 The Effect of the Geometry of the Anisotropy

Figure 7.10 shows the form of Wr(H) for different geometries of anisotropy, in the absence ofinteractions or any distribution in the strength of the anisotropy. It is apparent that as thenumber of easy axes increases the field at which the rotational hysteresis peaks decreases. It itexpected that this trend would hold for even larger numbers of easy axes, which may result fromshape anisotropy in irregularly shaped particles or grains.

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Figure 7.9: The value of the rotational hysteresis integral, RH , as a function of interactions (γ) andstandard deviation of the anisotropy field distribution (labeled σK here) for a system of uniaxial particlesthat change magnetisation by coherent rotation.

Figure 7.10:The rotational hysteresis loss, Wr(H), for systems with different geometries of anisotropy.

151


It is noted that planar and uniaxial anisotropy produce the same rotational hysteresis, since theyare simply the inverse of each other. In uniaxial anisotropy the magnetisation experiences a torquepulling it towards the easy axis and in planar anisotropy the torque pushes the magnetisationaway from the hard axis. Unidirectional anisotropy does not produce rotational hysteresis sincethe torque acting on the magnetisation for one half of the cycle is the opposite of that acting forthe other half of the cycle.

7.5.5 The Effect of Time Dependence

The model studied here does not simulate time dependence but its effect may be predicted. Timedependence indicates thermal activation over energy barriers within a system to allow evolution toa lower energy state. Magnetic viscosity serves to reduce hysteresis loss in alternating hysteresisand so may be expected to do the same in rotational hysteresis. Thus time dependence will reducethe magnitude of Wr over the entire field range over which it occurs. The shift in the field atwhich the rotational hysteresis peaks will depend on the field dependence of the viscosity. Thevalue of the rotational hysteresis integral will necessarily be reduced by viscosity since it will serveto reduce the values of Wr at all fields.

7.6 Conclusions

In the system modelled here the rotational hysteresis loss and the value of the rotational hysteresisintegral, RH , depend on both interactions within the system and on distributions in anisotropy.Demagnetising interactions serve to increase the fields at which rotational hysteresis occurs andconsequently to decrease the value of the rotational hysteresis integral. The opposite is true formagnetisating interactions.

The effects of distributions in activation energy (anisotropy fields) depends on the interactions. Itmay be seen in figure 7.9 that in the presence of magnetising interactions the effect is pronounced,serving to increase the value of RH . The effect is small when the interactions are demagnetising,with a distribution in activation energies serving to very slightly reduce the value of RH .

Both alternating and rotational hysteresis losses are also influenced by the geometry of the anisotropyof particles, or regions within a material. In this case, though, the dependence of the field at whichthe peak in Wr occurs and the coercivity is similar. As the number of easy axes in a particleincreases then both the field at which the peak in the rotational hysteresis losses occurs and thecoercivity decrease.

Clearly this simple model is not representative of many real materials or any of the materialsstudied within this work. It is likely that the model studied here would only be applicable to verysmall particle systems. Nevertheless it is reasonable to expect that the trends in the dependence ofthe rotational hysteresis observed here will also apply to other models of magnetisation change andso to real materials. Other mechanisms of magnetisation change have different state dependence

152


of activation energy but the effects of distributions in those activation energies and of interactionsmodifying the internal field by which activations are stimulated are likely to be similar. Thisis certainly the case with the dependence of the coercivity on interactions and distributions inanisotropy and so is likely to also be the case with the rotational hysteresis properties. It wouldbe interesting to apply such a study to other, more representative, models of magnetisation changeto confirm that this is the case.

The overwhelming evidence in the literature is that in real materials the field at which rotationalhysteresis losses peak is always greater than the coercivity (sometimes much greater). This maybe explained by negative (magnetostatic), demagnetising interactions in real materials decreasingthe coercivity (relative to the no interactions case) but increasing the field at which the rotationalhysteresis losses peak. This occurs because demagnetising interactions increase the field requiredfor irreversible magnetisation processes to take place and so increase the field at which rotationalhysteresis losses occur. Since interactions are demagnetising in most materials then the effectsof distributions in activation energies is likely to be less but will also serve to increase the fieldat which rotational hysteresis losses peak. A useful consequence of this is that in cases wherethe distribution of anisotropy fields is known to be very narrow then the difference between thecoercive field and the field at which the rotational hysteresis peaks may be an indicator of therelative strength of the interactions within a material.

It is likely that geometric demagnetisation effects in real materials will decrease the measuredvalue of the rotational hysteresis integral in the same manner as demagnetising interactions. Thisis particularly true for the system studied here since the interactions were simulated as beingproportional to a mean field. This is consistent with the differences between the values of therotational hysteresis integral when calculated by integration with respect to 1/Ha and those wherethe integration is performed with respect to 1/Hi, as observed in chapter 6. Thus it is likelythat most values of RH quoted in the literature are artificially low, since torque magnetometermeasurements do not allow Wr to be measured as a function of Hi.

Magnetic viscosity was not simulated here but can be expected to reduce the rotational hysteresislosses at all fields and so reduce the value of the rotational hysteresis integral. This effect is likely tobe small in the measurements of real materials made using the DABAAM and the RSCM since theyinvolve measurements made with high field rotation rates. Torque magnetometry measurements,however, are made over long time frames and so are likely to be more effected by time dependenceeffects, giving lower values of the rotational hysteresis integral.

The dependencies of the value of the rotational hysteresis integral on material parameters mean thatthey must be considered when interpreting its’ value in terms of the mechanisms of magnetisationoccurring within a material. This is consistent with the cautions of Shtrikman and Wohlfarth[74]but inconsistent with many such studies reported in the literature. It is expected that the values ofRH may be seen to be consistent with the mechanisms characterised by other means and so rota-tional hysteresis measurements may be useful in providing further evidence for a characterisation.Interpretation of measured values of the rotational hysteresis integral should take into accountthe possible effects of interactions, distributions in activation energies, demagnetisation effects and

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time dependence.

