Paleomagnetic Principles and Practice (Modern Approaches in Geophysics)

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We can feel the pull of the Earths 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. In classical electromagnetism, all magnetic fields are the result of electric currents.

We can define what we mean by magnetic fields in terms of the electric currents that generate them. Figure 1. If iron filings are sprinkled on the sheet, the filings would line up with the magnetic field produced by the current in the wire. A circle tangential to the field is shown to the right, which illustrates the right-hand rule, that is, if your right thumb points in the direction of current flow, your fingers will curl in the direction of the magnetic field.

The magnetic field H is at right angles to both the direction of current flow and to the radial vector r. The magnitude of H is proportional to the strength of the current In the simple case illustrated in Figure 1. The more general case known as the Biot-Savart law in which the wire need not be straight is given by:. The Biot-Savart law is equivalent to Ampres law and also to one of Maxwells equations of electromagnetism.

In a steady electrical field, where is the electric current density. In English, we say that 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. The flux of a vector field be it flowing water, wind, or a magnetic field is the integral of the vector over a given area.

Magnetic fields in free space generate magnetic flux. Magnetic Flux can be quantified when a source of flux passes through a closed circuit because it produces a current which can be measured. The density of flux lines is one measure of the strength of the magnetic field called the magnetic induction B. Magnetic induction can be thought of as something that creates an observable torque u B on a length of wire carrying an electric current Similarly, the torque m B is what causes the compass needle with magnetic moment m to jump when you turn on the current in a nearby wire and consequently produce a magnetic induction B.

A force of 1 newton per meter is generated by passing a current of 1 ampere perpendicular to the direction of a magnetic induction of one tesla. The weber is defined as the amount of magnetic flux which, when passed through a one-turn coil of conductor carrying a current of one ampere, produces an electromotive force of one volt. This definition suggests a means to measure the strength of magnetic induction and is the basis of the fluxgate magnetometer.

We noted that an electrical current in a wire produces a magnetic field that curls around the wire. A current loop surrounding an area and carrying a current as shown in Figure 1. This energy is given by or where and H are the magnitudes of m and H, respectively and is the magnetic permeability of free space see Table 1.

Magnetization M is a moment per unit volume units of or per unit mass 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 a magnetization that is proportional to them. The relationship between M in the material and the external field H is defined as:. The parameter is known as the bulk magnetic susceptibility of the material and can be a complicated function of orientation, temperature, state of stress and applied field see Chapter 5.

Certain materials can produce magnetic fields in the absence of external electric currents. As we shall see in Chapter 2, 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 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 the rest of this book. From the foregoing discussion, we see that B and H are closely related. 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:. Because SI units have only recently become the standard in paleomagnetic applications, the loose usage of B and H may perhaps be forgiven.

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, or magnetic dipoles. An isolated electrical charge produces electrical fields that begin at the source the charge and diverge outward. This property of the vector field can be quantified by the divergence As there is no equivalent magnetic source, the magnetic field has no divergence.

Thus, we have another of Maxwells equations:. Because the curl of the magnetic field is not generally zero, but depends on the current density and the time derivative of the electric field, magnetic fields cannot generally be represented as gradient of a scalar field.

However, in the special case away from currents and changing electric fields, the magnetic field can be written as the gradient of a scalar field that is known as the magnetic potential i. The presence of a magnetic moment m creates a magnetic field which is the gradient of a scalar field Figure 1. This scalar field is a function of distance from the moment. Given a dipole moment m, the potential of the magnetic field produced by m is:.

The units of H are moment per unit volume which reduce to as shown earlier. We have derived magnetic units in terms of the Systme International SI. However, you will quickly notice that in many laboratories and in the literature people frequently use what are known as cgs units. The conversions between the two systems are given in Table 1. One of the principal goals of paleomagnetism is to study ancient geomagnetic fields.

Here, we review the general properties of the Earths present magnetic field.

The geomagnetic field is generated by convection currents in the liquid outer core of the Earth which is composed of iron, nickel and some unkown lighter component s. Motions of the conducting fluid, which are partially controlled by the spin of the Earth about its axis, act as a self-sustaining dynamo and create an enormous magnetic field. To first order, the field is very much like one that would be produced by a gigantic bar magnet located at the Earths center and aligned with the spin axis. If the field were actually that of a geocentric axial dipole GAD , it would not matter which cross section we chose because such a field is rotationally symmetric about the axis going through the poles; in other words, the magnetic field lines would always point North.

