December 28, 2021

The material on this website is freely available for educational purposes.

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Proper citation of the material found in these pages is:

Tauxe, L, Banerjee, S.K., Butler, R.F. and van der Voo R, Essentials of Paleomagnetism, 5th Web Edition, 2018.

The printed version of this book appeared January, 2010. Order a printed version. (cheaper than printing it yourself!)

This book is intended to work with the companion software package described in PmagPy Cookbook.

This material is based upon work supported by the National Science Foundation.

Preface

1 Purpose of the book

The geomagnetic field acts both as an umbrella, shielding us from cosmic radiation and as a window, offering one of the few glimpses of the inner workings of the Earth. Ancient records of the geomagnetic field can inform us about geodynamics of the early Earth and changes in boundary conditions through time. Thanks to its essentially dipolar nature, the geomagnetic field has acted as a guide, pointing to the axis of rotation thereby providing latitudinal information for both explorers and geologists.

Human measurements of the geomagnetic field date to about a millenium and are quite sparse prior to about 400 years ago. Knowledge of what the field has done in the past relies on accidental records carried by geological and archaeological materials. Teasing out meaningful information from such materials requires an understanding of the fields of rock magnetism and paleomagnetism, the subjects of this book. Rock and paleomagnetic data are useful in many applications in Earth Science in addition to the study of the ancient geomagnetic field. This book attempts to draw together essential rock magnetic theory and useful paleomagnetic techniques in a consistent and up-to-date manner. It was written for several categories of readers:

1. Earth scientists who use paleomagnetic data in their research,
2. students taking a class with paleomagnetic content,
3. other professionals with an interest in evaluating or using paleomagnetic data, and
4. anyone with at least college level chemistry, physics and a cursory knowledge of Earth science with an interest in magnetism in the Earth.

There are a number of excellent references on paleomagnetism and on the related specialties (rock magnetism and geomagnetism). The ever popular but now out of print text by Butler (1992) has largely been incorporated into the present text. For in-depth coverage of rock magnetism, we recommend Dunlop and Özdemir (1997). Similarly for geomagnetism, please see Backus et al. [1996]. A rigorous analysis of the statistics of spherical data is given by Fisher et al. (1987). The details of paleomagnetic poles are covered in van der Voo (1993) and magnetostratigraphy is covered in depth by Opdyke and Channell (1996). The Treatise in Geophysics, vol. 5 (edited by Kono, 2007) and The Encyclopedia of Geomagnetism and Paleomagnetism (edited by Gubbins and Herrero-Bervera, 2007) have up to date reviews of many topics covered in this book. The present book is intended to augment or distill information from the broad field of paleomagnetism, complementing the existing body of literature.

An important part of the problems in this book is to teach students to write simple computer programs themselves and use programs that are supplied as a companion set of software (PmagPy). The programming language chosen for this is Python because it is free, cross platform, open source and well supported. There are excellent online tutorials for Python and many open source modules which make software development cheaper and easier than any other programming environment. The appendix provides a brief introduction to programming and using Python. The reader is well advised to peruse the PmagPy Cookbook for further help in gaining necessary skills with a computer. Also, students should have access to a relatively new computer (Windows, Mac OS 10.4 or higher are supported, but other computers may also work.) Software installation is described at: magician.ucsd.edu/Software/PmagPy.

2 What is in the book

This book is a collaborative effort with contributions from R.F. Butler (Chapters 1, 3, 4, 6, 7, 9, 11 and the Appendix), S.K Banerjee (Chapter 8) and R. van der Voo (Chapter 16). The MagIC database team designed and deployed the MagIC database which we have made liberal use of in providing data for problem sets and in the writing of the PmagPy Cookbook, so there were significant contributions to this book project from C.G. Constable, A.A.P. Koppers and Rupert Minnett.

At the beginning of most chapters, there are recommended readings which will help fill in background knowledge. There are also suggested readings at the end of most chapters that will allow students to pursue the subject matter in more depth.

The chapters themselves contain the essential theory required to understand paleomagnetic research as well as illustrative applications. Each chapter is followed by a set of practical problems that challenge the student’s understanding of the material. Many problems use real data and encourage students to analyze the data themselves. [Solutions to the problems may be obtained from LT by instructors of classes using this book as a text.] The appendix contains detailed derivations, assorted techniques, useful tables and a comprehensive explanation of the PmagPy set of programs.

Chapter 1 begins with a review of the physics of magnetic fields. Maxwell’s equations are introduced where appropriate and the magnetic units are derived from first principles. The conversion of units between cgs and SI conventions is also discussed and summarized in a handy table.

Chapter 2 reviews essential aspects of the Earth’s magnetic field, discussing the geomagnetic potential, geomagnetic elements, and the geomagnetic reference fields. The various magnetic poles of the Earth are also introduced.

Chaptes 3-8 deal with rock and mineral magnetism. The most important aspect of rock magnetism to the working paleomagnetist is how rocks can become magnetized and how they can stay that way. In order to understand this, Chapter 3 presents a discussion of the origin of magnetism in crystals, including induced and remanent magnetism. Chapter 4 continues with an explanation of anisotropy energy, magnetic domains and superparamagnetism. Magnetic hysteresis is covered in Chapter 5. Chapter 6 deals with specific magnetic minerals and their properties, leading up to the origin of magnetic remanence in rocks, the topic of Chapter 7. Finally Chapter 8 deals with applied rock magnetism and environmental magnetism.

Chapters 9-13 delve into the nuts and bolts of paleomagnetic data acquisition and analysis. Chapter 9 suggests ways of sampling rocks in the field and methods for treating them in the laboratory to obtain a paleomagnetic direction. Various techniques for obtaining paleointensities are described in Chapter 10. Once the data are in hand, Chapters 11 and 12 deal with statistical methods for analyzing magnetic vectors. Paleomagnetic tensors are introduced in Chapter 13, which explains measurement and treatment of anisotropy data.

Chapters 14-16 illustrate diverse applications of paleomagnetic data. Chapter 14 shows how they are used to study the geomagnetic field. Chapter 15 describes the development of the geomagnetic polarity time scale and various applications of magnetostratigraphy. Chapter 16 focuses on apparent polar wander and tectonic applications.

The appendix contains more detailed information, included for supplemental background or useful techniques. It is divided into several sections: Appendix A summarizes various definitions and detailed derivations including various mathematical tricks such as vector and tensor operations. Appendix B.1 describes some plots commonly employed by paleomagnetists. Appendix C.2 collects together methods and tables useful in directional statistics. Appendix D describes techniques specific to the measurement and analysis of anisotropy data. The PmagPy Cookbook provides an introduction to the Magnetics Information Consortium (MagIC) database, the current repository for rock and paleomagnetic data and summarizes essential computer skills including basic Unix commands, an introduction to Python programming and extensive examples of programs in the PmagPy software package used in the problems at the end of each chapter.

3 How to use the book

Each chapter builds on the principles outlined in the previous chapters, so the reader is encouraged to work through the book sequentially. There are recommended readings before and after every chapter selected to provide backgound information and supplemental reading for the motivated reader respectively. These are meant to be optional.

The reader is encouraged to study Chapter 6 in the PmagPy Cookbook before beginning to work on the problems at the end of each chapter. The utility of the book will be greatly enhanced by successfully installing and using the programs referred to in the problems. By conscientiously trying them out as they are mentioned, the reader will not only gain familiarity with PmagPy software package, but also with the concepts discussed in the chapters.

We have attempted to maintain a consistent notation throughout the book. Vectors and tensors are in bold face; other parameters, including vector components, are in italics. The most important physical and paleomagnetic parameters, acronyms and statistics are listed in Appendix A.

The problems in this book are intended to be solved with Jupyter Notebooks in conjunction with the PmagPy software of Tauxe et al., (2016). To get ready for this, you must do the following steps:

1. Install Anaconda Python 3, following the instructions in the PmagPy Cookbook in the Section on Full PmagPy Install. Be sure to include the non-default Python packages.
2. Install PmagPy following the instructions for pip install from the command line.
3. Move the data files to a directory convenient to you (e.g., your Desktop) following the instructions in the cookbook.
4. The data files used in the problems are in the PmagPy subdirectory data_files/notebooks/Essentials/, separated by directory.
5. Navigate to the PmagPy subdirectory data_files/notebooks/Essentials/ and launch the jupyter notebook environment. Make a new notebook for a problem set at the end of each chapter.
6. To see the Jupyter notebook solutions, navigate to the PmagPy subdirectory data_files/notebooks/Essentials/ and launch the jupyter notebook environment. Open each chapter’s solution in turn.

The problems at the end of each chapter assume proficiency in both Python programming and the use of Jupyter notebooks. See Python Programming for Earth Scientists for a complete course.

For examples on how to use PmagPy in a Jupyter notebook, see PmagPy.ipynb in your data_files directory.

4 Acknowledgements

LT is the primary author of this book who bears sole responsibility for all mistakes. There are significant contributions by RFB, SKB and RvdV. We are indebted to many people for assistance great and small. This book began life as a set of lecture notes based loosely on the earlier book by Tauxe (1998). Many pairs of eyes hunted down errors in the text and the programs each time the course was given. The course was also occasionally co-taught with Cathy Constable and Jeff Gee who contributed significantly to the development of the manuscript and the proof-reading there-of. Thanks go to the many “live” and “online” students who patiently worked through various drafts. Special thanks go to Kenneth Yuan, Liu Cy , Maxwell Brown and Michael Wack who provided many detailed comments and helpful suggestions. Reviews by Ken Kodama, Brad Clement, Scott Bogue and Cor Langereis improved the book substantially. Also, careful proof-reading by Newlon Tauxe of the first few chapters is greatly appreciated. And of course, I am deeply grateful to my mentors, Dennis V. Kent and Neil D. Opdyke who taught me how to do science.

I owe a debt of gratitude to the many sources of public domain software that ended up in the package PmagPy, including contributions by Cathy Constable, Monika Korte, Jeff Gee, Peter Selkin, Ron Shaar, Nick Swanson-Hysell, Ritayan Mitra and especially Lori Jonestrask, as well as the many dedicated contributors to the Numpy, Matplotlib, Basemap, Cartopy, and Pandas Python modules used extensively by PmagPy. Also, many illustrations were prepared with the excellent programs Magmap, Contour and Plotxy by Robert L. Parker, to whom I remain deeply grateful. I gratefully acknowledge the authors of many earlier books, too many to name but included in the Bibliography, which both educated and inspired me.

Finally, I am grateful to my husband, Hubert Staudigel, and my children, Philip and Daniel Staudigel who have long tolerated my obsession with paleomagnetism with grace and good humor and frequently good advice.

Contents

Chapter 1
The physics of magnetism

Paleomagnetism is the study of the magnetic properties of rocks. It is one of the most broadly applicable disciplines in geophysics, having uses in diverse fields such as geomagnetism, tectonics, paleoceanography, volcanology, paleontology, and sedimentology. Although the potential applications are varied, the fundamental techniques are remarkably uniform. Thus, a grounding in the basic tools of paleomagnetic data analysis can open doors to many of these applications. One of the underpinnings of paleomagnetic endeavors is the relationship between the magnetic properties of rocks and the Earth’s magnetic field.

In this chapter we will review the basic physical principles behind magnetism: what are magnetic fields, how are they produced and how are they measured? Although many find a discussion of scientific units boring, much confusion arose when paleomagnetists switched from “cgs” to the Système International (SI) units and mistakes abound in the literature. Therefore, we will explain both unit systems and look at how to convert successfully between them. There is a review of essential mathematical tricks in Appendix A to which the reader is referred for help.

1.1 What is a magnetic field?

Magnetic fields, like gravitational fields, cannot be seen or touched. We can feel the pull of the Earth’s gravitational field on ourselves and the objects around us, but we do not experience magnetic fields in such a direct way. We know of the existence of magnetic fields by their effect on objects such as magnetized pieces of metal, naturally magnetic rocks such as lodestone, or temporary magnets such as copper coils that carry an electrical current. If we place a magnetized needle on a cork in a bucket of water, it will slowly align itself with the local magnetic field. Turning on the current in a copper wire can make a nearby compass needle jump. Observations like these led to the development of the concept of magnetic fields.

Electric currents make magnetic fields, so we can define what is meant by a “magnetic field” in terms of the electric current that generates it. Figure 1.1a is a picture of what happens when we pierce a flat sheet with a wire carrying a current i. When iron filings are sprinkled on the sheet, the filings line up with the magnetic field produced by the current in the wire. A loop tangential to the field is shown in Figure 1.1b, which illustrates the right-hand rule (see inset to Figure 1.1b). If your right thumb points in the direction of (positive) current flow (the direction opposite to the flow of the electrons), your fingers will curl in the direction of the magnetic field.

The magnetic field H points at right angles to both the direction of current flow and to the radial vector r in Figure 1.1b. The magnitude of H (denoted H) is proportional to the strength of the current i. In the simple case illustrated in Figure 1.1b the magnitude of H is given by Ampère’s law:

where r is the length of the vector r. So, now we know the units of H: Am⁻¹.

Ampère’s Law in its most general form is one of Maxwell’s equations of electromagnetism: in a steady electrical field, ∇× H = J_f, where J_f is the electric current density (see Section A.3.6 in the appendix for review of the ∇ operator). In words, the curl (or circulation) of the magnetic field is equal to the current density. The origin of the term “curl” for the cross product of the gradient operator with a vector field is suggested in Figure 1.1a in which the iron filings seem to curl around the wire.

1.2 Magnetic moment

An electrical current in a wire produces a magnetic field that “curls” around the wire. If we bend the wire into a loop with an area πr² that carries a current i (Figure 1.2a), the current loop would create the magnetic field shown by pattern of the iron filings. This magnetic field is that same as the field that would be produced by a permanent magnet. We can quantify the strength of that hypothetical magnet in terms of a magnetic moment m (Figure 1.2b). The magnetic moment is created by a current i and also depends on the area of the current loop (the bigger the loop, the bigger the moment). Therefore, the magnitude of the moment can by quantified by m = iπr². The moment created by a set of loops (as shown in Figure 1.2c) would be the sum of the n individual loops, i.e.:

(1.1)

So, now we know the units of m: Am². In nature, magnetic moments are carried by magnetic minerals the most common of which are magnetite and hematite (see Chapter 6 for details).

1.3 Magnetic flux

The magnetic field is a vector field because at any point it has both direction and magnitude. Consider the field of the bar magnet in Figure 1.3a. The direction of the field at any point is given by the arrows while the strength depends on how close the field lines are to one another. The magnetic field lines represent magnetic flux. The density of flux lines is one measure of the strength of the magnetic field: the magnetic induction B.

Just as the motion of electrically charged particles in a wire (a current) create a magnetic field (Ampère’s Law), the motion of a magnetic field creates electric currents in nearby wires. The stronger the magnetic field, the stronger the current in the wire. We can therefore measure the strength of the magnetic induction (the density of magnetic flux lines) by moving a conductive wire through the magnetic field (Figure 1.3b).

Magnetic induction can be thought of as something that creates a potential difference with voltage V in a conductor of length l when the conductor moves relative to the magnetic induction B with velocity v (see Figure 1.3b): V = vlB. From this we can derive the units of magnetic induction: the tesla (T). One tesla is the magnetic induction that generates a potential of one volt in a conductor of length one meter when moving at a rate of one meter per second. So now we know the units of B: V ⋅ s ⋅ m⁻² = T.

Another way of looking at B is that if magnetic induction is the density of magnetic flux lines, it must be the flux Φ per unit area. So an increment of flux dΦ is the field magnitude B times the increment of area dA. The area here is the length of the wire l times its displacement ds in time dt. The instantaneous velocity is dv = ds∕dt so dΦ = BdA and the rate of change of flux is:

(1.2)

Equation 1.2 is known as Faraday’s law and in its most general form is the fourth of Maxwell’s equations. We see from Equation 1.2 that the units of magnetic flux must be a volt-second which is a unit in its own right: the weber (Wb). The weber is defined as the amount of magnetic flux which, when passed through a one-turn coil of a conductor carrying a current of one ampere, produces an electric potential of one volt. This definition suggests a means to measure the strength of magnetic induction and is the basis of the “fluxgate” magnetometer.

1.4 Magnetic energy

A magnetic moment m in the presence of a magnetic field B has a magnetostatic energy (E_m) associated with it. This energy tends to align compass needles with the magnetic field (see Figure 1.4). E_m is given by −m ⋅ B or −mB cosθ where m and B are the magnitudes of m and B, respectively (see Section A.3.4 in the appendix for review of vector multiplication). Magnetic energy has units of joules and is at a minimum when m is aligned with B.

1.5 Magnetization and magnetic susceptibility

Magnetization M is a normalized moment (Am²). We will use the symbol M for volume normalization (units of Am⁻¹) or Ω for mass normalization (units of Am²kg⁻¹). Volume normalized magnetization therefore has the same units as H, implying that there is a current somewhere, even in permanent magnets. In the classical view (pre-quantum mechanics), sub-atomic charges such as protons and electrons can be thought of as tracing out tiny circuits and behaving as tiny magnetic moments. They respond to external magnetic fields and give rise to an induced magnetization. The relationship between the magnetization induced in a material M_I and the external field H is defined as:

(1.3)

The parameter χ_b is known as the bulk magnetic susceptibility of the material; it can be a complicated function of orientation, temperature, state of stress, time scale of observation and applied field, but is often treated as a scalar. Because M and H have the same units, χ_b is dimensionless. In practice, the magnetic response of a substance to an applied field can be normalized by volume (as in Equation 1.3) or by mass or not normalized at all. We will use the symbol κ for mass normalized susceptibility and K for the raw measurements (see Table 1.1) when necessary.

Certain materials can produce magnetic fields in the absence of external magnetic fields (i.e., they are permanent magnets). As we shall see in later chapters, these so-called “spontaneous” magnetic moments are also the result of spins of electrons which, in some crystals, act in a coordinated fashion, thereby producing a net magnetic field. The resulting spontaneous magnetization can be fixed by various mechanisms and can preserve records of ancient magnetic fields. This remanent magnetization forms the basis of the field of paleomagnetism and will be discussed at length in subsequent chapters.

1.6 Relationship of B and H

B and H are closely related and in paleomagnetic practice, both B and H are referred to as the “magnetic field”. Strictly speaking, B is the induction and H is the field, but the distinction is often blurred. The relationship between B and H is given by:

(1.4)

where μ is a physical constant known as the permeability. In a vacuum, this is the permeability of free space, μ_o. In the SI system, μ has dimensions of henries per meter and μ_o is

4π × 10⁻⁷H ⋅ m⁻¹. In most cases of paleomagnetic interest, we are outside the magnetized body so M = 0 and B = μ_oH.

1.7 A brief tour of magnetic units in the cgs system

So far, we have derived magnetic units in terms of the Système International (SI). In practice, you will notice that people frequently use what are known as cgs units, based on centimeters, grams and seconds. You may wonder why any fuss would be made over using meters as opposed to centimeters because the conversion is trivial. With magnetic units, however, the conversion is far from trivial and has been the source of confusion and many errors. So, in the interest of clearing things up, we will briefly outline the cgs approach to magnetic units.

