Magnetic Properties of Rocks and Minerals

Author and Co-author Contact Information

Richard J. Harrison (corresponding author)

Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ,

U.K.

email: rjh40@esc.cam.ac.uk

Tel.: +44 1223 333380

Rafal E. Dunin-Borkowski

Ernst Ruska-Centre for Microscopy and Spectroscopy with Electrons, Peter Gruenberg Institute,

Research Centre Juelich, D-52425 Juelich, Germany.

email: rdb@fz-juelich.de

Tel.: +49 2461 61 9297

Takeshi Kasama

Center for Electron Nanoscopy, Technical University of Denmark, DK-2800 Kongens Lyngby,

Denmark.

email: takeshi.kasama@cen.dtu.dk

Tel.: +45 4525 6475

Edward T. Simpson

Department of Materials Science and Metallurgy, University of Cambridge, Pembroke Street,

Cambridge CB2 3QZ, U.K.

email: N/A

1

Tel.: N/A

Joshua M. Feinberg

Institute for Rock Magnetism, University of Minnesota, Minneapolis, Minnesota, 55455-0219,

U.S.A.

email: feinberg@umn.edu

Tel.: +001 612 624 8429

1. ABSTRACT

! This review describes the current state-of-the-art in the field of computational and

experimental mineral physics, as applied to the study of magnetic minerals. The review is divided

into four sections, describing new developments in the study of mineral magnetism at the atomic,

nanometer, micrometer, and macroscopic length scales. We begin with a description of how

atomistic simulation techniques are being used to study the magnetic properties of minerals surfaces

and interfaces, and to gain new insight into the coupling between cation and magnetic ordering in

Fe-Ti-bearing solid solutions. Next, we review the theory of off-axis electron holography, and its

application to the study of magnetotactic bacteria and minerals containing nanoscale

transformation-induced microstructures. Then, we review the theory and application of

micromagnetic simulations to the study of non-uniform magnetization states and magnetostatic

interactions in minerals at the micrometer length scale. Finally, we review recent developments in

the use of macroscopic magnetic measurements for characterizing and quantifying the microscopic

spectrum of coercivities and interaction fields present in rocks and minerals.

2. KEYWORDS

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Magnetism, Microstructure, Mineralogy, Magnetite, Hematite, Ilmenite, Atomistic simulations,

Cation ordering, Electron holography, Micromagnetic simulations, Magnetostatic interactions,

Exchange interactions, FORC diagrams.

3. INTRODUCTION

! Magnetic minerals are pervasive in the natural environment, and are present in all types of

rocks, meteorites, sediments and soils. These minerals retain a memory of the geomagnetic field

that was present during the rock’s formation. Palaeomagnetic recordings have been exploited for

more than fifty years to map the movements of the continental and oceanic plates on Earth, and

have also proved to be a powerful tool for reconstructing the geological history of other planets,

moons and asteroids (Connerney et al. 1999 and 2004; Acuna et al. 1999; Lawrence et al. 2008; Fu

et al. 2012; Shea et al. 2012; Tarduno et al. 2012). The variation in intensity of the geomagnetic

field, as determined from rocks and archaeological material, has been used to provide an

understanding of the behavior of the geodynamo and to constrain models of fluid motion in the

Earth’s core going back over 3.4 billion years (Labrosse and Macouin 2003; Gallet et al. 2005;

Valet et al. 2005; Tarduno et al. 2010; Granot et al. 2012). More recently, magnetism has been used

to trace changes in climate, as the concentration and size of magnetic particles incorporated into

sediments are highly sensitive to environmental factors (Evans and Heller 2003). The importance of

magnetic archives of paleoclimate has been highlighted by Maher (2008) and Maher and Thompson

(2012), who demonstrated that magnetic proxies offer a radically different interpretation of how the

Asian monsoon has changed throughout the Holocene than that obtained using geochemical

methods.

! Interpretations of rock magnetic measurements are completely reliant on an accurate

understanding of the physical processes by which a material acquires and maintains a faithful

record of the geomagnetic field. Since the pioneering work of Néel (Néel 1948 and 1949), rock

magnetists have attempted to develop a quantitative understanding of how assemblages of magnetic

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minerals in single-domain (SD), pseudo-single domain (PSD), or multi-domain (MD) states acquire

and maintain natural remanent magnetization (NRM) (see Dunlop and Özdemir 1997 for a detailed

overview). The theories work well in ideal cases, i.e., when magnetic grains are homogeneous,

defect-free, and sufficiently well separated from each other that magnetic interactions between them

can be neglected. They begin to fail, however, when the mineral is heterogeneous at the nanometer

scale, as is necessarily the case when the magnetic grains form part of a nanoscale intergrowth.

Recent studies have demonstrated that nanoscale microstructures are extremely common in

magnetic minerals, and that they have a significant impact on their macroscopic magnetic properties

(Harrison and Becker 2001; Harrison et al. 2002; McEnroe et al. 2001 and 2002; Robinson et al.

2002, 2004, and 2006; Harrison et al. 2005; Feinberg et al. 2004 and 2005; Kasama et al. 2010;

Brownlee et al. 2010 and 2011). These microstructures not only determine the intensity and stability

of macroscopic magnetism recorded in rocks – thereby controlling the fidelity of paleomagnetic

recordings at the global scale – but are often important in an industrial context, providing natural

analogues of magnetic phenomena (such as exchange bias) that are central to the design of new

magnetic recording materials (Skumryev et al. 2003; Puntes et al. 2004; McEnroe et al. 2007;

Fabian et al. 2008).

! This review describes the current state-of-the-art in the field of computational and

experimental mineral physics, as applied to the study of magnetic minerals. Particular emphasis is

placed on the relationship between nanoscale microstructure and macroscopic magnetic properties.

For a comprehensive review of the magnetic properties of specific rocks and minerals, the reader is

referred to Hunt et al. (1995) and Dunlop and Özdemir (1997). Arguably the most significant recent

advance is the application to mineral magnetism of off-axis electron holography, a transmission

electron microscopy (TEM) technique that yields a two-dimensional vector map of magnetic

induction with nanometer spatial resolution (Harrison et al. 2002). Electron holography is capable

of imaging the magnetization states of individual magnetic particles and the magnetostatic

interaction fields between neighboring particles: two factors that play a central role in the interplay

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between magnetism and microstructure. By combining this capability with electron tomography, it

is now possible to determine both the micromagnetic structures and the three-dimensional

morphologies of nanoscale magnetic particles directly and quantitatively as a function of

temperature and applied magnetic field. In tandem with these techniques, advances in the

application of atomistic and micromagnetic simulations to the study of magnetic ordering in

minerals have opened the way to novel interpretations and modeling of nanoscale magnetic

properties (Robinson et al. 2002). Only now are the sizes of systems that are accessible to

simultaneous experimental and computational study converging at the nanometer length scale

(Bryson et al. 2012). This convergence provides unique opportunities for tackling problems that lie

at the frontiers of rock magnetism.

! This review is organized in order of increasing length scale of magnetic interactions. Section

4 deals with magnetism at the atomic length scale. It contains a brief description of exchange

interactions and magnetic structure in Fe-bearing oxides, and the use of atomistic simulations of

magnetic ordering to the study magnetism at surfaces and interfaces. Section 5 deals with

magnetism at the nanometer length scale. Following a summary of the theory of electron

holography, recent applications of holography to the study of magnetic minerals are reviewed.

Section 6 deals with magnetism at the micrometer length scale, including advances in

micromagnetic simulations that allow the magnetic behavior of particles with realistic three-

dimensional morphologies to be modeled. In Section 7, we move to the macroscopic length scale,

with a description of how new approaches for the measurement of macroscopic magnetic properties

(i.e., FORC diagrams) are providing quantitative information about the spectrum of coercivities and

interaction fields that exist at the microscopic scale.

4. MAGNETISM AT THE ATOMIC LENGTH SCALE

4.1. Exchange interactions and magnetic structure in Fe-bearing oxides

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! The driving force for magnetic ordering in Fe-bearing oxides is the superexchange interaction

between neighboring transition metal cations via intermediate oxygen anions (Goodenough 1966).

The magnitudes and signs of superexchange interactions define the magnetic ground state and the

magnetic ordering temperature of the mineral, and play a fundamental role in determining its

macroscopic magnetic properties. The exchange interaction energy for classical spins can be

expressed in the form:

! Emag = �X

i 6=j

JijSi · Sj ! (1)

where Si and Sj and are the spins on atoms i and j, and Jij is the corresponding exchange integral.

Positive values of Jij lead to parallel (i.e., ferromagnetic) alignment of spins; negative values lead to

antiparallel (i.e., antiferromagnetic) alignment.

! Empirical values of Jij can be obtained from spin-wave dispersion curves measured using

inelastic neutron scattering (Samuelson 1969; Samuelson and Shirane 1970; Brockhouse 1957;

Watanabe and Brockhouse 1962; Glasser and Milford 1963; Phillips and Rosenberg 1966).

Alternatively, theoretical values can be obtained from first-principles calculations (Sandratskii

1998; Matar 2003; Pinney et al. 2009; Sadat Nabi et al. 2010). The simplest theoretical approach

involves calculating the total energies of several different collinear arrangements of spins, and then

determining values of Jij directly from Eqn. 1 (Sandratskii et al. 1996; Rollmann et al. 2004). This

approach is limited, however, by the small number of alternative structures that can be generated for

a given unit cell. The use of non-collinear magnetic structures (Sandratskii 1998) provides a more

general procedure for calculating exchange integrals out to arbitrary cation-cation separations (Uhl

and Siberchico 1995). This approach is based on the calculation of the total energies of spin-spiral

configurations, over a grid of wave vectors within the Brillouin zone. It is often found that

exchange interactions that are determined by spin-wave and first-principles methods overestimate

magnetic ordering temperatures significantly (Sandratskii et al. 1996; Uhl and Siberchico 1995). If

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necessary, the calculated values of Jij can be scaled or refined to provide better agreement with

experimental observations (Burton 1985; Harrison and Becker 2001; Harrison 2006).

! Exchange integrals for hematite (Fe2O3), ilmenite (FeTiO3), and magnetite (Fe3O4) are listed

in Table 1 and Fig. 1 (see also Sadat Nabi et al. 2010 for DFT-calculated values for Fe2O3, FeTiO3

and the Fe2O3-FeTiO3 solid solution). The crystal structures of hematite-ilmenite and magnetite are

compared in Fig. 2. Superexchange interactions are highly sensitive to the relative positions of the

two cations and the intermediate oxygen anion, varying in magnitude approximately as cos2ψ,

where ψ is the cation-oxygen-cation bond angle (Coey and Ghose 1987). In hematite (Fig. 2a), Fe3+

cations occupy two thirds of the octahedral interstices within a hexagonal close-packed oxygen

sublattice, forming symmetrically equivalent A and B layers parallel to the (001) basal plane (space

group R3c ). Each octahedron shares a face with an octahedron in the layer above or below, and

edges with three octahedra in its own layer. Since both face- and edge-sharing octahedral-octahedral

linkages have ψ ~ 90° (Table 1), both first- and second-nearest-neighbor interactions are weak.

Third- and fourth-nearest-neighbor interactions involve corner-sharing octahedra in adjacent layers

(ψ ~ 120° and 132°, respectively; Table 1). These interactions are large and negative, leading to an

antiferromagnetic ground state in which A-layer spins are antiparallel to B-layer spins. Above 260

K (the Morin transition), spins lie parallel to the (001) basal plane and point nearly perpendicular to

a <100> crystallographic axis of the hexagonal unit cell (Tanner et al. 1988). The antiferromagnetic

sublattices are canted by an angle of ~0.065°, leading to a weak parasitic moment lying within the

basal plane and directed along <100> (Dzyaloshinskii 1958). The choice of 6 possible <100>

directions for the canted moment leads to a complex magnetic and crystallographic domain

microstructure in hematite (Tanner et al. 1988), although its presence has often been overlooked in

magnetic studies. By considering arbitrary, non-collinear configurations of the atomic magnetic

moments, Sandratskii and Kübler (1996) used first-principles calculations to demonstrate that the

canted magnetic structure appears as a direct consequence of spin-orbit coupling (Fig. 3). Below the

Morin transition, the spin alignment switches to [001], and the canting is lost. A small residual of

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the weak parasitic moment is often observed in the basal plane even below the Morin transition

(Özdemir et al. 2008), although the origin of this so-called ‘defect’ moment is not fully understood.

! In ilmenite, Fe2+ and Ti4+ are ordered onto A and B (or B and A) layers, and the equivalency of

the layers is lost (space group R3 ). Since one layer is fully occupied by Ti4+, the strong interlayer

interactions that were present in hematite are eliminated. Second-nearest-neighbor interactions

extend across the intervening Ti4+ layers. These weak negative interactions result in

antiferromagnetic ordering below 60 K. The spins in one A layer are then aligned antiparallel to

those on the adjacent A layers. Spins are oriented parallel and antiparallel to [001].

! In magnetite (Fig. 2b), cations occupy tetrahedral and octahedral interstices within a cubic

closed-packed oxygen sublattice. Octahedra are occupied by Fe2+ and Fe3+ cations, whereas

tetrahedra are occupied exclusively by Fe3+. Octahedra share edges with adjacent octahedra and

corners with adjacent tetrahedra. There are no shared oxygens between adjacent tetrahedra. The

first-nearest-neighbor interaction is weak, due to the unfavorable cation-oxygen-cation bond angle

(Table 1). The dominant negative tetrahedal-octahedral interaction leads to a ferrimagnetic

structure, in which spins on the octahedral sites are antiparallel to those on the tetrahedral sites.

Exchange interactions are weak when the two cations are not linked directly by a common oxygen,

leading to a weak tetrahedral-tetrahedral interaction and a rapid decrease in interaction strength for

cation-cation separations that are greater than ~4 Å. Spins point parallel to <111> above 130 K and

parallel to <100> below 130 K. Below 120-125 K magnetite undergoes a first order phase transition

(the Verwey transition) to a monclinic structure (Walz 2002; Senn et al. 2012). The Verwey

transition causes a 15-fold increase in the magnetocrystalline anisotropy of magnetite (Abe et al.

1976) and leads to the development of extensive crystallographic twinning (Moloni et al. 1996;

Carter-Stiglitz et al. 2006; Kasama et al. 2010 and 2012; Bryson et al. 2012; Coe et al. 2012). Both

these effects have a dramatic impact on the magnetic properties of magnetite at low temperatures

(Muxworthy and McClelland 2000; Kosterov 2001; Smirnov and Tarduno 2002; Smirnov 2006a,

2006b, 2007; Kosterov and Fabian 2008). For further details of the crystal and magnetic structures

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of minerals, the reader is referred to Banerjee (1991), Dunlop and Özdemir (1997), and Harrison

(2000).

4.2. Atomistic simulations of magnetic ordering

4.2.1. Theory

! Magnetic ordering in minerals was first described using a mean-field model by Néel (1948).

For some applications, e.g. for estimating Néel temperatures from Jij, this macroscopic approach

remains extremely useful (Stephenson 1972a and 1972b). When studying materials that are

heterogeneous at the nanometer scale, however, the mean-field model is inappropriate, and the

atomistic nature of the magnetic interactions must be taken into account.

! Atomistic simulations are increasingly used to study magnetism at surfaces and interfaces

(Kodama 1999; Kodama and Berkowitz 1999; Kachkachi et al. 2000a, 2000b; Dimian and

Kachkachi 2002; Kachkachi and Dimian 2002; Garanin and Kachkachi 2003; Kachkachi and

Mahboub 2004; Harrison and Becker 2001; Robinson et al. 2002; Harrison 2006; Harrison et al.

2005; Harrison et al. 2007). They are also well suited to the study of disordered systems, such as

spin glasses, that require an atomistic approach to account for frustrated exchange interactions

(Harrison 2009; Charilaou et al. 2011). It is currently practical to describe systems containing ~104

magnetic atoms (Kodama and Berkowitz 1999). For magnetite, this limitation corresponds to a

spherical particle of diameter ~8 nm. If surface properties are not of interest, then an effectively

infinite (bulk) system can be simulated by creating a large supercell of the crystal structure and

applying periodic boundary conditions (Mazo-Zuluaga and Restrepo 2004; Harrison 2006 and

2009).