The angle between the field lines and the horizontal at the surface of the Earth, however, would vary between zero at the equator and 90 at the poles. Moreover, the magnetic field lines would be more crowded together at the poles than at the equator the magnetic flux is higher at the poles resulting in a polar field that would have twice the intensity of the equatorial field.

This so-called dipole model will serve as a useful crutch throughout our discussions of paleomagnetic data and applications, but as will be pointed out in more detail, it is a poor physical representation for what is actually producing the magnetic field. Before looking at global maps of the geomagnetic field, we will first consider the properties of the magnetic field at a point on the surface of the Earth.

The geomagnetic field is a vector, hence has both direction and intensity see Figure 1. A vector in three dimensions requires three parameters to define it fully no matter what coordinate system you choose. In cartesian coordinates these would be, for example, and Depending on the particular problem at hand, some coordinate systems are more suitable to use because they have the symmetry of the problem built. We will need to convert among them at will.

The three elements of a magnetic vector that will be used most frequently are magnitude B, declination D and inclination I, as shown in Figure 1. The convention used in this book is that axes are denoted while the components along the axes are In the geographic frame of reference, positive is to the north, is east and is vertically down; components of B, for example, can alternatively be designated.

From Figure 1. The horizontal component can also be projected onto the north and East axes the directions in which measurements are often made , i. Equations 1. Be careful of the sign ambiguity of the tangent function. You may end up in the wrong quadrant and have to add [See Example 1. Such a unit sphere is difficult to represent on a 2-D page. There are several popular projections, including the Lambert equal area projection; we will be making extensive use of this projection in later chapters. The principles of construction of the equal area projection are shown in Figure 1.

The point P corresponds to a D of 40 and I of D is measured around the perimeter of the equal area net and I is transformed as follows:. 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 use solid symbols, while upper hemisphere projections are open. For many purposes, it is useful to have a compact representation of the the spatial distribution of the geomagnetic field for a particular time. It is often handy to have a mathematical approximation for the field along with estimates for rates of change such that field vectors can be accurately estimated at a given place at a given time within a few hundred years at least. Because the magnetic field at the Earths surface can be approximated by a scalar potential field, a convenient mathematical representation for the magnetic field is in terms of spherical harmonics.

Such a representation is used for another potential field, gravity. The geomagnetic field is the gradient of the scalar potential as already mentioned, but the scalar potential is a more compact representation of the field. The formula for the scalar potential of the geomagnetic field at radius co-latitude longitude is often written:.

The and subscripts indicate fields of external or internal origin and is the radius of the Earth and the are proportional to the Legendre polynomials, normalized according to the convention of Schmidt see, for example, Chapman. The Schmidt polynomials are increasingly wiggly functions of the argument Examples are:.

Once the scalar potential is known, the components of the magnetic field can be calculated by the following relationships:. Here, is positive down and to the north, the opposite of and as defined in Figure 1. Note that equation 1. The Gauss coefficients are determined by fitting equations 1. The International or Definitive Geomagnetic Reference Field for a given time interval is an agreed upon set of values for a number of Gauss coefficients, and their time derivatives.

Using the values listed in, for example, Table 1. These maps demonstrate that the field is a complicated function of position on the surface of the Earth. Although the general trend in inclination shown in Figure 1. Finally, if the field were a GAD field, declination would be everywhere zero.

This is clearly not the case, as is shown by the plots of declination. Perhaps the most important result of spherical harmonic analysis for our purposes is that the field is dominated by the first order terms 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 and two equatorial axes that intersect the equator at the Greenwich meridian and at 90 East The vector sum of the geocentric dipoles is a dipole that is currently inclined by 11 to the spin axis.

The axis of this so-called best-fitting dipole pierces the surface of the Earth at the diamond in Figure 1. This point and its antipode are called geomagnetic poles. These poles are distinguished from the geographic poles where the spin axis of the Earth intersects its surface. The Northern. Averaging ancient magnetic poles over some 10, years gives what is known as a paleomagnetic pole. Because the geomagnetic field is axially dipolar to a first order approximation, we can write:.

Consider some latitude on the surface of the Earth in Figure 1. Using the equations for and we find that:. This equation is sometimes called the dipole formula or dipole equation which shows that the inclination of the magnetic field is directly related to the co-latitude for a field produced by a geocentric axial dipole or. This allows us to calculate the latitude of the measuring position from the inclination of the 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:. 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 unuseful for tectonic reconstructions. The dipole formula assumes that the magnetic field is exactly axial. Because there are more terms in the geomagnetic potential than just we know that this is not true. Because of the non-axial geocentric dipole terms, a given measurement of I will yield an equivalent magnetic co-latitude.