154

8. Interpretation of the Field Dependence

of Rotational Hysteresis Losses

THE rotational hysteresis losses, Wr(Ha), of the materials studied in chapter 4 are presented.The method by which the rotational hysteresis integral, RH , should be computed in order

to allow meaningful comparison with values obtained from models of hysteresis is discussed. Itis found that computation of the rotational hysteresis integral by integration of the data withrespect to 1/Ha produces artificially low values because of demagnetisation effects. A method bywhich the rotational hysteresis data may be corrected for demagnetisation effects is presented. Thelarger values of the rotational hysteresis integral computed using this method are interpreted givingconsideration to the interactions believed to occur within the materials. The values are found tobe consistent with the mechanisms of magnetisation change found to occur within the materialsstudied in chapter 4, but not in themselves indicative of them. Meaningful interpretation of therotational hysteresis integral is only possible where the effects of interactions may be excluded oraccounted for.

8.1 Introduction

8.1.1 The Effect of Sample Geometry on Rotational Hysteresis Losses

Rotational hysteresis measurements are most commonly made using torque magnetometry. Suchmeasurements do not allow determination of the magnetisation vector during the hysteresis and sodo not allow correction for sample demagnetisation effects, which depend on the sample geometry.Reporting measurements as a function of Ha, rather than Hi is often justified by reference toBozorth’s[168] argument that

Wr = 2π|τ |

= 2π|M ×Hi|

= 2π|M × (Ha −DM)|

= 2π|M ×Ha −D(M ×M)|

= 2π|M ×Ha|, (8.1)

155


which seems to imply that measurements of rotational hysteresis losses are independent of samplegeometry. What this actually implies, however, is that for all applied field strengths Wr(Ha) iscertainly equal to Wr(Hi) only when Hi = Ha−DM . That is, whilst the magnitudes of rotationalhysteresis losses are independent of sample geometry (for the same volumes) the applied fields atwhich those magnitudes of the rotational hysteresis occur are not. The functional form of Wr(Ha),then, does depend on the sample geometry. This has been verified by Elk, who showed that themagnitude of Wr is independent of demagnetisation effects but that the fields at which it occursis not[169].

The dependence of the functional form of Wr(Ha) on sample geometry means that the value of therotational hysteresis integral RaH , computed by integration with respect to 1/Ha also depends onsample geometry. This may be seen by comparison of the expansion of the rotational hysteresisintegrals computed with integration with respect to 1/Hi (field corrected for demagnetisationeffects) and 1/Ha (applied field):

RiH =1µ0

∫ ∞0

Wr

Msd

1|Hi|

=2π

µ0Ms

∫ ∞0

M ×Hid1|Hi|

=2π

µ0Ms

∫ ∞0

M × (Ha −DM)d1

|Ha −DM |

=2π

µ0Ms

∫ ∞0

M ×Had1

|Ha −DM |(8.2)

and

RaH =1µ0

∫ ∞0

Wr

Msd

1|Ha|

=2π

µ0Ms

∫ ∞0

M ×Had1|Ha|

(8.3)

During rotational hysteresis measurements sample demagnetisation effects must always be presentbecause of the requirement of rotational symmetry about the axis of rotation, thus D > 0 (exceptin the case of a thin cylindrical disk of infinite radius). This, together with the fact that M is notconstant with field, means that the integrals of equations 8.2 and 8.3 are not equal and so

RiH 6= RaH . (8.4)

In fact since M and Ha must have at least some component aligned then

RiH > RaH (8.5)

This is consistent with the values of RiH being greater than the values of RaH for the measurementsdescribed in chapter 6.

156

Interpretation of the Field Dependence of Rotational Hysteresis Losses

The inequality of RaH and RiH , demonstrating the dependence of RaH on sample geometry, meansthat values of RaH obtained from measurements of samples with significant demagnetisation effectsare unsuitable for comparison with values obtained from theoretical models. Instead, values of RiHmust be used for such comparisons.

8.1.2 Material Parameters Effecting Rotational Hysteresis Losses

The rotational hysteresis loss of a material depends on a number of parameters, of which themechanism by which the material changes its magnetisation is just one. In particles for magneticrecording Bottoni has concluded that the strength of anisotropy is the dominant factor affecting therotational hysteresis loss[114]. In chapter 7 it was found that in the case of coherent rotation thendistributions in anisotropy and the geometry of the anisotropy may effect the field dependence ofthe rotational hysteresis loss and the value of the rotational hysteresis integral. Where interactionswere positive then the value of RH was significantly effected by a distribution in anisotropy andwhen interactions were negative then the value of RH was barely effected by a distribution inanisotropy.

It was also shown in chapter 7 that in a system that changes magnetisation by coherent rotation therotational hysteresis loss is dependent on interactions within the material. It is very unlikely thatthis dependence of rotational hysteresis losses on interactions will be solely confined to systems thatchange magnetisation by coherent rotation. Interactions within a material may significantly effectthe actual mechanisms of magnetisation change occurring by altering the distribution of activationenergies within a material. More generally, though, interactions serve to alter the internal fieldsavailable to activate irreversible processes. Thus it is reasonable to expect that interactions willeffect rotational hysteresis losses in the more general case.

Dependence of rotational hysteresis loss on interactions implies that the values of the rotationalhysteresis integral will also depend on interactions. At first sight the form of equations 8.2 and8.3 may seem to suggest that the dependence of the rotational hysteresis integral on interactionsmay be similar to its dependence on demagnetisation effects, in cases where the interactions maybe approximated as a mean interaction field. The functional form of M , however, varies implicitlywith the interaction strength, even in the case of mean field interactions [5] and so an analyticexamination of the effect of interactions similar to equations 8.2 and 8.3 is not possible.

In chapter 7 it was shown that in a system that changes magnetisation by irreversible coherentrotation then the value of the rotational hysteresis integral decreases as interactions become morenegative (demagnetising) and increases as interactions become more positive (magnetising). It islikely that such a dependency will apply more generally to other mechanisms of magnetisationchange. Experimental data reported by Bottoni [72, 117] is consistent with this, as discussed inchapter 7.

Chapter 7 also contained a discussion of the possible effect of time dependence on rotationalhysteresis loss. Since the net effect of thermal activations is to decrease the energy of a systemthen it is likely that viscosity effects will serve to reduce rotational hysteresis losses at all fields.

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The value of the rotational hysteresis integral will be correspondingly decreased. Such effectsare likely to be small, particularly over the short time frames involved in DABAAM and RSCMmeasurements.