! The magnetic energy of such a system is a sum of exchange, anisotropy, magnetostatic, and

dipole-dipole interaction terms:

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! Emag = �X

i 6=j

JijSi · Sj �X

i

Ki(Si · ei)2 � (gµB)X

i

B · Si + Ed ! (2)

where g is the Landé factor, μB is the Bohr magneton, B is an externally applied magnetic field, K is

a uniaxial anisotropy constant, e is the corresponding uniaxial anisotropy axis and Ed is the

demagnetizing energy due to dipole-dipole interactions (Kodama and Berkowitz 1999; Kachkachi

et al. 2000a). Ed can be expressed in the form:

! Ed =(gµB)2

2

X

i 6=j

(Si · Sj)R2ij � 3(Si · Rij)(Rij · Sj)

R5ij

! (3)

where Rij is the vector joining atoms i and j. For ellipsoidal particles, Eqn. 3 simply generates a

macroscopic shape anisotropy (Kachkachi et al. 2000a). The large computational overhead involved

in summing Eqn. 3 over all pairs of atoms can be avoided, therefore, by describing this shape

anisotropy by a macroscopic approximation of the form:

! Ed

=1

2V(D

x

M2x

+ Dy

M2y

+ Dz

M2z

) ! (4)

where Dx, Dy, and Dz are demagnetizing factors and Mx, My, and Mz are the components of net

magnetization along the principal axes of the ellipsoid (Stacy and Banerjee 1974). Kachkachi et al.

(2000a) found that Eqn. 4 yielded identical results to Eqn. 3 for nanoparticles of maghemite (γ-

Fe2O3). Alternatively, Fourier methods for calculating dipolar interactions can significantly increase

computational efficiency (Beleggia 2004). Different values of K and e can be specified for atoms in

the core and at the surface of a particle. For a surface atom, e is given by the sum of vectors joining

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the atom to its nearest neighbors, and points approximately perpendicular to the surface (Kodama

and Berkowitz 1999; Garanin and Kachkachi 2003; Kachkachi and Mahboub 2004). In this way, the

anisotropy is enhanced when the local symmetry is lower than that of the bulk structure. Values for

surface anisotropy constants, K, of ~1-4 kB/cation (where kB is the Boltzmann constant) are

suggested by electron paramagnetic resonance measurements of dilute magnetic cations substituted

onto low-symmetry sites in non-magnetic oxides (Low 1960). Bulk anisotropies are at least two

orders of magnitude smaller.

! Monte Carlo methods provide an efficient way of determining the equilibrium spin

configuration for a given temperature and applied field (Kachkachi et al. 2000a; Mazo-Zuluaga and

Restrepo 2004; Harrison 2006; Harrison et al. 2007). An atom is chosen at random, and its spin

direction changed by a random amount. If the resulting energy change, ΔEmag, is negative, then the

change is accepted. If ΔEmag is positive, then the change is accepted with a probability of exp(-

ΔEmag/kBT). After a sufficient number of steps, the system reaches equilibrium. The equilibrium

configuration is obtained by averaging over a number of steps until the system converges to the

desired statistical significance. In many applications (e.g., for the simulation of hysteresis loops), it

is not only the equilibrium configuration that is important, but the transitional configurations

adopted during the approach to equilibrium. In these cases, a dynamic solution to Eqn. 3 is required.

One approach is to use the Landau-Lifshitz-Gilbert (LLG) equation to calculate the trajectory of

each spin (Brown 1963), in the form:

! dm

dt= �m⇥He � ⇥m⇥ (m⇥He) ! (5)

where m is the magnetic moment of a given atom, γ is the gyromagnetic ratio, λ is a damping

constant, and He is the effective magnetic field acting on that atom:

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! He = �dEmag

dm! (6)

The first term in Eqn. 5 describes the precession of the magnetic moment about the effective field

direction. The second term decreases the precession angle over time (damping), eventually orienting

the magnetic moment along the effective field direction. Although the LLG method has been

applied successfully to the study of magnetic nanoparticles (Dimian and Kachkachi 2002;

Kachkachi and Dimian 2002; Kachkachi and Mahboub 2004), the method takes many iterations to

converge, and can often predict unreasonably large coercivities in atomistic simulations due to the

large value of the effective exchange field relative to the applied field. Kodama and Berkowitz

(1999) adapted the two-dimensional conjugate direction algorithm of Hughes (1983) to provide a

more efficient method of energy minimization for three-dimensional atomistic simulations

(achieving convergence in 5-15 iterations). Whichever method is used, finite temperatures can be

modeled by applying random rotations to the spins between energy minimization steps. The

magnitudes of rotations are adjusted to give ΔEmag = NkBT, where N is the number of spins in the

particle. Random rotations of the individual spins can be combined with random uniform rotations

of all of the spins to model collective modes of thermal relaxation, such as superparamagnetism.

! The term ‘chemical ordering’ is used to describe changes in the distribution of magnetic and

non-magnetic atoms in a crystal, which may be brought about by both order-disorder and exsolution

processes. In homogeneous systems, the coupling between magnetic and chemical ordering can be

described by using established thermodynamic models (Inden 1981; Kaufman 1981; Burton and

Davidson 1988; Burton 1991; Ghiorso 1997; Harrison and Putnis 1997 and 1999). In heterogeneous

systems, however, the presence of internal interfaces and phase boundaries necessitates the use of

an atomistic approach. The chemical energy of a given atomic configuration can be written in terms

of chemical exchange interaction parameters (Bosenick et al. 2001) in the form:

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! Echem = E0 +X

p,q

Np,qJchemp,q ! (7)

where Jchemp,q is the energy associated with placing a pair of unlike cations (labeled p) at a given

separation (q) within the structure, and Np,q is the number of times that each type of pair appears in

the configuration. E0 is constant for a fixed bulk composition, and can be neglected. Values for

Jchemp,q can be obtained from first-principles or empirical-potential calculations (Becker et al. 2000;

Warren et al. 2000a and b; Dove 2001; Bosenick et al. 2001; Vinograd et al. 2004), or by calibration

with respect to known experimental cation ordering temperatures (Dang and Rancourt 1996). A

mixture of first-principles and empirical-potential calculations, combined with calibration with

respect to well constrained experimental cation distribution data, has been used to study the

systematics of chemical interactions in a range of spinel oxides, including the magnetite–ulvöspinel

(Fe3O4–Fe2TiO4) solid solution and its Mg analogue magnesioferrite–qandilite (MgFe2O4–

Mg2TiO4) (Palin and Harrison 2007a, b; Palin et al. 2008; Harrison et al. 2013). Harrison et al.

(2000a) used static-lattice calculations to estimate Jchemp,q for the ilmenite-hematite solid solution.

These estimates were then refined by fitting the model to cation distribution data obtained using

neutron diffraction (Harrison et al. 2000b; Harrison and Redfern 2001).

! In order to simulate coupled magnetic and chemical ordering, a combination of two different

Monte Carlo steps must be performed: spin flips and atom swaps. In the spin-flip step, the spin of a

randomly-chosen atom is changed by a random amount, and the change in magnetic energy, ΔEmag,

is used to determine whether this change is accepted or rejected. In the atom-swap step, two atoms

are chosen at random and their positions are exchanged. If either atom is magnetic, then the swap

will also change the configuration of the spins, and the total energy change, ΔE = ΔEchem + ΔEmag,

is used to determine whether the swap is accepted or rejected. Atom swaps preserve the net spin of

the system. Therefore, after a given number of atom swaps, an equal number of spin flips are

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performed in order to allow the spin configuration to equilibrate with respect to the new atomic

configuration. The alternation of atom swaps and spin flips is repeated until the system reaches a

state of global equilibrium with respect to both chemical and magnetic degrees of order.

4.2.2. Application to magnetic nanoparticles

! For magnetic nanoparticles with sizes of 1-10 nm, surface atoms (defined as those having

fewer nearest-neighbors than the bulk structure) make up at least 25% of the total number of atoms.

Finite-size and surface effects give rise to magnetic properties that deviate significantly from those

of the bulk material (Kodama 1999; Kachkachi et al. 2000b). The effect of surface anisotropy and

surface roughness on the spin structure of 2.5 nm-diameter NiFe2O4 particles is illustrated in Fig. 4

(Kodama 1999; Kodama and Berkowitz 1999). Whereas smooth nanoparticles may adopt uniform

spin configurations (Figs. 4a and b), rough surfaces (characterized by the presence of surface

vacancies and broken exchange interactions) typically display surface spin disorder (Figs. 4c and d).

A number of different surface spin configurations can be adopted, depending on the thermal and

field history of the particle. Energy barriers between different surface spin configurations can be

very high, leading to high-field irreversibility (Fig. 4c). In NiFe2O4, for example, hysteresis persists

in magnetic fields of up to 16 T (Kodama et al. 1996), implying effective anisotropy fields for

surface spins that are ~400 times larger than the bulk magnetocrystalline anisotropy field. The

observation of shifted hysteresis loops during field cooling (Kodama et al. 1997) implies that

certain surface configurations freeze in preferentially, and that there is strong exchange coupling

between surface and core spins.

! The effect of varying surface anisotropy and exchange coupling on the hysteresis properties

of SD ferromagnetic particles has been explored by Kachkachi and Dimian (2002). Significant

deviations from the classical Stoner-Wohlfarth (1948) model are observed when the surface

anisotropy and exchange constants are of similar magnitude (Fig. 5). These deviations are

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associated with non-uniform reversal mechanisms, which involve the successive switching of

surface and core spins.

! The magnetic properties of maghemite (γ-Fe2O3) nanoparticles have been investigated using

both Monte Carlo and conjugate direction methods (Kachkachi et al. 2000a; Kodama and

Berkowitz 1999). Mössbauer spectroscopy indicates that surface spins in maghemite nanoparticles

are highly canted (Coey 1971). Their magnetic properties are dominated by surface effects, which

result in high coercive fields, high-field irreversibility, and shifted hysteresis loops in field-cooled

samples (Kachkachi et al. 2000a; Tronc et al. 2000). Particles are not saturated in fields of up to 5.5

T (Fig. 6a), and an anomalous increase in magnetization appears below 70 K (Fig. 6b), which is

more pronounced in smaller particles (Fig. 6c). The simulated contribution to the magnetization

from core and surface spins is illustrated in Fig. 7 (Kachkachi et al. 2000a). Assuming that

exchange interactions between surface atoms are an order of magnitude weaker than interactions

between bulk atoms, an anomalous increase in magnetization observed at low temperatures can be

attributed to the ordering of surface spins.

4.2.3. Application to coupled magnetic and chemical ordering in solid solutions

! The magnetic properties of the ilmenite-hematite solid solution are influenced profoundly by

nanoscale microstructures resulting from chemical ordering. Slowly cooled rocks that contain finely

exsolved hematite-ilmenite have strong and extremely stable magnetic remanence, which may

account for some of the magnetic anomalies that are present in the deep crust and on planetary

bodies that no longer retain a magnetic field, such as Mars (McEnroe et al. 2001, 2002, 2004a, b,

and c; McEnroe et al. 2009; Brown and McEnroe 2012). This remanence has been attributed to the

presence of a stable ferrimagnetic substructure, which is associated with the coherent interface

between nanoscale ilmenite and hematite exsolution lamellae (the so-called ‘lamellar magnetism

hypothesis’; Harrison and Becker 2001; Robinson et al. 2002 and 2004; Harrison 2006; Kasama et

al. 2003, 2004 and 2009; Fabian et al. 2008; McCammon et al. 2009; Fig. 8a). Rapidly cooled

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members of the hematite-ilmenite series, on the other hand, are well known for their ability to

acquire self-reversed thermoremanent magnetization (i.e., they acquire a remanent magnetization on

cooling that is antiparallel to the applied field direction). This phenomenon is related to the

presence of fine-scale twin domains that form on cooling through the R3c -R3 cation ordering phase

transition (Ishikawa and Syono 1963; Nord and Lawson 1989 and 1992; Hoffman 1992; Bina et al.

1999; Prévot et al. 2001; Lagroix et al. 2004; Fabian et al. 2011; Robinson et al. 2012 and 2013;

Fig. 8b).

! Harrison (2006) used Monte Carlo simulations to investigate the consequences of coupling

between magnetic and chemical ordering in the ilmenite-hematite solid solution (Fig. 9). Key

features of the equilibrium phase diagram are reproduced successfully by the simulations: i) a

paramagnetic (PM) to antiferromagnetic (AF) transition in the hematite-rich, cation-disordered

(R3c ) solid solution; ii) a PM R3c to PM R3 cation ordering transition in the ilmenite-rich solid

solution; iii) a PM R3c + PM R3 miscibility gap developing below a tricritical point at x = 0.58 ±

0.02, T = 1050 ± 25 K; and iv) an AF R3c + PM R3 miscibility gap developing below a eutectoid

point at x = 0.18 ± 0.02, T = 800 ± 25 K.

! A snapshot of the simulated cation/spin configuration obtained at 100 K for a bulk

composition 30% Fe2O3 70% FeTiO3 (ilm70) is shown in Fig. 10. The supercell contains two

nanoscale precipitates of AF R3c hematite within a host of PM R3 ilmenite. The lower precipitate

has a thickness of 2 nm (~1.5 unit cells), corresponding to the lower size limit of exsolution

lamellae typically observed in natural samples (Robinson et al. 2002). The precipitate contains a

total of 11 Fe-bearing cation layers: 9 Fe3+ layers, which are bounded by mixed Fe3+-Fe2+ ‘contact’

layers. The natural tendency for hematite lamellae to form with an odd number of Fe-bearing layers

results in the formation of a ‘defect’ moment due to the presence of uncompensated spins (Fig.

10c). Under favorable conditions, this ‘lamellar magnetism’ far outweighs the spin-canted moment

of the hematite phase (Robinson et al. 2004). The upper precipitate has a less well-defined shape

and an atomically rough interface with the ilmenite host. It has a thickness of 0.7-1.4 nm (~0.5-1

16

unit cells), which corresponds to the length scale of the compositional clustering that is commonly

observed in natural samples in the vicinity of precipitate-free zones (McEnroe et al. 2002). The

more irregular shape and rough interface of the upper precipitate enhances the spin imbalance,

yielding a larger net magnetization (Fig. 10d).

! Samples that have cooled rapidly through the R3c- R3 transition develop a high degree of

short-range cation order, which is characterized by the formation of fine-scale twin domains

(Harrison and Redfern 2001; Nord and Lawson 1989 and 1992; Fabian et al. 2011; Fig. 8b).

Adjacent domains have an antiphase relationship with each other, in terms of the ordering of Fe and

Ti layers; an Fe-rich layer becomes a Ti-rich layer on crossing the twin wall and vice versa. In order

to highlight this relationship, twin domains and twin walls are often referred to as antiphase

domains (APDs) and antiphase domain boundaries (APBs), respectively. Fig. 11 shows the results

of Monte Carlo simulations of a 48-layer supercell of ilm70, with APBs at its centre and upper/

lower boundaries (Harrison 2006). The degree of cation order is defined by the order parameter:

! Q =NB

Ti �NATi

NATi + NB

Ti! (8)

where NATi and NB

Ti are the number of Ti4+ cations on A and B layers, respectively. The cation/spin

configuration after annealing the supercell in the simulation at 850 K is shown in Figs. 11a-d. The

APDs are cation-ordered (Q = ±1) and the APBs are cation-disordered (Q = 0) (Fig. 11a). The APBs

are enriched in Fe relative to the APDs, although the magnitude of this enrichment is enhanced by

the immiscibility of ilmenite and hematite at this temperature (Fig. 11b). The spin profile at 25 K

shows the presence of oppositely magnetized ferrimagnetic domains, a consequence of the switch

round of Fe-rich and Ti-rich layers at the APB (Fig. 11c). Experimental confirmation of such

negative exchange coupling, obtained using electron holography, is discussed in Section 5.3.4. The

17

APB shown in Fig. 11c is characterized by an asymmetric spin profile. Only the very centre of the

APB can be classed as antiferromagnetic. At 400 K, the majority of the supercell is magnetically

disordered, whereas the APBs retain a narrow region of magnetic order (Fig. 11d).

! After annealing at 1100 K, the supercell contains a well-ordered domain and a smaller, less

well (anti)ordered domain (Fig. 11e). This situation is reached as the system attempts to remove one

APD and establish an equilibrium state of homogenous long-range order. Fe-enrichment now occurs

at the APBs and across the less well (anti)ordered domain. The spin profile at 25 K indicates a

strong ferrimagnetic moment associated with the ordered domain and a weak ferrimagnetic moment

with the anti-ordered domain (Fig. 11g). At 375 K, magnetic order is lost in the ordered domain,

whereas weak magnetic order is retained across the anti-ordered domain and boundary regions (Fig.

11h). These properties lead to a self-reversal in the net magnetization on cooling (Fig. 12).

Magnetic ordering in the Fe-enriched anti-ordered domain sets in below 425 K, yielding a weak

positive ferrimagnetic moment. Magnetic order spreads to the ordered domain on cooling below

350 K. Below 250 K, the moment of the anti-ordered domain is outweighed by the oppositely-

oriented moment of the ordered domain, and the net magnetization reverses. In contrast, no net

reversal is observed in the 850 K simulation, which contains equally well ordered and anti-ordered

domains, despite the enhanced enrichment of Fe at the APB. The predictions of the atomistic

simulation have largely been confirmed by experimental obervations (Fabian et al. 2011) and have

been developed into a comprehensive theory of self reversal applicable to much larger

microstructural length scales by Robinson et al. (2012 and 2013).