The derivation of magnetic units in cgs is entirely different from SI. The approach we will take here follows that of Cullity (1972). We start with the concept of a magnetic pole with strength p instead of with current loops as we did for SI units. We will consider the force between two poles p₁,p₂ (see Figure 1.5) Coulomb’s law. This states that the force between two charges (q₁,q₂) is:

(1.5)

where r is the distance between the two charges. In cgs units, the proportionality constant k is simply unity, whereas in SI units it is 1 __ 4πϵ₀ where ϵ₀ = 10⁷ _ 4πc² and c is the speed of light in a vacuum (hence ϵ₀ = 8.859 ⋅ 10⁻¹² AsV⁻¹m⁻¹). [You can see why many people really prefer cgs but we are not allowed to publish in cgs in most of geophysical journals so we just must grin and bear it!]

For magnetic units, we use pole strength p₁,p₂ in units of electrostatic units or esu, so Equation 1.5 becomes

Force in cgs is in units of dynes (dyn), so

A magnetic pole, as an isolated electric charge, would create a magnetic induction μ_oH in the space around it. One unit of field strength (defined as one oersted or Oe) is the unit of field strength that exerts a force of one dyne on a unit of pole strength. The related induction (μ_oH) has units of gauss or G.

The relationship between force, pole and magnetic field is written as:

Returning to the lines of force idea developed for magnetic fields earlier, let us define the oersted to be the magnetic field which would produce an induction with one unit of induction per square centimeter. Imagine a sphere with a radius r surrounding the magnetic monopole. The surface area of such a sphere is 4πr². When the sphere is a unit sphere (r = 1) and the field strength at the surface is 1 Oe, then there must be a magnetic flux of 4π units of induction passing through it.

You will have noticed the use of the permeability of free space μ_o in the above treatment – a parameter missing in many books and articles using the cgs units. The reason for this is that μ_o is unity in cgs units and simply converts oersteds (H) to gauss (B = μ_oH). Therefore in cgs units, B and H are used interchangeably. We inserted it in this derivation to remind us that there IS a difference and that the difference becomes very important when we convert to SI because μ_o is not unity, but 4π x 10⁻⁷! For conversion between commonly used cgs and SI paramters, please refer to Table 1.1.

Proceeding to the notion of magnetic moment, from a cgs point of view, we start with a magnet of length l with two poles of strength p at each end. Placing the magnet in a field μ_oH, we find that it experiences a torque Γ proportional to p,l and H such that

(1.6)

Recalling our earlier discussion of magnetic moment, you will realize that pl is simply the magnetic moment m. This line of reasoning also makes clear why it is called a “moment”. The units of torque are energy, which are ergs in cgs, so the units of magnetic moment are technically erg per gauss. But because of the “silent” μ_o in cgs, magnetic moment is most often defined as erg per oersted We therefore follow convention and define the “electromagnetic unit” (emu) as being one erg ⋅ oe⁻¹. [Some use emu to refer to the magnetization (volume normalized moment, see above), but this is incorrect and a source of a lot of confusion.]

1.8 The magnetic potential

An isolated electrical charge produces an electrical field that begins at the source (the charge) and spread (diverge) outward (see Figure 1.6a). Because there is no return flux to an oppositely charged “sink”, there is a net flux out of the dashed box shown in the figure. The divergence of the electrical field is defined as ∇⋅ E which quantifies the net flux (see Appendix A.3.6 for more). In the case of the field around an electric charge, the divergence is non-zero.

Magnetic fields are different from electrical fields in that there is no equivalent to an isolated electrical charge; there are only pairs of “opposite charges” – magnetic dipoles. Therefore, any line of flux starting at one magnetic pole, returns to its sister pole and there is no net flux out of the box shown in Figure 1.6b; the magnetic field has no divergence (Figure 1.6b). This property of magnetic fields is another of Maxwell’s equations: ∇⋅ B = 0.

In the special case away from electric currents and magnetic sources (so B = μ_oH), the magnetic field can be written as the gradient of a scalar field that is known as the magnetic potential, ψ_m, i.e.,

The presence of a magnetic moment m creates a magnetic field which is the gradient of some scalar field. To gain a better intuitive feel about the relationship between scalar fields and their gradient vector fields, see Appendix A.3.6. Because the divergence of the magnetic field is zero, by definition, the divergence of the gradient of the scalar field is also zero, or ∇²ψ_m = 0. The operator ∇² is called the Laplacian and ∇²ψ_m = 0 is Laplace’s equation. This will be the starting point for spherical harmonic analysis of the geomagnetic field discussed briefly in Chapter 2.

The curl of the magnetic field (∇×H) depends on the current density and is not always zero and magnetic fields cannot generally be represented as the gradient of a scalar field. Laplace’s equation is only valid outside the magnetic sources and away from currents.

So what is this magnetic potential and how does it relate to the magnetic moments that give rise to the magnetic field? Whatever it is, it has to satisfy Laplace’s equation, so we turn to solutions of Laplace’s equation for help. One solution is to define the magnetic potential ψ_m as a function of the vector r with radial distance r and the angle θ from the moment. Given a dipole moment m, a solution to Laplace’s equation is:

(1.7)

You can verify this by making sure that∇²ψ_m = 0.

The radial (H_r) and tangential (H_θ) components of H at P (Figure 1.7) then would be:

(1.8)

respectively.

1.9 Origin of the geomagnetic field

Measurement and description of the geomagnetic field and its spatial and temporal variations constitute one of the oldest geophysical disciplines. However, our ability to describe the field far exceeds our understanding of its origin. All plausible theories involve generation of the geomagnetic field within the fluid outer core of the Earth by some form of magnetohydrodynamic dynamo. Attempts to solve the full mathematical complexities of magnetohydrodynamics succeeded only in 1995 (Glatzmaier and Roberts, 1995).

Quantitative treatment of magnetohydrodynamics is (mercifully) beyond the scope of this book, but we can provide a qualitative explanation. The first step is to gain some appreciation for what is meant by a self-exciting dynamo. Maxwell’s equations tell us that electric and changing magnetic fields are closely linked and can affect each other. Moving an electrical conductor through a magnetic field will cause electrons to flow, generating an electrical current. This is the principle of electric motors. A simple electromechanical disk-dynamo model such as that shown in Figure 1.8 contains the essential elements of a self-exciting dynamo. The model is constructed with a copper disk rotating attached to an electrically conducting (e.g., brass) axle. An initial magnetic induction field, B, is perpendicular to the copper disk in an upward direction. Electrons in the copper disk experience a push from the magnetic field known as the Lorentz force, F_L, when they pass through the field.

The Lorentz force is given by:

(1.9)

where q is the electrical charge of the electrons, and v is their velocity. The Lorentz force on the electrons is directed toward the axle of the disk and the resulting electrical current flow is toward the outside of the disk (Figure 1.8).

Brush connectors are used to tap the electrical current from the disk, and the current passes through a coil under the disk. This coil is cleverly wound so that the electrical current produces a magnetic induction field in the same direction as the original field. The electrical circuit is a positive feedback system that reinforces the original magnetic induction field. The entire disk-dynamo model is a self-exciting dynamo. As long as the disk keeps rotating, the electrical current will flow, and the magnetic field will be sustained even if the original field disappears.

With this simple model we encounter the essential elements of any self-exciting dynamo:

1. A moving electrical conductor is required and is represented by the rotating copper disk.
2. An initial magnetic field is required.
3. An interaction between the magnetic field and the conductor must take place to provide reinforcement of the original magnetic field. In the model, this interaction is the Lorentz force with the coil acting as a positive feedback (self-exciting) circuit.
4. Energy must be supplied to overcome electrical resistivity losses. In the model, energy must be supplied to keep the disk rotating.

More complicated setups using two disks whose fields interact with one another generate chaotic magnetic behavior that can switch polarities even if the mechanical motion remains steady. Certainly no one proposes that systems of disks and feedback coils exist in the Earth’s core. But interaction between the magnetic field and the electrically conducting iron-nickel alloy in the outer core can produce a positive feedback and allow the Earth’s core to operate as a self-exciting magnetohydrodynamic dynamo. For reasonable electrical conductivities, fluid viscosity, and plausible convective fluid motions in the Earth’s outer core, the fluid motions can regenerate the magnetic field that is lost through electrical resistivity. There is a balance between fluid motions regenerating the magnetic field and loss of magnetic field because of electrical resistivity. The dominant portion of the geomagnetic field detectable at the surface is essentially dipolar with the axis of the dipole nearly parallel to the rotational axis of the Earth. Rotation of the Earth must therefore be a controlling factor on the time-averaged fluid motions in the outer core. It should also be pointed out that the magnetohydrodynamic dynamo can operate in either polarity of the dipole. Thus, there is no contradiction between the observation of reversals of the geomagnetic dipole and magnetohydrodynamic generation of the geomagnetic field. However, understanding the special interactions of fluid motions and magnetic field that produce geomagnetic reversals is a major challenge.

As wise economists have long observed, there is no free lunch. The geomagnetic field is no exception. Because of ohmic dissipation of energy, there is a requirement for energy input to drive the magnetohydrodynamic fluid motions and thereby sustain the geomagnetic field. Estimates of the power (energy per unit time) required to generate the geomagnetic field are about 10¹³ W (roughly the output of 10⁴ nuclear power plants). This is about one fourth of the total geothermal flux, so the energy involved in generation of the geomagnetic field is a substantial part of the Earth’s heat budget.

Many sources of this energy have been proposed, and ideas on this topic have changed over the years. The energy sources that are currently thought to be most reasonable are a combination of cooling of the Earth’s core with attendant freezing of the outer core and growth of the solid inner core. The inner core is pure iron, while the liquid outer core is some 15% nickel (and probably has trace amounts of other elements as well). The freezing of the inner core therefore generates a bouyancy force as the remaining liquid becomes more enriched in the lighter elements. These energy sources are sufficient to power the fluid motions of the outer core required to generate the geomagnetic field.

SUPPLEMENTAL READINGS: Jiles (1991), Chapter 1; Cullity (1972), Chapter 1.

1.10 Problems

Problem 1

In axisymmetric spherical coordinates, ∇ (the gradient operator) is given by

We also know that

and that ψ_m is a scalar function of position:

Find the radial and tangential components of H if m is 80 ZAm², [remember that “Z” stands for Zeta which stands for 10²¹], r is 6 x 10⁶ m and θ is 45^o. What are these field values in terms of B (teslas)?

Write your answers in a markdown cell in a jupyter notebook using latex syntax.

Problem 2

a) In your Jupyter notebook, write Python functions to convert induction, moment and magnetic field quantities in cgs units to SI units. Use the conversion factors in Table 1.1. Use your function to convert the following from cgs to SI:

i) B = 3.5 x10⁵ G

ii) m = 2.78 x 10⁻²⁰ G cm³

iii) H = 128 oe

b) In a new code block, modify your function to allow conversion from cgs => SI or SI => cgs. Rerun it to convert your answers from a) back to cgs.

HINTS: Call the functions with the values of B, m and H and have the function return the converted values. In the modified functions, you can specify whether the conversion is from cgs or SI.

Problem 3

Figure 1.9 shows a meridional cross section through the Earth in the plane of a magnetic dipole source m. At the location directly above the dipole, the field from the dipole is directed vertically downward and has intensity 10 μT. The dipole source is placed at 3480 km from the center of the Earth. Assume a mean Earth radius of 6370 km. Adapt the geometry of Figure 1.7 and the equations describing the magnetic field of a dipole to the model dipole in Figure 1.9.

a) Calculate the magnetic dipole moment of the model dipole. Remember to keep track of your units!

b) Compare this field to the total field produced by a centered axial magnetic dipole moment (i.e., one that is pointing straight up and is in the center of the circles) equivalent to that of the present geomagnetic field (m ∼ 80 ZAm²; Z=10²¹). Assume a latitude for the point of observation of 60^∘. [HINT: the angle θ in Equation 1.10 is the co-latitude, not the latitude.]

(1.10)

Problem 4

Knowing that B = μ_oH, work out the fundamental units of μ_o in SI units. Prepare your answer in a markdown cell in your Jupyter notebook.

Chapter 2
The geomagnetic field

The part of the geomagnetic field of interest to paleomagnetists is generated by convection currents in the liquid outer core of the Earth which is composed of iron, nickel and some unknown lighter component(s). The source of energy for this convection is not known for certain, but is thought to be partly from cooling of the core and partly from the bouyancy of the iron/nickel liquid outer core caused by freezing out of the pure iron inner core. Motions of this conducting fluid are controlled by the bouyancy of the liquid, the spin of the Earth about its axis and by the interaction of the conducting fluid with the magnetic field (in a horribly non-linear fashion). Solving the equations for the fluid motions and resulting magnetic fields is a challenging computational task. Recent numerical models, however, show that such magnetohydrodynamical systems can produce self-sustaining dynamos which create enormous external magnetic fields.

2.1 Components of magnetic vectors

The magnetic field of a dipole aligned along the spin axis and centered in the Earth (a so-called geocentric axial dipole, or GAD) is shown in Figure 2.1a. [See Chapter 1 for a derivation of how to find the radial and tangential components of such a field.] By convention, the sign of the Earth’s dipole is negative, pointing toward the south pole as shown in Figure 2.1a and magnetic field lines point toward the north pole. They point downward in the northern hemisphere and upward in the southern hemisphere.

Although dominantly dipolar, the geomagnetic field is not perfectly modeled by a geocentric axial dipole, but is somewhat more complicated (see Figure 2.1b). At the point on the surface labeled ‘P’, the geomagnetic field points nearly north and down at an angle of approximately 60^∘. Vectors in three dimensions are described by three numbers and in many paleomagnetic applications, these are two angles (D and I) and the strength (B) as shown in Figure 2.1b and c. The angle from the horizontal plane is the inclination I; it is positive downward and ranges from +90^∘ for straight down to -90^∘ for straight up. If the geomagnetic field were that of a perfect GAD field, the horizontal component of the magnetic field (B_H in Figure 2.1b) would point directly toward geographic north. In most places on the Earth there is a deflection away from geographic north and the angle between geographic and magnetic north is the declination, D (see Figure 2.1c). D is measured positive clockwise from North and ranges from 0 → 360^∘. [Westward declinations can also be expressed as negative numbers, i.e., 350^∘ = -10^∘.] The vertical component (B_V in Figure 2.1b,c) of the geomagnetic field at P, is given by

(2.1)

and the horizontal component B_H (Figure 2.1c) by

(2.2)

B_H can be further resolved into north and east components (B_N and B_E in Figure 2.1c) by

(2.3)

Depending on the particular problem, some coordinate systems are more suitable to use because they have the symmetry of the problem built into them. We have just defined a coordinate system using two angles and a length (B,D,I) and the equivalent Cartesian coordinates of (B_N,B_E,B_V ). We will need to convert among them at will. There are many names for the Cartesian coordinates. In addition to north, east and down, they could also be x,y,z or even x₁,x₂ and x₃. The convention used in this book is that axes are denoted X₁,X₂,X₃, while the components along the axes are frequently designated x₁,x₂,x₃. In the geographic frame of reference, positive X₁ is to the north, X₂ is east and X₃ is vertically down in keeping with the right-hand rule. To convert from Cartesian coordinates to angular coordinates (B,D,I):

(2.4)

Be careful of the sign ambiguity of the tangent function. You may well end up in the wrong quadrant and have to add 180^∘; this will happen if both x₁ and x₂ are negative. In most computer languages, there is a function atan2 which takes care of this, but most hand calculators will not. Remember that most computer languages expect angles to be given in radians, not degrees, so multiply degrees by π∕180 to convert to radians. Note also that in place of B for magnetic induction with units of tesla as a measure of vector length, (see Chapter 1), we could also use H, M ( both Am⁻¹) or m (Am²) for magnetic field, magnetization or magnetic moment respectively.

2.2 Reference magnetic field

We can measure declination, inclination and intensity at different places around the globe, but not everywhere all the time. Yet it is often handy to be able to predict what these components are. For example, it is extremely useful to know what the deviation is between true North and declination in order to find our way with maps and compasses. In principle, magnetic field vectors can be derived from the magnetic potential ψ_m as we showed in Chapter 1. For an axial dipolar field, there is but one scalar coefficient (the magnetic moment m of a dipole source). For the geomagnetic field, there are many more coefficients, including not just an axial dipole aligned with the spin axis, but two orthogonal equatorial dipoles and a whole host of more complicated sources such as quadrupoles, octupoles and so on. A list of coefficients associated with these sources allows us to calculate the magnetic field vector anywhere outside of the source region. In this section, we outline how this might be done.

As we learned in Chapter 1, the magnetic field at the Earth’s surface can be calculated from the gradient of a scalar potential field (H = −∇ψ_m), and this scalar potential field satisfies Laplace’s Equation:

(2.5)

For the geomagnetic field (ignoring external sources of the magnetic field which are in any case small and transient), the potential equation can be written as:

(2.6)

where a is the radius of the Earth (6.371 × 10⁶ m). In addition to the radial distance r and the angle away from the pole θ, there is ϕ, the angle around the equator from some reference, say, the Greenwich meridian. Here, θ is the co-latitude and ϕ is the longitude. The g_l^ms and h_l^ms are the gauss coefficients (degree l and order m) for hypothetical sources at radii less than a calculated for a particular year. These are normally given in units of nT. The P_l^ms are wiggly functions called partially normalized Schmidt polynomials of the argument cosθ. These are closely related to the associated Legendre polynomials. [When m = 0 the Schmidt and Legendre polynomials are identical.] The first few of P_l^ms are:

To get an idea of how the gauss coefficients in the potential relate to the associated magnetic fields, we show three examples in Figure 2.3. We plot the inclinations of the vector fields that would be produced by the terms with g₁⁰,g₂⁰ and g₃⁰ respectively. These are the axial (m = 0) dipole (l = 1), quadrupole (l = 2) and octupole (l = 3) terms. The associated potentials for each harmonic are shown in the insets.

In general, terms for which the difference between the subscript (l) and the superscript (m) is odd (e.g., the axial dipole g₁⁰ and octupole g₃⁰) produce magnetic fields that are antisymmetric about the equator, while those for which the difference is even (e.g., the axial quadrupole g₂⁰) have symmetric fields. In Figure 2.3a we show the inclinations produced by a purely dipolar field of the same sign as the present day field. The inclinations are all positive (down) in the northern hemisphere and negative (up) in the southern hemisphere. In contrast, inclinations produced by a purely quadrupolar field (Figure 2.3b) are down at the poles and up at the equator. The map of inclinations produced by a purely axial octupolar field (Figure 2.3c) are again asymmetric about the equator with vertical directions of opposite signs at the poles separated by bands with the opposite sign at mid-latitudes.