Paleomagnetists often assume that is a reasonable estimate of and the validity of this assumption depends on several factors. We consider first what would happen if we took random measurements of the Earths present field see Figure 1. We randomly selected positions on the globe shown in Figure 1. These directions are plotted in Figure 1. We also plot the inclinations as a function of latitude on Figure 1. We see that, as expected from a predominantly dipolar field, inclinations cluster around the values expected for a geocentric axial dipolar field. 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 and longitude The. Paleomagnetists use the following conventions: is measured positive eastward from the Greenwich meridian and goes from is measured from the North pole and goes from Of course relates to latitude, by is the magnetic co-latitude and is given by equation 1.

Be sure not to confuse latitudes and co-latitudes. Also, be careful with declination. Declinations between and are equivalent. The first step in the problem of calculating a VGP is to determine the magnetic co-latitude by equation 1. Furthermore, the declination D is the angle from the geographic North Pole to the great circle joining S and P, and is the difference in longitudes between P and S, Now we need some tricks from spherical trigonometry.

In Figure 1. Two formulae from spherical trigonometry come in handy for the purpose of calculating VGP, the Law of Sines:. We can locate VGPs using these two relationships. To determine we first calculate the angular difference between the pole and site longitude. Now we can convert the directions in Figure 1. The grouping of points is much tighter in Figure 1. As pointed out earlier, magnetic intensity varies over the globe in a similar manner as inclination.

It is often convenient to express paleointensity values in terms of the equivalent geocentric dipole moment which would have produced the observed intensity at that paleolatitude. First, the magnetic paleoco-latitude is calculated as before from the observed inclination and the dipole formula of equation 1. Sometimes the site co-latitude as opposed to magnetic co-latitude is used in the above equation, giving a virtual axial dipole moment VADM. Earths ancient magnetic field 1. It is well known that magnetic field direction and intensity change with time.

The declination in San Diego, for example, has changed by more than a degree over the fifty year time-span. The constantly changing nature of the geomagnetic field is known as secular variation SV. There are observatory records of the magnetic field vector going back several centuries. Beyond that, we rely on so-called paleosecular variation or PSV records that are preserved in archeological and geological materials. The geomagnetic field oscillates around the GAD direction with an amplitude of some 30 over an interval of some 9 meters approximately 23 kyr in the lake sediments that surround Mono Lake Lund et al.

On rare occasions, the field departs drastically from what can be considered normal of secular variation and executes what is known as a geomagnetic excursion. When viewed over sufficient time, the geomagnetic field reverses its polarity, by which we mean that the sign of the axial dipole changes. An example of a so-called polarity reversal is shown in Figure 1. When the polarity is the same as the present polarity it is said to be normal. When it is in the opposite state, it is said to be reverse.

Rocks of both polarities have been documented from early in the Earths history, although the frequency of reversal has changed considerably through time see Opdyke and Channell. A list of dates of past geomagnetic polarity reversals is known as a geomagnetic polarity time scale GPTS. The first GPTS was calibrated for the last five million years by dating basalts of known polarity see the excellent book by Glen []. The polarity sequence is broken down into times of dominantly normal polarity and times of dominantly reverse polarity. These time units are known as chrons. The uncertainty in the dating of young basalts exceeded the average duration of polarity intervals for times prior to about five million years until the advent of high precision dating techniques.

The most complete historical record of paleomagnetic reversals at least for the last million years or so is retained in the ocean crust. Modern time-scales are all based on the template provided by magnetic field anomalies measured by magnetometers towed across the oceans see e. Magnetic anomalies are generated at oceanic ridges or spreading centers, where molten rock from the mantle solidifies and acquires a magnetization during cooling see Chapter 2. These strongly magnetized rocks are gradually carried away from the ridge by the process of seafloor spreading, and, as the polarity of the magnetic field changes, quasi-linear bands of oceanic crust with magnetizations of alternating polarity are generated.

These bands produce magnetic fields that alternately add to and subtract from the Earths ambient magnetic field, resulting in lineated magnetic anomalies. The anomalous magnetic field is obtained by subtracting the IGRF from the total magnetic field data Figure 1.

These data are processed in order to make the anomalies as square as possible. Then, a square-wave is fitted to the data which are interpreted in terms of changes in polarity. In practice, several profiles can be stacked in order to average out noise and to produce a template that best represents the reversal history of the geomagnetic field.