8.1.3 Interpretation of the Rotational Hysteresis Integral

The dependence of the value of the rotational hysteresis integral on sample demagnetisation effectsmeans that only values that are corrected for demagnetisation effects are suitable for comparisonwith theoretical models. Thus it is necessary either for experimental measurements of rotationalhysteresis to include full determination of the magnetisation vector, allowing RiH to be computedby integration with respect to 1/Hi, or for the measured form of Wr(Ha) to be corrected to Wr(Hi)by some other procedure. A method for correcting Wr(Ha) to Wr(Hi) is suggested in section 8.2.

It is not necessary to correct measured values of the rotational hysteresis integral for their de-pendence on material parameters such as anisotropy, interactions and viscosity since these areproperties of the material rather than the geometry of the sample. It is, however, necessary toaccount for the potential effects of these factors when interpreting the value of the rotationalhysteresis integral or comparing it with values obtained from theoretical models.

8.2 Correction of Rotational Hysteresis Measurements for

Demagnetisation Effects

Measurements made with the RSCM allow determination of the magnetisation vector during rota-tional hysteresis and so the results may be simply corrected for demagnetisation effects. With othermeasurement methods such as torque magnetometry or the DABAAM, however, the magnetisationvector during the rotational hysteresis is not known and so there is no direct means for correc-tion for demagnetisation effects. The results of the measurements performed using the RSCM,presented in chapter 6, suggest that the activated component of magnetisation during rotationalhysteresis is similar to that of the initial magnetisation curve at corresponding fields. Thus if itis assumed that the active component of the magnetisation during rotational hysteresis is equalin magnitude to that on the initial magnetisation curve then the phase of the magnetisation (theangle between the magnetisation and the applied field) may be determined as

φM (Ha) = arcsin(

Wr(Ha)2πµ0HaM(Ha)

). (8.6)

where Wr(Ha) is the measured form of the rotational hysteresis loss and M(Ha) is the magneti-sation along the initial magnetisation curve. It is necessary that both Wr(Ha) and M(Ha) aremeasured from the same zero magnetisation state. Correction for demagnetisation effects maythen be achieved by equations 6.26 and 6.27 and so the rotational hysteresis as a function of Hi

determined byWr(Hi) = 2πµ0HiM(Hi) sinφM (Hi) (8.7)

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It should be noted that this correction for demagnetisation effects is not exact since the resultsin chapter 6 (see for example figure 6.13) show that the activated component of magnetisationduring rotational hysteresis is slightly larger than that on the initial magnetisation curve. Thedifference between the magnetisations will cause the values of Hi(Ha) computed by equation 6.26to be smaller than their true values, meaning the rotational hysteresis loss will be slightly shiftedto lower fields. The value of the rotational hysteresis integral will be correspondingly slightly largerthan the true value. The significance of these errors may be assessed through comparison of RSCMmeasurements with DABAAM measurements corrected using the method described here.

8.3 Experimental Methods

Measurements of rotational hysteresis loss in the samples described in chapter 3 were made using theDABAAM (see chapter 5). The measurements were all performed at room temperature, with thesamples starting in a rotationally demagnetised state (see section 3.3.2). The field was progressivelyincreased, with the sample stationary, between measurements.

In order to assess the accuracy of the method of correcting rotational hysteresis measurementsfor demagnetisation effects, measurements of rotational hysteresis in AlNiCo 7 were also madeusing the RSCM. These measurements were again performed at room temperature from the samerotationally demagnetised state. A rotation frequency of 3 Hz was used, so that the measurementsare performed over similar time intervals as the DABAAM measurements.

Values of the rotational hysteresis integral were computed by Reimann summation as describedin section 2.3.4. The uncertainties in the quoted values reflect the uncertainty in the individualWr(H) values, the uncertainty of the Reimman summation and, in the case of incomplete rotationalhysteresis work data, the uncertainty introduced by the extrapolation against reciprocal field, asdescribed in section 2.3.4. In the case of RiH the uncertainty also reflects that introduced by themethod of correction for demagnetisation effects described in section 8.2. Such uncertainties arerepresentative of the mathematical extremes of the computation process and so it is anticipatedthat the quoted values of RH will be much closer to the true values than the uncertainties suggest.


8.4.1 Rotational Hysteresis Losses

The rotational hysteresis losses as a function of applied field strength for each of the materialsmeasured are presented in figures 8.1 to 8.3.

159


Figure 8.1: The rotational hysteresis loss per unit volume per cycle, Wr, versus the applied field strength,

Ha, for AlNiCo 7.


Ha, for MnAlC.

160



Ha, for the NdFeB samples: MQ1, MQ2, SN1 and SN2.

The figures reveal a similar form of Wr(H) for all of the materials, in that Wr monotonically rises toa maximum value before monotonically decreasing. In chapter 4 the mechanisms of magnetisationchange in AlNiCo 7 and MnAlC were characterised as being similar (see section 4.2.3). The shapeof the curves of Wr(Ha), however, are quite distinct. For AlNiCo the applied field distribution ofthe rotational hysteresis appears to be quite symmetric but in MnAlC the distribution is skewed,rising rapidly and decaying slowly.

It is also somewhat surprising that the rotational hysteresis losses are so similar in all four of theNdFeB materials. The results of chapter 4 suggest that in the two sintered materials magnetisationchange (in the alternating hysteresis case) is controlled by domain nucleation and domain wallpinning within the boundary regions of the grains. In the melt-quenched materials the mechanismswere characterised as being predominately (particularly in the case of MQ1) incoherent rotationmechanisms. Despite the difference in the mechanisms that control these materials their rotationalhysteresis properties are clearly similar.

The data in figures 8.1 to 8.3 may be corrected for demagnetisation effects using the proceduredescribed in section 8.2. This corrected data is presented in figures 8.4 to 8.6.

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Figure 8.4: The rotational hysteresis loss per unit volume per cycle, Wr, versus the intrinsic field strength,Hi, for AlNiCo 7. Data from both RSCM measurements (actual) and DABAAM measurements (corrected)are presented.

Figure 8.5: The rotational hysteresis loss per unit volume per cycle, Wr, versus the intrinsic field strength,

Hi, for MnAlC.

162


Figure 8.6: The rotational hysteresis loss per unit volume per cycle, Wr, versus the intrinsic field strength,

Hi, for the NdFeB samples: MQ1, MQ2, SN1 and SN2.