As noted before, there is not one, but three dipole terms in Equation 2.6, the axial term (g₁⁰) and two equatorial terms (g₁¹ and h₁¹). Therefore, the total dipole contribution is the vector sum of these three or . The total quadrupole contribution (l = 2) combines five coefficients and the total octupole (l = 3) contribution combines seven coefficients.

So how do we get this marvelous list of gauss coefficients? If you want to know the details, please refer Langel (1987). We will just give a brief introduction here. Recalling Chapter 1, once the scalar potential ψ_m is known, the components of the magnetic field can be calculated from it. We solved this for the radial and tangential field components (H_r and H_θ) in Chapter 1. We will now change coordinate and unit systems and introduce a third dimension (because the field is not perfectly dipolar). The north, east, and vertically down components are related to the potential ψ_m by:

(2.7)

where r, θ, ϕ are radius, co-latitude (degrees away from the North pole) and longitude, respectively. Here, B_V is positive down, B_E is positive east, and B_N is positive to the north, the opposite of H_r and H_θ as defined in Chapter 1. Note that Equation 2.7 is in units of induction, not Am⁻¹ if the units for the gauss coefficients are in nT, as is the current practice.

Going backwards, the gauss coefficients are determined by fitting Equations 2.7 and 2.6 to observations of the magnetic field made by magnetic observatories or satellite for a particular time. The International (or Definitive) Geomagnetic Reference Field or I(D)GRF, for a given time interval is an agreed upon set of values for a number of gauss coefficients and their time derivatives. IGRF (or DGRF) models and programs for calculating various components of the magnetic field are available on the internet from the National Geophysical Data Center; the address is http://www.ngdc.noaa.gov. there is also a program igrf.py included in the PmagPy package (see igrf.py documentation).

In practice, the gauss coefficients for a particular reference field are estimated by least-squares fitting of observations of the geomagnetic field. You need a minimum of 48 observations to estimate the coefficients to l = 6. Nowadays, we have satellites which give us thousands of measurements and the list of generation 10 of the IGRF for 2005 goes to l = 13.

In order to get a feel for the importance of the various gauss coefficients, take a look at Table 2.1, which has the Schmidt quasi-normalized gauss coefficients for the first six degrees from the IGRF for 2005. The power at each degree is the average squared field per spherical harmonic degree over the Earth’s surface and is calculated by R_l = ∑ _m(l + 1)[(g_l^m)² + (h_l^m)²] (Lowes, 1974). The so-called Lowes spectrum is shown in Figure 2.4. It is clear that the lowest order terms (degree one) totally dominate, constituting some 90% of the field. This is why the geomagnetic field is often assumed to be equivalent to a magnetic field created by a simple dipole at the center of the Earth.

2.3 Geocentric axial dipole (GAD) and other poles

The beauty of using the geomagnetic potential field is that the vector field can be evaluated anywhere outside the source region. Using the values for a given reference field in Equations 2.6 and 2.7, we can calculate values of B,D and I at any location on Earth. Figure 2.1b shows the lines of flux predicted from the 2005 IGRF from the core-mantle boundary up. We can see that the field becomes simpler and more dipolar as we move from the core mantle boundary to the surface. Yet, there is still significant non-dipolar structure in the geomagnetic field even at the Earth’s surface.

We can recast the vectors at the surface of the Earth into maps of components as shown in Figure 2.5a,b. We show the potential in Figure 2.5c for comparison with that of a pure dipole (inset to Figure 2.3a). These maps illustrate the fact that the field is a complicated function of position on the surface of the Earth. The intensity values in Figure 2.5a are, in general, highest near the poles (∼ 60 μT) and lowest near the equator (∼ 30 μT), but the contours are not straight lines parallel to latitude as they would be for a field generated strictly by a geocentric axial dipole (GAD) (e.g, Figure 2.1a). Similarly, a GAD would produce lines of inclination that vary in a regular way from -90^∘ to +90^∘ at the poles, with 0^∘ at the equator; the contours would parallel the lines of latitude. Although the general trend in inclination shown in Figure 2.5b is similar to the GAD model, the field lines are more complicated, which again suggests that the field is not perfectly described by a geocentric bar magnet.

Perhaps the most important result of spherical harmonic analysis for our purposes is that the field at the Earth’s surface is dominated by the degree one terms (l = 1) and the external contributions are very small. The first order terms can be thought of as geocentric dipoles that are aligned with three different axes: the spin axis (g₁⁰) and two equatorial axes that intersect the equator at the Greenwich meridian (h₁⁰) and at 90^∘ East (h₁¹). The vector sum of these geocentric dipoles is a dipole that is currently inclined by about 10^∘ to the spin axis. The axis of this best-fitting dipole pierces the surface of the Earth at the circle in Figure 2.6. This point and its antipode are called geomagnetic poles. Points at which the field is vertical (I = ±90^∘ shown by a square in Figure 2.6) are called magnetic poles, or sometimes dip poles. These poles are distinguishable from the geographic poles, where the spin axis of the Earth intersects its surface. The northern geographic pole is shown by a star in Figure 2.6.

It turns out that when averaged over sufficient time, the geomagnetic field actually does seem to be approximately a GAD field, perhaps with a pinch of g₂⁰ thrown in (see e.g., Merrill et al., 1996). The GAD model of the field will serve as a useful crutch throughout our discussions of paleomagnetic data and applications. Averaging ancient magnetic poles over enough time to average out secular variation (thought to be 10⁴ or 10⁵ years) gives what is known as a paleomagnetic pole; this is usually assumed to be co-axial with the Earth’s geographic pole (the spin axis).

Because the geomagnetic field is axially dipolar to a first approximation, we can write:

(2.8)

Note that g₁⁰ is given in nT in Table 2.1. Thus, from Equation 2.8,

(2.9)

Given some latitude λ on the surface of the Earth in Figure 2.1a and using the equations for B_V and B_N, we find that:

(2.10)

This equation is sometimes called the dipole formula and shows that the inclination of the magnetic field is directly related to the co-latitude (θ) for a field produced by a geocentric axial dipole (or g₁⁰). The dipole formula allows us to calculate the latitude of the measuring position from the inclination of the (GAD) magnetic field, a result that is fundamental in plate tectonic reconstructions. The intensity of a dipolar magnetic field is also related to (co)latitude because:

(2.11)

The dipole field intensity has changed by more than an order of magnitude in the past and the dipole relationship of intensity to latitude turns out to be not useful for tectonic reconstructions.

2.4 Plotting magnetic directional data

Magnetic field and magnetization directions can be visualized as unit vectors anchored at the center of a unit sphere. Such a unit sphere is difficult to represent on a 2-D page. There are several popular projections, including the Lambert equal area projection which we will be making extensive use of in later chapters. The principles of construction of the equal area projection are covered in the Appendix B.1.

In general, regions of equal area on the sphere project as equal area regions on this projection, as the name implies. Plotting directional data in this way enables rapid assessment of data scatter. A drawback of this projection is that circles on the surface of a sphere project as ellipses. Also, because we have projected a vector onto a unit sphere, we have lost information concerning the magnitude of the vector. Finally, lower and upper hemisphere projections must be distinguished with different symbols. The paleomagnetic convention is: lower hemisphere projections (downward directions) use solid symbols, while upper hemisphere projections are open.

The dipole formula allows us to convert a given measurement of I to an equivalent magnetic co-latitude θ_m:

(2.12)

If the field were a simple GAD field, θ_m would be a reasonable estimate of θ, but non-GAD terms can invalidate this assumption. To get a feel for the effect of these non-GAD terms, we consider first what would happen if we took random measurements of the Earth’s present field (see Figure 2.7). We evaluated the directions of the magnetic field using the IGRF for 2005 at 200 positions on the globe (shown in Figure 2.7a). These directions are plotted in Figure 2.7b using the paleomagnetic convention of open symbols pointing up and closed symbols pointing down. In Figure 2.7c, we plot the inclinations as a function of latitude. As expected from a predominantly dipolar field, inclinations cluster around the values for a geocentric axial dipolar field but there is considerable scatter and interestingly the scatter is larger in the southern hemisphere than in the northern one. This is related to the low intensities beneath South America and the Atlantic region seen in Figure 2.5a.

2.4.1 D′,I′ transformation

Often we wish to compare directions from distant parts of the globe. There is an inherent difficulty in doing so because of the large variability in inclination with latitude. In such cases it is appropriate to consider the data relative to the expected direction (from GAD) at each sampling site. For this purpose, it is useful to use a transformation whereby each direction is rotated such that the direction expected from a geocentric axial dipole field (GAD) at the sampling site is the center of the equal area projection. This is accomplished as follows:

Each direction is converted to Cartesian coordinates (x_i) by:

(2.13)

These are rotated to the new coordinate system (x′_i, see Appendix A.3.5) by:

where I_d = the inclination expected from a GAD field (tanI_d = 2tanλ), λ is the site latitude, and α is the inclination of the paleofield vector projected onto the N-S plane (α = tan⁻¹(x₃∕x₁)). The x′_i are then converted to D′,I′ by Equation 2.4.

In Figure 2.8a we show the geomagnetic field vectors evaluated at random longitudes along a latitude band of 45^∘N. The vectors are shown in their Cartesian coordinates of North, East and Down. In Figure 2.8b we show what happens when we rotate the coordinate system to peer down the direction expected from an axial dipolar field at 45^∘N (which has an inclination of 63^∘). The vectors circle about the expected direction. Finally, we see what happens to the directions shown in Figure 2.7b after the D′,I′ transformation in Figure 2.8. These are unit vectors projected along the expected direction for each observation in Figure 2.7a. Comparing the equal area projection of the directions themselves (Figure 2.7b) to the transformed directions (Figure 2.8c), we see that the latitudal dependence of the inclinations has been removed.

2.4.2 Virtual geomagnetic poles

We are often interested in whether the geomagnetic pole has changed, or whether a particular piece of crust has rotated with respect to the geomagnetic pole. Yet, what we observe at a particular location is the local direction of the field vector. Thus, we need a way to transform an observed direction into the equivalent geomagnetic pole.

In order to remove the dependence of direction merely on position on the globe, we imagine a geocentric dipole which would give rise to the observed magnetic field direction at a given latitude (λ) and longitude (ϕ). The virtual geomagnetic pole (VGP) is the point on the globe that corresponds to the geomagnetic pole of this imaginary dipole (Figure 2.9a).

Paleomagnetists use the following conventions: ϕ is measured positive eastward from the Greenwich meridian and ranges from 0 → 360^∘; θ is measured from the North pole and goes from 0 → 180^∘. Of course θ relates to latitude, λ by θ = 90 − λ. θ_m is the magnetic co-latitude and is given by Equation 2.12. Be sure not to confuse latitudes and co-latitudes. Also, be careful with declination. Declinations between 180^∘ and 360^∘ are equivalent to D - 360 ^∘ which are counter-clockwise with respect to North.

The first step in the problem of calculating a VGP is to determine the magnetic co-latitude θ_m by Equation 2.12 which is defined in the dipole formula (Equation 2.12). The declination D is the angle from the geographic North Pole to the great circle joining the observation site S and the pole P, and Δϕ is the difference in longitudes between P and S, ϕ_p −ϕ_s. Now we use some tricks from spherical trigonometry as reviewed in Appendix A.3.1.

We can locate VGPs using the law of sines and the law of cosines. The declination D is the angle from the geographic North Pole to the great circle joining S and P (see Figure 2.9) so:

(2.14)

which allows us to calculate the VGP co-latitude θ_p. The VGP latitude is given by:

To determine ϕ_p, we first calculate the angular difference between the pole and site longitude Δϕ.

(2.15)

If cosθ_m ≥ cosθ_s cosθ_p, then ϕ_p = ϕ_s + Δϕ. However, if cosθ_m < cosθ_s cosθ_p then ϕ_p = ϕ_s + 180 − Δϕ.

Now we can convert the directions in Figure 2.7b to VGPs (see Figure 2.9c). The grouping of points is much tighter in Figure 2.9c than in the equal area projection because the effect of latitude variations in dipole fields has been removed. If a number of VGPs are averaged together, the average pole position is called a “paleomagnetic pole”. How to average poles and directions is the subject of Chapters 11 and 12.

The procedure for calculating a direction from a VGP is a similar procedure to that for calculating the VGP from the direction. Magnetic colatitude θ_m is calculated in exactly the same way as before and yields inclination from the dipole formula. The declination can be calculated by solving for D in Equation 2.14 as:

This equation works most of the time, but breaks down under some circumstances, for example, when the pole latitude is further to the south than the site latitude. The following algorithm works in the more general case:

2.4.3 Virtual dipole moment

As pointed out earlier, magnetic intensity varies over the globe in a similar manner to inclination. It is often convenient to express paleointensity values in terms of the equivalent geocentric dipole moment that would have produced the observed intensity at a specific (paleo)latitude. Such an equivalent moment is called the virtual dipole moment (VDM) by analogy to the VGP (see Figure 2.9a). First, the magnetic (paleo)co-latitude θ_m is calculated as before from the observed inclination and the dipole formula of Equation 2.10. Then, following the derivation of Equation 2.11, we have

(2.16)

Sometimes the site co-latitude as opposed to magnetic co-latitude is used in the above equation, giving a virtual axial dipole moment (VADM; see Figure 2.9d).

SUPPLEMENTAL READINGS: Merrill et al. (1996), Chapters 1 & 2

2.5 Problems

For this and future problem sets, you will need the PmagPy package (see section in the Preface at the beginning of the book). After you have installed this and properly set your path, you can import the functions from PmagPy using these commands:

Please consult the Jupyter notebook PmagPy.ipynb for more help on using PmagPy functions within a notebook.

Problem 1

a) Write a python script in an Jupyter notebook that converts declination, inclination and intensity to North, East, and Down. Read in the data in the file Chapter_2/ps2_prob1_data.txt. For this the loadtxt function in the Numpy module will come in handy.

b) Choose 10 random spots on the surface of the earth. You can use the pmag.get_unf to generate a list for you. Then use the ipmag.igrf function to evaluate the declination, inclination and intensity at each of these locations in January 2006. As with all PmagPy programs, and functions, you can find out what they do by printing out the doc string: you can find out what they do by getting the help message:

Calls like these generates help messages which will help you to call the function properly.

c) Take the vectors from the output of Problem 1b and convert them to cartesian coordinates, using the script you wrote in Problem 1a.

Problem 2

a) Plot the IGRF directions from Problem 1b on an equal area projection by hand. Use the equal area net provided in the Appendix. Remember that the outer rim is horizontal and the center of the diagram is vertical. Azimuth goes around the rim with clockwise being positive. Put a thumbtack through the equal area (Schmidt) net and place a piece of tracing paper on the thumbtack. Mark the top of the stereonet with a tick mark on the tracing paper.

To plot a direction, rotate the tick mark of the tracing paper around counter clockwise until the top of the paper is rotated by the declination of the direction. Then count tick marks toward the center from the outer rim (the horizontal) to the inclination angle, plot the point, and rotate back so that the tick is North again. Put all your points on the diagram.

b) Now use the ipmag functions plot_net and plot_di. or write your own! Both plots should look the same....

Problem 3

You went to Wyoming (112^∘ W and 36^∘ N) to sample some Cretaceous rocks. You measured a direction with a declination of 345^∘ and an inclination of 47^∘.

a) What direction would you expect from the present (GAD) field?

b) What is the virtual geomagnetic pole position corresponding to the direction you actually measured? [Hint: Use the function pmag.dia_vgp in the PmagPy module or for a challenge, write your own! ]

Chapter 3
Induced and remanent magnetism

Scientists in the late 19th century thought that it might be possible to exploit the magnetic record retained in accidental records to study the geomagnetic field in the past. Work in the mid 20th century provided the theoretical and experimental basis for presuming that such materials might retain a record of past geomagnetic fields. There are several books and articles that describe the subject in detail (see e.g., the supplemental readings). We present here a brief overview of theories on how rocks get and stay magnetized. We will begin with magnetism at the atomic level caused by electronic orbits and spins giving rise to induced magnetizations. Then we will see how electronic spins working in concert give rise to permanently magnetized substances (like magnetic minerals) making remanent magnetization possible.

3.1 Magnetism at the atomic level

We learned in Chapter 1 that magnetic fields are generated by electric currents. Given that there are no wires leading into or out of permanent magnets, you may well ask, “Where are the currents?” At the atomic level, the electric currents come from the motions of the electrons. From here quantum mechanics quickly gets esoteric, but some rudimentary understanding is helpful. In this chapter we will cover the bare minimum necessary to grasp the essentials of rock magnetism.

In Chapter 1 we took the classical (pre-quantum mechanics) approach and suggested that the orbit of an electron about the nucleus could be considered a tiny electric current with a correspondingly tiny magnetic moment. But quantum physics tells us that this “planetary” view of the atom cannot be true. An electron zipping around a nucleus would generate radio waves, losing energy and eventually would crash into the nucleus.

Apparently, this does not happen, so the classical approach is fatally flawed and we must turn to quantum mechanics.

In quantum mechanics, electronic motion is stabilized by the fact that electrons can only have certain energy states; they are quantized. The energy of a given electron can be described in terms of solutions, Ψ, to something called Schrödinger’s wave equation. The function Ψ(r,θ,ϕ) gives the probability of finding an electron at a given position. [Remember from Chapter 2 that r,θ,ϕ are the three spherical coordinates.] It depend on three special quantum numbers (n,l,m):

(3.1)

The number n is the so-called “principal” quantum number. The R_n^l(r) are functions specific to the element in question and the energy state of the electron n. It is evaluated at an effective radius r in atomic units. The Y _l^m are a fully normalized complex representation of the spherical harmonics introduced in Section 2.2. For each level n, the number l ranges from 0 to n-1 and m from l backwards to −l.

The lowest energy of the quantum wave equations is found by setting n equal to unity and both l and m to zero. Under these conditions, the solution to the wave equation is given by:

(3.2)

where Z is the atomic number and ρ is 2Zr∕n. Note that at this energy level, there is no dependence of Y on ϕ or θ. Substituting these two equations into Equation 3.1 gives the probability density Ψ for an electron as a function of radius of r. This is sketched as the line in Figure 3.1. Another representation of the same idea is shown in the inset, whereby the density of dots at a given radius reflects the probability distribution shown by the solid curve. The highest dot density is found at a radius of about one atomic unit, tapering off the farther away from the center of the atom. Because there is no dependence on θ or ϕ the probability distribution is a spherical shell. All the l,m = 0 shells are spherical and are often referred to as the 1s, 2s, 3s shells, where the numbers are the energy levels n. A surface with equal probability is a sphere and example of one such shell is shown in Figure 3.2a.

For l = 1, m will have values of -1, 0 and 1 and the Y _l^m(ϕ,θ)s are given by:

As might be expected, the shells for l = 2 are even more complicated that for l = 1. These shells are called “d” shells and two examples are shown in Figure 3.2c and d.