The template of reversals obtained from marine magnetic anomaly data is in terms of kilometers from the ridge crest and covers the period of time for which there is sea floor remaining since the Jurassic. The numerical calibration of the time-scale is frequently updated. All time-scale calibrations rely on the tying of numerical ages to known reversals see e.

Ages for other reversals are interpolated or extrapolated. Numerical age information in recent time scales comes from both decay of radioactive isotopes and from calculations of longterm variations in the Earths orbit see e. Hilgen []. The details of the history of reversals for times older than the oldest sea floor magnetic anomaly record about Ma are sketchy, but will eventually be documented using sedimentary records of the magnetic field see Opdyke and Channell [].

Examination of the reversal history shown in Figure 1. Furthermore, the frequency of reversals appears to change see for example, Merrill et al. Above the polarity pattern in Figure 1. The reversal frequency is relatively high in the interval Ma, but appears to drop gradually to zero at the beginning of the so-called Cretaceous. Normal Superchron CNS , a period of some 40 m. Since the end of the CNS at about 84 Ma, the frequency of reversals has increased to the present average rate of about four per million years.

The magnetization in a rock, as well as retaining a record of the direction of the magnetic field when cooled from high temperature, has an intensity that is also a function of field magnitude. It is sometimes possible to estimate the magnitude of the Earths field from geological samples see e. We plot compilations of such data. For the Tanaka et al. Much of the Mesozoic had a rather low. The frequency of polarity reversals changes dramatically from the CNS to the present.

The sparse paleointensity data in Figure 1. The link between reversal frequency and paleointensity is more strongly made by the sedimentary paleointensity data of Tauxe and Hartl [] Figure 1. These data indicate that the field is generally higher in the early part of the Oligocene when there are fewer reversals about 1. If we replot the polarity interval averages diamonds in Figure 1. It is often supposed that when averaged over some time interval, the directions of the magnetic field will average to those generated by a geocentric axial dipole.

That is, all terms but will cancel. This is the GAD hypothesis. Because this hypothesis is central to many paleomagnetic applications, it is worthwhile considering its validity. If the GAD hypothesis is true, then the average declinations of in situ rock formations should point directly toward the Earths spin axis and the average inclinations should relate to the site latitude by the dipole formula equation 1.

The solid lines are the inclination expected from the dipole formula. It seems that the data fit the GAD model to first order, but data from reversed polarity intervals seem to be shifted to somewhat more positive values than expected from the GAD hypothesis Schneider and Kent []. The error in estimating paleolatitude that results from assuming a GAD model is about While this is not large, it should be kept in mind that the GAD hypothesis may not strictly be true see also Johnson and Constable []. Solution The programs described in this book are listed in the Appendix, with instructions on their use.

Example 1. Solution First enter the data ex1. The output should look like Figure1. N , long. E ] Scientists in the late 19th century considered that it might be possible to exploit the magnetic record retained in rocks in order to study the geomagnetic field in the past. Early work in rock magnetism provided the theoretical and experimental basis for presuming that rocks might retain a record of past geomagnetic fields. There are several books and articles that describe the subject in detail see e.

We present here a brief overview of theories on how rocks become and stay magnetized. Substances generally respond to magnetic fields; a few generate them. Therefore it is convenient to separate the magnetization of a material M into two contributions: that which exists only in the presence of an external magnetic field induced magnetization and that which exists in zero external magnetic field remanent magnetization.

As stated in Chapter 1, most of the magnetic behavior of solids results from electronic spin. Classical physics suggests that the moment generated by an orbiting electron is proportional to its angular momentum. Quantum physics tells us that the angular momentum must be quantized. The fundamental unit of magnetic moment of electrons is termed the Bohr magneton and has a value of The magnetic moments of electrons respond to externally applied magnetic fields, which creates an induced magnetization that is observable outside the substance.

From Chapter 1, we see that is the magnetic susceptibility. At its simplest, can be treated as a scalar and is referred to as the bulk magnetic susceptibility 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. 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. We will explore the origin of magnetic susceptibility only briefly here. The orbit of an electron can be characterized as a moving charge with velocity and charge see Figure 2.

What keeps the. The attracive force between the proton and the electron is given by Coulombs law:. Balancing these two competing forces and solving for gives a fundamental orbital frequency The tiny current generated by the electronic orbit creates a magnetic moment. In the presence of an external field H, there is a torque on the electron. The new balance of forces changes by some increment which is known as the Larmor frequency.