Presented in figure 8.4 are both the corrected form of Wr(Hi) and Wr(Hi) as determined by RSCMmeasurements. Since the RSCM actually measures the magnetisation vector during rotational hys-teresis then it allows determination of the true form of Wr(Hi). The good agreement of the actualand corrected forms of Wr(Hi) in figure 8.4 suggests the method of correction for demagnetisationeffects suggested in section 8.2 is valid. Demagnetisation effects in AlNiCo are particularly largeand so validity in this case is likely to mean the method is more generally applicable.

It is apparent that the correction for demagnetisation effects significantly alters the form of rota-tional hysteresis loss vs field in AlNiCo, where geometric demagnetisation effects are large, becauseof the high magnetisation relative to field. In MnAlC the correction is much less significant be-cause of the much lower magnetisation to field ratio. The correction has changed the shape ofthe distribution of rotational hysteresis loss vs field in AlNiCo from being symmetric to skewed,making it similar to the distribution for MnAlC, which has similar mechanisms of magnetisationchange. It appears that the initially symmetric distribution against applied field, then, is largely aresult of sample demaganetisation effects rather than any distribution in anisotropy.

Correction for demagnetisation effects in the NdFeB materials shifts the rotational hysteresis lossesto lower fields but does not significantly alter the shape of the distributions. The rotationalhysteresis loss as a function of field is highly skewed in these samples, with the loss very slowlydecaying to zero at seemingly high fields. This is evidence of the very high (magneto crystalline)anisotropy in NdFeB.

163


The method of Paige et al. [65] may be used to estimate the mean anisotropy field, HK , of thesamples studied by determination of the root of the linear extrapolation of Wr plotted against1/H. For AlNiCo this yields HK = (4.2 ± 0.5) × 106 Am−1 (implying K1 = (1.9 ± 0.3) × 105

Jm−3) when the Wr(Hi) data is considered or HK = (6.3 ± 0.7) × 106 Am−1 when the Wr(Hi)data is used, again showing the importance of correcting for demagnetisation effects. In MnAlCvalues of HK = (2.5 ± 0.3) × 106 Am−1 (vs 1/Hi) and HK = (2.6 ± 0.3) × 106 Am−1 (vs 1/Ha)are obtained. The data for the NdFeB materials is not sufficiently complete at high fields for themethod to be reasonably applied. It is necessary for the decay of Wr(H) to be concave upwardsbefore the linear extrapolation against reciprocal field is possible.

8.4.2 Interpretation of the Rotational Hysteresis Integral

The data of figures 8.1 to 8.6 may be used to calculate the rotational hysteresis integral for eachmeasurement. The values of the rotational hysteresis integral are presented in table 8.1. Thosevalues computed with the integration performed with respect to 1/Ha are denoted as RaH and thosewhere the integration is performed by integration with respect to 1/Hi as RiH .

Since the value of the rotational hysteresis integral may be effected by the presence of interactionswithin a material these have also been assessed. Values of the interaction field factor (IFF),defined as IFF = 100(H ′r −Hr)/Hc, where MDCD(Hr) = 0 and MIRM (H ′r) = 1

2Mr(∞) [91, 87],are presented in table 8.1. The convention that the IFF is negative where the interactions aredemagnetising and positive where they are magnetising has been adopted. Values of the extremain ∆M [42] are also presented in table 8.1. The ∆M plots for these materials have been previouslypresented in chapter 4. Both the values of the IFF and ∆M are believed to be broadly representativeof interactions occurring within a material.

Also presented in table 8.1 are the dominant mechanisms of magnetisation change that occur inthese materials, as characterised in chapter 4 by measurements in alternating fields.

It is immediately apparent that the value of the rotational hysteresis integral is effected by demag-netisation effects in the manner predicted in chapter 7 and section 8.1.1. In the case of AlNiCo,where demagnetisation effects are large, the value of RH is altered from 0.58±0.06 to 1.02±0.12 bycorrection for demagnetisation effects, whereas in MnAlC where demagnetisation effects are less thecorrection is less significant. In the case of the NdFeB samples the correction for demganetisationeffects increases the value of the rotational hysteresis integral by between 5 - 20%.

The values of RiH , then, are those that may be considered in comparison with those obtainedfrom theoretical models. In making such comparison, though, the effects of interactions within thematerials must be considered. If the effect of interactions were ignored then the very similar valuesof RiH for MQ1 and MnAlC would seem to imply that similar mechanisms of magnetisation changeoccur in these materials. The very different microstructures of these materials, though, means thatthis is patently not the case.

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Sample Dominant Mechanisms of IFF(%) ∆M extrema RaH RiHMagnetisation Change

MQ1 Incoherent rotation 4.2±0.1 +0.31±0.01 1.55±0.33 1.62±0.37MQ2 Incoherent rotation and -9.1±0.1 -0.39±0.03 0.89±0.18 0.98±0.21

domain nucleation

SN1 Domain nucleation -6.5±0.1 -0.30±0.01 1.00±0.19 1.10±0.23SN2 Domain nucleation -13.2±0.1 -0.35±0.01 1.00±0.18 1.18±0.22

AlNiCo Domain wall pinning -21.8±0.1 -0.41±0.01 0.58±0.06 1.02±0.12MnAlC Domain wall pinning -14.6±0.1 -0.23±0.03 1.62±0.11 1.67±0.13

Table 8.1: Parameters representative of interactions (IFF and ∆M) within the materials studied together

with the values of their rotational hysteresis integrals. W ar is computed from the Wr(Ha) data and W i

r from

the Wr(Hi) data. The dominant mechanisms of magnetisation change presented are those characterised

by measurements in alternating fields.

In MQ1 the value of RiH is 1.62±0.37 and the interactions are indicated by the IFF and ∆M plot(see figure 4.16) as being positive. The origin of these positive interactions is unclear but is likelyto be exchange interactions. It is noted that the positive ∆M plot has not been caused by incorrectdemagnetisation, as described by El-Hilo et al. [93], since the demagnetised state was produced byrotational demagnetisation, which produces a low energy state suitable for the characterisation ofinteractions. Thus in keeping with the conclusions of chapter 7 it is expected that the measuredvalue of the rotational hysteresis integral would have been increased by the positive interactions.The value of 1.62±0.37 is, then, an upper bound on the zero interactions value for this material.Comparison with the theoretical values presented in table 1.1 shows that this is consistent withthe magnetisation change in MQ1 being controlled by an incoherent rotation mechanism.