Returning to the tiny circuit idea, somehow the motion of the electrons in their shells acts like an electronic circuit and creates a magnetic moment. In quantum mechanics, the angular momentum vector of the electron L is quantized, for example as integer multiples of ℏ, the “reduced” Planck’s constant (or h _ 2π where h = 6.63 x 10⁻³⁴ Js). The magnetic moment arising from the orbital angular momentum is given by:

(3.3)

This is known as the Bohr magneton.

So far we have not mentioned one last quantum number, s. This is the “spin” of the electron and has a value of ±1 2. The spin itself produces a magnetic moment which is given by 2sm_b, hence is numerically identical to that produced by the orbit.

Atoms have the same number of electrons as protons in order to preserve charge balance. Hydrogen has but one lonely electron which in its lowest energy state sits in the 1s electronic shell. Helium has a happy pair, so where does the second electron go? To fill in their electronic shells, atoms follow three rules:

1. No two electrons may have the same set of quantum numbers. This is Pauli’s exclusion principle. Because spin (s) can be ±1 2, two electrons fit in one orbital. When a single electron occupies a given orbital, it is called “unpaired” and has a magnetic moment of 1 m_b.
2. Orbitals are filled in order of increasing energy. The energy state of a given orbital is dependent on the context (whether the atom is bound to other atoms or not), but in general they will be filled according to the scheme shown in Figure 3.3.
3. Electrons are added so that the spins remain as parallel as possible (Hund’s rule). Notice in Figure 3.3 that when filling the third energy level (n = 3), all five d shells are filled with one kind of spin (say, all up, or +1 2), before the electrons begin to pair up. Also, because the energies of the shells change somewhat according to the context they are in, the 4s shell will actually give up an electron to a d shell, before the d shells begin to pair up. Hund’s rule gives the atoms with some d shell electrons (the so-called “transition elements”, e.g., Cr, Mn, Fe, Co and Ni) the possibility of large magnetic moments.

Each unpaired spin has a moment of one Bohr magneton m_b. The elements with the most unpaired spins are the transition elements which are responsible for most of the paramagnetic behavior observed in rocks. For example, in Figure 3.3 we see that Mn has a structure of: (1s²2s²2p⁶3s²3p⁶)3d⁵4s², hence has five unpaired spins and a net moment of 5 m_b. Fe has a structure of (1s²2s²2p⁶3s²3p⁶)3d⁶4s² with a net moment of 4 m_b, In minerals, the transition elements are in a variety of oxidation states. Fe commonly occurs as Fe²⁺ and Fe³⁺. When losing electrons to form ions, transition metals lose the 4s electrons first, so we have for example, Fe³⁺ with a structure of (1s²2s²2p⁶3s²3p⁶)3d⁵, or 5 m_b. Similarly Fe²⁺ has 4 m_b and Ti⁴⁺ has no unpaired spins. Iron is the main magnetic species in geological materials, but Mn²⁺ (5 m_b) and Cr³⁺ (3 m_b) occur in trace amounts.

3.2 Induced magnetization

We have learned that there are two sources of magnetic moments in electronic motions: the orbits and the (unpaired) spins. These moments respond to external magnetic fields giving rise to an induced magnetization, a phenomenon alluded to briefly in Chapter 1. We will consider first the contribution of the electronic orbits.

3.2.1 Orbital contribution and diamagnetism

The angular momentum of electrons is quantized in magnitude but also has direction (see L in Figure 3.4). The angular momentum vector has an associated magnetic moment vector m_b. A magnetic field H exerts a torque on the moment, which nudges it (and the momentum vector associated with it) to the side (ΔL). L therefore will precess around the magnetic field direction, much like a spinning top precesses around the direction of gravity. The precession of L is called Larmor precession.

The changed momentum vector from Larmor precession in turn results in a changed magnetic moment vector Δm. The sense of the change in net moment is always to oppose the applied field. Therefore, the response of the magnetic moments of electronic orbitals creates an induced magnetization M_I that is observable outside the substance; it is related to the applied field by:

We learned in Chapter 1 that the proportionality between induced magnetization and the applied field is known as the magnetic susceptibility. The ratio M_I∕H for the response of the electronic orbitals is termed the diamagnetic susceptibility χ_d; it is negative, essentially temperature independent and quite small. This diamagnetic response is a property of all matter, but for substances whose atoms possess atomic magnetic moments, diamagnetism is swamped by effects of magnetic fields on the atomic magnetic moments. In the absence of unpaired electronic spins, diamagnetic susceptibility dominates the magnetic response. Common diamagnetic substances include quartz (SiO₂), calcite (CaCO₃) and water (H₂O). The mass normalized susceptibility of quartz is -0.62 x 10⁻⁹ m³kg⁻¹ to give you an idea of the magnitudes of these things.

3.2.2 Role of electronic spins and paramagnetism

In many geological materials, the orbital contributions cancel out because they are randomly oriented with respect to one another and the magnetization arises from the electronic spins. We mentioned that unpaired electronic spins behave as magnetic dipoles with a moment of one Bohr magneton. In the absence of an applied field, or in the absence of the ordering influence of neighboring spins which are known as exchange interactions, the electronic spins are essentially randomly oriented. An applied field acts to align the spins which creates a net magnetization equal to χ_pH where χ_p is the paramagnetic susceptibility. For any geologically relevant conditions, the induced magnetization is linearly dependent on the applied field. In paramagnetic solids, atomic magnetic moments react independently to applied magnetic fields and to thermal energy. At any temperature above absolute zero, thermal energy vibrates the crystal lattice, causing atomic magnetic moments to oscillate rapidly in random in orientations. In the absence of an applied magnetic field, atomic moments are equally distributed in all directions with a resultant magnetization of zero.

A useful first order model for paramagnetism was worked out by P. Langevin in 1905. (Of course in messy reality things are a bit more complicated, but Langevin theory will work well enough for us at this stage.) Langevin theory is based on a few simple premises:

1. Each unpaired spin contributes a dipole moment.
2. In the absence of an applied field, the moments are essentially randomly oriented, i.e., all directions are equally likely to occur.
3. Application of a magnetic field exerts an aligning torque on the atomic magnetic moments. The magnetic energy E_m (see also Section 1.4 in Chapter 1) of a magnetic moment m at an angle θ with an external magnetic field B = μ_oH is given by:

   (3.4)

   Magnetic energy is at a minimum when the magnetic moment is lined up with the magnetic field.

4. There is competition between the magnetic energy E_m and the thermal energy kT where k is Boltzmann’s constant (1.38 x 10⁻²³ m² kg s⁻² K⁻¹) and T is temperature in kelvin).

Consider an atomic magnetic moment, (m = 2m_b = 1.85×10⁻²³ Am²), in a magnetic field of 10⁻² T, (for reference, the largest geomagnetic field at the surface is about 65 μT – see Chapter 2). The aligning energy is therefore mB = 1.85 × 10⁻²⁵ J). However, thermal energy at 300K (traditionally chosen as a temperature close to room temperature providing easy arithmetic) is Boltzmann’s constant times the temperature, or about 4 x 10⁻²¹ J. So thermal energy is several orders of magnitude larger than the aligning energy and the net magnetization is small even in this rather large (compared to the Earth’s field) magnetizing field.

Using the principles of statistical mechanics, we find that the probability density of a particular magnetic moment having a magnetic energy of E_m is given by:

(3.5)

From this we see that the degree of alignment depends exponentially on the ratio of magnetic energy to thermal energy. The degree of alignment with the magnetic field controls the net magnetization M. When spins are completely aligned, the substance has a saturation magnetization M_s. The probability density function leads directly to the following relation (derived in Appendix A.2.1):

(3.6)

where a = mB∕kT. The function enclosed in square brackets is known as the Langevin function (ℒ).

Equation 3.6 is plotted in Figure 3.5a and predicts several intuitive results: 1) M = 0 when B = 0 and 2) M∕M_s = 1 when the applied magnetic field is infinite. Furthermore, M is some 90% of M_s when mB is some 10-20 times kT. When kT >> mB,ℒ(a) is approximately linear with a slope of ∼ 1∕3. At room temperature and fields up to many tesla, ℒ(a) is approximately mB∕3kT. If the moments are unpaired spins (m = m_b), then the maximum magnetization possible (M_s) is given by the number of moments N, their magnitude (m_b) normalized by the volume of the material v or M_s = Nm_b∕v, and

Please note that we have neglected all deviations from isotropy including quantum mechanical effects as well as crystal shape, lattice defects, and state of stress. These complicate things a little, but to first order the treatment followed here provides a good approximation. We can rewrite the above equation as:

(3.7)

To first order, paramagnetic susceptibility χ_p is positive, larger than diamagnetism and inversely proportional to temperature. This inverse T dependence (see Figure 3.5b) is known as Curie’s law of paramagnetism. The paramagnetic susceptibility of, for example, biotite is 790 x 10⁻⁹ m³ kg⁻¹, or about three orders of magnitude larger than quartz (and of the opposite sign!).

We have considered the simplest case here in which χ can be treated as a scalar and is referred to as the bulk magnetic susceptibility χ_b. In detail, magnetic susceptibility can be quite complicated. The relationship between induced magnetization and applied field can be affected by crystal shape, lattice structure, dislocation density, state of stress, etc., which give rise to possible anisotropy of the susceptibility. Furthermore, there are only a finite number of electronic moments within a given volume. When these are fully aligned, the magnetization reaches saturation. Thus, magnetic susceptibility is both anisotropic and non-linear with applied field.

3.3 Ferromagnetism

Some substances give rise to a magnetic field in the absence of an applied field. This magnetization is called remanent or spontaneous magnetization, also loosely known as ferromagnetism (sensu lato). Magnetic remanence is caused by strong interactions between neighboring spins that occur in certain crystals.

The so-called exchange energy is minimized when the spins are aligned parallel or anti-parallel depending on the details of the crystal structure. Exchange energy is a consequence of the Pauli exclusion principle (no two electrons can have the same set of quantum numbers). In the transition elements, the 3d orbital is particularly susceptible to exchange interactions because of its shape and the prevalence of unpaired spins, so remanence is characteristic of certain crystals containing transition elements with unfilled 3d orbitals.

In oxides, oxygen can form a bridge between neighboring cations which are otherwise too far apart for direct overlap of the 3d orbitals in a phenomenon known as superexchange. In Figure 3.6 the 2p electrons of the oxygen are shared with the neighboring 3d shells of the iron ions. Pauli’s exclusion principle means that the shared electrons must be antiparallel to each of the electrons in the 3d shells. The result is that the two cations are coupled. In the case shown in Figure 3.6 there is an Fe²⁺ ion coupled antiparallel to an Fe³⁺ ion. For two ions with the same charge, the coupling will be parallel. Exchange energies are huge, equivalent to the energy associated with the same moment in a field of the order of 1000 T. [The largest field available in the Scripps paleomagnetic laboratory is about 2.5 T, and that only fleetingly.]

As temperature increases, crystals expand and exchange becomes weaker. Above a temperature characteristic of each crystal type (known as the Curie temperature T_c), cooperative spin behavior disappears entirely and the material becomes paramagnetic.

While the phenomenon of ferromagnetism results from complicated interactions of neighboring spins, it is useful to think of the ferromagnetic moment as resulting from a quasi-paramagnetic response to a huge internal field. This imaginary field is termed the Weiss molecular field H_w. In Weiss theory, H_w is proportional to the magnetization of the substance, i.e.,

where β is the constant of proportionality. The total magnetic field that the substance experiences is:

where H is the external field. By analogy to paramagnetism, we can substitute a = μ_om_b(H_tot)∕kT) for H in Langevin function:

(3.8)

For temperatures above the Curie temperature T_c (i.e. T − T_c > 0) there is by definition no internal field, hence βM is zero. Substituting Nm_b∕v for M_s, and using the low-field approximation for ℒ(a), Equation 3.8 can be rearranged to get:

(3.9)

Equation 3.9 is known as the Curie-Weiss law and governs ferromagnetic susceptibility above the Curie temperature (dashed line in Figure 3.7).

Below the Curie temperature H_w >> H; we can neglect the external field H and get:

Substituting again for M_s and rearranging, we get:

(3.10)

where T_c is the Curie temperature and is given by:

We have treated ferromagnetism from a classical point of view and this is strictly incorrect because ferromagnetism results primarily from quantum mechanical phenomena. The primary difference between the classical derivation and the quantum mechanical one lies in the fact that in quantum mechanics, only certain angles of the magnetic moments are allowed, as opposed to all directions in Langevin theory. In the end, the predictions of magnetization as a function of temperature are different in detail. The end product of the quantum mechanical treatment (see Dunlop and Özdemir, 1997) is that the variation of saturation magnetization as a function of temperature can be reasonably well approximated (near the Curie Temperature, T_c) by a normalized power law variation:

(3.11)

where γ is 0.5 from simple molecular field theory and T_o is absolute zero (in kelvin). Dunlop and Özdemir (1997) cite a value of around 0.43 for γ, but the data sets cited by Dunlop and Özdemir (1997; e.g., Figure 3.5 on page 52) are actually best-fit with values for γ of about 0.36 – 0.39 (see Figure 3.8). These curves have been normalized by their inferred curie Temperatures which are around 565^∘C (data of B. Moskowitz, cited in Banerjee, 1991).

As we have seen, below the Curie temperature, certain crystals have a permanent (remanent) magnetization resulting from the alignment of unpaired electronic spins over a large area within the crystal. Spins may be either parallel or anti-parallel; the sense of spin alignment is controlled entirely by crystal structure. The energy term associated with this phenomenon is the exchange energy. There are three categories of spin alignment: ferromagnetism (sensu stricto), ferrimagnetism and antiferromagnetism (see Figure 3.9).

In ferromagnetism (sensu stricto, Figure 3.9a), the exchange energy is minimized when all the spins are parallel, as occurs in pure iron. When spins are perfectly antiparallel (antiferromagnetism, Figure 3.9b), there is no net magnetic moment, as occurs in ilmenite. Occasionally, the antiferromagnetic spins are not perfectly aligned in an antiparallel orientation, but are canted by a few degrees. This spin-canting (Figure 3.9c) gives rise to a weak net moment, as occurs in hematite, a common magnetic mineral (see Chapter 6). Also, antiferromagnetic materials can have a net moment if spins are not perfectly compensated owing to defects in the crystal structure, as occurs in fine-grained hematite. The uncompensated spins result in a so-called defect moment (Figure 3.9d). We note in passing that the temperature at which spins become disordered in antiferromagnetic substances is termed the Néel temperature. In ferrimagnetism, spins are also aligned antiparallel, but the magnitudes of the moments in each direction are unequal, resulting in a net moment (Figure 3.9e).

In figures like Figure 3.9, electronic spins are depicted as being simply aligned with some minimum energy direction (aligned with the field, or along some easy axis). Yet we already know about the paramagnetic effect of misalignment through random thermal fluctuations. We learned that an external magnetic field generates a torque on the electronic spins, and in isolation, a magnetic moment will respond to the torque in a manner similar in some respects to the way a spinning top responds to gravity: the magnetic moment will precess about the applied field direction, spiraling in and come to a rest parallel to it (Figure 3.10a). Because of the strong exchange coupling in ferromagnetic phases, spins tend to be aligned parallel (or antiparallel) to one another and the spiralling is done in a coordinated fashion, with neighboring spins as parallel as possible to one another (Figure 3.10b). This phenomenon is known as a spin wave.

SUPPLEMENTAL READINGS: O’Reilly (1984), Chapter 3.1; Dunlop and Özdemir (1997), Chapter 2.1 to 2.7.

3.4 Problems

Problem 1

a) Given one Bohr magneton (m_b) in the Earth’s field (40 μT), write a program using Python that calcuates magnetostatic interaction energy (-m_bB cosθ) for angles 0→ 180^∘. Make a plot of this with the matplotlib module in Python.

b) Calculate the thermal energy at room temperature (300K). How does this compare with the interaction energy?

Problem 2

Fayalite (Fe₂SiO₄) is a paramagnetic solid with magnetic susceptibility χ = 4.4 x 10⁻⁴ (cgs units) at 0^∘C (= 273K). A single crystal of fayalite has a volume of 2 cm³. This crystal is placed in a magnetic field, H = 10 oe at 0^∘C. What is the resulting induced magnetic moment m of this crystal?

a) Do this problem first in cgs units. Then convert your answer to SI using the conversion factors in Table 1.1 in Chapter 1.

b) Do the problem again by first converting all the parameters into SI units. Check your answer by converting the SI answer that you get back to cgs. You should get the same answer (but you would be surprised how many people do this wrong).

Problem 3

If fayalite is placed in a magnetic field H= 100 oe at a temperature of 500^∘C (= 773K), what is the resulting magnetization, M?

Problem 4

MnS is a paramagnetic solid. At 300K there are 4 x 10²⁸ molecules of MnS per m³. Look up the number of unpaired spins for the cationic magnetic moment of Mn²⁺ in the text and find the paramagnetic susceptibility, χ, of MnS at 300K?

Problem 5

a) Read into a Pandas DataFrame the datafile Chapter_3/BMoskinBan91.txt provided. Make a plot of magnetization versus temperature. What is the Curie temperature of the material?

b) Using this Equation 3.11 from the chapter, find the value for γ between 0.35 and 0.43 at intervals of 0.01 that fits the best. Plot the data as in Figure 3.8 in the chapter, i.e. M_s(T)∕Ms(T_o) against T∕T_c.

Chapter 4
Magnetic anisotropy and domains

We will start with the second part of the question: what fixes magnetizations in particular directions? A basic principle is that ferromagnetic particles have various contributions to the magnetic energy which controls their magnetization. No matter how simple or complex the combination of energies may become, the grain will seek the configuration of magnetization which minimizes its total energy. The short answer to our question is that certain directions within magnetic crystals are at lower energy than others. To shift the magnetization from one “easy” direction to another requires energy. If the barrier is high enough, the particle will stay magnetized in the same direction for very long periods of time – say billions of years. In this chapter we will address the causes and some of the consequences of these energy barriers for the magnetization of rocks. Note that in this chapter we will be dealing primarily with energy densities (volume normalized energies), as opposed to energy and will distinguish the two by the convention that energies are given with the symbol E and energy densities with ϵ.

In Chapter 6, we will discuss the behavior of common magnetic minerals, but to develop the general theory, it is easiest to focus on a single mineral. We choose here the most common one, magnetite. It has a simple, cubic structure and has been the subject intensive study. However, we will occasionally introduce concepts appropriate for other magnetic minerals where appropriate.

The simplest permanently magnetized particles are quasi-uniformly magnetized. These so-called single domain (SD) particles have spins that act in concert, staying as parallel (or anti-parallel) as possible. As particles get larger, the external energy can be minimized by allowing neighboring spins to diverge somewhat from strict parallelism; these particles are referred to as pseudo-single domain or PSD. Eventually, the spins organize themselves into regions with quasi-uniform magnetization (magnetic domains) separated by domain walls and are called multi-domain (MD) particles. These more complicated spin structures are very difficult to model and most paleomagnetic theory is based on the single domain approximation. Therefore we begin with a discussion of the energies of uniformly magnetized (single-domain) particles.