The change in orbital frequency results in a changed magnetic moment. The change in net magnetization is inversely proportional to H. The ratio is the diamagnetic susceptibility it is negative, essentially temperature independent, and quite small. Unpaired electronic spins also behave as magnetic dipoles. 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 spins are essentially randomly oriented.

An applied field acts to align the spins which creates a net magnetization equal to is the paramagnetic susceptibility. Each unpaired spin has a moment of one Bohr magneton The elements with the most unpaired spins are the transition elements. These are responsible for most of the paramagnetic behavior observed in rocks. A useful model for paramagnetism see e. In the absence of an applied field, the moments are essentially randomly oriented, i.

An applied field acts to align the spins which creates a net moment. There is competition between thermal energy T is temperature in kelvin and the magnetic energy of a magnetic moment m at an angle with an external magnetic field H is given by:. Magnetic energy is at a minimum when the magnetic moment is parallel to the magnetic field.

Using the principles of statistical mechanics, we find that the probability density of a given moment having energy is:. Because we have made the assumption that there is no preferred alignment within the substance, we can assume that the number of moments between angles and with respect to H is proportional to the solid angle and the probability density function, i.

When we measure the induced magnetization, we really measure only the component of the moment parallel to the applied field see Section 2. The total saturation moment of a given population of N individual magnetic moments is The saturation value of magnetization is thus normalized by the volume Therefore, the magnetization expressed as the fraction of saturation is:.

The function enclosed in square brackets is known as the Langevin function and is shown in Figure 2. At room temperature and fields up to many tesla, L a is approximately If the moments are unpaired spins then and:. We have neglected all deviations from isotropy including quantum mechanical effects as well as crystal shape, lattice defects, and state of stress. We can rewrite the above equation as:. To first order, paramagnetic susceptibility is: positive, larger than diamagnetism and inversely proportional to temperature.

This inverse T dependence is known as Curies law of paramagnetism. Some substances give rise to a magnetic field in the absence of an applied field. This magnetization is called remanent or spontaneous magnetization, and constitutes the phenomenon which is 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 quantum mechanical principle which states that 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. As temperature increases, the scatter in spin directions also increases.

Above a temperature characteristic of each crystal type known as the Curie temperature cooperative spin behavior disappears 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 here the Weiss molecular field In Weiss theory, is proportional to the magnetization of the substance,.

The total magnetic field that the substance experiences is:. By analogy to paramagnetism, we can substitute for H in equation 2. For temperatures above the Curie temperature we set to zero. Substituting for and using the low-field approximation for equation 2. Equation 2. Below the Curie temperature, we can neglect the external field H and get:. Below the Curie temperature, exhange interactions are strong.

Above the Curie temperature, it follows the Curie-Weiss law equation 2. 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 2. In ferromagnetism sensu stricto, Figure 2.

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When spins are perfectly antiparallel antiferromagnetism, Figure 2. Occasionally, the antiferromagnetic spins are not perfectly aligned in an antiparallel orientation, but are canted by a few degrees. This spin-canting Figure 2. Also, antiferromagnetic mate-. The uncompensated spins result in a so-called defect moment Figure 2. Also, the temperature at which spins become disordered in antiferromagnetic substances is termed the Nel 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 2. Magnetic anisotropy energy. Single crystals may have net magnetic moments which remain in the absence of an applied field. However, the direction of the net moment is free to rotate within the crystal, if it is not blocked by some other factor. Such a remanence would not have a long memory of ancient fields and would be useless for paleomagnetic purposes. The direction that a particular moment will have within a crystal will tend to lie in a direction that minimizes the magnetic energy.

Magnetic anisotropy energy see also OReilly [] and Dunlop and zdemir [] is responsible for blocking magnetic moments in particular directions within a crystal. By magnetic anisotropy, we mean that magnetic grains have easy directions of magnetization. A grain tends to be magnetized along these easy directions and energy is required to move the magnetic moment through the intervening hard directions.

An example of anisotropy energy resulting from crystal shape is illustrated in Figure 2. As anyone will remember from playing with magnets in school or on the refrigerator, magnetic moments prefer to be aligned head to tail, rather than with heads abutting. The magnetic energy of moments aligned along the length of a crystal is lower than those in which the moment is aligned crosswise.

The direction along the length of the grain is easy and across the length is hard. The energy required to move the moment from one easy direction to the other through the hard direction is the anisotropy energy. The case illustrated in Figure 2. Other sources of anisotropy energy are crystal structure magnetocrystalline energy and the state of stress within the crystal magnetostrictive energy. The magnitude of the magnetic field that supplies sufficient energy to overcome the anisotropy energy is called the switching or coercive field Consider a particle with volume whose easy axis makes an angle with the magnetic field H see Figure 2.