In the remaining materials the interactions are shown to be negative by both the values of the IFFand ∆M . These negative interactions are almost certainly a result of dipolar interactions. Thusin these cases the measured values of RiH are lower bounds on the zero interactions values.

The value of RiH in MQ2 is likely to be a composite of that produced by the incoherent rotationmechanism believed to occur in about 60% the grains and the domain nucleation believed to controlthe magnetisation change in the rest. Since irreversible rotation mechanisms typically producelower values of the rotational hysteresis integral than domain processes then it is to be expectedthat the value of RiH in MQ2 will be lower than that in SN1 and SN2, where the magnetisationchange is almost solely due to domain processes. Table 8.1 shows that this is the case, althoughit is noted that there is considerable overlap in the uncertainties of the measurements. The lowerbound of RiH = 0.98±0.21 in MQ2, though, certainly implies that the rotation mechanism in thesmaller grains must be incoherent.

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The values of RiH in SN1 and SN2 are considerably lower than those of ∼ 3 expected for domainwall processes, particularly given that there is evidence of considerable overlap in the activationenergy distributions for domain nucleation and domain wall pinning in these materials. The lowvalues of RiH are most likely a result of the strong demagnetising interactions in these materials.Even so the lower bounds on RiH of ∼1.15 certainly precludes the possibility of coherent rotation orfanning occurring in these materials, which is consistent with the domain wall processes believed tooccur in these materials. It is difficult to gain further insight into the mechanisms of magnetisationchange occurring within these materials on the basis of the values of RiH .

Similarly in AlNiCo and MnAlC the values of RiH are much lower than is expected for materialswhere the magnetisation change is controlled by domain wall pinning. This is again consistentwith the negative interactions in these materials serving to decrease the value of RiH . Neverthelessthe lower bound on the value of RiH = 1.67±0.13 in MnAlC precludes most rotation mechanismsand is suggestive of domain wall processes.

It is noted that the strength of the demagnetisation interactions in AlNiCo are greater than thosein MnAlC (where the sample is a pressed powder) and that the corresponding value of RiH issignificantly less. Given that similar magnetisation mechanisms occur within these materials thenthis is consistent with the predictions of chapter 7 and further evidence that the value of therotational hysteresis integral depends on interactions for mechanisms other than coherent rotation.

8.5 Conclusions

The form of the rotational hysteresis loss as a function of applied field, Wr(Ha), is effected bysample demagnetisation effects in the same way as are magnetisation measurements in made inalternating fields. The fact that the value Wr = 2π|M ×H| is independent of whether applied orintrinsic field is used does not imply independence from sample demagnetisation effects. Ratherit shows that the magnitudes of loss achieved during rotational hysteresis loss are independent ofsample demagnetisation effects but the fields at which those losses occur are not. This is analogousto the magnitudes of M achieved on the initial magnetisation curve being independent of sampledemagnetisation effects but the fields at which those magnitudes occur not being.

The dependence of Wr(H) on sample demagnetisation effects implies that the values of the rota-tional hysteresis integral are also. The prediction that the values computed by integration withrespect to 1/Ha are always less than those computed by integration with respect to 1/Hi wasfound to be true for all of the materials examined in this chapter and in chapter 6. Values of therotational hysteresis integral computed from data of loss vs applied field, such as is obtained fromtorque magnetometer measurements, will be artificially low.

The method for correcting data of WR(Ha) to Wr(Hi) suggested in section 8.2 relies on the acti-vated component of magnetisation during rotational hysteresis being similar to that on the initialmagnetisation curve, measured from the same zero magnetisation state. The method was foundto be reliable in AlNiCo 7 and is likely to be more generally applicable. It would be interesting to

166


compare further RSCM measurements with corrected DABAAM measurements in order to verifythat this is the case.

The rotational hysteresis loss properties of the six permanent magnet materials were seen to besimilar, particularly when corrected for sample demagnetisation effects. The initially symmetricdistribution of WR(Ha) in AlNiCo 7 was seen to become skewed once corrected for sample de-magnetisation effects, indicating that the shape of the distribution in this case was significantlyaltered by the demagnetisation effects. In all of the materials correction for demagnetisation effectsshifted the rotational hysteresis losses to lower fields and correspondingly increased the value ofthe rotational hysteresis integral.

The values of the rotational hysteresis integral for each of the materials were interpreted takinginto account the interactions occuring within the materials, as characterised by IFF and ∆Mmeasurements. It is observed that the values here are consistent with the findings of chapter 7 thatRiH is decreased by demagnetising interactions and increased by magnetising interactions. Wheninterpreted in this context the measured values of the rotational hysteresis integral for the materialsstudied here were found to be consistent with, but not necessarily indicative of, the mechanisms ofmagnetisation change determined by measurements in alternating fields.

Interpretation of the value of RiH purely in terms of the mechanisms of magnetisation change isdifficult in permanent magnets because of the significant effect of interactions. This is not to saythat RiH can not provide any insight into the magnetisation mechanisms occurring in permanentmagnets but the effects of interactions make the interpretation less clear. Where interactions aredemagnetising then the measured value of RiH will be a lower bound on the zero interactions valueand where they are magnetising then it will be an upper bound on the zero interactions value.Since dipolar interactions produce negative interactions in most materials then experimentallydetermined values of the rotational hysteresis integral can be expected to be lower than the valuesexpected for particular mechanisms in the absence of interactions. This is observed to be the casein the the values observed here and in values reported in the literature for sintered NdFeB [120].

In particulate materials the situation is likely to be somewhat different. In these cases the effectsof interactions may be more easily assessed and controlled by variation of the packing density andso interpretation of the value of the rotational hysteresis integral is easier. Evolution of the valueof the rotational hysteresis integral with different heat treatments such as in [117] are certainlyvery suggestive of evolution of the mechanisms of magnetisation change, particularly where theinteractions remain similar. Changes in the value with different packing density such as in [114, 117]are less easy to interpret. Both changes in the mechanisms of magnetisation change and the effectof interactions themselves are likely to effect the value in these cases, as discussed in chapter 7.

In interpreting the value of the rotational hysteresis integral, then, it is necessary to correct forsample demagnetisation and to consider the effects of interactions within a material.