5. MAGNETISM AT THE NANOMETER LENGTH SCALE

! Off-axis electron holography is an advanced TEM technique that allows a two-dimensional

projection of the in-plane component of the magnetic induction in a specimen to be mapped with a

spatial resolution approaching the nanometer scale. The high spatial resolution of this technique

makes it ideal for the study of magnetic particles that are in the SD to PSD size range, as well as for

18

magnetic minerals that are structurally and/or chemically heterogeneous. Its ability to provide

images of stray magnetic fields also makes electron holography an ideal technique for the study of

magnetostatic interactions between magnetic nanoparticles. We begin by reviewing theoretical and

practical aspects of electron holography. We then describe its recent application to several different

magnetic minerals.

5.1. Theory of off-axis electron holography of magnetic materials

5.1.1. Amplitude and phase of a TEM image

! The formation of a TEM image can be described in terms of the electron wavefunction in the

image plane of the microscope:

! ⇥(r) = A(r)exp[i�(r)] ! (9)

where A is amplitude, φ is phase shift (with respect to a wave that has travelled through vacuum

alone), and r is a vector in the plane of the sample (Cowley 1995). As an electron passes through the

microscope, it experiences a phase shift that is associated with both the electrostatic potential of the

sample (Fig. 13a) and the in-plane components of the magnetic induction (Fig. 13b). In a

conventional TEM image, only the spatial distribution of the image intensity,

! I(r) = �(r)�⇤(r) = A2(r) ! (10)

is recorded, and all information about the phase shift is lost. Electron holography is an

interferometric technique that allows phase information to be recovered. After subtraction of the

electrostatic contribution to the phase shift (see below), a phase image can be converted directly

19

into a quantitative two-dimensional map of the in-plane magnetic induction in the sample

(Tonomura 1992; Völkl et al. 1998; Dunin-Borkowski et al. 2004; Midgely 2001).

! The phase shift (measured relative to that of an electron that has passed through vacuum

alone) is given by the expression:

! �(x) = CE

ZV0(x, z)dz �

⇣ e

~

⌘ ZZB?(x, z)dxdz ! (11)

where x is a direction in the plane of the sample, z is the incident electron beam direction, V0 is the

mean inner potential, and B? is the component of magnetic induction perpendicular to both x and z

(Reimer 1991). CE is a constant that depends on the accelerating voltage of the TEM, in the form:

! CE =✓

2⇥

�

◆ ✓E + E0

E(E + 2E0)

◆! (12)

where λ is the electron wavelength, and E and E0 are the kinetic and rest mass energies of the

incident electrons, respectively. Values of CE for a range of accelerating voltages are listed in Table

2. If neither V0 nor B? varies with z inside the specimen, and both parameters are zero outside the

specimen, Eqn. 11 can be expressed more simply in the form:

! �(x) = CEV0(x)t(x)�⇣ e

~

⌘ ZB?(x)t(x)dx ! (13)

where t is the thickness of the sample. Differentiating with respect to x then leads to the expression:

20

! d�(x)dx

= CEd

dx{V0(x)t(x)}�

⇣ e

~

⌘B?(x)t(x) ! (14)

For constant V0 and t, the first term in Eqn. 14 is zero, and the gradient of the phase shift is

proportional to the desired in-plane component of the magnetic induction in the specimen (Fig.

13b). The phase gradient can then be written in the form:

!d�(x)

dx= �

✓et

~

◆B?(x) ! (15)

Unfortunately, in most cases, both V0 and t vary across the specimen, and careful separation of the

magnetic and mean inner potential contributions to the measured phase shift is required before

quantitative analysis of the magnetic induction is possible (see Section 5.1.5).

5.1.2. Calculation of the mean inner potential

! For a specimen that has a single composition and crystallographic orientation, the mean inner

potential contribution to the phase shift is proportional to the specimen thickness (Fig. 13a). Direct

measurement of V0 using electron holography is possible if an independent measurement of the

specimen thickness is available. Such measurements are rare, however, and it is often necessary to

calculate theoretical values of V0. An estimate for V0 can be obtained by assuming that the specimen

can be described as a collection of neutral free atoms (the ‘non-binding’ approximation), and by

using the expression:

! V0 =✓

h2

2�me�

◆ X

�

fel(0) ! (16)

21

where fel(0) is the electron scattering factor at zero scattering angle (with dimensions of length), Ω

is the unit cell volume, and the sum is performed over all atoms in the unit cell (Reimer 1991).

Calculated values for fel(0) have been tabulated by Doyle and Turner (1968) and Rez et al. (1994).

Eqn. 16 leads to an overestimation of V0 by approximately 10%, because the redistribution of

electrons due to bonding (which typically results in a contraction of the electron density around

each atom) is neglected. Calculated upper limits of V0 for common magnetic oxide minerals are

listed in Table 3.

5.1.3. Formation of an electron hologram

! The microscope setup for electron holography is illustrated schematically in Fig. 14. A field-

emission gun (FEG) is used to provide a highly coherent source of electrons. For studies of

magnetic materials, a Lorentz lens (a high-strength minilens located below the lower objective pole-

piece) allows the microscope to be operated at high magnification with the objective lens switched

off and the sample in magnetic-field-free conditions. Since a large amount of off-line image

processing must be carried out to process holograms and in particular to remove the mean inner

potential contribution to the phase shift, there are great advantages in recording holograms digitally,

using a charge-coupled device (CCD) camera.

! For off-axis electron holography, the sample is typically placed half-way across the field of

view, so that part of the electron wave passes through the sample (the sample wave) and part passes

through vacuum (the reference wave). A voltage of 50-200 V is applied to an electrostatic biprism

wire (typically a < 1 μm-diameter quartz wire coated in Pt or Au) mounted in place of one of the

selected area diffraction apertures. This voltage deflects the sample and reference waves, causing

them to overlap. If the electron source is sufficiently coherent, then an interference fringe pattern

(an electron hologram) is formed in the overlap region. The sample and reference waves can then be

considered as originating from two virtual sources, S1 and S2 (Fig. 14). The angles of the sample

22

and reference waves differ by a small amount, which is proportional to the biprism voltage, and can

be described by the wavevector qc . The intensity in the overlap region is then given by the

expression:

!

Ihol

(r) = |⇥(r) + exp[2�iqc

· r]|2

= 1 + A2(r) + 2A(r)cos[2�iqc · r + ⇥(r)] .! (17)

Hence, an electron hologram consists of a sum of the intensities of the sample and reference waves,

onto which is superimposed a set of cosinusoidal fringes with local amplitude A and phase shift φ.

An example hologram, acquired from a sample containing maghemite inclusions in a matrix of

hematite, is shown in Fig. 15a. Local shifts in the positions of the holographic interference fringes,

visible in the inset to Fig. 15a, are directly proportional to the phase shift of the electron wave,

which results in turn from variations in the thickness, mean inner potential, and magnetic induction

of the inclusion and the host. A broader set of Fresnel fringes are also visible at the edges of the

hologram in Fig. 15a. These fringes are caused by the edge of the biprism wire.

5.1.4. Processing of the electron hologram

! The sequence of processing steps that is required to extract a phase map, �(r), from an

electron hologram is illustrated in Fig. 15. First a hologram of the region of interest is acquired (Fig.

15a). The sample is then moved away from the field of view and a reference hologram is acquired

from a region of vacuum (Fig. 15b). Next, both the sample and the reference holograms are Fourier

transformed (Fig. 15c). The Fourier transform of Eqn. 17 comprises a central peak at q = 0 and two

sidebands at q = ± qc:

23

!

FT [Ihol

(r)] = �(q) + FT [A2(r)]

+�(q + qc)⌦ FT [A(r)exp(i⇥(r)]

+�(q� qc)⌦ FT [A(r)exp(�i⇥(r)]! (18)

The sidebands contain the Fourier transforms of either the complex image wave or its conjugate.

Both amplitude and phase information is recovered by isolating one sideband digitally (Fig. 15d)

and performing an inverse Fourier transform of this part of the Fourier transform alone. The

diagonal streak at the lower left of Fig. 15d results from the presence of Fresnel fringes visible in

the raw holograms (Figs. 15a and b). This streak can lead to artifacts in the reconstructed phase

map, and is normally masked out (i.e., replaced by pixels with values of zero) before the inverse

Fourier transform is performed. The complex image waves that are derived from the sample and

reference holograms are divided by each other to remove phase shifts caused by inhomogeneities in

the charge and thickness of the biprism wire, and distortions caused by aberrations of the

microscope lenses and the recording system (de Ruijter and Weiss 1993). The phase shift is then

obtained by evaluating the arctangent of the ratio of the imaginary and real components of the

corrected complex image wave (Fig. 15e). The initial phase map is presented modulo 2π. The 2π

discontinuities can be removed by using one of a number of automated phase unwrapping

algorithms (Ghiglia and Pritt 1998) to produce an ‘unwrapped’ final phase image (Fig. 15f).

5.1.5. Removing the mean inner potential contribution

! If the direction of magnetization in the sample can be reversed exactly, for example by

applying a large magnetic field to the specimen, then the magnetic contribution to the phase shift

changes sign in Eqn. 11. If phase images that have been acquired before and after magnetization

reversal are added together digitally, then the magnetic contribution to the phase shift cancels out,

24

leaving twice the mean inner potential contribution. Magnetization reversal can be performed in situ

in the TEM by using the magnetic field of the TEM objective lens (Fig. 16). The sample is typically

tilted to an angle of ±30° to the horizontal. The objective lens is then turned on to provide a chosen

vertical magnetic field of up to 2 T. The objective lens is then turned off and the sample tilted back

to the horizontal prior to acquisition of the hologram. In practice, the two saturation remanent states

may not be exactly equal and opposite to each other. It is then necessary to repeat the switching

process several times, so that non-systematic differences between switched pairs of phase images

average out. Systematic differences between switched pairs, which can lead to artifacts in the final

magnetic induction map, are often identified by inspection. Once the mean inner potential

contribution to the phase shift has been determined in this way, it can be subtracted from each

individual phase image of the same region of the sample. By varying the magnitude of the applied

field, it is possible to record a series of images that correspond to any desired point on the remanent

hysteresis loop.

! When interpreting the subsequent remanent hysteresis loop, it is important to remember that

holography measures the coercivity of remanence (Hcr) rather than the coercivity (Hc) of the

sample. For SD particles, however, Hcr/Hc ~ 1, allowing an estimate of the coercivity to be made. A

sample with uniaxial anisotropy constant, K, and saturation magnetization, MS, switches when the

vertical field reaches 0.52 BK (assuming a tilt angle of 30° to the horizontal), where BK = 2K/MS is

the coercivity for fields applied along the anisotropy axis (Stoner and Wohlfarth 1948).

5.2. Interpretation of electron holographic phase images

5.2.1. Quantification of the magnetic induction

! A quantitative measure of B? , integrated in the electron beam direction, can be obtained from

the gradient of the magnetic contribution to the phase shift (Eqn. 15). This measurement includes

25

contributions from the internal magnetization of the sample, the internal demagnetizing field, and

stray magnetic fields created by the sample in the vacuum surrounding it. For this reason, there is

not always an intuitive relationship beween the magnetisation of the sample and its holographic

phase shift. Routine simulation of the phase shift based on a proposed magnetisation model is now

considered an essential part of holography studies (Beleggia and Zhu 2003; Beleggia et al. 2003a, b;

Bryson et al. 2012; Section 5.2.4). A simulation of the contributions to the phase shift associated

with the presence of a uniformly magnetized 200 nm-diameter spherical particle of magnetite is

shown in Fig. 17. The total phase shift (Fig. 17c) is the sum of mean inner potential (Fig. 17a) and

magnetic (Fig. 17b) contributions. An analytical expression that describes the phase shift shown in

Fig. 17c, along a line passing through the centre of the particle in a direction perpendicular to B? ,

is:

!�(x)|

xa

= 2CE

V0

pa2 � x2 +

⇣ e

~

⌘B?

"a3 � (a2 � x2) 3

2

x

#! (19)

! �(x)|x>a

=⇣ e

~

⌘B?

✓a3

x

◆

! (20)

where a is the radius of the particle (de Graef et al. 1999). The mean inner potential and magnetic

contributions to this phase profile are shown in Figs. 17d-f. The difference between the minimum

and maximum values of the magnetic contribution to the phase shift in Fig. 17e is:

! ��mag = 2.044⇣ e

~

⌘B?a2 ! (21)

For a uniformly magnetized cylinder of radius a, the equivalent expression is:

26

! �⇥mag = �⇣ e

~

⌘B?a2 ! (22)

! To a good approximation, ion-beam thinned TEM specimens of magnetic materials can often

be described locally as plates or wedges of semi-infinite extent in the horizontal plane. If such a

sample is magnetized uniformly parallel to its edge, then the effect of demagnetizing and stray

fields on the measured phase shift may be negligible, and it may then be possible to determine B?

directly by using Eqn. 15. This approach requires, however, that the local specimen thickness is

known. A measure of the sample thickness can be obtained using energy filtered imaging (Egerton

1996). Two images of the sample are acquired: an unfiltered image (formed using both elastically

and inelastically scattered electrons) and a zero-loss energy-filtered image. The log of the ratio

between the unfiltered and zero-loss energy-filtered images yields the quantity t/λin, where λin is a

mean free path for inelastic scattering. Values of λin can be calculated or measured experimentally

(Egerton 1996; Golla-Schindler et al. 2005). Harrison et al. (2002) determined a value for λin = 170

nm for magnetite. Care is required when using this approach, as the effective magnetic thickness of

a sample may be significantly smaller than its physical thickness, due to the presence of

magnetically ‘dead’ layers on its surfaces, resulting from specimen preparation techniques such as

ion-beam thinning. For a sample of known B?and V0, an estimate of the thickness of the

magnetically dead layers can be obtained by comparing the physical thickness of the specimen

derived from the mean inner potential contribution to the phase shift (the first term in Eqn. 13) with

the magnetic thickness derived from Eqn. 15. For an Ar-ion milled synthetic sample of ilm70,

assuming a value for V0 = 19.6 V (calculated using Eqn. 16) and a saturation induction of 0.225 T,

the average difference between the magnetic and physical specimen thickness was found to be ~40

nm in total.

!

27

5.2.2. Visualization of the magnetic induction

! The in-plane component of the integrated magnetic induction can be visualized by adding

contours to the magnetic contribution to the phase shift, as shown in Fig. 17h in the form of the

cosine of the phase image. The spacing of the contours can be varied by multiplying the phase map

by a constant before calculating its cosine (an ‘amplification’ factor of 4 was used in Fig. 17). By

calculating the horizontal and vertical derivatives of the magnetic contribution to the phase shift

(dφmag/dx and dφmag/dy), a vector field can be determined and displayed in the form of either an

arrow map or a color map (Fig, 17j), whereby the direction and magnitude of the projected in-plane

magnetic induction are represented by the hue and intensity of a color, respectively, according to the

color wheel shown in Fig. 17l. Color can also be added to the cosine image if desired (Fig. 17k).

5.2.3 Quantification of holography images in the general case

! Quantitative analysis of the magnetic contribution to the phase shift can, in principle, be used

to determine an absolute value for the magnetic moment of a particle. Until recently, such analysis

was restricted to uniformly-magnetised particles with well-defined shapes (e.g. spheres), for which

analytical expressions for the magnetic phase shift can be derived (Eqns. 19-22; Fig. 17). However,

a general method for measuring the magnetic moment of a particle has now been developed, that

does not depend on the particle’s shape or magnetization state (Beleggia et al. 2010). The

measurement scheme is based on a loop integral of the phase image around a circular boundary

containing the structure of interest:

! Mx

=

~Rc

eµ0

Z 2⇡

0�(R

c

cos ✓, Rc

sin ✓) cos ✓d✓ ! (23)

! My =

~Rc

eµ0

Z 2⇡

0��(Rc cos ✓, Rc sin ✓) sin ✓d✓ ! (24)

28

where Mx and My are the in-plane components of the magnetic moment vector, ħ is the reduced

Planck’s constant, Rc is the radius of the integration circle, e is the electron charge, μ0 is the

permeability of free space, and φ(Rccosθ, Rcsinθ) is the phase shift at a given angle θ around the

integration circle.

! The above method was successfully employed by Lappe et al. (2011) to measure the magnetic

moments of irregularly shaped metallic Fe nanoparticles embedded in a silicate matrix (olivine),

thus demonstrating that these particles would have the thermal stability necessary to retain pre-

accretionary palaeomagnetic remamence in primitive chondritic meteorites. An example of the

analysis is shown in Fig. 18. The mean inner potential (Fig. 18a) and magnetic contributions to the

phase shift (Fig. 18b) were determined from reversed pairs of holograms, as described in section

5.1.5. The phase shift around the circumference of a circle enclosing the particle of interest was

extracted from the magnetic phase map, and integrated to yield Mx and My via Eqns. 23 and 24 (Fig.