The magnetic moment m is drawn away from the easy axis, making an angle with the easy axis. The component of m parallel to H is given by The energy of such a particle is governed by two competing sources Stoner and Wohlfarth []. The anisotropy energy encourages alignment of m with the easy axis, and the magnetostatic energy acts to align m with H. Anisotropy of the type depicted in Figure 2.


The anisotropy energy density for a uniaxial material is given by:. Most often, uniaxial magnetic anisotropy results from anisotropic shape as in Figure 2. In this case, where is a dimensionless, shape dependent factor called the demagnetizing factor. This type of anisotropy energy is termed magnetostatic energy or shape anisotropy. Magnetic anisotropy caused by magnetocrystalline sources can lead to several easy axes, depending on the symmetry of the crystal structure. In the case of a cubic crystal whose easy axis is aligned along the body diagonal as in magnetite , the energy equation is somewhat more complicated than equation 2.

One must take into account the relationship of m to the three crystal axes. Thus the magnetocrystalline anisotropy energy for a cubic mineral is:. In magnetite at room temperature, and so the easy axis is along the body diagonal [] direction. The magnetocrystalline anisotropy constants are strongly dependent on temperature, and changes in sign or relative magnitudes result in diagnostic features in thermomagnetic curves, as discussed later. Another source of magnetic anisotropy is stress. Magnetic crystals change shape as a result of the ordering of magnetic moments below the Curie temperature, a phenomenon known as magnetostriction.

As the spins move about in the crystal in response to applied fields, the crystal undergoes further deformation. The fractional change in length, is termed Mag-. Because the response of a magnetic substance to an applied field depends strongly on the physical properties of the material, it is rapidly becoming routine to measure what is known as hysteresis loops in rock and paleomagnetic studies see e. While the interpretation of these loops is not simple, much can be learned in a short amount of time by analyzing the loops in an informed way.

Hysteresis loops are generated by subjecting a small sample to a large magnetic field The magnetization is monitored as the applied field decays to zero, switches polarity and approaches then returns through zero to Before describing the analysis of loops in detail, we must first consider what controls the shape of loops in a simple system. If we imagine a particle similar to that illustrated in Figure 2.

In the uniaxial case is given by equation 2. The magnetic energy from the external field is Thus, the energy density of the magnetic grain depicted in Figure 2. As shown in Figure 2. The angle and the component of m parallel to equals the magnitude of m, see the square loop in Figure 2. As the field decreases to zero and then to remains unchanged until the field is sufficiently large to cause m to switch through the intervening hard direction to the other direction along the easy axis contributing to the resulting loop.

This field is known as the switching field and is a function of and Fo For we define the switching field to be the intrinsic coercivity which is related to and by:. The calculation of for other than 0 is more difficult. The portion of the loop from to 0 the descending loop is calculated by first numerically evaluating the smallest for which E is at a minimum. Then The portion back to the ascending loop must be calculated in two parts.

We begin by assuming that the moments are in a state of saturation from exposure to and evaluate as a function of H for increasing H until the intrinsic coercive field is reached. At this point the ascending loop joins the descending loop. In Figure 2. In the case of the loop is a line. The moment is entirely bent into the direction of H for large H. As the field decreases, the moment relaxes back into the easy direction and is zero at zero field.

In rocks, there are many individual magnetic particles whose contributions sum to produce the observed hysteresis loops. After exposure to fields in excess of the intrinsic coercivities of all the grains in the sample the saturating field a sample will have a saturation remanent magnetization In the uniaxial case, is 0. In some geologically important cases, minerals may have an anisotropy energy that is controlled by magnetocrystalline sources; hence they may not be uniaxial, but instead may be, say, cubic in symmetry see e.

In magnetite at room temperature , the easy axis is along the body diagonal. There are four easy axes within a given crystal and the maximum angle between an easy axis and any H direction is A random assemblage of particles with cubic anisotropy will have a much higher saturation remanence. The theoretical ratio of for such assemblages is 0. Hysteresis loops are rich in information, but require a large number of parameters to describe them a typical loop can consist of hundreds of measurements. It would be useful to characterize the main features of the loop.