4.1 The magnetic energy of particles

4.1.1 Exchange energy

We learned in Chapter 3 that some crystalline states are capable of ferromagnetic behavior because of quantum mechanical considerations. Electrons in neighboring orbitals in certain crystals “know” about each other’s spin states. In order to avoid sharing the same orbital with the same spin (hence having the same quantum numbers – not allowed by Pauli’s exclusion principle), electronic spins in such crystals act in a coordinated fashion. They will be either aligned parallel or antiparallel according to the details of the interaction. This exchange energy density (ϵ_e) is the source of spontaneous magnetization and is given for a pair of spins by:

We define here a parameter that we will use later: the exchange constant A = J_eS²∕a where a is the interatomic spacing. A = 1.33 x 10⁻¹¹ Jm⁻¹ for magnetite, a common magnetic mineral.

Recalling the discussion in Chapter 3, while s orbitals which are spherical, the 3d electronic orbitals “poke” in certain directions. Hence spins in some directions within crystals will be easier to coordinate than in others. We can illustrate this using the example of magnetite, a common magnetic mineral (Figure 4.1a). Magnetite octahedra (Figure 4.1a), when viewed at the atomic level (Figure 4.1b) are composed of one ferrous (Fe²⁺) cation, two ferric (Fe³⁺) cations and four O²⁻ anions. Each oxygen anion shares an electron with two neighboring cations in a covalent bond.

In Chapter 3 we mentioned that in some crystals, spins are aligned anti-parallel, yet there is still a net magnetization, a phenomenon we called ferrimagnetism. This can arise from the fact that not all cations have the same number of unpaired spins. Magnetite, with its ferrous (4 m_b) and ferric (5 m_b) states is a good example. There are three iron cations in a magnetite crystal giving a total of 14 m_b to play with. Magnetite is very magnetic, but not that magnetic! From Figure 4.1b we see that the ferric ions all sit on the tetrahedral (A) lattice sites and there are equal numbers of ferrous and ferric ions sitting on the octahedral (B) lattice sites. The unpaired spins of the cations in the A and B lattice sites are aligned anti-parallel to one another because of superexchange (Chapter 3) so we have 9 m_b on the B sites minus 5 m_b on the A sites for a total of 4 m_b per unit cell of magnetite.

4.1.2 Magnetic moments and external fields

We know from experience that there are energies associated with magnetic fields. Just as a mass has a potential energy when it is placed in the gravitational field of another mass, a magnetic moment has an energy when it is placed in a magnetic field. We have seen this energy briefly in Sections 1.4 and Equation 3.4. This energy has many names (magnetic energy, magnetostatic energy, Zeeman energy, etc.). Here we will work with the volume normalized magnetostatic interaction energy density (ϵ_m). This energy density essentially represents the interaction between the magnetic lines of flux and the magnetic moments of the electronic spins. It is energy that aligns magnetic compass needles with the ambient magnetic field. We find the volume normalized form (in units of Jm⁻³) by substituting |M| = |m|v^{1 2} (see Chapter 1) into Equation 3.4:

(4.1)

ϵ_m is at a minimum when the magnetization M is aligned with the field B. Single-domain particles have a quasi-uniform magnetization and the application of a magnetic field does not change the net magnetization, which remains at saturation (M_s). The direction of all the magnetic spins could swing coherently toward the applied field. Yet the magnetizations in many particles do not rotate freely toward the magnetic field (or we would not have paleomagnetism!). There is another contribution to the energy of the magnetic particle associated with the magnetic crystal itself. This energy depends on the direction of magnetization in the crystal – it is anisotropic – and is called anisotropy energy. Anisotropy energy creates barriers to free rotation of the magnetization within the magnetic crystal, which lead to energetically preferred directions for the magnetization within individual single-domain grains.

There are many causes of anisotropy energy. The most important ones derive from the details of crystal structure (magnetocrystalline anisotropy energy), the state of stress within the particle (magnetostriction), and the shape of the particle, (shape anisotropy). We will consider these briefly in the following subsections.

4.1.3 Magnetocrystalline anisotropy energy

For equant single-domain particles or particles with low saturation magnetizations, the crystal structure dominates the magnetic energy. In such cases, the so-called easy directions of magnetization are crystallographic directions along which magnetocrystalline energy is at a minimum. The energy surface shown in Figure 4.1c represents the magnetocrystalline anisotropy energy density, ϵ_a for magnetite at room temperature. The highest energy bulges are in directions perpendicular to the cubic faces ([001, 010, 100]). The lowest energy dimples are along the body diagonals ([111]). Magnetite (above about 120K) has a cubic structure with direction cosines α₁,α₂,α₃. These direction cosines are the angles between a given direction and the crystallographic axes [100, 010, 001] – see Appendix A.3.5 for review of direction cosines). For such a crystal the magnetocrystalline anisotropy energy density is given by:

(4.2)

where K₁ and K₂ are empirically determined magnetocrystalline anisotropy constants. In the case of (room temperature) magnetite, K₁ is -1.35 x 10⁴ Jm⁻³. Note that the units of the K_i are in Jm⁻³, so ϵ_a is in units of energy per unit volume (an energy density). If you work through the magnetocrystalline equation, you will find ϵ_a is zero parallel to the [100] axis, K₁∕4 parallel to the [110] and K₁∕3 + K₂∕27 parallel to the [111] direction (the body diagonal). So when K₁ is negative, the [111] direction (body diagonal) has the minimum energy. This is the reason that there is a dimple in the energy surface along that direction in Figure 4.1c.

As a consequence of the magnetocrystalline anisotropy energy, once the magnetization is aligned with an easy direction, work must be done to change it. In order to switch from one easy axis to another (e.g. from one direction along the body diagonal to the opposite), the magnetization has to traverse a path over an energy barrier which is the difference between the energy in the easy direction and that in the intervening hard direction. In the case of magnetite at room temperature, we have this energy barrier as ϵ[111]-ϵ[110] or to first order K₁∕3 − K₁∕4 = K₁∕12.

Because electronic interactions depend heavily on inter atomic spacing, magnetocrystalline anisotropy constants are a strong function of temperature (see Figure 4.2). In magnetite, K₁ changes sign at a temperature known as the isotropic point. At the isotropic point, there is no large magnetocrystalline anisotropy. The large energy barriers that act to keep the magnetizations parallel to the body diagonal are gone and the spins can wander more freely through the crystal. Below the isotropic point, the energy barriers rise again, but with a different topology in which the crystal axes are the energy minima and the body diagonals are the high energy states.

At room temperature, electrons hop freely between the ferrous and ferric ions on the B lattice sites, so there is no order. Below about 120 K, there is an ordered arrangement of the ferrous and ferric ions. Because of the difference in size between the two, the lattice of the unit cell becomes slightly distorted and becomes monoclinic instead of cubic. This transition occurs at what is is known as the Verwey temperature (T_v). Although the isotropic point (measured magnetically) and the Verwey transition (measured electrically) are separated in temperature by about 15^o, they are related phenomena (the ordering and electron hopping cause the change in K₁).

The change in magnetocrystalline anisotropy at low temperature can have a profound effect on the magnetization. In Figure 4.3 we show a typical (de)magnetization curve for magnetite taken from the “Rock magnetic bestiary” web site maintained at the Institute for Rock Magnetism: http://irm.umn.edu/bestiary. There is a loss of magnetization at around 100 K. This loss is the basis for low-temperature demagnetization (LTD). However, some portion of the magnetization always remains after low temperature cycling (called the low temperature memory), so the general utility of LTD may be limited.

Cubic symmetry (as in the case of magnetite) is just one of many types of crystal symmetries. One other very important form is the uniaxial symmetry which can arise from crystal shape or structure. The energy density for uniaxial magnetic anisotropy is:

(4.3)

Here the magnetocrystalline constants have been designated K_u1,K_u2 to distinguish them from K₁,K₂ used before. In this equation, when the largest uniaxial anisotropy constant, K_u1, is negative, the magnetization is constrained to lie perpendicular to the axis of symmetry. When K_u1 > 0, the magnetization lies parallel to it.

An example of a mineral dominated by uniaxial symmetry is hematite, a mineral with hexagonal crystal symmetry. The magnetization of hematite is quite complicated, as we shall learn in Chapters 6 and 7, but one source is magnetization is spin-canting (see Chapter 3) within the basal plane of the hexagonal crystal. Within the basal plane, the anisotropy constant is very low and the magnetization wanders fairly freely. However, the anisotropy energy away from the basal plane is strong, so the magnetization is constrained to lie within the basal plane.

4.1.4 Magnetostriction - stress anisotropy

Exchange energy depends strongly on the details of the physical interaction between orbitals in neighboring atoms with respect to one another, hence changing the positions of these atoms will affect that interaction. Put another way, straining a crystal will alter its magnetic behavior. Similarly, changes in the magnetization can change the shape of the crystal by altering the shapes of the orbitals. This is the phenomenon of magnetostriction. The magnetic energy density caused by the application of stress to a crystal be approximated by:

4.1.5 Magnetostatic (shape) anisotropy

There is one more important source of magnetic anisotropy: shape. To understand how crystal shape controls magnetic energy, we need to understand the concept of the internal demagnetizing field of a magnetized body. In Figure 4.4a we show the magnetic vectors within a ferromagnetic crystal. These produce a magnetic field external to the crystal that is proportional to the magnetic moment (see Chapter 1). This external field is identical to a field produced by a set of free poles distributed over the surface of the crystal (Figure 4.4b). The surface poles do not just produce the external field, they also produce an internal field shown in Figure 4.4c. The internal field is known as the demagnetizing field (H_d). H_d is proportional to the magnetization of the body and is sensitive to the shape. For a simple sphere in Figure 4.4a and applied field condition shown in Figure 4.4d, the demagnetizing field is given by:

where N is a demagnetizing factor determined by the shape. In fact, the demagnetizing factor depends on the orientation of M within the crystal and therefore is a tensor (see Appendix A.3.5 for review of tensors). The more general equation is H_d = N ⋅ M where H_d and M are vectors and N is a 3 x 3 tensor. For now, we will simplify things by considering the isotropic case of a sphere in which N reduces to the single value scalar quantity N.

For a sphere, the surface poles are distributed over the surface such that there are none at the “equator” and most at the “pole” (see Figure 4.4d). Potential field theory shows that the external field of a uniformly magnetized body is identical to that of a centered dipole moment of magnitude m = vM (where v is volume). At the equator of the sphere as elsewhere, H_d = −NM. But the external field at the equator is equal to the demagnetizing field just inside the body because the field is continuous across the body. We can find the equatorial (tangential) demagnetizing field at the equator by substituting in the equatorial colatitude θ = 90^∘ into H_θ in Equation 1.8 from Chapter 1), so:

so substituting and solving for H_d we get H_d = −1 3M, hence N = 1 3.

Different directions within a non-spherical crystal will have different distributions of free poles (see Figures 4.4e,f). In fact the surface density of free poles is given by σ_m = M ⋅ . Because the surface pole density depends on the direction of magnetization, so too will N. In the case of a prolate ellipsoid magnetized parallel to the elongation axis a (Figure 4.4e), the free poles are farther apart than across the grain, hence, intuitively, the demagnetizing field, which depends on 1∕r², must be less than in the case of a sphere. Thus, N_a <1 3. Similarly, if the ellipsoid is magnetized along b (Figure 4.4e), the demagnetizing field is stronger or N_b > 1 3.

Getting back to the magnetostatic energy density, ϵ_m = M⋅B, remember that B includes both the external field B_e = −μ_oH_e and the internal demagnetizing field μ_oN ⋅ M. Therefore, magnetostatic energy density from both the external and internal fields is given by:

(4.4)

The two terms in Equation 4.4 are the by now familiar magnetostatic energy density ϵ_m, and the magnetostatic self energy density or the demagnetizing energy density ϵ_d. ϵ_d can be estimated by “building” a magnetic particle and considering the potential energy gained by each incremental volume dv as it is brought in (−μ_oMdv ⋅ H_d) and integrating. The 1 2 appears in order to avoid counting each volume element twice and the v disappears because all the energies we have been discussing are energy densities – the energy per unit volume.

For the case of a uniformly magnetized sphere, we get back to the relation H_d = −NM and ϵ_d simplifies to:

(4.5)

In the more general case of a prolate ellipsoid, M can be represented by the two components parallel to the a and b axes (see Figure 4.4f) with unit vectors parallel to them â, . So, M = M cosθâ + M sinθ . Each component of M has an associated demagnetizing field H_d = −N_aM cosθâ − N_bM sinθ where N_a,N_b are the eigenvalues of the tensor N (the values of the demagnetizing tensor along the principal axes a and b). In this case, the demagnetizing energy can be written as:

(4.6)

In an ellipsoid with three unequal axes a,b,c, N_a + N_b + N_c = 1 (in SI; in cgs units the sum is 4π). For a long needle-like particle, N_a ≃ 0 and N_b = N_c ≃ 1 2. A useful approximation for nearly spherical particles is N_a = 1 3[1 − 2 5(2 − b a − c a)] (Stacey and Banerjee, 1974). For more spheroids, see Nagata (1961, p. 70) and for the general case, see Dunlop and Özdemir (1997). In the absence of an external field, the magnetization will be parallel to the long axis (θ = 0) and the magnetostatic energy density (also known as the ‘self’ energy is given by:

(4.7)

Note that the demagnetizing energy in Equation 4.6 has a uniaxial form, directionally dependent only on θ, with the constant of uniaxial anisotropy K_u = 1 2ΔNμ_oM². ΔN is the difference between the largest and smallest values of the demagnetizing tensor N_c − N_a.

For a prolate ellipsoid N_c = N_b and choosing for example c∕a = 1.5 we find that N_a −N_c =∼ 0.16. The magnetization of magnetite is 480 kAm⁻¹, so K_u ≃ 2.7 x 10⁴ Jm⁻³. This is somewhat larger than the absolute value of K₁ for magnetocrystalline anisotropy in magnetite (K₁= -1.35 x 10⁴ Jm⁻³), so the magnetization for even slightly elongate grains will be dominated by uniaxial anistropy controlled by shape. Minerals with low saturation magnetizations (like hematite) will not be prone to shape dominated magnetic anisotropy, however.

4.1.6 Magnetic energy and magnetic stability

Paleomagnetists worry about how long a magnetization can remain fixed within a particle and we will begin to discuss this issue later in the chapter. It is worth pointing out here that any discussion of magnetic stability will involve magnetic anisotropy energy because this controls the energy required to change a magnetic moment from one easy axis to another. One way to accomplish this change is to apply a magnetic field sufficiently large that its magnetic energy exceeds the anisotropy energy. The magnetic field capable of flipping the magnetization of an individual uniformly magnetized particle (at saturation, or M_s) over the magnetic anisotropy energy barrier is the microscopic coercivity H_k. For uniaxial anisotropy (K = K_u) and for cubic magnetocrystalline anisotropy (K = K₁), microscopic coercivity is given by:

(4.8)

respectively (see Dunlop and Özdemir, 1997 for a more complete derivation). For elongate particles dominated by shape anisotropy, H_k reduces to ΔNM. [Note that the units for coercivity as derived here are in Am⁻¹, although they are often measured using instruments calibrated in tesla. Technically, because the field doing the flipping is inside the magnetic particle and B (measured in tesla) depends on the magnetization M as well as the field H (Equation 1.4), coercivity should be written as μ_oH_k if the units are quoted in tesla. Microscopic coercivity is another parameter with many names: flipping field, switching field, intrinsic coercivity and also more loosely, the coercive field and coercivity. We will come back to the topic of coercivity in Chapter 5. ]

4.2 Magnetic domains

So far we have been discussing hypothetical magnetic particles that are uniformly magnetized. Particles with strong magnetizations (like magnetite) have self energies that quickly become quite large because of the dependence on the square of the magnetization. We have been learning about several mechanisms that tend to align magnetic spins. In fact in very small particles of magnetite (< 40 nm), the spins are essentially lined up. The particle is uniformly magnetized and we called it single domain (SD). In larger particles (∼80 nm) the self energy exceeds the other exchange and magnetocrystalline energies and crystals have distinctly non-uniform states of magnetization.

There are many strategies possible for magnetic particles to reduce self energy. Numerical models (called micromagnetic models) can find internal magnetization configurations that minimize the energies discussed in the preceding sections. Micromagnetic simulations for magnetite particles (e.g. Schabes and Bertram, 1988) allow us to peer into the state of magnetization inside magnetic particles. These simulations give a picture of increasing complexity from so-called flower to vortex (Figure 4.5) remanent states. These particles share many properties of the uniformly magnetized single domain particles and are called pseudo-single domain (PSD) particles.

As particles grow larger (>∼200 nm), they break into multiple magnetic domains, separated by narrow zones of rapidly changing spin directions called domain walls. Magnetic domains can take many forms. We illustrate a few in Figure 4.6. The uniform case (single domain) is shown in Figure 4.6a. The external field is very large because the free poles are far apart (at opposite ends of the particle). When the particle organizes itself into two domains (Figure 4.6b), the external field is reduced by about a factor of two. In the case of four lamellar domains (Figure 4.6c), the external field is quite small. The introduction of closure domains as in Figure 4.6d reduces the external field to nothing.

As you might already suspect, domain walls are not “free”, energetically speaking. If, as in Figure 4.7a, the spins simply switch from one orientation to the other abruptly, the exchange energy cost would be very high. One way to get around this to spread the change over several hundred atoms, as sketched in Figure 4.7b. The wall width δ is wider and the exchange energy price is much less. However, there are now spins in unfavorable directions from a magnetocrystalline point of view (they are in “hard” directions). Exchange energy therefore favors wider domain walls while magnetocrystalline anisotropy favors thin walls. With some work (see e.g., Dunlop and Özdemir, 1997, pp. 117-118), it is possible to come up with the following analytical expressions for wall width (δ_w) and wall energy density (ϵ_w):

(4.9)

where A is the exchange constant (see Section 4.1.1) and K is the magnetic anisotropy constant (e.g., K_u or K₁). Note that ϵ_w is the energy density per unit wall area, not per volume. Plugging in values for magnetite given previously we get δ_w = 90 nm and ϵ_w = 3x 10⁻³Jm⁻².

In Figure 4.8 we plot the self energy (Equation 4.12) and the wall energy (ϵ_w from Equation 4.9) for spheres of magnetite. We see that the wall energy in particles with diameters of some 50 nm is less than the self energy, yet the width of the walls about twice as wide as that. So the smallest wall is really more like the vortex state and it is only for particles larger than a few tenths of a micron that true domains separated by discrete walls can form. Interestingly, this is precisely what is predicted from micromagnetic modelling (e.g., Figure 4.5).

How can we test the theoretical predictions of domain theory? Do domains really exist? Are they the size and shape we expect? Are there as many as we would expect? In order to address these questions we require a way of imaging magnetic domains. Bitter (1931) devised a way for doing just that. Magnetic domain walls are regions with large stray fields (as opposed to domains in which the spins are usually parallel to the sides of the crystals to minimize stray fields). In the Bitter technique magnetic colloid material is drawn to the regions of high field gradients on highly polished sections allowing the domain walls to be observed (see Figure 4.9a).