18b). Bellegia et al. (2010) show that the likely sources of error in this method scale with Rc2, and

hence a measurement of the true moment (M0) can be obtained by fitting a function of the form M =

M0 + aRc2 to data obtained over a range of Rc values (inset to Fig. 18b). If the particles are assumed

to be uniformly magnetised with a saturation magnetization Ms = 1750 kA/m (Garrick-Bethell and

Weiss 2010) then the total magnetic moment can be used to calculate the particle volume (V = M0/

Ms) and generate a projected thickness model (Fig. 18c). First the mean inner potential of the

particle is isolated using particle analysis and the background mean inner potential contribution

from the surrounding silicate matrix is subtracted (Fig. 18a). The isolated mean inner potential

signal (which is proportional to the particle thickness) is then multiplied by a calibration factor so

that the integral of the mean inner potential is equal to V (Fig. 18c). Given Mx, My and t, the

holographic phase shift can be then be simulated using the methods outlined in 5.2.4 (Fig. 18d).

5.2.4. Simulation of holography images in the general case

29

! General Fourier methods for the calculation of the magnetic phase shift are derived by

Beleggia and Zhu (2003) and Beleggia et al. (2003a, b). These methods have been incorporated into

a convenient software package ATHLETICS by Bryson et al. (2012), which automates the

simulation of electron holography images (http://www.esc.cam.ac.uk/research/research-groups/

athletics) for non-uniformly magnetised samples. The real space variation in magnetic moment,

M(r), can be written in the form:

! M(r) = M0m(r) ! (25)

where m(r) is a unit vector. Beleggia and Zhu (2003) derive the magnetic phase shift in reciprocal

space from the Fourier transform of m(r):

!�m

(k) =i⇡B0

�0k2?(m(k

x

, ky

, 0)⇥ k)|z

! (26)

where B0 = µ0M0 is the saturation induction, φ0 is the flux quantum (h/2e = 2.07 x 103 T nm3), m(k)

is the Fourier transform of m(r) and k⊥ = (kx2 + ky2)1/2. The phase of an electron wave exiting a

sample is evaluated as follows. The x and y components of m(r) are multiplied by the thickness to

create two 2D ‘magnetic thickness’ matrices mx(x, y) and my(x, y). The FFT of mx and my are

calculated and φm(k) is evaluated via Eqn. 26. The inverse FFT of φm(k) then yields φm(x, y). The

application of this method is shown in Fig. 18d, demonstrating excellent quantitative agreement

between oberved and simulated magnetic signals.

5.3. Experimental Results

5.3.1. Electron holography of isolated magnetite crystals

30

! An experimental study of isolated magnetic nanoparticles allows the effects of particle size,

shape, and magnetocrystalline anisotropy on their magnetic state to be assessed without the

complicating influence of magnetostatic interactions. Magnetotactic bacteria provide a convenient

source of high-purity, relatively defect-free magnetite crystals with varying morphologies, aspect

ratios, and sizes in the range 10-200 nm (Devouard et al. 1998; Bazylinski and Frankel 2004; Arató

et al. 2005). Although magnetotactic bacteria normally grow crystals in closely spaced chains, the

preparation of bacteria for TEM examination by air drying inevitably leads to cell damage and a

degree of chain break-up. Fig. 19a shows a high-resolution TEM image of an isolated 50 nm

magnetite crystal from a bacterial cell. This crystal was separated by at least 500 nm from adjacent

crystals. The three-dimensional morphology and orientation of the crystal were determined by using

electron tomography, from a series of two-dimensional high-angle annular dark-field (HAADF)

images taken over an ultra-high range of tilt angles (Fig. 19b). The tomographic reconstruction

reveals that the particle is elongated slightly in the [111] direction in the plane of the specimen (as

indicated by the white arrow in Fig. 19a). The crystallographic orientation of the particle is shown

in the form of a stereogram in Fig. 19e.

! Electron holography of the magnetite crystal was performed both at room temperature (Fig.

19c) and at 90 K (Fig. 19d). The magnetic contribution to the phase shift was isolated by

performing a series of in situ magnetization reversal experiments, as described in Section 5.1.5. The

direction of the in-plane component of the applied field is indicated by the black double arrow. Both

images show uniformly magnetized SD states, including the characteristic return flux of an isolated

magnetic dipole (Fig. 17k). In both cases, the remanent magnetization direction appears to make a

large angle to the applied field direction. At room temperature, the phase contours in the crystal

make an angle of ~30° to the [111] elongation direction (Fig. 19c). The contours are parallel to the

[111] elongation direction at 90 K (below the Verwey transition; Fig. 19d).

! Fig. 20a shows a profile of the magnetic contribution to the phase image that was used to

create Fig. 19c, taken along a line passing through the centre of the crystal in a direction

31

perpendicular to the phase contours. A least-squares fit of the experimental profile to Eqns. 19 and

20 yielded a value for B? of 0.6 ± 0.12 T. This value is equal to the room temperature saturation

induction of magnetite, suggesting that the magnetization direction of the particle lies exactly in the

plane of the specimen, close to the [131] crystallographic direction (Fig. 19e). This direction

corresponds to the longest diagonal dimension of the particle, which is consistent with shape

anisotropy dominating the magnetic state of the crystal at room temperature. The 90 K phase profile

(Fig. 20b) yielded a value for B? of 0.46 ± 0.09 T. This value is lower than the saturation induction

of magnetite at 90 K, suggesting that, at remanence, the magnetization direction in the crystal is

tilted out of the plane by ~40° to the horizontal. This direction is close to either [210] or [012] (Fig.

19e). Below the Verwey transition, the magnetocrystalline anisotropy of magnetite is known to

increase considerably in magnitude (Abe et al. 1976; Muxworthy and McClelland 2000), and to

switch from <111>cubic to [001]monoclinic. The [001]monoclinic easy axis can lie along any one of the

original <100>cubic directions. Both the [100] and [001] directions of the original cubic crystal lie

close to the observed remanence direction, suggesting that magnetocrystalline anisotropy has a

more significant impact on the remanence direction than shape anisotropy at 90 K. The fact that the

remanence direction in Fig. 19d is perpendicular to the applied field direction suggests that this

choice may be influenced by the morphology of the crystal.

! Theoretical predictions of the effect of particle size and shape on the magnetic state of

magnetite are shown in Fig. 21 (Butler and Banerjee 1975; Muxworthy and Williams 2006). The

upper solid line shows the theoretical boundary between SD and two-domain states (Butler and

Banerjee 1975). The dashed line shows the boundary between SD and single vortex (SV) states

predicted by micromagnetic simulations (Muxworthy and Williams 2006; see Section 6). For

equidimensional particles, the equilibrium SD/SV transition is predicted to occur at a particle size

of 70 nm (Fabian et al. 1996; Williams and Wright 1998) and the transition to a superparamagnetic

(SP) state is observed to occur below 25-30 nm (Dunlop and Özdemir 1997). The observation of a

32

stable SD state for the roughly equidimensional 50 nm crystal in Fig. 19 is in agreement with the

expected behavior.

5.3.2. Electron holography of chains of closely spaced magnetite crystals

! Figs. 22a and b show a bright-field TEM image and a three-dimensional tomographic

reconstruction, respectively, of a double chain of magnetite crystals from a magnetotactic bacterial

cell (Simpson et al. 2005). Each crystal has its [111] crystallographic axis aligned accurately (to

within 4°) of the chain axis, as shown by the arrows in Fig. 22a, but is rotated by a random angle

about this axis (like beads on a string). From a magnetic perspective, the alignment of the crystals

ensures that their room temperature magnetocrystalline easy axes are closely parallel to the chain

axis. The largest crystals are elongated slightly along [111], with lengths of 92-94 nm and widths of

82-88 nm. Isolated crystals of this size would be expected to adopt SV states at equilibrium (Fig.

21). Electron holography of similar chains revealed, however, that such crystals are magnetized

uniformly parallel to each chain axis (Fig. 22c). For such highly aligned chains of crystals,

magnetostatic interactions move the boundary between SD and SV states to larger particle sizes,

and promote the stability of SD states. This effect, which is also predicted by micromagnetic

simulations (Muxworthy et al. 2003a; Muxworthy and Williams 2006; see Section 6), enables

bacteria to grow SD crystals to much larger sizes than would otherwise be possible, thereby

optimizing the overall magnetic moment of the chain.

! In Fig. 22, strong magnetostatic interactions between crystals, combined with their high

degree of alignment, result in uniform magnetic phase contours that are constrained tightly along

the chain axis. In this case, the magnetic induction of the chain can be quantified by assuming that it

has approximately a cylindrical geometry (Eqn. 22), yielding a value for B? of 0.62 T for one of

the central crystals. This value is close to that predicted for magnetite (B0 = 0.6 T), suggesting that

the crystals are magnetized parallel to their length and to the chain axis. Interacting chains of

33

closely spaced crystals are always magnetized along the chain axis, thus providing a reliable

magnetic moment for magnetotaxis. This is confirmed by hysteresis (Pan et al. 2005; see Section 7)

and remanence experiments (Hanzlik et al. 2002) on magnetotactic bacteria, which demonstrate that

chains of magnetosomes act like a single elongated particle (with uniaxial anisotropy) and switch as

a single unit.

! When the alignment of crystals in a chain is less than perfect, it is possible to produce a non-

uniform magnetization state by the application of a suitably oriented magnetic field (Fig. 23). Fig.

23a illustrates the magnetization state of two double chains of magnetite crystals (from the same

bacterial cell) after the application of a magnetic field with an in-plane component of 1 T parallel to

the chain axes. Despite the imperfect alignment of the crystals in the upper double chain, all four

chains are magnetized along their length. Some flux divergence is evident where large gaps occur

between crystals in the upper double chain, indicating that magnetostatic interactions are weakened

slightly at these positions. Fig. 23b illustrates the magnetization state after application of a similar

magnetic field perpendicular to the chain axes. Although three out of the four chains are unaffected

by the change in the applied field direction, the uppermost chain is split into two halves, each of

which has a small component of its magnetization in the direction of the applied field. The two

crystals in the centre of the chain are now arranged in an energetically unfavorable opposing

configuration, and the magnetization of one of the crystals is deflected so that it points at an angle

of ~60° to the chain axis.

! Fig. 24 shows magnetic induction maps that have been acquired from a similar pair of

magnetite chains both at room temperature and at 116 K (close to the Verwey transition). Whereas

the contours are highly parallel to each other and to the chain axes at room temperature (Fig. 24a),

their direction is far more variable and irregular at low temperature (Fig. 24b). This behavior is

most pronounced in some crystals in Fig. 24b that show S-shaped magnetic configurations. As

mentioned above, in magnetite the change in structure from cubic to monoclinic at the Verwey

transition is associated with a change in easy axis from <111> to <100>. Although particle

34

interactions and shape anisotropy result in the preservation of the overall magnetic induction

direction in Fig. 24b along the chain axis at low temperature, it is likely that the undulation of the

contours along the chain axes results from a competition between the effects of magnetocrystalline

anisotropy, shape anisotropy and magnetostatic interactions, which are only mutually favorable at

room temperature.

5.3.3. Electron holography of two-dimensional magnetite nanoparticle arrays

! In contrast to the sizes of crystals in magnetotactic bacteria, the grain sizes of primary

magnetic minerals in most igneous and metamorphic rocks exceed the MD threshold. Such rocks

are less likely to maintain strong and stable natural remanent magnetization (NRM) over geological

times than those containing SD grains. It has long been proposed, however, that solid state

processes such as sub-solvus exsolution can transform an MD grain into a collection of SD grains,

thus increasing the stability of the NRM (Davis and Evans 1976). This transformation is brought

about by the formation of intersecting paramagnetic exsolution lamellae, which divide the host

grain into a three-dimensional array of isolated magnetic regions that have SD-PSD sizes.

! An excellent example of this phenomenon occurs in the magnetite-ulvöspinel (Fe3O4-

Fe2TiO4) solid solution (Davis and Evans 1976; Price 1980 and 1981). This system forms a

complete solid solution at temperatures above ~450 °C but unmixes at lower temperatures (Ghiorso

1997). Intermediate bulk compositions exsolve during slow cooling to yield an intergrowth of SD-

or PSD-sized magnetite-rich blocks that are separated by non-magnetic ulvöspinel-rich lamellae.

Fig. 25a illustrates the typical microstructure observed in a natural sample of exsolved

titanomagnetite (Harrison et al. 2002). This image is a composite chemical map, obtained using

energy-filtered TEM imaging, showing the distribution of Fe in blue (magnetite) and Ti in red

(ulvöspinel). Ulvöspinel lamellae form preferentially parallel to {100} planes of the cubic

magnetite host lattice. In TEM sections that are oriented parallel to {100}, this symmetry generates

a rectangular array of cuboidal magnetite blocks. Profiles of the Fe and Ti distribution along the line

35

marked C (Fig. 25b) demonstrate that the blocks are essentially free of Ti, i.e., that they are nearly

pure magnetite.

! Harrison et al. (2002) used electron holography to determine the magnetic remanence states of

region B in Fig. 25a. The magnetite blocks were found to be primarily in SD states (Fig. 26). The

dimensions of the blocks, which are plotted on Fig. 21 for reference, indicate that the vast majority

would display SV states at remanence if they were isolated and at equilibrium. Micromagnetic

simulations of isolated cuboidal particles (Section 6) indicate that SD states can exist in metastable

form up to a certain size above the equilibrium SD-SV threshold (Fabian et al. 1996; Williams and

Wright 1998; Witt et al. 2005). The majority of the blocks in Fig. 26 fall within the limits of

metastability for SD states calculated by Witt et al. (2005). It appears that the presence of

magnetostatic interactions favors the adoption of metastable SD states over equilibrium SV states.

This behavior results from the fact that the demagnetizing energy – which destabilizes the SD state

with respect to the vortex state in isolated particles – is reduced greatly in an array of strongly

interacting SD particles.

! Transitions between different magnetic states in an individual block can be seen in Fig. 26.

For example, block 8 (labeled in Fig. 25a) in Fig. 26e is magnetized NNW (blue), whereas in

Fig. 26f it is magnetized SSE (yellow). It contains an off-centered vortex in Fig. 26b, suggesting

that magnetization reversal in this block may occur via the formation, displacement and subsequent

annihilation of a vortex (Enkin and Williams 1994; Pike and Fernandez 1999; Guslienko et al.

2001; Dumas et al. 2007), rather than by the coherent rotation of the SD moment.

! Several blocks are observed to act collectively to form magnetic ‘superstates’ that would

normally be observed in a single, larger magnetized region. One example is where two or more

blocks interact to form a single vortex superstate. Two-, three-, and five-block vortex superstates are

visible in Fig. 26 (e.g., blocks 1 and 2 in Fig. 26g and blocks 1, 2, 3, 5 and 6 in Fig. 26e). A similar

superstate involving three elongated blocks is shown in Figs. 27a and b, and schematically in

Fig. 28a. The absence of closely-spaced contours between the superstate and the adjacent single

36

vortex in Fig. 27b shows that stray interaction fields are eliminated in the intervening ulvöspinel.

Flux closure is achieved with considerably less curvature of magnetization within the three-

component assembly than is required in the adjacent conventional vortex, reducing the exchange

energy penalty associated with the non-uniform magnetization configuration (Evans et al. 2006).

! A second example of collective behavior involves the interaction of a chain of blocks to form

an SD superstate that is magnetized parallel to the chain axis but perpendicular to the easy axes of

the individual blocks. This behavior is illustrated schematically in Fig. 28b and can be found in

several places in Fig. 26 (e.g., blocks 16, 17 and 18 in Figs. 26a, b, d, f and h). An extreme example

of this behavior is shown in Fig. 29, which shows saturation isothermal remanent states in an

exsolved titanomagnetite inclusion within clinopyroxene (Feinberg et al. 2004 and 2005). These

states were recorded after tilting the sample by angles of ±30° and applying a 2 T vertical field (Fig.

16). The in-plane component of the field was parallel to the elongation direction of the central

blocks. Nevertheless, strong interactions between the blocks (which are separated by ~15 nm of

ulvöspinel) constrain the remanence to lie almost perpendicular to the elongation direction of the

individual blocks and to the applied field direction. The expected remanent state of such a system

might have been expected to involve adjacent blocks being magnetized in an alternating manner

along their elongation directions, as is seen in blocks 16, 17 and 18 and blocks 9, 10 and 11 in

Figs. 26c, e and g and shown schematically in Fig. 28c. It should be noted at this point that the role

of stress-induced anisotropy has not been accounted for, and this may yet turn out to play a

significant role in determining the magnetisation directions in such intergrowths.

! A further example of magnetostatic interactions between blocks is shown in Fig. 27d. The two

largest blocks (colored green) are both magnetized in the same direction. The small block between

them (colored red) is magnetized in the opposite direction, apparently because it follows the flux

return paths of its larger neighbors.