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168

9. Conclusions and Suggestions for

Further Study

DURING the course of this work equipment was successfully developed to allow the precisemeasurement of rotational hysteresis in permanent magnets. The Dual Air Bearing Angular

Acceleration Magnetometer (DABAAM), described in chapter 5, allows determination of rotationalhysteresis loss by measurements of the angular acceleration of a sample set spinning in a field. Amethod of separating the torques due to friction and eddy currents from that due to rotationalhysteresis, by consideration of the angular velocity dependence of the torque has been described.The device was shown to be useful for measurements of rotational hysteresis losses and of resistivity.

The second device developed for the measurement of rotational hysteresis, the Rotating Sampleand Coil Magnetometer (RSCM) is the subject of chapter 6. This device employs search coils tomeasure the magnetisation of a sample during rotational hysteresis. The device is novel in thatthe search coils rotate with the sample during measurements. This configuration means that thedevice measures only that component of the total magnetisation that contributes to the rotationalhysteresis loss. Measurements with the RSCM were found to be in good agreement with those ofthe DABAAM but were limited to lower fields because of the high rotational hysteresis torquesthat are exhibited by the materials studied here.

Characterisation of the mechanisms of magnetisation change in linearly applied fields in the sixmaterials studied has been presented in chapter 4. The characterisation was performed usinga combination of remanent magnetisation measurements, magnetic viscosity measurements andmeasurements of the dependence of the reversible magnetisation on the irreversible magnetisation.The interpretation of ∆M plots and plots of η along the hysteresis loop measured from a varietyof demagnetised states provided clear insight into the magnetisation mechanisms occurring in thematerials studied. Measurements of the variation of the fluctuation field were consistent with thecharacterised mechanisms and with the materials having considerable distribution in parameterseffecting the distribution of activation energies.

It was determined that in the sintered rare-earth iron materials the magnetisation is controlled bya combination of domain nucleation and domain wall pinning within the boundary regions of thegrains of the materials. The interior of the grains are defect free and allow the domain walls pinnedat the boundaries to bow through them unimpeded prior to irreversible changes. There appears to

169


be considerable overlap in the nucleation and pinning field distributions in these materials, withthe nucleation field distribution having the higher mean value (Hn > Hp) and so magnetisationreversal in these materials is predominately controlled by domain nucleation.

The opposite appears to be the case in the samples of MnAlC and AlNiCo 7 studied. In thesematerials the mean pinning field was found to be greater than the mean nucleation field (Hp > Hn)and so their magnetisation change is controlled by domain wall pinning within the interior of thegrains.

The data for the melt-quenched rare-earth iron materials is consistent with magnetisation changein these materials being controlled by an incoherent rotation mechanism. The data also suggeststhat another mechanism, involving domain walls, occurs in the larger multi-domain grains in thesematerials. It is estimated that 40% of the volume of grains in MQ2 are multi-domain and a fewpercent are multi domain in MQ1. The magnetisation mechanisms in these grains are similar tothose that occur in the sintered materials.

The effects that interactions, distributions in anisotropy field and the geometry (type) of anisotropymay have on rotational hysteresis loss were studied via consideration of a simple model in chapter7. Data from the model suggests that interactions serve to change the fields at which rotationalhysteresis losses occur and consequently the value of the rotational hysteresis integral. Experimen-tal data in the literature was reinterpreted according to the trends predicted by the modelling andfound to be consistent with it. This suggests that despite the simplicity of the model its predic-tions concerning the effect of interactions on the value of the rotational hysteresis integral may beapplicable to real materials.

Measurements of rotational hysteresis losses in the materials studied were presented in chapter 8.It was shown that the value of the rotational hysteresis integral is effected by geometric demag-netisation effects and a method by which correction for such effects may be performed has beenproposed. After correction for such effects the rotational hysteresis data for each of the materialsstudied was found to be consistent with the characterisations presented only when the interactionswithin the materials were considered. Interpretation of the value of the rotational hysteresis inte-gral in permanent magnets, then, requires that it is corrected for demagnetisation effects and thatthe interactions within the material are considered.

Each of the measurement techniques performed in this work are able to independently give someinsight into the mechanisms of magnetisation change that occur within materials. When employedtogether they allow for considerable confidence in the characterisation of the mechanisms of mag-netisation change in permanent magnet materials.

9.1 Suggestions for Further Study

Ideally characterisation of permanent magnets would be carried out by direct (visual) observation ofthe magnetic structure of a material as it undergoes hysteresis (both alternating and rotational).

170

Conclusions and Suggestions for Further Study

It is difficult to know how such measurements may be performed. Magnetic force microscopyis not suited as it does not allow for observation of the actual magnetisation change processesbecause of the times required for data collection. Kerr microscopy allows for dynamic observationof domain structures but it is normally difficult with such apparatus to apply sufficiently strongfields to observe significant magnetic change in permanent magnets. It would be interesting if suchapparatus could be developed to allow high field observations.

A number of improvements could be made to the RSCM. Specifically:

• Variable resistors could be connected in series with each of the sense and compensationcoils so that the magnitude of the field signal from each of the compensation coils couldbe better matched with that from its corresponding sense coil, by altering the resistances.This would greatly reduce the magnitude of the net field signal and possibly eliminate theneed for its subtraction from sample measurements altogether. More symmetric and regularwinding of the coils would also help in this regard by ensuring that the true axis of the sense-compensation coil pairs are aligned. It is difficult to know how to go about this because theextremely thin gauge wire used (52 µm) is very difficult to work with.

• An alternative to improving the winding of the coils may be to devise a system of betteraligning the sense and compensation coils after they have been matched. This would involveslight physical rotation of either of the coils so as to minimise the magnitude of the signalproduced by a changing field. It is likely that this would be most readily achieved in anoscillating applied field, where the magnitude of the signal could still be measured via lock-in techniques. Alignment of the coils would eliminate the need for the φcs correction (andcalibration) described in chapter 6.

• The use of a stronger resin to hold the samples should be investigated, so that slippingduring the measurement is eliminated. The alternative would be to improve the sensitivityof the RSCM and so allow the use of smaller samples (as was the case with the DABAAM).Improving the sensitivity of the device could be achieved through the use of smaller coils witha greater number of turns, or pre-amplification of the signal prior to its passing through theslip rings. Any pre-amplifier used would need to be battery powered to avoid the coherentnoise that would be generated in its power supply, if it was passed through the slip rings.