There are by now other ways of imaging magnetic domains. We will not review them all here, but will just highlight the ways that are more commonly used in rock and paleomagnetism. The magneto-optical Kerr effect or MOKE uses the interaction between polarized light and the surface magnetic field of the target. The light interacts with the magnetic field of the sample which causes a small change in the light’s polarization and ellipticity. The changes are detected by reflecting the light into nearly-crossed polarizers. The longitudinal Kerr effect can show the alignment of magnetic moments in the surface plane of the sample. Domains with different magnetization directions show up as lighter or darker regions in the MOKE image (see Figure 4.9b.)

Another common method for imaging magnetic domains employs a technique known as magnetic force microscopy. Magnetic force microscopy (MFM) uses a scanning probe microscope that maps out the vertical component of the magnetic fields produced by a highly polished section. The measurements are made with a cantilevered magnetic tip that responds to the magnetic field of the sample. In practice, the measurements are made in two passes. The first establishes the topography of the sample (Figure 4.9c). Then in the second pass, the tip is raised slightly above the surface and by subtracting the topographic only signal the attraction of the magnetic surface can be mapped (Figure 4.9d). Figure 4.9e shows an interpretation of the magnetic directions of different magnetic domains.

4.3 Thermal energy

We have gone some way toward answering the questions posed at the beginning of the chapter. We see now that anisotropy energy, with contributions from crystal structure, shape and stress, that inhibits changes in the magnetic direction thereby offering a possible mechanism whereby a given magnetization could be preserved for posterity. We also asked the question of what allows the magnetization to come into equilibrium with the applied magnetic field in the first place; this question requires a little more work to answer. The key to this question is to find some mechanism which allows the moments to “jump over” magnetic anisotropy energy barriers. One such mechanism is thermal energy E_T , which was given in Chapter 3 as:

We know from statistical mechanics that the probability P of finding a grain with a given thermal energy sufficient to overcome some anisotropy energy E_a and change from one easy axis to another is P = exp(−E_a∕E_T ). Depending on the temperature, such grains may be quite rare, and we may have to wait some time t for a particle to work itself up to jumping over the energy barrier.

Imagine a block of material containing a random assemblage of magnetic particles that are for simplicity uniformly magnetized and dominated by uniaxial anisotropy. Suppose that this block has some initial magnetization M_o and is placed in an environment with no ambient magnetic field. Anisotropy energy will tend to keep each tiny magnetic moment in its original direction and the magnetization will not change over time. At some temperature, certain grains will have sufficient energy to overcome the anisotropy energy and flip their moments to the other easy axis. As the energy surface is spherical, with no dimples or protruberances, there is no preferred direction and, over time, the magnetic moments will become random. Therefore, the magnetization as a function of time in this simple scenario will decay to zero. The equation governing this decay is:

(4.10)

where t is time and τ is an empirical constant called the relaxation time. Relaxation time is the time required for the remanence to decay to 1∕e of M_o. This equation is the essence of what is called Néel theory (see, e.g., Néel, 1955). The value of τ depends on the competition between magnetic anisotropy energy and thermal energy. It is a measure of the probability that a grain will have sufficient thermal energy to overcome the anisotropy energy and switch its moment. Therefore in zero external field:

(4.11)

where C is a frequency factor with a value of something like 10¹⁰ s⁻¹. The anisotropy energy is given by the dominant anisotropy parameter K (either K_u,K₁, or λ) times the grain volume v.

Thus, the relaxation time is proportional to anisotropy constant and volume, and is inversely related to temperature. Relaxation time τ varies rapidly with small changes in v and T. To see how this works, we can take K_u for slightly elongate cuboids of magnetite (length to width ratio of 1.3 to 1) and evaluate relaxation time as a function of particle width (see Figure 4.10). There is a sharp transition between grains with virtually no stability (τ is on the order of seconds) and grains with stabilities of billions of years.

Grains with τ ≃ 10² − 10³ seconds have sufficient thermal energy to overcome the anisotropy energy frequently and are unstable on a laboratory time scale. In zero field, these grain moments will tend to rapidly become random, and in an applied field, also tend to align rapidly with the field. The net magnetization is related to the field by a Langevin function (see Section 3.2.2 in Chapter 3). Therefore, this behavior is quite similar to paramagnetism, hence these grains are called superparamagnetic (SP). Such grains can be distinguished from paramagnets, however, because the field required to saturate the moments is typically much less than a tesla, whereas that for paramagnets can exceed hundreds of tesla.

4.4 Putting it all together

We are now in a position to pull together all the threads we have considered in this chapter and make a plot of what sort of magnetic particles behave as superparamagnets, which should be single domain and which should be multi-domain according to our simple theories. We can estimate the superparamagnetic to single domain threshold for magnetite as a function of particle shape by finding for the length (2a) that gives a relaxation time of 100 seconds as a function of width to length ratio (b∕a) for parallelopipeds of magnetite (heavy blue line in Figure 4.11). To do this, we follow the logic of Evans and McElhinny (1969) and Butler and Banerjee (1975). In this Evans diagram, we estimated relaxation time using Equation 4.11, plugging in values of K as either the magnetocrystalline effective anisotropy constant ( 1_ 12K₁) or the shape anisotropy constant (1 2ΔNμ_oM²), whichever was less. We also show the curve at which relaxation time is equal to 1 Gyr, reinforcing the point that very small changes in crystal size and shape make profound differences in relaxation time. The figure also predicts the boundary between the single domain field and the two domain field, when the energy of a domain wall is less than the self energy of a particle that is uniformly magnetized. This can be done by evaluating wall energy with Equation 4.9 for a wall along the length of a parallelopiped and area (4ab) as compared to the self energy (1 2μ_oN_aM²v) for a given length and width to length ratio. When the wall energy is less than the self energy, we are in the two domain field.

Figure 4.11 suggests that there is virtually no SD stability field for equant magnetite; particles are either SP or MD (multi-domain). As the width to length decreases (the particle gets longer), the stability field for SD magnetite expands. Of course micromagnetic modelling shows that there are several transitional states between uniform magnetization (SD) and MD, i.e. the flower and vortex remaent states (see Fabian et al., 1996), but Figure 4.11 has enormous predictive power and the version of Butler and Banerjee (1975), (which is slightly different in detail) continues to be used extensively. It is worth pointing out however, that the size at which domain walls appear in magnetite is poorly constrained because it depends critically on the exact shape of the particle, its state of stress and even its history of exposure to past fields. Estimates in the literature range from as small as 20 nm to much larger (up to 100 nm) depending on how the estimates are made. Nonetheless, it is probably true that truly single domain magnetite is quite rare in nature, yet more complicated states are difficult to treat theoretically. Therefore most paleomagnetic studies rely on predictions made for single domain particles.

SUPPLEMENTAL READING: Dunlop and Özdemir (1997), Chapters 2.8 and 5.

4.5 Problems

Problem 1

Assume that the magnetization of magnetite is about 4.8 x 10⁵ Am⁻¹. Using values for other parameters from the text, write a Python program to calculate the following:

a) Self energy (or magnetostatic energy) for a sphere 1, 10 and 100 μm in diameter. [Hint: see Equation 4.12 below for the ‘self’ energy density. Also, remember the difference between energy and energy density!]

(4.12)

b) Magnetostatic (shape) anisotropy energy for an ellipsoid whose principal semi-axis is 1 μm and whose major and minor semi-axes are each 0.25 μm. You may use the “nearly spherical” approximation in the text.

c) The critical radius of a sphere at which wall energy equals self energy.

Problem 2

Calculate grain diameter for magnetite spheres with τs of 10⁻¹, 10, 10², 10³, 10⁵, 10⁹, 10¹⁵ seconds. Use values for Boltzmann’s constant, C (the frequency factor) and |K₁| at room temperature (300K).

Problem 3 [From Jeff Gee]

a) Consider a highly elongate rod (needle-shaped grain) of magnetite. Explain why the demagnetizing factor along the long axis of the rod is about zero while that across the long axis is about one half.

b) The file Chapter_4/prolate.txt gives the values of demagnetizing factors for a prolate ellipsoid (with axes a> b=c). For an elongate rod of magnetite with range of aspect ratios (AR = c:b) provided in the table, plot the magnetostatic self energy density in the absence of an external field. Use this plot to estimate the aspect ratio at which shape anisotropy will be equal that of magnetocrystalline anisotropy (use a value of K₁ at room temperature (300K) of -1.43 x 10⁴ J/m³).

c) What is the maximum microscopic coercivity (H_k) for such an elongate grain of magnetite (assume an infinitely long grain)? Coercivities are more commonly reported in units of T so provide this corresponding value as well.

Chapter 5
Magnetic hysteresis

The ease with which particles can be coerced into changing their magnetizations in response to external fields can tell us much about the overall stability of the particles and perhaps also something about their ability to carry a magnetic remanence over the long haul. The concepts of long term stability, incorporated into the concept of relaxation time and the response of the magnetic particles to external magnetic fields are therefore linked through the anisotropy energy constant K (see Chapter 4) which dictates the magnetic response of particles to changes in the external field. This chapter will focus on the response of magnetic particles to changing external magnetic fields.

5.1 The “flipping” field

Magnetic remanence is the magnetization in the absence of an external magnetic field. If we imagine a particle with a single “easy” axis – a so-called “uniaxial” particle with magnetic anisotropy constant K_u, the magnetic energy density (energy per unit volume) of a particle whose magnetic moment makes an angle θ to the easy axis direction (Figure 5.1a) can be expressed as:

As the moment swings around with angle θ to the easy axis, the anisotropy energy density ϵ_a will change as sketched in Figure 5.1b. The energy minima are when θ is aligned parallel to the easy axis (an axis means either direction along the axis, so we pick one direction as being 0 and the other as 180^∘). In the absence of a magnetic field, the moment will lie along one of these two directions. [In reality, thermal energy will perturb this direction somewhat, depending on the balance of anisotropy to thermal energy, but for the present discussion, we are assuming that thermal energy can be neglected.]

When an external field is applied at an angle ϕ to the easy axis (and an angle ϕ−θ with the magnetic moment; see Figure 5.1a), the magnetostatic interaction energy density ϵ_m given by the dot product of the magnetization and the applied field (Equation 4.1 in Chapter 4) or:

The two energy densities (ϵ_a and ϵ_m) are shown as the thin solid and dashed lines in Figure 5.1c for an applied field of 30 mT aligned with an angle of 45^∘ to the easy axis. There is a competition between the anisotropy energy (tending to keep the magnetization parallel to the easy axis) and the interaction energy (tending to line the magnetization up with the external magnetic field). Assuming that the magnetization is at saturation, we get the total energy density of the particle to be:

(5.1)

The total energy density ϵ_t is shown as the heavy solid line in Figure 5.1c.

The magnetic moment of a uniaxial single domain grain will find the angle θ that is associated with the minimum total energy density (ϵ_min; see Figure 5.1b,c). For low external fields, θ will be closer to the easy axis and for higher external fields (e.g., 30 mT; Figure 5.1c), θ will be closer to the applied field direction (ϕ).

When a magnetic field that is large enough to overcome the anisotropy energy is applied in a direction opposite to the magnetization vector, the moment will jump over the energy barrier and stay in the opposite direction when the field is switched off. The field necessary to accomplish this feat is called the flipping field (μ_oH_f) (also sometimes the “switching field”). [Note the change to the use of H for internal fields where M cannot be considered zero.) We introduced this parameter in Chapter 4 (see Equation 4.8) as the microscopic coercivity. Stoner and Wohlfarth (1948) showed that the flipping field can be found from the condition that dϵ_t∕dθ = 0 and d²ϵ_t∕dθ² = 0. We will call this the “flipping condition”. The necessary equations can be found by differentiating Equation 5.1:

(5.2)

and again

(5.3)

Solving these two equations for B and substituting μ_oH for B, we get after some trigonometric trickery:

(5.4)

where t = tan^{1 3} ϕ. In this equation, ϕ is the angle between the applied field and the easy axis direction opposite to m.

Now we can derive the so-called “microscopic coercivity” (H_k) introduced in Section 4.1.6 in Chapter 4. Microscopic coercivity is the maximum flipping field for a particle. When magnetic anisotropy of a particle is dominated by uniaxial anisotropy constant K_u and ϕ is zero (antiparallel to the easy direction nearest the moment), μ_oH_k = 2K_u M_s. Using the values appropriate for magnetite (K_u = 1.4 x 10⁴ Jm⁻³ and M_s = 480 mAm⁻¹ we get μ_oH_k = 58 mT. To see why this would indeed result in a flipped moment, we plot the behavior of Equations 5.1 - 5.3 in Figure 5.2. The minimum in total energy ϵ_t occurs at an angle of θ = 180^∘ (Figure 5.2a) and the first and second derivatives satisfy the flipping condition by having a common zero crossing (θ = 0 in Figure 5.2b). There is no other applied field value for which this is true (see, e.g., the case of a 30 mT field in Figure 5.2c,d).

The flipping condition depends not only on the applied field magnitude but also on the direction that it makes with the easy axis (see μ_oH_f versus ϕ in Figure 5.3). When ϕ is parallel to the easy axis (zero) (and anti-parallel to m, μ_oH_f is 58 mT as we found before. μ_oH_f drops steadily as the angle between the field and the easy axis increases until an angle of 45^∘ when μ_oH_f starts to increase again. According to Equation 5.4, μ_oH_f is undefined when ϕ = 90^∘, so when the field is applied at right angles to the easy axis, there is no field sufficient to flip the moment.

5.2 Hysteresis loops

In this section we will develop the theory for predicting the response of substances to the application of external fields, in experiments that generate hysteresis loops. We will define a number of parameters which are useful in rock and paleomagnetism. For computational details in estimating these parameters from hysteresis data, see Appendix C.1.

Let us begin by considering what happens to single particles when subjected to applied fields in the cycle known as the hysteresis loop. From the last section, we know that when a single domain, uniaxial particle is subjected to an increasing magnetic field the magnetization is gradually drawn into the direction of the applied field. If the flipping condition is not met, then the magnetization will return to the original direction when the magnetic field is removed. If the flipping condition is met, then the magnetization undergoes an irreversible change and will be in the opposite direction when the magnetic field is removed.

5.2.1 Uniaxial anisotropy

Imagine a single domain particle with uniaxial anisotropy. Because the particle is single domain, the magnetization is at saturation and, in the absence of an applied field is constrained to lie along the easy axis. Now suppose we apply a magnetic field in the opposite direction (see track # 1 in Figure 5.4a). When B reaches μ_oH_f in magnitude, the magnetization flips to the opposite direction (track #2 in Figure 5.4) and will not change further regardless of how high the field goes. The field then is decreased to zero and then increased along track #3 in Figure 5.4 until μ_oH_f is reached again. The magnetization then flips back to the original direction (track #4 in Figure 5.4a).

Applying fields at arbitrary angles to the easy axis results in loops of various shapes (see Figure 5.4b). As ϕ approaches 90^∘, the loops become thinner. Remember that the flipping fields for ϕ = 22^∘ and ϕ = 70^∘ are similar (see Figure 5.3) and are lower than that when ϕ = 0^∘, but the flipping field for ϕ = 90^∘ is infinite, so that “loop” is closed and completely reversible.

Before we go on, it is useful to consider for a moment how hysteresis measurements are made in practice. Measurements of magnetic moment m as a function of applied field B are made on a variety of instruments, such as a vibrating sample magnetometer (VSM) or alternating gradient force magnetometer (AGFM). In the latter, a specimen is placed on a thin stalk between pole pieces of a large magnet. There is a probe mounted behind the specimen that measures the applied magnetic field. There are small coils on the pole pieces that modulate the gradient of the applied magnetic field (hence alternating gradient force). The specimen vibrates in response to changing magnetic fields and the amplitude of the vibration is proportional to the moment in the axis of the applied field direction. The vibration of the specimen stalk is measured and calibrated in terms of magnetic moment. The magnetometer is only sensitive to the induced component of m parallel to the applied field B_o, which is m_|| = mcosϕ (because the off axis terms are squared and very small, hence can be neglected.) In the hysteresis experiment, therefore, the moment parallel to the field m_|| is measured as a function of applied field B.

In rocks with an assemblage of randomly oriented particles with uniaxial anisotropy, we would measure the sum of all the millions of tiny individual loops. A specimen from such a rock would yield a loop similar to that shown in Figure 5.5a. If the field is first applied to a demagnetized specimen, the initial slope is the (low field) magnetic susceptibility (χ_lf) first introduced in Chapter 1. From the treatment in Section 5.1 it is possible to derive the equation χ_lf = μ_oM_s²∕3K_u for this initial (ferromagnetic) susceptibility (for more, see O’Reilly 1984).

If the field is increased beyond the flipping field of some of the magnetic grains and returned to zero, the net remanence is called an isothermal remanent magnetization (IRM). If the field is increased to +B_max, all the magnetizations are drawn into the field direction and the net magnetization is equal to the sum of all the individual magnetizations and is the saturation magnetization M_s. When the field is reduced to zero, the moments relax back to their individual easy axes, many of which are at a high angle to the direction of the saturating field and cancel each other out. A loop that does not achieve a saturating field (red in Figure 5.5a is called a minor hysteresis loop, while one that does is called the outer loop.

The net remanence after saturation is termed the saturation remanent magnetization M_r (and sometimes the saturation isothermal remanence sIRM). For a random assemblage of single domain uniaxial particles, M_r∕M_s = 0.5. The field necessary to reduce the net moment to zero is defined as the coercive field (μ_oH_c) (or coercivity).

The coercivity of remanence μ_oH_cr is defined as the magnetic field required to irreversibly flip half the magnetic moments (so the net remanence after application of a field equal to −μ_oH_cr to a saturation remanence is 0). The coercivity of remanence is always greater than or equal to the coercivity and the ratio H_cr∕H_c for our random assemblage of uniaxial SD particles is 1.09 (Wohlfarth, 1958). Here we introduce two ways of estimating coercivity of remanence, illustrated in Figure 5.5. If, after taking the field up to some saturating field +B_max, one first turned the field off (the descending curve), then increased the field in the opposite direction to the point labeled μ_oH′_cr, and one were to then switch the field off again, the magnetization would follow the dashed curve up to the origin. For single domain grains, the dashed curve would be parallel to the lower curve (the ascending curve). So, if one only measured the outer loop, one could estimate the coercivity of remanence by simply tracing the curve parallel to the lower curve (dashed line) from the origin to the point of intersection with the upper curve (circled in Figure 5.5a). This estimate is only valid for single domain grains, hence the prime in μ_oH_cr′.

An alternative means of estimating coercivity of remanence is to use a so-called ΔM curve (Jackson et al., 1990) which is obtained by subtracting the ascending loop from the descending loop (see Figure 5.5b). When all the moments are flipped into the new field, the ascending and descending loops join together and ΔM is 0. ΔM is at 50% of its initial value at the field at which half the moments are flipped (the definition of coercivity of remanence); this field is here termed μ_oH_cr.