For example, the maximum magnetization achieved is the saturation magnetization In order to estimate this, data must be carefully adjusted to remove any para- or diamagnetic contributions. The maximum saturation remanence acquired is estimated by the value at the intercept of the descending curve see Figure 2. The bulk or average coercivity is the intercept of the ascending loop. Another useful parameter is the field required to achieve the saturation remanence, the saturating field In a saturation hysteresis loop, the field required to close the loop whereby the ascending and descending curves join, is and is indicated by in Figure 2.

The prime indicates that this value was deterimined from a hysteresis loop, as opposed to direct measurement as described later. This remanence is termed an isothermal remanent magnetization IRM. In this case, is around 0. The saturation IRM is the same as calculated from hysteresis loops, as shown in Figure 2. The value of the field necessary to remagnetize half the moments aligned at thereby reducing the net magnetization to 0, is termed the coercivity of remanence or There are several ways of estimating It can be measured directly by subjecting the saturation remanence to an increasingly strong field in the opposite direction and measuring remanence.

The field required to reduce to zero is as illustrated in Figure 2. Alternatively, it can be calculated from the hysteresis loop, as shown in Figure 2. Imagine that, instead of continuing from 0 to the maximum field on the ascending loop, one switches the field off at some value of H larger than and allows the magnetization to relax back to some remanent value.

The value of H which results in a net remanence of 0 provides another estimate of the coercivity of remanence which we call This parameter can be estimated numerically by sliding the descending loop down by the value of so that it has a zero intercept and determining the field at which the axis intercepts the ascending loop of the adjusted curve.

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This method gives is a reasonable estimate of A third way to estimate is illustrated in Figure 2. The difference between the descending and ascending loops in Figure 2. This curve was termed by Tauxe et al. The derivative of the curve called is shown in Figure 2. The field required to reduce to half its initial value is another measure of the coercivity of remanence and is termed here The derivative of the curve see Figure 2.

Until now, we have assumed that a magnetic crystal behaves as a single isolated magnetic dipole. Such grains are termed single domain SD grains. In nature, this condition is rarely met. The free poles at the grains surface create a magnetic energy which increases with grain volume. At some size, it becomes energetically more favorable to break the magnetization into several uniformly magnetized regions, or magnetic domains, as this reduces the associated magnetic field. Magnetic domains are separated by domain walls.

Such grains are termed multi-domain MD grains. Magnetic grains with few domains behave much like single domain grains in terms of magnetic stability and saturation remanence. These grains have earned the name pseudo-single domain PSD grains e. The field produced by MD grains could be reduced in several ways Figure 2.


Each configuration has a penalty with respect to one or more of the various energy terms. For example, the circular spin option Figure 2. It requires a great deal of energy for magnetic grains to nucleate domain walls. Within the wall, the spins must change from one easy direction to another see Figure 2. The narrower the wall, the greater the exchange energy because the spins are not parallel. The wider the wall, the greater the magnetocrystalline energy, because the spins will orient at some angle. The number of walls in a given grain will depend on its size, distribution of defects, state of stress and shape, to mention a few factors.

The reader is referred to Dunlop and zdemir [] for a more complete discussion of domain theories and observations. Because moving domain walls is easier than switching the entire moment of SD grains, MD grains have lower coercivities and lower saturation remanences than SD grains. A typical hysteresis loop for a population of MD magnetite grains is shown in Figure 2. The ratio for such a population is typically less than about 0.

Day et al. When plotted on a log-log plot see Figure 2. As we shall see later, physical interpretation of hysteresis loops is more complex and simple plots of the ratios alone are virtually meaningless. Mechanisms of remanence acquisition. We turn now to the subject of how geological materials become magnetized in nature. In order for rocks to stay magnetized in a particular direction, we must consider the role of the competition between exchange energy, anisotropy, and thermal energies in particles.

There are a number of mechanisms by which magnetic remanence can be acquired. These are important to paleomagnetists because the mode of magnetic remanence acquisition is critical. In the following, we will discuss the most important of these. Appendix 2 contains a list and short definition of many of the common forms of magnetic remanence. The essence of paleomagnetic stability can be illustrated with a discussion of magnetic viscosity, or the change in magnetization with time at constant temperature.

The following ideas are explained in more detail by Nel [, ] see also Stacey and Banerjee [] and Dunlop and zdemir []. Imagine a block containing an assemblage of randomly oriented, non-interacting, uniformly magnetized particles. Let us further suppose that each particle has a single easy axis and that the magnetization lies in either direction along that axis. Occasionally, a particular particle has sufficient thermal energy to overcome the magnetic anisotropy energy associated with the intervening hard directions and the moment switches its direction along the easy axis.