5.3.4. Exchange interactions across antiphase boundaries in ilmenite-hematite

37

! Harrison et al. (2005) used electron holography to study the nature of the exchange coupling

at APBs in ilmenite-hematite. A sample of ilm70 was synthesized from the oxides under controlled

oxygen fugacity at 1573 K, quenched through the cation-ordering phase transition and annealed for

10 hours at 1023 K. Representative magnetic induction maps are shown in Figs. 30a-c. Each figure,

which is derived from the gradient of the magnetic contribution to the recorded phase shift, shows a

magnetic remanent state obtained at a different stage of the switching process. The direction and

magnitude of the in-plane magnetic flux are defined by the hue and intensity of the color,

respectively.

! The magnetization is constrained by shape and magnetocrystalline anisotropy to lie either

parallel or antiparallel to the intersection of the specimen plane with the (001) crystallographic

plane (indicated by the double black arrow in Fig. 30). As a result, regions with strong in-plane

magnetization appear either blue or green. Regions that have no in-plane magnetization appear as

dark bands. Analysis (see Fig. 31 below) shows that the dark bands in Fig. 30 are associated with

three distinct types of magnetic wall (Robinson et al. 2012 and 2013). A finger-like region of

reversed magnetization (labeled ‘1’ in Fig. 30a) enlarges by the movement of its left-hand boundary

as the applied field is increased (Fig. 30b). This left-hand boundary is a conventional free-standing

180° Bloch wall. In contrast, in regions where a 180° reversal in magnetization direction coincides

exactly with the position of an APB (e.g., at regions labeled ‘2’), the reversal results from negative

exchange coupling across the APB, as predicted by Monte Carlo simulations (Fig. 11). This type of

boundary is referred to as a 180° ‘chemical’ wall, and occurs without any out-of-plane rotation of

the magnetic moments. A third type of magnetic wall appears as thick dark bands, which are also

coincident with the positions of APBs (e.g., at regions labeled ‘3’). Such walls form when the

negative exchange coupling between adjacent APDs is overcome at sufficiently large fields, forcing

the magnetization direction on either side to point in the same direction. These walls are referred to

as 0° walls.

38

! For a 180° Bloch wall, the in-plane component of the magnetic induction is generally

described by an expression of the form:

!B?(x) = B0tanh

⇣x

w

⌘! (27)

where B0 is the saturation induction and 2w is the wall width. By substituting Eqn. 27 into Eqn. 13,

the magnetic phase profile across a 180° Bloch wall (assuming a constant thickness, t) is:

! �(x) = B0twln

⇣cosh

⇣ x

w

⌘⌘! (28)

Eqn. 28 provides an excellent fit to the phase profile of a 180° Bloch wall for 2w = 19 nm (Fig.

31a). In contrast, the phase profile on either side of a 180° chemical wall is non-linear (see below),

and the reversal in the slope of the phase profile at the centre of the wall occurs much more abruptly

(Fig. 31b). A fit to the central portion of this wall yields 2w = 7 nm (dashed line in Fig. 31b). This

value is close to the resolution limit of the measurements, and provides an upper limit for the width

of the chemical wall. A 0° wall can be considered as the superposition of a 180° Bloch wall and a

180° chemical wall. Assuming that both types of wall can be described by Eqn. 27 with the same

value of w, the in-plane component of magnetic induction is of the form:

!B?(x) = B0tanh2

⇣x

w

⌘! (29)

By substituting Eqn. 29 into Eqn. 13, the magnetic phase profile across a 0° wall is:

39

! �(x) = B0t[x� 10tanh⇣ x

w

⌘]! (30)

Eqn. 30 provides an excellent fit to the phase profile of a 0° Bloch wall (Fig. 31c). An average of 13

measurements yielded 2w = (50 ± 14) nm for 0° walls. Previous studies demonstrated that self-

reversed thermoremanent magnetization (SR-TRM) was observed only when APDs were below

80-100 nm in size (Nord and Lawson 1989 and 1992). This limit is imposed by the formation of 0°

walls, which allow negative exchange coupling between adjacent domains to be overcome when the

APDs are much larger than 50 nm in size. A detailed exploration of the interaction between

chemical and magnetic boundaries, and its role in the acquisition of self-reversed thermremanent

magnetisation in ilmenite-hematite, is given by Fabian et al. (2011) and Robinson et al. (2012 and

2013).

6. MAGNETISM AT THE MICROMETER LENGTH SCALE

! Due to the large number of atoms and spins involved, atomistic simulations, which describe

the discrete arrangement of atoms and spins on a crystalline lattice, are currently unsuitable for

systems larger than ~10 nm (Section 4). The magnetization states of larger particles (or collections

of particles) are more efficiently described using micromagnetic simulations (Brown 1963).

Micromagnetics is the study of magnetization at the nm to μm length scale (i.e., a length scale that

is much larger than that of the crystalline lattice but smaller than that of a magnetic domain).

Micromagnetic models treat magnetization as a classical, continuous vector field in space. The

energy of the system is described by a number of macroscopic constants, whose values can be

derived from their microscopic equivalents (Eqn. 2). Micromagnetic simulations of magnetic

minerals have progressed rapidly from one-dimensional (Moon and Merrill 1984 and 1985; Moon

1991) to two-dimensional (Newell et al. 1993; Xu et al. 1994), and finally three-dimensional

(Schabes and Bertram 1988a and b; Williams and Dunlop 1989 and 1990; Wright et al. 1997;

40

Fabian et al. 1996; Williams and Wright 1998) models of homogeneous isolated single crystals with

simple geometric shapes. The recent application of finite element/boundary element methods

(FEM/BEM) to micromagnetic simulations now allows the simulation of heterogeneous,

polycrystalline systems that have complex and realistic morphologies (Fidler and Schrefl 2000;

Williams et al. 2006, 2010, 2011; Chang et al. 2012). In addition, substantial progress is being made

in the application of micromagnetic simulations to the study of magnetostatic and exchange

interactions between arrays of closely-spaced magnetic particles (Muxworthy et al. 2003a;

Muxworthy et al. 2004; Carvallo et al. 2003; Muxworthy and Williams 2005, 2006; Evans et al.

2006), and to the application of micromagnetics to the study of the interaction between magnetic

domains and ferroelastic twin domains below the Verwey transition in magnetite (Kasama et al.

2010, 2012; Bryson et al. 2012).

6.1. Theory

! We begin by summarizing the basic principles of micromagnetics, as applied to rock magnetic

problems. For detailed reviews of the technical aspects of micromagnetic simulations, the reader is

referred to Brown (1963), Wright et al. (1997), Fabian et al. (1996), and Fidler and Schrefl (2000).

6.1.1. The micromagnetic energy

! Micromagnetism is a continuum approximation, in which the magnetization of a particle is

taken to be a continuous function of position (c.f. Eqn. 25):

! m(r) =

M(r)

MS=

0

@�⇥⇤

1

A=

0

@cos(⇧)sin(⌅)sin(⇧)sin(⌅)

cos(⌅)

1

A ! (31)

41

where m is a unit vector parallel to the magnetization direction M at position r, and MS is the

saturation magnetization. The direction of m is defined either in terms of direction cosines α, β, and

γ or in terms of the polar coordinates φ and θ. M represents the local average of many thousands of

individual spins.

! Minimization of the microscopic exchange energy (Eqn. 1) requires m to be uniform

throughout a grain. Deviations from uniform magnetization at the macroscopic length scale impose

deviations on the angles between adjacent spins at the atomic scale. On the assumption that these

angular deviations are small, the macroscopic exchange energy can be expressed in the form of a

truncated Taylor expansion of Eqn. 1:

! Eex

= A

Z

V

(rm)2dV ! (32)

where the exchange constant A is related to the atomistic exchange integrals and V is the volume of

the particle. The exchange energy is positive wherever gradients in the macroscopic magnetization

occur (e.g., within domain walls) and zero wherever the magnetization is uniform (e.g., within

domains).

! Since the angular relationship between the atomic spins and the net magnetization is fixed, the

macroscopic expression for the magnetocrystalline anisotropy energy is equivalent to that in Eqn. 2.

For unixial anisotropy this expression is:

! Ea = �Z

VK(m · e)2dV ! (33)

while for cubic magnetocrystalline anisotropy the expression is:

42

! Ea =Z

V[K1(�2⇥2 + ⇥2⇤2 + ⇤2�2) + K2�

2⇥2⇤2]dV ! (34)

Similarly, the macroscopic expression for the magnetostatic energy is equivalent to that in Eqn. 2:

! Eh

= �µ0MS

Z

V

Hext

· m dV ! (35)

where Hext is the applied magnetic field.

! Calculation of the demagnetizing energy is the most challenging and computationally

intensive part of any micromagnetic simulation. A macroscopic expression for the demagnetizing

energy (Eqn. 3) can be formulated in terms of the demagnetizing field, Hd(r), which is the sum of

the magnetic fields at position r created by all of the magnetic moments in the particle:

! Ed = �12µ0MS

Z

Vm · HddV .! (36)

A general method for calculating Hd follows from Maxwell’s equations in a current-free region with

static electric and magnetic fields:

! r⇥H = 0 ! (37)

! � · B = µ0� · (H + M) = 0! (38)

From Eqn. 37, it follows that the magnetic field, H = Hext + Hd, can be described as the gradient of

a magnetic scalar potential, φ:

43

! H = �r� ! (39)

By rearranging Eqn. 38, one obtains Poisson’s equation:

! �2� = � · M! (40)

Outside the particle, M is zero, and Eqn. 40 reduces to the Laplace equation:

! r2� = 0 ! (41)

The general solution to Eqn. 40 is of the form:

! ⌅(r) =14�

Z

V

⇥(r0)|r� r’|dV 0 +

Z

S

⇤(r0)|r� r’|dS0

�! (42)

where ⇢(r) = �r · M is the density of magnetic volume charges due to non-zero divergence of the

magnetization within the interior of the particle, and �(r) = M · n is the density of magnetic

surface charges due to the component of magnetization normal to the particle surface (n).

! The total energy of the system is the sum of exchange, anisotropy, magnetostatic, and

demagnetizing energies:

! Etot

= Eex

+ Ea

+ Eh

+ Ed ! (43)

Important energy terms that are missing from Eqn. 43 include the magnetoelastic energy arising

from the stress fields surrounding dislocations and other lattice defects and the magnetostrictive

44

self-energy associated with the elastic strain of an inhomogeneously magnetized particle. For a

mineral such as magnetite, magnetostriction can be neglected for particles that are smaller than ~6

μm (Huber 1967). The inclusion of magnetostriction into micromagnetic models of titanomagnetite

is discussed by Fabian and Heider (1996).

6.1.2. Discretization of the micromagnetic energy

! Although micromagnetism is a continuum approach, numerical calculation and minimization

of the total energy (Eqn. 43) requires discretization of the volume that describes the object of

interest. The most common approach is to divide the object into a three-dimensional mesh of cubic

elements. Each element is assigned a magnetization vector at its centre (Eqn. 31) and is assumed to

be magnetized homogneously. The elements must be large enough to average out the discrete

effects of the crystalline lattice (i.e., they should be significantly larger than the unit cell size), yet

small enough that the angular differences between the magnetization directions of adjacent cubes

are smaller than ~15° (Williams and Wright 1998). This upper limit is imposed by the use of a

truncated Taylor expansion for the exchange energy (Eqn. 32), which assumes that the gradient of

the magnetization is small. Assuming that the magnetization varies most rapidly at domain walls

(which have a width of ~100 nm wide in magnetite), and that approximately 4-10 elements are

required over this distance to obtain an accurate value for the exchange energy, the maximum

element size is of the order 10-25 nm. Larger element sizes can be used in cases where the

magnetization remains fairly uniform, and also if the primary interest is in examining the effects of

magnetostatic interactions between particles rather than the magnetization states of individual

particles (Muxworthy et al. 2003a). A minimum of two elements per exchange length, l =p

A/Kd ,

where Kd = µ0M2S/2, is usually recommended (Rave et al. 1998).

! The exchange, anisotropy, and magnetostatic contributions to the total energy (Eqn. 43) are

functions of the local magnetization and its derivatives. After discretization of the particle volume,

45

these terms are readily calculated using finite difference (FD) methods (Wright et al. 1997).

Calculating the demagnetizing energy, however, involves summing over contributions from all

elements in the system. The assumption that each element is magnetized uniformly (i.e., that

r · M = 0) eliminates the volume-charge contribution to the magnetic scalar potential (the first

term in Eqn. 42) and reduces the problem to summing the surface-charge contributions from the

faces of each element. The calculation can be simplified further by transforming Eqn. 42 into a

product of spatial terms (i.e., terms that depend only on the geometric relationship between pairs of

elements) and angular terms (i.e., terms that depend on the direction of magnetization within each

element). The demagnetizing energy can then be expressed in the form:

! Ed =µ0M2

S

8⇤

NX

l=1

NX

m=1

W�⇥l�m�l⇥m ! (44)

where W↵�l�m are spatial coefficients (evaluated using the method of Rhodes and Rowlands 1954), α

and β are angular terms corresponding to the charges of different faces of each element, and N is the

number of elements (Wright et al. 1997). The spatial terms can be evaluated once at the start of the

simulation and stored in a look-up table. The summation can be accelerated greatly using fast

fourier transform (FFT) methods, whereby Eqn. 44 is rewritten as a convolution and summed in

frequency space (Fabian et al. 1996; Wright et al. 1997).

! The total energy must be minimized in order to obtain the equilibrium magnetization state of

the object. Dynamic approaches make use of the Landau-Lifschitz-Gilbert equation of motion (Eqn.

5), and are particularly suitable for the study of magnetization reversal processes. Alternatively,

conjugate gradient (Fabian et al. 1996; Wright et al. 1997), Monte Carlo (Kirschner et al. 2005), or

simulated annealing (Thomson et al. 1994; Winklhofer et al. 1997) methods may be used. Whereas

simulated annealing and Monte Carlo methods are typically used to find the magnetic domain state

46

that corresponds to the absolute energy minimum (AEM) of the object, the use of LLG and

conjugate gradient techniques typically results in the determination of magnetic domain states that

represent local energy minima (LEM). The LEM state that is obtained depends on the initial state of

the particle. Simulations typically start with the smallest particle size, which is initialized with a

uniform magnetization state in a chosen direction. The final magnetic structure obtained for that

particle size serves as the initial guess for the next, slightly larger, particle size. In this way,

systematic changes in domain structure as a function of particle size and shape can be determined.

6.1.3. Finite element discretization

! Most naturally occurring magnetic particles have irregular morphologies. Discretization using

a regular array of cubes provides a poor description of non-cuboidal grain shapes (Fig. 32). Curved

boundaries are approximated simply by assigning a value of M = 0 to certain elements of the

regular cubic array (‘cell blanking’). The finite-difference discretization of a sphere shown in Fig.

32a has a highly stepped surface, which may result in magnetostatic artifacts that can drastically

alter its predicted magnetic domain structure, behavior, and stability. Improvements to this approach

can be made by assigning values of M according to the volume fraction of each element that is

enclosed by the true particle volume (Witt et al. 2005). In this way, elements that occur entirely

within the particle have M = MS, elements entirely outside the particle have M = 0, and those at the

boundary have 0 < M < MS. State-of-the-art micromagnetic simulations involve the use of finite-

element methods (FEM) to simulate magnetic domain structure in complex geometries (Fidler and

Schrefl 2000). Efficient discretization is then carried out using a combination of triangles, squares,

and rectangles (in two dimensions) or tetrahedra, cubes, and hexahedra (in three dimensions). An

FEM discretization using 60 tetrahedral elements (Fig. 32b) provides a much better representation

of a sphere than the finite-difference discretization using 343 cubic elements (Fig. 32a). FE models

reduce magnetostatic artifacts that originate on grain surfaces drastically. In order to determine the

demagnetizing energy when using FE methods, each node of the FE mesh is associated with a value

47

of the magnetic scalar potential. Values of φ are determined by solving Poisson’s equation (Eqn. 40)

inside the particle and Laplace’s equation (Eqn. 41) outside the particle, subject to the following

boundary conditions (Fidler and Schrefl 2000):

! �int

= �ext ! (45)

! (⇥�int

�⇥�ext

) · n = M · n .! (46)

!

! Because FE methods do not require the use of a regular periodic array of nodes, it is possible

to adapt the mesh to better suit a given pattern of non-uniform magnetization. For example, it is

more efficient to have a high density of nodes in regions where the magnetization varies rapidly and

a low density of nodes in regions where the magnetization remains uniform. Adaptive mesh

algorithms actively modify the FE mesh in response to the changing magnetization state of the

system and guarantee that accurate solutions are obtained near magnetic inhomogeneities or domain

walls, while keeping the number of elements to a minimum (Fidler and Schrefl 2000; Scholz et al.

1999).