5.2.2 Magnetic susceptibility

Figure 5.5a is the loop created in the idealized case in which only uniaxial ferromagnetic particles participated in the hysteresis measurements; in fact the curve is entirely theoretical. In “real” specimens there can be paramagnetic, diamagnetic AND ferromagnetic particles and the loop may well look like that shown in Figure 5.6. The initial slope of a hysteresis experiment starting from a demagnetized state in which the field is ramped from zero up to higher values is the low field magnetic susceptibility or χ_lf (see Figure 5.6). If the field is then turned off, the magnetization will return again to zero. But as the field increases passed the lowest flipping field, the remanence will no longer be zero but some isothermal remanence. Once all particle moments have flipped and saturation magnetization has been achieved, the slope relating magnetization and applied field reflects only the non-ferromagnetic (paramagnetic and/or diamagnetic) susceptibility, here called high field susceptibility, χ_hf. In order to estimate the saturation magnetization and the saturation remanence, we must first subtract the high field slope. So doing gives us the blue dashed line in Figure 5.6 from which we may read the various hysteresis parameters illustrated in Figure 5.5b.

5.2.3 Cubic anisotropy

In the case of equant grains of magnetite for which magnetocrystalline anisotropy dominates, there are four easy axes, instead of two as in the uniaxial case (see Chapter 4). The maximum angle ϕ between an easy axis and an applied field direction is 55^∘. Hence there is no individual loop that goes through the origin (see Figure 5.7). A random assemblage of particles with cubic anisotropy will therefore have a much higher saturation remanence. In fact, the theoretical ratio of M_r∕M_s for such an assemblage is 0.87, as opposed to 0.5 for the uniaxial case (Joffe and Heuberger, 1974).

5.2.4 Superparamagnetic particles

In superparamagnetic (SP) particles, the total magnetic energy E_t = ϵ_tv (where v is volume) is balanced by thermal energy kT. This behavior can be modeled using statistical mechanics in a manner similar to that derived for paramagnetic grains in Section 3.2.2 in Chapter 3 and summarized in Appendix A.2.2. In fact,

(5.5)

where γ = M_sBv kT and N is the number of particles of volume v, is a reasonable approximation. The end result, Equation 5.5, is the familiar Langevin function from our discussion of paramagnetic behavior (see Chapter 3); hence the term “superparamagnetic” for such particles.

The contribution of SP particles for which the Langevin function is valid with given M_s and d is shown in Figure 5.8a. The field at which the population reaches 90% saturation B₉₀ occurs at γ ∼ 10. Assuming particles of magnetite (M_s = 480 mAm⁻¹) and room temperature (T = 300∘K), B₉₀ can be evaluated as a function of d (see Figure 5.8b). Because of its inverse cubic dependence on d, B₉₀ rises sharply with decreasing d and is hundreds of tesla for particles a few nanometers in size, approaching paramagnetic values. B₉₀ is a quick guide to the SP slope (the SP susceptibility χ_sp) contributing to the hysteresis response and was used by Tauxe et al. (1996) as a means of explaining distorted loops sometimes observed for populations of SD/SP mixtures. B₉₀ (and χsp) is very sensitive to particle size with very steep slopes for the particles at the SP/SD threshold. The exact threshold size is still rather controversial, but Tauxe et al. (1996) argue that it is ∼ 20 nm.

For low magnetic fields, the Langevin function can be approximated as ∼1 3γ . So we have:

(5.6)

We can rearrange Equation 4.11 in Chapter 4 to solve for the volume at which a uniaxial grain passes through the superparamagnetic threshold we find:

(5.7)

Comparing this expression with that derived for ferromagnetic susceptibility in Section 5.2.1, we find that χ_sp is a factor of ln(Cτ) ≃ 27 larger than the equivalent single domain particle.

5.2.5 Particles with domain walls

Moving domain walls around is much easier than flipping the magnetization of an entire particle coherently. The reason for this is the same as the reason that it is easier to move a rug by lifting up a small wrinkle and pushing that through the rug, than to drag the whole rug by the same amount. Because of the greater ease of changing magnetic moments in multidomain (MD) grains, they have lower coercive fields and saturation remanence is also much lower than for uniformly magnetized particles (see typical MD hysteresis loop in Figure 5.9a.)

The key to understanding multi-domain hysteresis is the reduction in multi-domain magnetic susceptibility χ_md from “true” magnetic susceptibility (χ_i) because of self-demagnetization. The true susceptibility would be that obtained by measuring the magnetic response of a particle to the internal field H_i (applied field minus the demagnetizing field −NM – see Section  4.1.5; see Dunlop 2002a). Recalling that the demagnetizing factor is N, the so-called screening factor f_s is (1 + Nχ_i)⁻¹ and χ_md = f_sχ_i. If we assume that χ_md is linear for fields less than the coercivity, then by definition χ_md = M_r H_c (see Figure 5.9b). From this, we get:

By a similar argument, coercivity of remanence (H_cr) is suppressed by the screening factor which gives coercivity so:

Putting all this together leads us to the remarkable relationship noted by Day et al. (1977; see also Dunlop 2002a):

(5.8)

When χ_i H_c M_s is constant, Equation 5.8 is a hyperbola. For a single mineralogy, we can expect M_s to be constant, but H_c depends on grain size and the state of stress which are unlikely to be constant for any natural population of magnetic grains. Dunlop (2002a) argues that if the main control on susceptibility and coercivity is domain wall motion through a terrain of variable wall energies, then χ_i and H_c would be inversely related and gives a tentative theoretical value for χ_iH_c in magnetite of about 45 kAm⁻¹. This, combined with the value of M_s for magnetite of 480 kAm⁻¹ gives a value for χ_i H_c M_s ∼ 0.1. When anchored by the theoretical maximum for uniaxial single domain ratio of M_r∕M_s = 0.5, we get the curve shown in Figure 5.9c. The major control on coercivity is grain size, so the trend from the SD limit down toward low M_r∕M_s ratios is increasing grain size.

There are several possible causes of variability in wall energy within a magnetic grain, for example, voids, lattice dislocations, stress, etc. The effect of voids is perhaps the easiest to visualize, so we will consider voids as an example of why wall energy varies as a function of position within the grain. We show a particle with lamellar domain structure and several voids in Figure 5.10. When the void occurs within a uniformly magnetized domain (left of figure), the void sets up a demagnetizing field as a result of the free poles on the surface of the void. There is therefore, a self-energy associated with the void. When the void is traversed by a wall, the free pole area is reduced, reducing the demagnetizing field and the associated self-energy. Therefore, the energy of the void is reduced by having a wall bisect it. Furthermore, the energy of the wall is also reduced, because the area of the wall in which magnetization vectors are tormented by exchange and magnetocrystalline energies is reduced. The wall gets a “free” spot if it bisects a void. The wall energy E_w therefore is lower as a result of the void.

In Figure 5.11, we show a sketch of a hypothetical transect of E_w across a particle. There are four LEMs labelled a-d. Domain walls will distribute themselves through out the grain in order to minimize the net magnetization of the grain and also to try to take advantage of LEMs in wall energy.

Domain walls move in response to external magnetic fields (see Figure 5.11b-g). Starting in the demagnetized state (Figure 5.11b), we apply a magnetic field that increases to saturation (Figure 5.11c). As the field increases, the domain walls move in sudden jerks as each successive local wall energy high is overcome. This process, known as Barkhausen jumps, leads to the stair-step like increases in magnetization (shown in the inset of Figure 5.11g). At saturation, all the walls have been flushed out of the crystal and it is uniformly magnetized. When the field decreases again, to say +3 mT (Figure 5.11d), domain walls begin to nucleate, but because the energy of nucleation is larger than the energy of denucleation, the grain is not as effective in cancelling out the net magnetization, hence there is a net saturation remanence (Figure 5.11e). The walls migrate around as a magnetic field is applied in the opposite direction (Figure 5.11f) until there is no net magnetization. The difference in nucleation and denucleation energies was called on by Halgedahl and Fuller (1983) to explain the high stability observed in some large magnetic grains.

5.3 Hysteresis of mixtures of SP, SD and MD grains

Day et al. (1977) popularized the use of diagrams like that shown in Figure 5.9c which are known as Day diagrams. They placed quasi-theoretical bounds on the plot whereby points with M_r∕M_s ratios above 0.5 were labelled single domain (SD), and points falling in the box bounded by 0.5 > M_r∕M_s > 0.05 and 1.5 < H_cr∕H_c < 5 were labelled pseudo-single domain (PSD). Points with M_r∕Ms below 0.05 were labelled multi-domain (MD). This paper has been cited over 800 times in the literature and the Day plot still serves as the principle way that rock and paleomagnetists determined domain state and grain size.

The problem with the Day diagram is that virtually all paleomagnetically useful specimens yield hysteresis ratios that fall within the PSD box. In the early 90s, paleomagnetists began to realize that many things besides the trend from SD to MD behavior that control where points fall on the Day diagram. Pick and Tauxe (1994) pointed out that mixtures of SP and SD grains would have reduced M_r∕M_s ratios and enhanced H_cr∕H_c ratios. Tauxe et al. (1996) modelled distributions of SP/SD particles and showed that the SP-SD trends always fall above those observed from MD particles (modelled in Figure 5.9c).

Dunlop (2002a) argued that because M_r for SP grains is zero, the suppression of the ratio M_r∕M_s is directly proportional to the volume fraction of the SP particles. Moreover, coercivity of remanence remains unchanged, as it is entirely due to the non-SP fraction. Deriving the relationship of coercivity, however, is not so simple. It depends on the superparamagnetic susceptibility (χ_sp), which in turn depends on the size of the particle and also the applied field (see Section 5.2.4). In his simplified approach, Dunlop could only use a single (small) grain size, whereas in natural samples, there will always be a distribution of grain sizes. It is also important to remember that volume goes as the cube of the radius and for a mixture to display any SP suppression of M_r∕M_s almost all of the particles must be SP. It is impossible that these would all be of a single radius (say 10 or 15 nm); there must be a distribution of sizes. Moreover, Dunlop (2002a) neglected the complication in SP behavior as the particles reach the SD threshold size, whereas it is expected that many (if not most) natural samples containing both SP and SD grain sizes will have a large volume fraction of the larges SP sizes, making their neglect problematic.

Hysteresis ratios of mixtures of SD and MD particles will also plot in the “PSD” box. Dunlop (2002a) derived the theoretical behavior of such mixtures on the Day diagram. The key equations are 1) Equation 9 from Dunlop (2002) which governs the behavior of the ratio M_r∕M_s as a function of the volume fraction of single domain material (f_SD) and multi-domain material (f_MD):

2) Equation 10 from Dunlop (2002a) which governs the behavior of coercivity:

and 3) Equation 11 from Dunlop (2002a) which governs the behavior of coercivity of remanence in SD/MD mixtures:

where χ_SD and χ_MD are the susceptibilities of the SD and MD fractions respectively and (χ_r)_SD and (χ_r)_MD are the M_r vs H_cr slopes of the SD and MD remanences respectively. What we need to calculate the SD/MD mixing curve are values for the various parameters for single domain and multi domain end-members. These were measured empirically for the MV1H bacterial magnetosomes (see Chapter 6) and commercial magnetite (041183 of Wright Company) by Dunlop and Carter-Stiglitz (2006) and shown in Table ??. Using the linear mixing model of Dunlop (2002a), we plot the theoretical mixing curve predicted for these empirically constrained end-members as the heavy red line in Figure 5.9c.

If a population of SD particles are so closely packed as to influence one another, there will be an effect of particle interaction. This will also tend to suppress the M_r∕M_s ratio, drawing the hysteresis ratios down into the PSD box. Finally, the PSD box could be populated by pseudo-single domain grains themselves. Here we will dwell for a moment on the meaning of the term “pseudo-single domain”, which has evolved from the original posed by Stacey (1961; see discussion in Tauxe et al. 2002). In an attempt to explain trends in TRM acquisition Stacey envisioned that irregular shapes caused unequal domain sizes, which would give rise to a net moment that was less than the single domain value, but considerably higher than the very low efficiency expected for large MD grains. The modern interpretation of PSD behavior is complicated micromagnetic structures that form between classic SD (uniformly magnetized grains) and MD (domain walls) such as the flower or vortex remanent states (see, e.g., Figure 4.5 in Chapter 4). Taking all these factors into account means that interpretation of the Day diagram is far from unique. The simple calculations of Dunlop (2002a) are likely to be inappropriate for almost all natural samples.

5.4 First order reversal curves

Hysteresis loops can yield a tremendous amount of information yet much of this is lost by simply estimating the set of parameters M_r,M_s,H_cr,H_c,χ_i,χ_hf, etc. Mayergoyz (1986) developed a method using what are known as First Order Reversal Curves or FORCs to represent hysteresis data. The most recent way of dealing with FORCs is that of Harrison and Feinberg (2008) which is illustrated in Figure 5.12. In the FORC experiment, a specimen is subjected to a saturating field, as in most hysteresis experiments. The field is lowered to some field μ_oH_a, then increased again through some value μ_oH_b to saturation (see Figure 5.12a). The magnetization curve between μ_oH_a and μ_oH_b is a “FORC”. A series of FORCs (see Figure 5.12b) can be generated to the desired resolution.

To transform FORC data into some useful form, Harrison and Feinberg (2008) use a locally-weighted regression smoothing technique (LOESS). For a given measurement point P LOESS fits a second-order polynomial function of the form

(5.9)

to the measured magnetization surface in a specified region (for example the circle shown in Figure 5.12b) where the a_i are fitted coefficients. The LOESS technique takes a user defined number of the nearest neighbors (see inset to Figure 5.12b) for an arbitrary shaped region over which the data are smoothed. The coefficient −a₆(H_a,H_b) is the FORC density at the point. A FORC diagram is the contour plot of the FORC densities, rotated such that μ_oH_c = μ_o(H_b − H_a)∕2 and μ_oH_u = μ_o(H_a + H_b)∕2. Please note that because H_a < H_b, data are only possible for positive H_c.

Imagine we travel down the descending magnetization curve (dashed line in Figure 5.12a) to a particular field μ_oH_a less than the smallest flipping field in the assemblage. If the particles are single domain, the behavior is reversible and the first FORC will travel back up the descending curve. It is only when |μ_oH_a| exceeds the flipping field of some of the particles that the FORC will trace a new curve on the inside of the hysteresis loop. In the simple single domain, non-interacting, uniaxial magnetite case, the FORC density in the quadrants where H_a and H_b are of the same sign must be zero. Indeed, FORC densities will only be non-zero for the range of flipping fields because these are the bounds of the flipping field distribution. So the diagram in Figure 5.12c is nearly that of an ideal uniaxial SD distribution.

Consider now the case in which a specimen has magnetic grains with non-uniform magnetizations such as vortex structures or domain walls. Walls and vortices can move much more easily than flipping the moment of an entire grain coherently. In fact, they begin to move in small jumps (from LEM to LEM) as soon as the applied field changes. If a structure nucleates while the field is decreasing and the field is then ramped back up, the magnetization curve will not be reversible, even though the field never changed sign or approached the flipping field for coherent rotation. The resulting FORC for such behavior would have much of the “action” in the region where H_a is positive. When transformed to H_u and H_c, the diagram will have the high densities for small H_c but over a range of ±H_u. The example shown in Figure 5.13 is of a specimen that has been characterized as “pseudo-single domain”. The FORC diagram in Figure 5.13b has some of the FORC densities concentrated along the H_c axis characteristic of single domain specimens (e.g., Figure 5.12c), but there is also concentration along the H_u axis characteristic of PSD and MD specimens.

In many cases the the most interesting thing one learns from FORC diagrams is the degree to which there is irreversible behavior when the field is reduced to zero then ramped back up to saturation (see Figure 5.14). Such irreversible behavior in what Yu and Tauxe (2005) call the “Zero FORC” or ZFORC can arise from particle interactions, domain wall jumps or from the formation and destruction of vortex structures in the magnetic grains.

Fabian (2003) defined a parameter called “transient hysteresis” which is the area between the ascending and descending loops of a ZFORC (shaded area in Figure 5.14). This is defined as:

SUPPLEMENTAL READING: Dunlop and Özdemir (1997), chapters 5 and 11; O’Reilly (1984), pp 69-87; Dunlop (2002a,b)

5.5 Problems

Problem 1

For a grain with uniaxial anisotropy in an external field, the direction of magnetization in this grain will be controlled entirely by the uniaxial anisotropy energy density ϵ_a and the magnetic interaction energy ϵ_m. The total energy can be written:

Problem 2 [From Jeff Gee]

In this problem, we will begin to use some real data. The data files used with this book are part of the PmagPy distribution, which you should have already downloaded and installed. [See Preface for instructions.]

The file hysteresis.txt) in the Chapter_5 directory contains data for a single hysteresis loop. Note that the units are as measured: H (Oe), moment (emu) and it is fine to leave them in these units.

a) Read the data into a Pandas DataFrame. Determine the high field slope at |H| > 4000 Oe. Typically one calculates separate slopes for the +H data and -H data and averages these. A general least squares polynomial fit (numpy.polyfit) should do the trick.

b) Use the slope you determined to plot both the original hysteresis loop and the slope-corrected loop (i.e. removing the high field paramagnetic slope).

c) What is the ratio M_r∕M_s (saturation remanence/saturation magnetization) for this sample? The coercivity of remanence (H_cr) for this sample was estimated at 264 Oe. Based on the M_r∕M_s and H_cr∕H_c ratios, is this sample more likely to contain single domain or multidomain grains?

d) This small sample has a mass of 10.6 mg. Assuming the magnetic material is magnetite, estimate the mass fraction of magnetite (92 Am²/kg; note 1 emu/gm is equivalent to 1 Am²/kg).

Chapter 6
Magnetic mineralogy

An essential part of every paleomagnetic study is a discussion of what is carrying the magnetic remanence and how the rocks got magnetized. For this, we need some knowledge of what the important natural magnetic phases are, how to identify them, how they formed, and what their magnetic behavior is. In this chapter, we will cover a brief description of geologically important magnetic phases. Useful magnetic characteristics of important minerals can be found in Table 6.1 at the end of this chapter.

Iron is by far the most abundant transition element in the solar system, so most paleomagnetic studies depend on the magnetic iron bearing minerals: the iron-nickels (which are particularly important for extra-terrestrial magnetic studies), the iron-oxides such as magnetite, maghemite and hematite, the iron-oxyhydroxides such as goethite and ferrihydrite, and the iron-sulfides such as greigite and pyrrhotite. We are concerned here with the latter three as iron-nickel is very rare in terrestrial paleomagnetic studies.

6.1 Iron-oxides

The minerals we will be discussing are mostly solid solutions which the American Heritage dictionary defines as:

In iron oxides, titanium commonly substitutes for iron in the crystal structure. Because the titanium ion Ti⁴⁺ has no unpaired spins (see Chapter 3) and is a different size, the magnetic properties of titano-magnetite are different from magnetite with no titanium.