• It would be particularly desirable for the RSCM to be able to be used in higher applied fieldsand over a greater frequency range. Higher fields would allow the measurement of Mact(Ha)in modern permanent magnets. A greater frequency range would allow the study of possiblerate (time) dependence effects of rotational hysteresis. Comparison of time dependence duringrotational hysteresis with that during alternating hysteresis may provide further insight intothe relationship between the mechanisms of magnetisation change occurring in each case.Higher measurement frequencies could be simply achieved by using gearing of the motor drive.Achieving higher field strengths without further minaturisation of the RSCM is unlikely tobe possible using an electromagnet because of the gap and size of field uniformity required.

The simple model presented in chapter 7 is not not representative of any of the materials studied

171


here and serves only to hint at trends that may be more widely applicable. Models of rotationalhysteresis loss in which domain wall processes control the magnetisation change would also be ofinterest. Aside from the work of Elk et al. [170] little investigation has been carried out in thisarea. Micro-magnetic modelling of such materials and an examination of the effects of variousparameters on the rotational hysteresis loss may provide further evidence for the dependenciesobserved in this work.

Further insight into the relationships between rotational and alternating hysteresis may be gainedthrough the study of very small particle systems. In many such materials once the particle size isless than a critical dimension then the only possible mechanism of magnetisation change is thatof coherent rotation. Low aspect ratio Fe3O4 particles of size less than 54 nm are an exampleof such a system [13, 159]. Facilities to measure rotational hysteresis at low temperatures maybe required to study such systems in order to prevent the particles becoming superparamagnetic.Such measurements would be of interest as they would be of a system in which the magnetisationmechanism was certain to be the same in both the alternating and rotating field cases. By studyingthe rotational hysteresis losses of the particles at different packing densities then measurementsanalogous to those modelled in chapter 7 could be performed.

All of the experimental measurements of rotational hysteresis made during this work were per-formed with the material starting in a rotationally demagnetised state. Similar measurementswith materials initially in a thermally demagnetised or never previously magnetised state shouldalso be performed. In the case of linearly applied fields the initial zero magnetisation state cansignificantly effect the form of the initial magnetisation curve. This is because of differences in themechanisms controlling the magnetisation of a material from different zero magnetisation states.It is likely that similar effects will be observed in rotational hysteresis measurements. If this isthe case then studies of this effect may provide further insight into the relationship between ro-tational hysteresis loss and the mechanisms that control magnetisation change in a material. Itmay be instructive, for example, to compare the form of Wr(H) measured from thermally androtationally demagnetised states in a material in which domain nucleation is believed to controlthe magnetisation change.

172

Appendix A. Derivation of Selected

Equations

A.1 Equation 8 in Chapter 5

This derivation is of the power dissipated by eddy currents in a sphere made to rotate in a magneticfield.

Consider the case of a sinusoidally varying field, of frequency f , applied to a current loop withresistivity ρ, thickness dx, inner radius r and outer radius r+dr, as shown in figure A.1. The emfinduced in the loop is given by

emf = −πr2ωB cos(ωt) (A.1)

where ω = 2πf . The loop’s resistance, R, is

R =2πrρdrdx

(A.2)

Figure A.1:A current loop.

173


Equations A.1 and A.2 may be combined to give the power, Ploop, dissipated in the loop by theinduced eddy current:

Ploop =emf2

R=πr3ω2B2 cos2(ωt)drdx

2ρ(A.3)

This expression may be integrated with r varying from 0 to rdisk to give the power, Pdisk dissipatedin a disk of radius rdisk and thickness, dx:

Pdisk =∫ rdisk

0

Ploop =πr4diskω

2B2 cos2(ωt)dx8ρ

(A.4)

After making the substitution rdisk =√r2 − x2, as suggested by figure A.1 we obtain

Pdisk =π(r2 − x2)2ω2B2 cos2(ωt)dx

8ρ(A.5)

which may in turn be integrated with x varying from −r to r to give the power dissipated in asphere of radius r:

Psphere =2πω2B2r5 cos2(ωt)

15ρ(A.6)

This is the power dissipated by eddy currents induced by a sinusoidally varying field, which is onecomponent of a rotating field (the other being that due to a field oscillating at right angles with aπ2 phase shift). Thus the total power dissipated by eddy currents, Pe, induced by a rotating fieldis

Pe =2πω2B2r5 cos2(ωt)

15ρ+

2πω2B2r5 sin2(ωt)15ρ

(A.7)

which after the trigonometric identity sin2 θ + cos2 θ = 1 has been applied gives

Pe =2πω2B2r5

15ρ(A.8)

This is the same as equation 5.8 of chapter 5.

A.2 Equation 9 in Chapter 6

This derivation is of the magnetic moment induced by eddy currents in a sphere made to rotate ina magnetic field. The derivation follows very similar lines as the derivation of the power dissipated

174

Derivation of Selected Equations

by eddy currents in section A.1.

Consider the case of a sinusoidally varying field, of frequency f , applied to a current loop withresistivity ρ, thickness dx, inner radius r and outer radius r+dr, as shown in figure A.1. The emfinduced in the loop is given by equation A.1 and the resistance of the loop is given by equationA.2 and so the current flowing in the loop is

Iloop =emf

R= −rωB cos(ωt)drdx

2ρ. (A.9)

The magnitude of the magnetic moment induced by this current, and therefore the magneticmoment of the loop, is given by multiplying this current by the area of the loop, A, to give

µloop = IloopA = −πr3ωB cos(ωt)drdx

2ρ(A.10)

This expression may be integrated with r varying from 0 to rdisk to give the magnitude of themoment, µdisk induced in a disk of radius rdisk and thickness, dx:

µdisk =∫ rdisk

0

µloop = −πr4diskωB cos(ωt)dx

8ρ(A.11)

After making the substitution rdisk =√r2 − x2, as suggested by figure A.1 we obtain

µdisk = −π(r2 − x2)2ωB cos(ωt)dx8ρ

(A.12)

which may in turn be integrated with x varying from −r to r to give the magnitude of the momentinduced in a sphere of radius r:

µsphere = −2πωBr5 cos(ωt)15ρ

(A.13)

This is the magnitude of the moment induced by eddy currents in a sphere subject to a sinusoidallyvarying field. The negative sign implies that the moment is induced opposing the field (i.e. in theopposite direction). A sinusoidally varying field is one component of a rotating field, the otherbeing that due to a field oscillating at right angles with a π

2 phase shift. Thus the magnitude ofthe moment induced by eddy currents, µe, in a sphere in a rotating field is given by

µe =

√(−2πωBr5 cos(ωt)

15ρ

)2

+(−2πωBr5 sin(ωt)

15ρ

)2

(A.14)

which after the trigonometric identity sin2 θ + cos2 θ = 1 has been applied gives

175


µe =2πωBr5

15ρ(A.15)

The direction of the induced moment is opposite to the direction of the field at all times.