6.2. Applications of micromagnetic simulations

! !

6.2.1. Equilibrium domain states in isolated magnetite particles

! High-resolution micromagnetic studies of isolated cuboidal magnetite particles in the size

range 10 nm to 4 μm have been performed by Williams and Wright (1998), Fabian et al. (1996),

Witt et al. (2005) and Fukuma and Dunlop (2006) using FFT-accelerated finite-difference methods

combined with conjugate gradient energy minimization (Wright et al. 1997). The domain states that

are found to be stable in cubic particles in the size range 10 to 400 nm are i) a uniformly-

magnetized SD state; ii) a flower (F) state (Fig. 33a); and iii) an SV state (Fig. 33b). Fabian et al.

48

(1996), Winklhofer et al. (1997) and Witt et al. (2005) demonstrated that a double-vortex (DV) state

(Fig. 33c) exists as an LEM state in cubes that are larger than 300 nm, although the appearance of

this state appears to be sensitive to the precision used in the simulations. Both F and SV states

reduce the component of magnetization normal to the particle surface, thereby reducing the

demagnetizing energy (Eqn. 42). Although F states are not magnetized uniformly, they still obey the

Néel (1949) SD theory of thermoremanent and viscous remanent magnetization. For this reason, SD

and F states are often referred to interchangeably.

! The variation in the total micromagnetic energy of a magnetite cube with particle size is

illustrated in Fig. 34 (Muxworthy et al. 2003b). The starting configuration was a 50 nm cube with

uniform magnetization. This SD state relaxes to an F state, which is then used as the starting

configuration for the next largest particle. The F state remains (meta)stable up to a particle size of

96 nm. Above this critical size, it relaxes spontaneously to an SV state that has a much lower

energy. If the SV state is studied as a function of gradually decreasing particle size, then its energy

intersects that of the F state at a particle size of 64 nm. Hence, 64 and 96 nm correspond to the

lower and upper limits for the sizes of isolated magnetite cubes that can support metastable F states.

Above a particle size of 64 nm, the SV state represents the stable AEM state of the particle. The F

state can exist, however, as a metastable LEM state up to a particle size of 96 nm. Direct

experimental observation of metallic Fe nanoparticles that display both stable SV and metastable F

states have been reported by Lappe et al. (2011) using electron holography. On decreasing the

particle size, the SV to F state transition is continuous, corresponding to a gradual ‘unwinding’ of

the vortex (Williams and Wright 1998). There is close agreement between different micromagnetic

studies regarding the lower limit of SV stability (64-70 nm) but significant variation regarding the

upper limit (96-220 nm) (Fabian et al. 1996; Williams and Wright 1998; Muxworthy et al. 2003b;

Witt et al. 2005). The lower limit is defined strictly as the particle size at which the absolute

energies of the two alternative states become equal, whereas the upper limit is determined by the

disappearance of the energy barrier separating two states that have very different energies. The

49

latter transition is sensitive to the precision of the calculation and the method used to determine the

energy minimum, and is therefore subject to more variation from study to study.

! A gradual transition to classical MD states occurs for particle sizes in the range 1-4 μm. This

transition, which is described in Fig. 35, is characterized by (Williams and Wright 1998): i) an

alignment of the near-surface magnetization parallel to the particle surface; ii) an alignment of the

magnetization with the magnetocrystalline easy axes (or the projection of the easy axes on the

particle surface); iii) an increase in the fraction of the particle volume occupied by regions of

uniform magnetization; iv) a decrease in the size, together with a more domain-wall-like

appearance, of the non-uniformly magnetized regions; v) tilting of vortex cores away from [001],

allowing larger regions of magnetization to point along the magnetocrystalline easy axes; and vi)

vortex cores in larger particles becoming nucleation centers for domain walls.

! The magnetic structures of non-cuboidal particles have been investigated using a modified

version of the cell-blanking technique by Witt et al. (2005) and by FE methods by Williams et al.

(2006). Williams et al. (2006) demonstrated that non-cuboid particles can develop significant

‘configurational anisotropy’ as the F state adjusts itself to accommodate the grain shape as the

magnetisation direction is rotated. The effect was most pronounced in SD grains with tetrahedral

morphology, where coercivities up to 120 mT were predicted. The equilibrium SD-SV threshold

size in isolated particles with octahedral morphology was found by Witt et al. (2005) to be d = 88

nm, identical to that observed for cubic particles (d is defined here as the the diameter of a sphere

with the same volume as the particle). The similarity between the equilibrium SD-SV threshold size

of cubes and octahedra is not surprising, since they have identical demagnetizing factors. There is a

large difference, however, in the upper size limit for metastable SD states (d = 320 nm for octahedra

versus d = 160 nm for cubes). This difference is illustrated further in Fig. 36, in which the lower and

upper limit of stability for F states in cuboid (Fig. 36a) and non-cuboid (Fig. 36b) particles are

compared. The particle morphologies that were used to produce Fig. 36b are similar to those

observed in many magnetotactic bacteria (Fig. 36c), and are elongated along a <111>

50

crystallographic direction. The shaded areas in Figs. 36a and b show the range in the size and aspect

ratio of magnetite crystals in natural magnetotactic bacteria that have this morphology (Petersen et

al. 1989). A significant proportion of these particles lies above the upper limit of stability for F

states predicted for cuboidal particles (Fig. 36a). In cuboidal particles, magnetostatic interactions

along the bacterial chain would be required to prevent the formation of vortex states. All of the

particles, however, lie within the stability limit for F states for the more realistic particle

morphologies (Fig. 36b), implying that magnetostatic interactions may not be required to stabilize

such states in large naturally occurring bacterial magnetosomes. The stabilization of F states in

magnetosomes results in part from the elongation of the particles along <111> (so that

magnetocrystalline and shape anisotropies act in unison) and in part from the more rounded ends of

the crystals (which inhibit flowering and reduce the tendency to de-nucleate the F state). Non-

uniform magnetization states have been observed in large magnetite magnetosomes using electron

holography (McCartney et al. 2001).

6.2.2. Temperature-dependence of domain states in isolated particles

! The temperature-dependence of magnetic domain states, and the thermal relaxation properties

of SD and PSD particles, are of central importance to the theories of thermoremanent and viscous

remanent magnetization (Néel 1949). Most micromagnetic simulations are designed to minimize

the internal energy of the system (Eqn. 43) rather than its Gibb’s free energy. Consequently, the

effective temperature of the simulations is 0 K, and the effects of entropy and thermal fluctuations

are neglected. There are several ways of incorporating temperature into micromagnetic simulations.

A basic approach, which neglects thermal fluctuations, is to use temperature-dependent values of A,

K, and MS in the calculation of the internal energy (Muxworthy and Williams 1999; Muxworthy et

al. 2003b). The temperature-dependencies of A, K, and MS in magnetite are given by Heider and

Williams (1988), Fletcher and O’Reilly (1974), Abe et al. (1976), Bickford et al. (1957), Pauthenet

and Bochirol (1951), Belov (1993), and Muxworthy and McClelland (2000). This approach allows

51

the temperature-dependence of equilibrium domain states to be determined, but may incorrectly

predict the existence of LEM states that would be unstable in the presence of thermal fluctuations.

Thermal fluctuations can be incorporated into micromagnetic simulations by adding a random

thermal field to the effective field (Eqn. 6) and then determining the dynamic response of the

system using the LLG equation (Eqn. 5) (Fidler and Schrefl 2000; Scholz et al. 2001). Alternatively,

Monte Carlo methods can be used (Kirschner et al. 2005). Atomistic Monte Carlo simulations (see

Section 4.2.1) are first used to determine the equilibrium spin configuration at a given temperature.

The value of MS to be used in the micromagnetic simulations is then obtained by averaging the spin

configuration over the volume of one micromagnetic element. Thereafter, non-atomistic Monte

Carlo techniques, analogous to those described in Section 4.2.1, are used to determine the

equilibrium domain state of the micromagnetic model.

! The lower and upper limits for the sizes of metastable F states in cubic magnetite particles at

high temperatures are illustrated in Fig. 37 (Muxworthy et al. 2003b). The equilibrium SD-SV

threshold size increases from 70 nm at room temperature to approximately 90 nm close to the Néel

temperature. The upper SD-SV threshold size increases from 96 to ~200 nm close to the Néel

temperature, considerably extending the size range over which metastable F states can exist. The

SD-SV threshold size at low temperature (below the Verwey transition) has been explored by

Muxworthy and Williams (1999). The large increase in the magnitude of the magnetocrystalline

anisotropy at the Verwey transition (Abe et al. 1976; Muxworthy and McClelland 2000) stabilizes

the SD state with respect to the SV state, increasing the lower SD-SV threshold size to 140 nm at

110 K. However, the increase in magnetocrystalline anisotropy also increases considerably the

height of the energy barrier that separates the SD and SV states, increasing the probability of

particles becoming trapped in a metastable SV state on cooling below the Verwey transition. On the

basis of experimental observations and micromagnetic modelling, Kasama et al. (2012) and Bryson

et al. (2012) demonstrate that ferroelastic twinning plays an important role in the stabilisation of

non-uniform magnetisation states below the Verwey transition, and suggest that such twinning is

52

responsible for the field-memory effect, whereby the low-temperature hysteresis loop displays an

inflection at a field equal to that which was applied during cooling (Smirnov and Tarduno 2002;

Smirnov 2006a, b; 2007).

! In order to calculate the thermal relaxation properties of SD and PSD particles, it is necessary

to determine the magnitudes of the energy barriers that separate different LEM states. For SD

particles this can be done analytically (Newell 2006a and b). For PSD states energy barriers can be

calculated using constrained micromagnetic simulations (Enkin and Williams 1994; Winklhofer et

al. 1997; Muxworthy et al. 2003b). In an unconstrained simulation, the magnetic moments of all of

the elements are allowed to vary, so that the system evolves towards the nearest LEM state (Fig.

34). In a constrained simulation, the system is forced to adopt a non-LEM state by fixing the

orientations of the magnetic moments in some of the elements during the simulation (Fig. 38). For

example, by constraining the moments on opposite faces of a cuboidal particle to be either parallel

or antiparallel, it can be forced to adopt an F or an SV state, respectively (Fig. 38a). Figs. 38b and c

show the calculated energy of a particle with an aspect ratio 1.2 as the moments on opposite faces

are rotated independently of each other through 360° (Muxworthy et al. 2003b). At 27 °C (Fig. 38b)

the SV state is the AEM state, and there are two non-degenerate SD LEM states at 90° to each other.

The SD state with the lower energy is magnetized parallel to the elongation direction of the particle.

This state becomes the AEM state at 567 °C (Fig. 38c). The energy barriers that separate degenerate

AEM states are illustrated in Fig. 39 for two different particle sizes and aspect ratios (Muxworthy et

al. 2003b). The relaxation time of such a particle is related to the height of the energy barrier, EB

(Winklhofer et al. 1997):

! � = �0exp

✓EB(T )

kBT

◆! (47)

53

where ⌧�10 (~ 109-1010 Hz; McNab et al. 1968) is the frequency at which the particle attempts to

switch its magnetization direction. The dashed lines in Fig. 39 represent the blocking of remanent

magnetization on laboratory (EB ~ 25 kBT) and a geological (EB ~ 60 kBT) timescales. The figure

illustrates that blocking is more a function of the rapidly increasing energy barrier height on

cooling, rather than of the decrease in thermal energy.

6.2.3. Field-dependence of domain states

! The effect of an external field can be included in micromagnetic simulations via Eqn. 35, and

used to study hysteresis properties (Williams and Dunlop 1995) and reversal mechanisms (Enkin

and Williams 1994) of individual PSD particles, the acquisition of saturation isothermal remanent

magnetization (SIRM) and thermoremanent magnetization (TRM) (Winklhofer et al. 1997;

Muxworthy and Williams 1999; Muxworthy et al. 2003b), and to calculate the first-order reversal

curves (FORCs) for both isolated grains and arrays of particles (Carvallo et al. 2003; Muxworthy et

al. 2004; Muxworthy and Williams 2005; see Section 7). Hysteresis loops are typically obtained by

calculating a succession of quasi-static magnetic states, as the field is increased and decreased in a

stepwise manner (Williams and Dunlop 1995). This approach is valid so long as the damping of

gyromagnetic precession (Eqn. 5) is much more rapid than the rate of increasing/decreasing field.

PSD particles containing vortex states are observed to reverse their magnetization directions by a

combination of gradual rotations of the outer moments and discontinuous reversals of the core

moments (Williams and Dunlop 1995).

6.2.4. Magnetostatic interactions between particles

! Electron holographic observations of closely-spaced particles (see Section 5.3.3) highlight the

fundamental importance of magnetostatic interactions in determining the macroscopic properties of

rocks and minerals. The complex problem of determining the collective behavior of interacting

54

arrays of magnetic particles has recently been tackled using micromagnetic simulations

(Muxworthy et al. 2003a). Muxworthy et al. (2003a) performed a systematic study of saturation

magnetization (Ms), saturation remanence (Mrs), coercivity (Hc), and coercivity of remanence (Hcr)

as a function of particle size and spacing for regular three-dimensional arrays of cubic particles. The

simulations were performed for different anisotropy schemes (uniaxial versus cubic, aligned versus

random) to model a range of scenarios that are likely to be observed in natural systems. The results

can be summarized on a ‘Day plot’ (Day et al. 1977; Dunlop 2002a, b) of Mrs/Ms versus Hcr/Hc (Fig.

40). For widely spaced particles (i.e., when the distance between particles is greater than 5 times

their diameter), the effect of magnetostatic interactions is negligible, and the ratios of Mrs/Ms and

Hcr/Hc converge to the ideal values for non-interacting particles (Mrs/Ms = 0.5 and 0.87 for

randomly-oriented uniaxial and cubic anisotropies, respectively; Hcr/Hc = 1-1.5). As the spacing

between the particles is decreased, there is a consistent decrease in Mrs/Ms and increase in Hcr/Hc,

which moves the system gradually from the SD to the PSD, and ultimately to the MD, regions of

the Day plot (Fig. 40). This prediction is consistent with the observation of interaction ‘superstates’

using electron holography (Fig. 28), which are responsible for the PSD- and MD-like behavior of

closely-spaced SD particles in nanoscale intergrowths (Harrison et al. 2002; Evans et al. 2006). The

effect of interactions on the properties of larger PSD particles is much more complex, and can cause

the system to adopt either more SD-like or more MD-like behavior, depending on the particle size,

shape and spacing, and on the style of anisotropy. This behavior results in part from a shift of the

SD/SV threshold with increasing interactions: particles that would adopt SV states in the absence of

interactions are able to adopt SD states when they are interacting strongly with neighboring

particles. Micromagnetic simulations suggest that this effect occurs when the easy axes of

neighboring particles are well aligned, as is the case for chains of magnetite particles in

magnetotactic bacteria (Fig. 22) and for arrays of magnetic blocks formed by exsolution from an

ulvöspinel host (Fig. 26).

55

7. MAGNETISM AT THE MACROSCOPIC LENGTH SCALE

! In this final section, we review recent developments in the use of FORC diagrams to

characterize the magnetic properties of rocks and minerals (Pike et al. 1999). Until recently,

hysteresis loops were the most widely used method of characterizing bulk magnetic properties

(Roberts et al. 2000). However, parameters determined from hysteresis loops represent bulk

averages, and provide little information about the spectrum of coercivities and interaction fields that

exist at the microscopic scale. The FORC diagram is a generalization of the well-known Preisach

diagram (Preisach 1935). The method requires the acquisition of many thousands of individual

magnetization measurements, and has only been made possible by the advent of fully automated

vibrating-sample and alternating-gradient magnetometers (Flanders 1988), which allow the rapid

acquisition of magnetization data over a large range of temperatures and applied fields.

7.1. Theory

7.1.1. First-order reversal curves and the FORC distribution

! The definition of a first-order reversal curve is illustrated in Fig. 41a (Pike et al. 1999;

Roberts et al. 2000). Each FORC measurement begins by saturating the sample in a positive field.

The external field is then decreased to some value, Ha (the reversal field), and the magnetization of

the sample is measured as a function of increasing field, Hb, until positive saturation is reached

again. A large number of FORCs are acquired for different reversal fields, in order to sample the

entire area enclosed by a standard hysteresis loop (Fig. 41b). Values of Ha and Hb are chosen to

cover a regular grid in Ha-Hb space (Fig. 41c), resulting in a magnetization matrix, M(Ha, Hb). The

FORC distribution is defined as the mixed second derivative of M(Ha, Hb) with respect to Ha and

Hb:

56

! �(Ha,Hb) = �⇥2M(Ha,Hb)⇥Ha⇥Hb

! (48)

Note that in some studies, Eqn. 48 is multiplied by a factor of 1/2 (e.g., Pike 2003; Newell 2005). It

is customary (see Section 7.1.2) to define a new set of axes, Hc = (Hb-Ha)/2 and Hu = (Ha+Hb)/2, as

illustrated in Fig. 41c. The FORC diagram itself (Fig. 41d) is a contour plot of ⇢(Ha,Hb) , with Hc

and Hu on the horizontal and vertical axes, respectively (covering the region of the Ha-Hb plane

enclosed by the pink rectangle in Fig. 41c).