Two solid solution series are particularly important in paleomagnetism: the ulvöspinel-magnetite and ilmenite-hematite series. Both titanomagnetites and hemoilmenites crystallize at about 1300^∘C. Above about 600^∘C, there is complete solid solution between magnetite and ulvöspinel and above about 800^∘C between hematite and ilmenite. This means that all compositions are “allowed” in the crystal structure at the crystallization temperature. As the temperature decreases, the thermodynamic stability of the crystals changes. If a mineral has a given composition, say 60% titanium substitution (green dot in Figure 6.1a), when the temperature cools to intersect the red line, that composition is no longer thermodynamically stable and the two phases to either side are the equilibrium compositions. By 400^∘C the two equilibrium phases are ∼0.25 and ∼0.9 Ti substitution. To achieve the separation, the cations diffuse through the crystal leaving titanium richer and titanium poorer bands called lamellae (see Figure 6.2). Exsolution is inhibited if the crystals cool rapidly so there are many metastable crystals with non-equilibrium values of titanium substitution in nature.

Exsolution is important in paleomagnetism for two reasons. First, the different compositions have very different magnetic properties. Second, the lamellae effectively reduce the magnetic crystal size which we already know has a profound influence on the magnetic stability of the mineral. An example of this is shown in Figure 6.2b in which the larger crystal is several microns in width, too large to have single domain-like magnetization, yet the smaller magnetite lamellae are indeed small enough and carry a strong stable magnetization (Feinberg et al. 2005).

Compositions of minerals are frequently plotted on ternary diagrams like the one shown in Figure 6.3. [For help in reading ternary diagrams, please see the Appendix B.1.4] The apices of the ternary diagram are Fe²⁺ on the lower left, Fe³⁺ on the lower right and Ti⁴⁺ on the top. The oxides with these species are FeO (wüstite), Fe₂O₃ (hematite or maghemite depending on structure) and TiO (rutile). Every point on the triangle represents a cation mixture or solution that adds up to one cation (hence the fractional formulae).

Each of the solid arrows in Figure 6.3 (labelled titanomagnetite and hemoilmenite) represent increasing substitution of titanium into the crystal lattices of magnetite and hematite respectively. The amount of Ti substitution in titanomagnetites is denoted by “x”, while substitution in the hemoilmenites is denoted by “y”. Values for x and y range from 0 (magnetite or hematite) to 1 (ulvöspinel or ilmenite).

6.1.1 Titanomagnetites Fe_3−xTi_xO₄

In earlier chapters on rock magnetism, we learned a few things about magnetite. As mentioned in Chapter 4, magnetite (Fe₃O₄) has an inverse spinel structure (AB₂O₄). The oxygen atoms form a face-centered cubic lattice into which cations fit in either octahedral or tetrahedral symmetry. For each unit cell there are four tetrahedral sites (A) and eight octahedral sites (B). To maintain charge balance with the four oxygen ions (O²⁻), there are two Fe³⁺ ions and one Fe²⁺ ion. Fe³⁺ has five unpaired spins, while Fe²⁺ has four. As discussed in Chapter 3, each unpaired spin contributes a moment of one Bohr magneton (m_b). The divalent iron ions all reside in the octahedral lattice sites, whereas the trivalent iron ions are split evenly between octahedral and tetrahedral sites: Fe³⁺|Fe³⁺Fe²⁺|O₄. The A and B lattice sites are coupled with antiparallel spins and magnetite is ferrimagnetic. Therefore, the net moment of magnetite is (9-5=4) m_b per molecule (at 0 K).

Titanomagnetites can occur as primary minerals in igneous rocks. Magnetite, as well as various members of the hemoilmenite series, can also form as a result of high temperature oxidation. In sediments, magnetite often occurs as a detrital component. It can also be produced by bacteria or authigenically during diagenesis.

Substitution of Ti⁴⁺, which has no unpaired spins (see Chapter 3), has a profound effect on the magnetic properties of the resulting titanomagnetite. Ti⁴⁺ substitutes for a trivalent iron ion. In order to maintain charge balance, another trivalent iron ion turns into a divalent iron ion. The end members of the solid solution series are:

Ulvöspinel is antiferromagnetic because the A and B lattice sites have the same net moment. When x is between 0 and 1, the mineral is called a titanomagnetite. If x is 0.6, for example, the mineral is called TM60 (green dot in Figure 6.3).

The profound effect of titanium substitution on the intrinsic properties of titanomagnetite is illustrated in Figure 6.4. Because Ti⁴⁺ has no unpaired spins, the saturation magnetization decreases with increasing x (Figure 6.4a). The cell dimensions increase with increasing x (Figure 6.4b). As a result of the increased cell dimension, there is a decrease in Curie Temperature (Figure 6.4c). There is also a slight increase in coercivity (not shown).

The large M_s of magnetite (see Table 6.1) means that for deviations from equant grains as small as 10%, the magnetic anisotropy energy becomes dominated by shape. Nonetheless, aspects of the magnetocrystalline anisotropy provide useful diagnostic tests. The magnetocrystalline anisotropy constants are a strong function of temperature. On warming to ∼-100^∘C from near absolute zero, changes in these constants can lead to an abrupt loss of magnetization, which is known loosely as the Verwey transition (see Chapter 4). Identification of the Verwey transition suggests a remanence that is dominated by magnetocrystalline anisotropy. As we shall see, the temperature at which it occurs is sensitive to oxidation and the transition can be completely supressed by maghemitization (see Dunlop and Özdemir [1997]).

It should be noted that natural titanomagnetites often contain impurities (usually Al, Mg, Cr). These impurities also affect the magnetic properties. Substitution of 0.1 Al³⁺ into the unit cell of titanomagnetite results in a 25% reduction in M_s and a reduction of the Curie temperature by some 50^∘C. Substitution of Mg²⁺ into TM60 also results in a lower saturation magnetization with a reduction of some 15%.

6.1.2 Hematite-Ilmenite Fe_2−yTi_yO₃

Hematite has a corundum structure (see Figure 6.5). It is rhombohedral with a pseudocleavage (perpendicular to the c axis) and tends to break into flakes. It is antiferromagnetic, with a weak parasitic ferromagnetism resulting from either spin-canting or defect ferromagnetism (see Chapter 3). Because the magnetization is a spin canted antiferromagnetism, the temperature at which this magnetization disappears is called the Néel Temperature instead of the Curie Temperature which is sensu strictu only for ferromagnetic minerals. The Néel temperature for hematite is approximately 685^∘C.

Above about -10^∘C (the Morin transition), the magnetization is constrained by aspects of the crystal structure to lie perpendicular to the c axis or within the basal plane. Below the Morin transition, spin-canting all but disappears and the magnetization is parallel to the c axis. This effect could be used to demagnetize the grains dominated by spin-canting: it does not affect those dominated by defect moments. Most hematites formed at low-temperatures have magnetizations dominated by defect moments, so the remanence of many rocks will not display a Morin transition.

Hematite occurs widely in oxidized sediments and dominates the magnetic properties of red beds. It occurs as a high temperature oxidation product in certain igneous rocks. Depending on grain size, among other things, it is either black (specularite) or red (pigmentary). Diagnostic properties of hematite are listed in Table 6.1.

The substitution of Ti into the lattice structure of αFe₂O₃ has an even more profound influence on magnetic properties than for magnetite. For y = 0 the magnetization is spin-canted antiferromagnetic, but when y = 0.45, the magnetization becomes ferrimagnetic (see Figure 6.6a). For small amounts of substitution, the Ti and Fe cations are distributed equally among the cation layers. For y > 0.45, however, the Ti cations preferentially occupy alternate cation layers. Remembering that the Ti⁴⁺ ions have no net moment, we can imagine that antiparallel coupling between the two sub-lattices results in ferrimagnetic behavior, as opposed to the equal and opposite style of anti-ferromagnetism.

Titanohematite particles with intermediate values of y have interesting properties from a paleomagnetic point of view. There is a solid solution at high temperatures, but as the temperatures drop the crystals exsolve into titanium rich and poor lamellae (see Figure 6.2d). Figure 6.6 shows the variation in saturation magnetization and Néel temperature with Ti substitution. For certain initial liquid compositions, the exolution lamellae could have Ti-rich bands alternating with Ti-poor bands. If the Ti-rich bands have higher magnetizations, yet lower Curie temperatures than the Ti-poor bands, the Ti-poor bands will become magnetized first. When the Curie temperature of the Ti-rich bands is reached, they will become magnetized in the presence of the demagnetizing field of the Ti-poor bands, hence they will acquire a remanence that is antiparallel to the applied field. Because these bands have higher magnetizations, the net NRM will also be anti-parallel to the applied field and the rock will be self-reversed. This is fortunately very rare in nature.

6.1.3 Oxidation of (titano)magnetites to (titano)maghemites

Many minerals form under one set of equilibrium conditions (say within a cooling lava flow) and are later subjected to a different set of conditions (sea-floor alteration or surface weathering). They will tend to alter in order to come into equilibrium with the new set of conditions. The new conditions are often more oxidizing than the original conditions and compositions tend to move along the dashed lines in Figure 6.3. The degree of oxidation is represented by the parameter z.

While the solid solution between magnetite and ulvöspinel exists in principle, intergrowths of these two minerals are actually quite rare in nature because the titanomagnetites interact with oxygen in the melt to form intergrowths of low Ti magnetite with ilmenite. This form of oxidation is known as deuteric oxidation.

Low temperature oxidation will tend to transform a single phase spinel (titanomagnetite) into a new single phase spinel (titanomaghemite) by diffusion of Fe²⁺ from the lattice structure of the (titano)magnetite to the surface where it is converted to Fe³⁺; titanomaghemite is a “cation-deficient” inverse spinel. The inset to Figure 6.7c shows a magnetite crystal in the process of becoming maghemite. The conversion of the Fe²⁺ ion means a loss in volume which results in characteristic cracking of the surface. There is also a loss in magnetization, a shrinkage of cell size and, along with the tightening unit cell, an increase in Curie Temperature. These trends are shown for TM60 in Figure 6.7. Maghematization results in a much reduced Verwey transition (see Figure 6.8).

The (titano)maghemite structure is metastable and can invert to form the isochemical, but more stable structure of (titano)hematite, or it can be reduced to form magnetite. The two forms of Fe₂O₃ are distinguished by the symbols γ for maghemite and α for hematite. Inversion of natural maghemite is usually complete by about 350^∘C, but it can survive until much higher temperatures (for more details, see Dunlop and Özdemir, 1997). Also, it is common that the outer rim of the magnetite will be oxidized to maghemite, while the inner core remains magnetite.

6.2 Iron-oxyhydroxides and iron-sulfides

Of the many iron oxyhydroxides that occur in any abundance in nature, goethite (αFeOOH; Figure 6.9a,b) is the most common magnetic phase. It is antiferromagnetic with what is most likely a defect magnetization. It occurs widely as a weathering product of iron-bearing minerals and as a direct precipitate from iron-bearing solutions. It is metastable under many conditions and dehydrates to hematite with time or elevated temperature. Dehydration is usually complete by about 325^∘C. It is characterized by a very high coercivity but a low Néel temperature of about 100–150^∘C. Diagnostic properties of goethite are listed in Table 6.1.

There are two iron-sulfides that are important to paleomagnetism: greigite (Fe₃S₄; Figure 6.9c,d) and pyrrhotite (Fe₇S₈-Fe₁₁S₁₂; Figure 6.9e,f). These are ferrimagnetic and occur in reducing environments. They both tend to oxidize to various iron oxides leaving paramagnetic pyrite as the sulfide component.

The Curie temperature of monoclinic pyrrhotite (Fe₇S₈) is about 325^∘C (see Figure 6.10b; Table 6.1). Monoclinic pyrrhotite undergoes a transition at ∼ 35 K, so low temperature measurements can be diagnostic for this phase (see Figure 6.10a). Hexagonal pyrrhotite undergoes a structural transition from an imperfect antiferromagnet to a ferromagnet with much higher saturation magnetization at about 200^∘C. During a thermomagnetic experiment, the expansion of the crystal results in a large peak in magnetization just below the Curie Temperature (see Figure 6.10c). Mixtures of monoclinic and hexagonal pyrrhotite result in the behavior sketched in Figure 6.10d. The maximum unblocking temperature of greigite is approximately 330^∘C. Other diagnostic properties of greigite and pyrrhotite are listed in Table 6.1.

6.3 FeTi oxides in igneous rocks

The composition and relative proportions of FeTi oxides, crystallizing from a silicate melt depend on a number of factors, including the bulk chemistry of the melt, oxygen fugacity and the cooling rate. The final assemblage may be altered after cooling. FeTi oxides are generally more abundant in mafic volcanic rocks (e.g. basalts) than silicic lavas (e.g., rhyolites). FeTi oxides can be among the first liquidus phases (∼ 1000^∘C) in silicic melts, but in mafic lavas they generally are among the last phases to form (∼1050^∘C), often with plagioclase and pyroxene.

Although there is considerable variability, the Ti (ulvöspinel) content of the titanomagnetite crystallizing from a melt generally is lower in more silicic melts (see solid black lines in Figure 6.11). Titanomagnetites in tholeiitic lavas generally have 0.5 < x < 0.8 with an initial composition near TM60 (x=0.6) characteristic for much of the oceanic crust. The range of rhombohedral phases (dashed red lines) crystallizing from silicate melts is more limited, 0.05 < y < 0.3 for most lavas.

The final magnetic mineral assemblage in a rock is often strongly influenced by the cooling rate and oxygen fugacity during initial crystallization. As a first approximation, we distinguish slowly cooled rocks (which may undergo solid state exsolution and/or deuteric oxidation) from those in which the oxide minerals were rapidly quenched. As mentioned before, FeTi oxides in slowly cooled igneous rocks can exhibit exolution lamellae with bands of low and high titanium magnetites if the oxygen fugacity remains unoxidizing. This reaction is very slow, so its effects are rarely seen in nature.

The typical case in slowly cooled rocks is that the system becomes more oxidizing with increasing differentiation during cooling and crystallization. For example, both the dissociation of magmatic water and the crystallization of silicate phases rich in Fe will act to increase the oxidation state. This will drive compositions to higher z values (see Figure 6.3). The final assemblage typically consists of ilmenite lamellae and a nearly pure magnetite host because adding O₂ drives the reaction Fe₂TiO₄ + O₂ ⇌ Fe₃O₄ to the right. This process is known as oxyexsolution. Under even more oxidizing conditions, these phases may ultimately be replaced by their more oxidized counterparts (e.g., hematite, pseudobrookite).

Weathering at ambient surface conditions or mild hydrothermal alteration may lead to the development of cation deficient (titano)maghemites. This can either occur by addition of oxygen to the spinel phase with a corresponding oxidation of the Fe²⁺ to Fe³⁺ to maintain charge balance, or by the removal of some of the octahedral iron from the crystal structure.

6.4 Magnetic mineralogy of soils and sediments

Igneous (and metamorphic) rocks are the ultimate source for the components of sedimentary rocks, but biological and low-temperature diagenetic agents work to modify these components and have a significant effect on magnetic mineralogy in sediments. As a result there is a virtual rainbow of magnetic mineralogies found in sediments. (Titano)magnetite coming into the sedimentary environment from an igneous source may experience a change in pH and redox conditions that make it no longer the stable phase, hence it may alter. Also, although the geochemistry of seawater is generally oxidizing with respect to the stability field of magnetite, pronounced changes in the redox state of sediments often occur with increasing depth as a function of the breakdown of organic carbon. Such changes may result in locally strongly reducing environments where magnetite may be dissolved and authigenic sulfides produced. Indeed, changes down sediment cores in the ferrimagnetic mineral content and porewater geochemistry suggest that this process is active in some (most?) marine sedimentary sequences. For example, dissolution of magnetite and/or production of non-magnetic sulfides may be responsible for the oft-seen decrease in various bulk magnetic parameters (e.g., magnetic susceptibility, IRM, ARM, etc.) with depth.

Some of the more spectacular magnetic minerals found in sediments are biogenic magnetites produced by magnetotactic bacteria (see recent review by Kopp and Kirsch- vink, 2008 and Figure 6.12). The sizes and shapes of bacterial magnetite, when plotted on the Evans diagram from Chapter 4, suggest that magnetotactic bacteria form magnetite in the single domain grain size range – otherwise extremely rare in nature. It appears that bacterial magnetites are common in sediments, but their role in contributing to the natural remanence is still poorly understood.

SUPPLEMENTAL READINGS: Dunlop and Özdemir (1997), Chapter 3; Kopp and Kirschvink (2008).

6.5 Problems

Problem 1

You measured Curie Temperature curves for two samples A and B as shown in Figure 6.13. Based on your knowledge of Curie Temperatures, what is the likely magnetic mineralogy for each sample?

Problem 2

The data in demag.dat in the Chapter_6 data directory (see Preface for instructions) are thermal demagnetization data for a specimen that had a 2 T field exposed along x, a 0.4 T field exposed along y and a 0.12 T field exposed along z. The sample was then heated to a particular temperature step (^∘C) and cooled in zero magnetic field, allowing all grains that become superparamagnetic at temperatures lower than the treatment temperature to become randomized. After each treatment step, the magnetic vector was measured. The column headings are: Treatment temperature (C), Intensity, Declination, Inclination.

a) Write a python program to read the data in and convert the declination, inclination and intensity to cartesian components.

b) Modify your program to normalize the intensities to that measured at 20^∘C.

c) Extend the program to plot the x and y components as a function of temperature.

d) Based on your understanding of coercivity and Curie temperatures, what is carrying the x and y components?

Problem 3

Ferromagnetic minerals in two rock samples are known to be FeTi oxides and are found to have the properties described below. Using this information and looking up the properties of FeTi oxides described in the text, identify the ferromagnetic minerals. For titanomagnetite or titanohematite, approximate the compositional parameter x.

a.) Strong-field thermomagnetic analysis indicates a dominant Curie temperature T_c = 420^∘C. Subjecting the specimen to increasingly larger fields to measure successive isothermal remanences (see Chapter 5) reveals a coercivity spectrum with a coercivity of less than 300 mT. What is this ferromagnetic mineral?

b) Strong-field thermomagnetic analysis (used for measuring the Curie temperature) shows the behavior in Figure 6.14a with Curie temperature T_c = 200^∘C. In addition, electron microprobe data indicates abundances of FeO, Fe₂O₃, and TiO shown in Figure 6.14b. Unfortunately, electron microprobe data are not very effective in determining the Fe₂O₃:FeO ratio (placement from left to right in the TiO-FeO-Fe₂O₃ ternary diagram). Accordingly, there is much uncertainty in the Fe₂O₃:FeO ratio indicated by the microprobe data. But microprobe data are effective in determining the TiO:(Fe₂O₃ + FeO) ratio (placement from bottom to top in the TiO-FeO-Fe₂O₃ ternary diagram). With this information, identify the ferromagnetic mineral.