Equation A.15 is the same as equation 6.9 of chapter 6.

176

Appendix B. Description of the Other

Samples Studied

Results for rotational hysteresis measurements of materials other than those described in chapter3 are presented in chapters 5 and 6. These results are presented as a means of demonstrating thevalidity and wider applicability of the measurement methods described in those chapters. Thesematerials are not subjects of the wider study of this work and so have not been previously described.Descriptions of the materials are included here for completeness.

B.1 Ex1

Exchange spring materials consist of two distinct magnetic phases exchanged coupled at the inter-phase boundaries. By coupling a (magnetically) soft phase of high moment (usually α − Fe orFe3B) with a hard magnetic phase (typically NdFeB) then a material with both high coercivityand remanence may be produced. Provided the two phases are exchange coupled then the magneticproperties of the material are those of a single phase with a hysteresis loop exhibiting a highMr/Ms ratio (typically ∼0.75) and a very high component of reversible magnetisation on thedemagnetisation curve. The coercivity of such materials is usually significantly less than that ofthe hard phase but dramatically greater than that of the soft phase since, because of the exchangecoupling, the Gerlach principle [99] does not apply to these materials.

The sample designated Ex1 had a composition Nd4.5Fe73B18.5Co3Ga. The material was producedby melt spinning of amorphous ribbons which were then briefly heat treated to allow small graingrowth. The manufacture was conducted by Sumitomo Special Metals Co. Ltd, Japan. The precisedetails of the techniques involved in the manufacture have been described by Kanekiyo et al. [171].

Powder X-Ray diffraction measurement conducted by Folks et al. [172] have revealed that thematerial consists primarily of a (soft) Fe3B phase, a (hard) Nd2Fe14B phase and a small amount of(soft) α−Fe. The Co was determined to be concentrated in the Nd2Fe14B phase. In order for theexchange coupling between grains to be maximised then it is necessary for the grains to be verysmall, so that their surface area to volume ratio is maximized. TEM measurements by Folks et al.[172] found that the average grain size in Ex1 was around 20 nm. This size is about equal to the

177


Figure B.1:The hysteresis loop of Ex1.

domain wall width of the hard phase, which is the grain size suggested as giving optimal magneticproperties in such materials [173].

The hysteresis loop of Ex1 is shown in figure B.1. The evidence for the exchange coupling betweenthe hard and soft phases in Ex1 is the single phase behaviour of the hysteresis loop, the highMr/MS ratio and the very steep recoil curves on the demagnetisation curve. The fact that there isexchange coupling across grain boundaries complicates the consideration of the expected domainstructure. Whilst the individual grains are of single domain size it is possible that individualdomains may form over multiple grains. Modelling of such materials[174] suggests that the hardphase grains are generally single domain, whereas the soft phase grains may contain considerablevariation in the orientation of the magnetisation within the grain, without the presence of domainwalls (see, for example, figure 5 of [174]). Thus the magnetisation mechanisms in such materials,may differ significantly from those found in conventional single phase materials.

B.2 SrFe

The strontium ferrite sample studied was also obtained from Oriel Pty. Ltd., NSW Australia. Nei-ther the processing method nor the chemical composition of the material were divulged, althoughit is believed that the material is close to the composition SrFe12O19. Kerr microscopy studies re-vealed that the material was composed of grains of approximately cubic shape and length ∼ 1µm,with little distribution in grain size. No domains were observed within the grains, suggesting thatthey were too small to support domain walls. The grains appeared to be closely spaced, with no

178

Description of the Other Samples Studied

Figure B.2:The hysteresis loop of the strontium ferrite material.

observable inter-granular impurities. The material was observed to be isotropic.

B.3 Co-γ-Fe2O3

The Co-γ-Fe2O3 material was manufactured by BASF and the details of the manufacture havenot been disclosed. It is understood that the sample consists of γ-Fe2O3 particles that have beensurfaced modified with cobalt to increase the coercivity. Transmission Electron Microscopy (TEM)studies revealed that the particles were rounded cubes about 150 nm across [175].

The critical size, Dcr, below which a spherical γ-Fe2O3 particle is single domain is 54nm < Dcr <

74nm[13]. For a material consisting of many such particles the effects of interactions is to increasethe critical dimension[14]. Experimentally single domain behaviour has been observed in 120 nmγ-Fe2O3 particles at high packing densities[14]. Such behaviour has also been observed in Fe3O4,which has a critical dimension of Dcr ≈ 54nm for isolated particles[176] and Dcr ≈ 200nm forparticles packed in a material[177]. The sample prepared here was prepared as a resin bondedpacked material, which is likely to mean there are significant interactions between the particles.Thus it is likely that the particles in this case (D ∼ 150 nm) are at about the critical dimensionand are mostly single domain, with deviation of the spins in the face regions[178].

The hysteresis loop of the Co-γ-Fe2O3 material studied is shown in figure B.3. The hysteresisloop has been plotted as Mr/Ms vs Ha since the demagnetisation effects are unclear. This formis frequently used in the literature in reports of the magnetic properties of particulate recording

179


Figure B.3:The hysteresis loop of the Co-γ-Fe2O3 material.

materials.

The squareness ratio in figure B.3, Mr/Ms, is almost exactly 0.5, which is the theoretical value forStoner-Wohlfarth type particles. Morales et al. observed that in materials consisting of sphericalparticles of γ-Fe2O3 then Mr/Ms ∼ 0.5 when the particles are single domain and is significantlyreduced when they are multi-domain[14]. This is consistent with the conclusion that the particlesare single domain in this case.

180

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