! In order to calculate the FORC distribution at any point P, a least-squares fit to M(Ha, Hb) is

performed over a grid of points surrounding P (illustrated by the blue square in Fig. 41c). The most

common method used is that of Pike et al. (1999), in which the magnetization is fitted using a

second-order polynomial function:

! M(Ha,Hb) = a1 + a2Ha + a3H2a + a4Hb + a5H

2b + a6HaHb ! (49)

The value of the FORC distribution at P is then equal to -a6. The size of the grid is defined by a

smoothing factor, SF, such that the grid extends over (2SF + 1)2 points in the Ha-Hb plane. This

method becomes inefficient as the total number of points in the M(Ha, Hb) matrix increases. Heslop

and Muxworthy (2005) describe an alternative algorithm, based on the convolution method of

Savitzky and Golay (1964), which yields identical results but is a factor of 500 times faster.

Harrison and Feinberg (2008) adopt a method based on locally-weighted regression smoothing,

which allows automatic extrapolation across missing data. An increase in the SF leads to a

smoothing of the FORC diagram. While some smoothing is necessary to reduce experimental noise,

too much smoothing may unduly affect the form of the distribution. Heslop and Muxworthy (2005)

describe a numerical test, based on examination of the autocorrelation function of the residual of

57

observed and fitted values of M(Ha, Hb), to determine the optimum value of SF. The optimum value

depends on the resolution of the M(Ha, Hb) matrix, but values in the range 2-5 are typically

employed. The statistical significance of the FORC signal for a given smoothing factor can be

calculated from an analysis of the fitting residuals (Heslop and Roberts 2012). This information can

guide measurement protocols and provides a more quantitative framework for interpretation of

FORC distributions. Because the M(Ha, Hb) matrix does not extend to the Hc < 0 region (Fig. 41c),

increasing SF leads to an increase in the number of points close to the Hu axis that must be

extrapolated (Carvallo et al. 2005). The need to extrapolate data can be overcome by the use of

‘extended’ FORCs (Pike 2003), as described in Section 7.1.3.

!

7.1.2 Interpretation of the FORC diagram

! FORC diagrams provide an alternative method of measuring the Preisach distribution, which

yields information about the spectrum of coercivity and interaction fields within a sample (Preisach

1935; Mayergoyz 1991; Carvallo et al. 2005). The mathematical justification for using the Preisach

model for interpreting FORC diagrams is described by Pike et al. (1999) and illustrated

schematically in Figs. 42 and 43. The system is assumed to consist of a collection of particles with

either an irreversible (Fig. 42a) or a reversible (Fig. 42b) hysteresis loop (referred to as a

‘hysteron’). In the absence of an interaction field, irreversible particles switch their magnetization

direction at the coercive field ± Hc. In the presence of an interaction field, the hysteron is shifted to

either the left or right by an amount Hu, and switching now occurs at fields Ha and Hb (Fig. 42c). Ha

and Hb are related to the coercivity of the particle and the interaction field acting on it via Hc = (Hb-

Ha)/2 and Hu = (Ha+Hb)/2. Each irreversible particle contributes to the FORC distribution at the

corresponding point in Hc-Hu space (Fig. 43). It is often assumed that the FORC distribution can be

factorized into the product of two independent distributions:

58

! �(Hc,Hu) = g(Hc)f(Hu) ! (50)

where g(Hc) describes the distribution of coercivities and f(Hu) describes the distribution of

interaction fields. Carvallo et al. (2004) and (2005) measured FORC diagrams for a series of well-

characterized SD and PSD particles and found Eqn. 50 to be valid. Muxworthy et al. (2004) and

Muxworthy and Williams (2005) performed a similar test using FORC diagrams derived from

micromagnetic simulations. Although they observed a slight variation in Hc as a function of

interaction strength, they concluded that Eqn. 50 provides a reasonable approximation for

collections of SD particles with weak to moderate interactions. Analytical solutions for the effect of

weak interactions on FORC diagrams are given by Egli (2006).

7.1.3. Extended FORCs and the reversible ridge

! Reversible magnetization of the form shown in Fig. 42b is normally absent from the FORC

distribution, as its contribution disappears on taking the second derivative of M(Ha, Hb). This

problem can be overcome by the use of ‘extended FORCs’ (Pike 2003; Pike et al. 2005). The

magnetization matrix M(Ha, Hb) is normally defined only in the region Hb ≥ Ha (as shown by the

grid points in Fig. 41c). However, M(Ha, Hb) can be mathematically extended to cover the whole

Ha-Hb plane:

! M⇤(Ha,Hb) =⇢

M(Ha,Hb), if Hb > Ha

M(Ha,Ha), if Hb Ha! (51)

By using M* rather than M in Eqn. 48 the standard FORC diagram is obtained for Hc > 0, and a

‘reversible ridge’ is added to the Hu axis, describing the distribution of reversible magnetization in

the form:

59

! ⇥(Ha,Ha) =12�(Hb �Ha)

✓limHb!Ha

⇤M(Ha,Hb)⇤Hb

◆! (52)

Eqn. 52 describes the slope of the FORC with reversal field Ha, calculated at the point at which the

FORC joins the major hysteresis loop (Pike et al. 2005). An example of an extended FORC diagram

for a floppy disk recording material, including a profile of the reversible ridge, is shown in Fig. 44.

Since both the reversible and irreversible components of magnetization contribute, the extended

FORC distribution is properly normalized, such that the integral with respect to Ha and Hb equals

the saturation magnetization of the sample (Pike 2003). The finite resolution of real measurements,

however, often shifts the reversible ridge into FORC space, where it over-shadows low-coercivity

FORC contributions (e.g., Fig. 44a). This effect is particularly evident in samples where reversible

magnetization processes are dominant. Alternative processing methods (e.g. Harrison and Feinberg

2008) and extrapolation methods, such as the ‘slope extended’ FORC method (Egli et al. 2010),

have been proposed to avoid the generation of a reversible ridge.

7.2. FORC diagrams as a function of grain size

!

7.2.1. SP particles

! The expected form of the FORC diagram for SP particles is discussed by Pike et al. (2001a).

Particles that are far above their blocking temperature have a reversible magnetization of the form

shown in Fig. 42b, and do not contribute to a normal FORC diagram (although they would

contribute to the reversible ridge of an extended FORC diagram). Particles that are closer to their

blocking temperatures show thermal relaxation of their magnetization state on a timescale similar to

that of each FORC measurement step. This leads to contributions to the FORC distribution that

peak around the origin and extend along the negative Hu axis (Fig. 45). The form of the FORC

60

diagram can be predicted using Néel’s theory of thermal relaxation in SD particles (Pike et al.

2001a).

7.2.2. SD particles

! An analytical solution for the FORC diagram of ideal non-interacting particles with uniaxial

anisotropy has been derived by Newell (2005) (Fig. 46). For a collection of randomly oriented

identical particles, the FORC function consists of two main contrbutions: a ‘central ridge’ signal

that appears as a delta function along the Hc axis (Fig. 46a), and a continuous signal that appears

below the Hc axis only (Fig. 46b). The central ridge is associated with the symmetrical irreversible

switching of the particles, whereas the continuous part is associated with the reversible component

of magnetisation. The equivalent functions for the case of randomly oriented particles with a log-

normal distribution of aspect ratios (i.e. a distribution of switching fields) are shown in Fig. 46c and

d. The negative part of the continuous FORC function can be understood in terms of the

‘curvilinear’ form of the hysteron (Fig. 42d; Pike 2003; Newell 2005). In the Preisach model, the

irreversible and reversible components of magnetization (Fig. 42a and b) are completely

independent. For a curvilinear hysteron, however, the reversible component of magnetic

susceptibility decreases significantly as the particle switches from the upper to the lower branch of

the loop. This coupling between the irreversible and reversible components gives rise to a

systematic decrease, for a given Hb < 0, in the slopes of the FORCs as Ha decreases (Muxworthy et

al. 2004). This, in turn, translates to a negative contribution to the FORC distribution.

! A good source of uniaxial non-interacting single domain particles is sediments containing

fossil magnetotactic bacteria (Egli et al. 2010; Roberts et al. 2012). Fig. 47 shows a high-resolution

FORC diagram from a Lake Ely sediment sample (Egli et al. 2010) that displays all the predicted

features of the Newell (2005) model, including a narrow central ridge with around one order of

magnitude higher amplitude than the continuous signal. An analytical method for calculating the

61

broadening of the central ridge caused by inter-particle magnetostatic interactions is described by

Egli (2006).

!

7.2.3. PSD particles

! FORC diagrams for PSD size magnetite particles are described by Muxworthy and Dunlop

(2002). FORC diagrams were measured for a series of synthetic magnetites with grain sizes varying

from 0.3-11 μm (Fig. 48). Small PSD particles have SD-like FORC diagrams, characterized by a

closed positive peak at Hc > 0 and Hu = 0 (Fig. 48a). With increasing grain size, the position of the

peak shifts to lower Hc values (Fig. 48b), and eventually moves to the origin (Fig. 48c). This shift in

peak position is accompanied by a spreading of the distribution in the Hu direction for small Hc

(Fig. 48d). Similar changes are seen as a function of temperature for particles of a fixed size. A

diagnostic feature of the FORC diagram for samples dominated by single vortex (SV) states is the

so-called ‘butterfly’ structure identified by Pike and Fernandez (1999) and Dumas et al. (2007) (Fig.

49). This feature is associated with the nucleation and annihilation fields for vortices nucleating on

opposite sides of a particle. Lappe et al. (2011) identified the butterfly structure in the FORC

diagrams of metallic Fe nanoparticles that had been shown to contain SV states via electron

holography. Based on the theory of Pike and Fernandez (1999) they were then able to determine the

range of nucleation and annihilation fields for the samples (HN = 58 ± 55 mT and HA =170 ±

55mT). Such information is important in determining the paleomagnetic properties of Fe

nanoparticles in meteorites.

7.2.4. MD particles

! Theoretical predictions and experimental measurements of the FORC diagrams for non-

interacting MD particles are described by Pike et al. (2001b) and Church et al. (2011). One-

dimensional models of domain-wall pinning predict FORC diagrams consisting of perfectly vertical

contours, with the value of the FORC distribution decreasing smoothly with increasing Hc. This

62

model agrees well with experimental measurements on annealed (i.e. stress-free) magnetite samples

(Fig. 50). The vertical spread of the FORC function results from the fact that each particle contains

a large number of pinning sites at which a domain can be trapped during the FORC measurement.

These different pinning sites can be represented by an equivalent number of hysterons, which are

spread out along the Hu axis by the self-demagnetizing field (Pike et al. 2001b). FORC diagrams for

unannealed MD samples are similar to those observed at the upper end of the PSD range (compare,

for example, Fig. 50c with Fig. 48d).

7.3. Mean-field interactions and FORC diagrams

! In the most basic form of the Preisach model, the distribution of interaction fields is assumed

to be static. In reality, however, the field acting on a each particle is the sum of the stray fields

created by all the other particles in the system, and will vary as the overall magnetization of the

system changes (Egli 2006). In general, both the mean value and the standard deviation of the

interaction field distribution (IFD) are functions of the net magnetization of the system (the

‘variable-variance moving Preisach’ model). For example, if a collection of particles is fully

saturated in a large magnetic field, each particle experiences the roughly the same mean interaction

field and the standard deviation of the IFD tends to zero. In the demagnetized state, each particle

will experience a different interaction field; the mean value of the IFD is now zero and the standard

deviation is maximum. The constant of proportionality relating the mean interaction field to the net

magnetization of the system is referred to as the ‘moving parameter’, α, which can be either

positive or negative, depending on the geometry of the system (Stancu et al. 2001 and 2003).

Positive α implies that the mean field has a magnetizing effect, and leads to a spontaneous mutual

alignment of the particles. This case applies, for example, to the chains of magnetite particles in

magnetotactic bacteria (Fig. 22). Negative α implies that the mean field has a demagnetizing effect.

This case applies, for example, to perpendicular recording media (i.e., planar arrays of SD particles

which have their easy axes perpendicular to the plane). The FORC diagram for such a system,

63

composed of a perpendicular array of Ni pillars, is shown in Fig. 51 (Pike et al. 2005). The

‘wishbone’ form of the FORC diagram has two main peaks: one occurring at low Hc and Hu > 0,

and one occurring at high Hc and Hu = 0. The distance between these two peaks in the Hc direction

yields information about the range of coercivities in the system. The displacement of the first peak

in the positive Hu direction yields information about the strength of the mean-field demagnetizing

interaction.

7.4. Practical applications of FORC diagrams

! The FORC method has been applied in rock magnetism as a method of characterizing the

magnetic mineralogy of natural samples (Roberts et al. 2000), identifying mixtures of soft and hard

magnetic minerals (Muxworthy et al. 2005), and identifying magnetostatic interactions as a

preselection tool for paleointensity studies (Wehland et al. 2005). Pan et al. (2005) have used FORC

diagrams to determine the strength of magnetostatic interactions in concentrated samples of

magnetotactic bacteria (Fig. 52). The FORC distribution has large SD-like peak centered on Hc ~ 40

mT and displaced slightly in negative Hu direction. The vertical spread of the IFD has a FWHM of

just 6.3 mT, much lower than the ideal intra-chain interaction field of 60 mT. This observation

demonstrates that the magnetosome chains are effectively behaving as elongated SD particles, and

switch as a single unit. In such cases, the interaction fields measured by the FORC method provide

an indication of inter-chain and inter-cellular interactions. The small peak in the FORC distribution

about the origin can be attributed to the smaller magnetosomes that commonly occur at the ends of

the chain. Observation of a narrow central ridge (i.e. one with a width this is limited by the

resolution of the measurement itself) is diagnostic of non-interacting particles, which in turn is often

associated with intact magnetosome chains. FORC diagrams are becoming an increasingly powerful

method of identifying fossil magnetotactic bacteria in sediments and demonstrate that

magnetofossils are much more prevalent in nature than previously thought (Roberts et al. 2012).

FORC diagrams are increasingly being used to derive quantitative information about the

64

fundamental magnetic mineralogy of a sample – information that can be used as empirical input

into physical simulations of the paleomagnetic remanence acquisition process (Muxworthy and

Heslop 2011; Muxworthy et al. 2011). If the sample carries a TRM, a comparison of observed

versus simulated remanence can be used to estimate the intensity of the magnetising field, without

the need to heat the sample (as is the case for traditional paleointensity measurements). Such non-

heating paleointensity methods are principally of interest in the analysis of extraterrestrial materials,

where heating of the sample is either not permitted on curational grounds, or would lead to

excessive thermal alteration of the sample. Although in their infancy, such methods have the

potential to improve our understanding of the principles and limitations of current paleomagnetic

theories, and, in combination with the other techniques outlined in this review, to extend these

beyond the limits of single-domain theory.

8. SUMMARY

! Now is a very exciting time for the field of rock and mineral magnetism. The discovery of

large-amplitude magnetic anomalies on Mars (Connerney et al. 1999 and 2004; Acuna et al. 1999)

has ignited a general interest in the effect of nanoscale microstructures on the origin and stability of

planetary scale magnetic anomalies. Conventional wisdom – that these anomalies are due to the

induced magnetization of multi-domain magnetite – is now being challenged in light of the Mars

magnetic survey. Mars no longer generates its own magnetic field; the anomalies are purely

remanent in origin – faithfully recorded by magnetic minerals over 4 billion years ago (at a time

when Mars did generate a field) and maintained without significant decay until the present day. The

minerals responsible for the anomalies on Mars – and how they maintain such strong remanence

over time – is currently the subject of intense speculation.

! The techniques described in this review allow such problems to be tackled from both

experimental and theoretical viewpoints, encompassing the entire range of length scales of interest,

from atomistic interactions to planetary-scale magnetic anomalies. Since the dominant carriers of

65

stable natural remanent magnetization are SD particles with sizes in the range 30-200 nm,

techniques such as electron holography (Section 5) have the potential to revolutionize the way rock

magnetic measurements are made in the future. By using the three-dimensional morphologies of

magnetic nanoparticles, provided by electron tomography, as the input for finite element

micromagnetic simulations (Section 6), it is now possible to compare experimental observations

and theoretical predictions directly. Differences between observed and calculated behaviors are

likely to be the result of atomistic effects at surfaces, interfaces, and defects. Ultimately, the

application of atomistic simulations (Section 4) will permit the influence of such atomic-scale

features on standard rock magnetic analysis (Section 7) to be determined.