In the absence of an applied field, the moments of an assemblage of particles will tend to become randomly oriented and any initial magnetization will decay away according to the following equation:. The value of is a function of 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.

Relaxation time varies rapidly with small changes in and T. There is a sharp transition between grains with virtually no stability is on the order of seconds and grains with stabilities of years. Grains with 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. The net magnetization is related to the field by a Langevin function. 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 less than a tesla, whereas that for paramagnets can exceed hundreds of tesla. The magnetization which is acquired by viscous processes is called a viscous remanent magnetization or VRM.

With time, more and more grains will have sufficient thermal energy to overcome anisotropy energy barriers and will switch their magnetizations to an angle that is more in alignment with the external field. If a specimen with zero initial remanence is put into a magnetic field, the magnetization will grow to the equilibrium magnetization by the complement of equation 2. The more general case, in which the initial magnetization of a specimen is non-zero, can be written as see Kok and Tauxe [a] :. Some data sets appear to follow the relation see e.

Shimizu []. Such a relation suggests infinite remanence as which cannot be true over a long period of time. Such behavior can generally only be observed over a restricted time interval, long-term observations rarely show a strict 2. From equation 2. According to Nels theory for single domain thermal remanence Nel [, ] , there is a sharply defined range of temperatures over which increases from geologically short to geologically long time-scales.

The temperature at which is equal to about seconds is defined as the blocking temperature At or above the blocking temperature, but below the Curie temperature, a grain will be superparamagnetic. Further cooling. Consider a lava flow which has just been extruded. First, the molten lava solidifies into rock. While the rock is above the Curie temperature, there is no remanent magnetization; thermal energy dominates the system.

As the rock cools through the Curie temperature of its magnetic phase, exchange energy becomes more important and the rock acquires a remanence. In the superparamagnetic state, the magnetization is free to track the prevailing magnetic field because magnetic anisotropy energy is still less important than the thermal energy. The magnetic moments in the lava flow tend to flop from one easy direction to another, with a slight statistical bias toward the direction with the minimum angle to the applied field.

Thus, the equilibrium magnetization of superparamagnetic grains is only slightly aligned, and the degree of alignment is a linear function of the applied field for low fields such as the Earths. Now imagine that the lava continues to cool. The thermal energy will decrease until the magnetic anisotropy energy becomes important enough to freeze in the magnetic moment wherever it happens to be.

As the particles cool through their blocking temperatures, the magnetic moments become fixed because reaches a time that is geologically meaningful. From the preceeding discussion, we can make several predictions about the behavior of a TRM. The remanence of an assemblage of randomly oriented particles, acquired by cooling through the blocking temperature in the presence of a field, should be parallel to the orientation of that field.

The intensity of thermal remanence should be linearly related to the intensity of the magnetic field applied during cooling for weak fields such as the Earths. In a rock, each grain has its own blocking temperature and moment. Therefore, by cooling a rock between two temperatures, only a portion of the grains will be blocked; the rock thus acquires a partial thermal remanent magnetization or pTRM.

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An essential assumption in paleomagnetic applications is that each pTRM is independent of all others and that a pTRM acquired by cooling through two temperatures can be removed by exposure to the same peak temperature and cooling in zero field. Experimental results tend to substantiate the pTRM theory outlined above although the behavior of non-SD grains appears to be quite different. Although not a naturally occurring remanence, it is worthwhile at this point to introduce a type of remanence that is closely analogous to TRM, but which is acquired in gradually declining oscillating magnetic fields instead of during cooling.

Examination of equation 2. If, instead of raising the temperature, we subject a grain to an alternating field sufficient to overcome the anisotropy energy, the magnetization of the grain will follow the field. If we have a population of grains with a range of coercivities and we lower the peak field reached in each successive oscillation, the magnetic moments will get stuck in whatever direction they were pointing when the field went below their coercive fields. In zero field, the net magnetization will be zero. If there is a small DC bias field, then there will be a statistical preference for the direction of the bias field, which is analogous to the aquisition of TRM acquired during cooling.

This net magnetization is termed the anhysteretic remanent magnetization or ARM. Inspection of equation 2. Operations Manual: Bourke, P. Australian Natural History Territory University. Brockwell, S. Munson, , Geophysical survey as an mobility strategies on the lower Adelaide River, Northern approach to the ephemeral campsite problem: Case studies Australia. Unpublished PhD Thesis, Department of from the northern plains. Plains Anthropology Anthropology, Northern Territory University. Linford, N.

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