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FIGURE CAPTIONS

Fig. 1. Magnetic superexchange integrals, Jij (K), as a function of cation-cation distance (Å) for

hematite (blue), ilmenite (black), and magnetite (red). Data for hematite and ilmenite were

measured using inelastic neutron scattering (Samuelsen and Shirane 1970; Ishikawa et al. 1985).

Data for magnetite were calculated using first-principles methods (Uhl and Siberchico 1995).

98

Fig. 2. Comparison of the local structural topology of (a) hematite/ilmenite and (b) magnetite. In

ilmenite, A layers (red) are occupied by Fe2+ and B layers (blue) are occupied by Ti4+ (or vice

versa). Hematite has the same structural topology, but all layers are occupied by Fe3+. In magnetite,

tetrahedral sites (blue) are occupied by Fe3+ cations and octahedral sites (red) are occupied by both

Fe2+ and Fe3+ cations.

Fig. 3. Schematic illustration of the variation in total energy of canted antiferromagnetic hematite

with the canting angle, φ, as determined by first-principles calculations. Without the inclusion of

spin-orbit coupling (dashed line), the energy minimum occurs when the spins are exactly

antiparallel to each other (φ = 0). With spin-orbit coupling included (solid line), the energy

minimum occurs when the spins are slightly canted with respect to each other (φ > 0). Reproduced

with permission from Sandratskii and Kübler (1996).

Fig. 4. Calculated hysteresis loops and spin configurations of a 2.5 nm-diameter spherical particle

of NiFe2O4. Simulations were performed using an atomistic model of magnetic ordering (Kodama

1999; Kodama and Berkowitz 1999). Arrows in (b) and (d) show the spins on individual Fe atoms.

Light and dark circles correspond to tetrahedral and octahedral sites, respectively. A particle with no

broken bonds and low surface roughness has a low coercivity (a) and no surface spin disorder (b).

A particle with a higher broken-bond density and surface roughness displays high coercivity and

high-field irreversibility (c), resulting from the presence of significant surface spin disorder (d).

Reproduced with permission from Kodama (1999).

Fig. 5. Atomistic simulation of non-uniform magnetic switching in a ferromagnetic nanoparticle

with strong surface anisotropy. The spherical particle contains 176 surface spins (dark arrows) and

184 core spins (grey arrows). The easy axis for core spins is parallel to the z direction (indicated).

The easy axis for surface spins is approximately normal to the particle surface. In this example, the

99

core and surface anisotropy constants were chosen to be equal in magnitude and the exchange

integral between neighboring spins was one tenth the magnitude of the core/surface anisotropy

constant. Each spin configuration was obtained at a different value of the applied magnetic field,

starting with a saturating field in the negative z direction (a) and ending with a saturating field in the

positive z direction (f). The surface spins are observed to switch magnetization direction before the

core spins (c). Core spins are observed to switch magnetization direction in a cluster-like fashion (d

and e). Adapted with permission from Kachkachi and Dimian (2002).

Fig. 6. (a) Magnetization of a diluted assembly of γ-Fe2O3 nanoparticles, with a mean diameter of

2.7 nm, as a function of magnetic field at different temperatures. (b) Temperature-dependence of

magnetization at 55 kOe, extracted from (a), showing an anomalous increase in magnetization

below 70 K. (c) Temperature-dependence of magnetization in a field of 55kOe for three samples

with different mean diameters (2.7, 4.8, 7.1 nm). The anomalous increase in magnetization at low

temperatures is more pronounced in the smaller particles, consistent with a surface effect.

Reproduced with permission from Kachkachi et al. (2000a).

Fig. 7. (a-c) Temperature-dependence of the surface and core magnetization (per site) and (d) mean

magnetization, as obtained from atomistic Monte Carlo simulations of an ellipsoidal maghemite

nanoparticle. The exchange interactions on the surface are taken to be 1/10 times those in the core,

leading to an increase in the surface magnetization contribution at low temperatures. Temperature is

given in reduced units (⌧c

= T/T core

c

, where T core

c

is the critical temperature of the core spins).

Reproduced with permission from Kachkachi et al. (2000a).

Fig. 8. Examples of nanoscale microstructures in the ilmenite-hematite solid solution. (a) A natural

haemo-ilmenite containing abundant nanoscale exsolution lamellae of hematite in an ilmenite host

100

(McEnroe et al. 2002). Scale bar = 100 nm. (b) A synthetic sample of the ilmenite-hematite solid

solution (ilm70) containing curved antiphase domains (APDs) and antiphase boundaries (APBs),

formed after cooling through the R3c to R3 cation ordering phase transition. The APDs are crosscut

by two titanomagnetite lamellae, the result of annealing the sample under slightly reducing

conditions. Scale bar = 100 nm.

Fig. 9. Summary of the equilibrium phase relations in the ilmenite-hematite solid solution

determined by Monte Carlo simulation (Harrison 2006). Dashed and dotted lines show the

metastable magnetic ordering temperatures for the cation-disordered and cation-ordered solid

solution, respectively.

Fig. 10. Snapshots of a combined simulation of cation and magnetic ordering in the ilmenite-

hematite solid solution (ilm70) at 100 K (Harrison 2006). (a) Distribution of Fe3+, Fe2+, and Ti (red,

green, and blue, respectively). (b) Local chemical composition (red = hematite, black = ilmenite)

calculated by averaging the number of Ti cations within the first four coordination shells around

each site. (c) Magnitude and direction of spin on each site (red = negative, blue = positive, symbol

size proportional to magnitude of spin). (d) Local ferrimagnetic moment (blue = positive, red =

negative), calculated by averaging the spin values within the first 4 coordination shells around each

site. The blue regions highlight the ferrimagnetism associated with local spin imbalance at the

interface between ilmenite and hematite precipitates.

Fig. 11. Average values of order parameter, composition, and spin on each of the 48 layers of an

8x8x8 supercell of ilm70, pre-annealed at (a-d) 850 K and (e-h) 1100 K (Harrison 2006; Harrison

et al. 2005). A starting configuration with APBs at the bottom and the centre of the supercell was

chosen in each case. (a) The order parameter profile at 850 K shows two fully ordered/antiordered

APDs (Q = 1 and Q = -1, respectively) separated by APBs (Q = 0). (b) The composition profile at

101

850 K shows that unmixing has taken place within the PM R3c + PM R3 miscibility gap, with the

PM R3 phase corresponding to the APDs and the PM R3c phase corresponding to the APBs.

Dashed line indicates the bulk composition, x = 0.7. (c) The spin profile at 25 K shows that the

APDs are strongly ferrimagnetic. The APD centered on layer 14 has a net negative spin, whereas

the APD centered on layer 40 has a net positive spin (indicated by the arrows). (d) The spin profile

at 400 K shows that the Fe-rich APBs remain magnetically ordered, whereas the Fe-poor APDs are

magnetically disordered. The APBs are associated with a small net spin (see Fig. 12). (e) The order

parameter profile at 1100 K shows a fully ordered APD (Q ~ 1) and a less well (anti)ordered APD

(Q ~ -0.75). (f) The composition profile at 1100 K shows that the well ordered APD has x > 0.7,

whereas the less well ordered APD has x < 0.7. Evidence for Fe enrichment at the APBs is also

seen. (g) The spin profile at 25 K shows that the well ordered APD is strongly ferrimagnetic,

whereas the ferrimagnetic spin of the less well ordered APD is decreased by the influence of the

boundary regions. (h) The spin profile at 375 K shows that the less well ordered APD and boundary

regions are magnetically ordered, whereas the well ordered APD is magnetically disordered. The

magnetically ordered regions carry a small net spin that is opposite to the net spin of the well

ordered APD (see Fig. 12).

Fig. 12. Temperature-dependence of sublattice spins SA and SB (circles) and net spin |SA|-|SB|

(squares) for the simulations pre-annealed at (a) 850 K and (b) 1100 K. Insets show expanded view

of the variation in net spin. A self-reversal of the net magnetization occurs in (b) due to the negative

exchange coupling between poorly ordered and well ordered APDs (see Fig. 11e-h).

Fig. 13. Schematic illustration of the phase shift experienced by electrons passing through a

specimen in the TEM. (a) The mean inner potential contribution to the phase shift of electrons

passing through a sample of uniform structure and chemical composition reflects changes in the

specimen thickness (first term in Eqn. 13). (b) The magnetic contribution to the phase shift, given

102

by the second term in Eqn. 13, reflects the in-plane component of the magnetic induction, integrated

along the electron beam direction. For a sample containing two uniformly magnetized domains, one

magnetized out of the plane of the diagram (crosses) and one magnetized into the plane of the

diagram (dots), the gradient of the phase shift is constant within the domains and changes sign at

the domain wall.

Fig. 14. Schematic illustration of the setup used for generating off-axis electron holograms. The

sample occupies approximately half the field of view. Essential components are the field-emission

electron gun source, which provides coherent illumination, and the positively charged electrostatic

biprism, which overlap of the sample and (vacuum) reference waves. The Lorentz lens allows

imaging of magnetic materials in close-to-field-free conditions,

Fig. 15. Sequence of image processing steps required to convert an electron hologram into a phase-

shift image. (a) Original electron hologram of the region of interest (a natural sample of hematite

containing nanoscale inclusions of maghemite). Broad Fresnel fringes, caused by the edges of the

biprism wire, are visible in the upper right and lower left. The inset is a magnified image of the

outlined region, showing the change in position of the fine-scale holographic fringes as they pass

through an inclusion. (b) A reference hologram recorded over a region of vacuum. (c) Fourier

transform of the electron hologram shown in (a), comprising a central peak, two side bands, and a

diagonal streak due to the Fresnel fringes. (d) A mask is applied to the Fourier transform in (c) in

order to isolate one side band. The Fresnel streak is removed by assigning a value of zero to pixels

falling inside the region shown by the dashed line. (e) Inverse Fourier transform of (d) yields the

complex image wave, which in turn yields a modulo 2π image of the holographic phase shift. (f)

Automated phase unwrapping algorithms are used to remove the 2π phase discontinuities from (e)

to yield the final phase shift image.

103

Fig. 16. Schematic illustration of magnetic switching in the TEM. A uniaxial particle with

anisotropy constant K and saturation magnetision Ms, initially magnetized to the right, is tilted to an

angle of 30° to the horizontal. A chosen current is passed through the objective lens of the TEM,

exposing the sample to a downward pointing magnetic field of up to 2 T. The direction of

magnetization switches when the vertical field reaches 0.52 BK, where BK = 2K/Ms. The objective

lens is then switched off and the sample is tilted back to the horizontal.

Fig. 17. Simulation of the holographic phase shift associated with a 200 nm-diameter spherical

particle of magnetite. The particle is uniformly magnetized in the vertical direction. The mean inner

potential contribution to the phase shift is shown in (a), the magnetic contribution is shown in (b),

and the sum of the two is shown in (c). (d-f) Profiles of (a-c), taken horizontally through the centre

of the particle (i.e., in a direction normal to the magnetization direction). The analytical form of

these curves is given by Eqns. 19 and 20. (g-i) Cosine of 4 times the phase shift shown in (a-c). (j)

Color map derived from the gradient of the magnetic contribution to the phase shift (b). The hue

and intensity of the color indicates the direction and magnitude of the integrated in-plane

component of magnetic induction, according to the color wheel shown in (l). The color can be

combined with the contour map, as shown in (k).

Fig. 18. (a) Mean inner potential of an olivine sample containing a metallic Fe inclusion (outlined in

black). (b) Magnetic contribution to the phase shift of the metallic Fe inclusion, determined from

the difference in the holographic phase shift for two reversed magnetic states (see Section 5.1.5.).

Note the closely spaced phase contours inside the particle, indicating that it is uniformly magnetized

along its length, and the dipolar stray fields outside the particle that can be used to determine its

magnetic moment. The dashed circle indicates the integration loop used to calculate the total

magnetic moment of the particle using the methods outlined in Section 5.2.3 (Eqns. 23 and 24).

Inset shows calculations of the x and y component of magnetic moment (in µB) for a range of

104

difference integration radii (Rc). Solid lines are quadratic fits to the data, allowing the measurement

to be extrapolated to Rc = 0. (c) Thickness of the metallic particle (nm) projected along the electron

beam direction. This image was obtained by first subtracting the backround MIP signal from the

surrounding olivine, then creating a mask from the particle outline in (a), and finally multiplying the

image by a scale factor so that the total sum of the thickness image multiplied by the area per pixel

is equal to the volume of the particle (determined by dividing the total magnetic moment of the

particle by the known saturation magnetisation of Fe). (d) Calculated magnetic contribution to the

phase shift based on the projected thickness model from (c), using the method outlined in Section

5.2.4 (Eqn. 26).

Fig. 19. (a) High resolution image of a 50 nm diameter magnetite crystal from a magnetotactic

bacterium (image courtesy of M. Pósfai). (b) Three-dimensional reconstruction of the same particle,

obtained using electron tomography (image courtesy of R. Chong). (c) and (d) Remanent states of

the particle at room temperature and 90 K, respectively. The remanent states were obtained after

tilting the sample to ± 30° in the vertical 2 T field of the TEM objective lens. The in-plane

component of the applied field was directed along the black double arrow. (e) Stereographic

projection showing the crystallographic orientation of the sample. At room temperature, the

remanent magnetization direction is close to [131]. At 90 K, the remanent direction is close to either

[210] or [012].

Fig. 20. Profiles of the magnetic contribution to the phase shift across the magnetite particle shown

in Fig. 19 at (a) room temperature and (b) 90 K (closed circles). Profiles were taken through the

centre of the particle in a direction normal to the contours shown in Figs. 18c and d. Solid lines are

least-squares fits to the data using Eqns. 19 and 20, yielding B? = 0.6 ± 0.12 T at room

temperature, and B? = 0.46 ± 0.09 T at 90 K.

105

Fig. 21. Equilibrium threshold sizes for SP, SD, SV, and two domain magnetic states as a function

of particle length and axial ratio. Upper solid line shows the calculated boundary between SD and

two-domain states (Butler and Banerjee 1975). Lower solid lines show the sizes for SP behavior

with relaxation times of 4 x 109 years and 100 seconds (Butler and Banerjee 1975). Dashed line

shows the boundary between SD and SV states for uniaxial ellipsoidal particles, calculated using

finite-element micromagnetic methods (Muxworthy and Williams 2006). Open circles show the

sizes and aspect ratios of the magnetite blocks from region B in Fig. 25.

Fig. 22. (a) Bright-field TEM image of a double chain of magnetite magnetosomes, acquired at 400

kV using a JEOL 4000EX TEM (image courtesy of M. Pósfai). The white arrows are approximately

parallel to [111] in each crystal. (b) Electron tomographic reconstruction of the 3D morphology of

the double magnetosome chain shown in (a) (image courtesy of R Chong). (c) Magnetic phase

contours measured using electron holography from two pairs of bacterial magnetite chains at 293 K,

after magnetizing the sample parallel and antiparallel to the direction of the white arrow. Figs. 21a

and c adapted with permission from Simpson et al. (2005).

Fig. 23. Illustration of the effect of changing the applied magnetic field direction on magnetic

induction maps measured from two pairs of magnetite chains at 293 K. The applied field directions

are indicated using white arrows. In (a) the chains are magnetized in the same direction. In (b) the

top most chain is partially magnetized antiparallel to the other chains in the figure. Adapted with

permission from Simpson et al. (2005).

Fig. 24. Magnetic induction maps acquired from two pairs of bacterial magnetite chains at (a) 293

K and (b) 116 K. In the room-temperature holograms, the contours are parallel to each other within

the crystals and only deviate as a result of their morphologies and positions. At 116 K, this

106

regularity is less evident. The field lines undulate to a greater degree within the crystals, as well as

at kinks in the chains. The small vortex in the lower chain in b) is likely to be an artifact resulting

from diffraction contrast in this crystal. Adapted with permission from Simpson et al. (2005).

Fig. 25. (a) Chemical map of a titanomagnetite sample, acquired by using electron spectroscopic

imaging (Harrison et al. 2002). Blue and red correspond to Fe and Ti concentrations, respectively.

The blue regions are magnetic and are rich in magnetite (Fe3O4), whereas the red regions are

nonmagnetic and rich in ulvöspinel (Fe2TiO4). The numbers refer to individual magnetite-rich

blocks, which are discussed in the text. (b and c) Line profiles obtained from the Fe and Ti chemical

maps, respectively, along the line marked C in (a). The short arrows mark the same point in the

three pictures.

Fig. 26. Magnetic microstructure of region B in Fig. 25a measured by using electron holography

(Harrison et al. 2002). Each image corresponds to a different magnetic remanent state, acquired

with the sample in field-free conditions. The outlines of the magnetite-rich regions are marked in

white, while the direction of the measured magnetic induction is indicated both using arrows and

according to the color wheel shown at the bottom. Images a, c, e, and g were obtained after

applying a large field toward the top left of each picture, then the indicated field toward the bottom

right, after which the external magnetic field was removed for hologram acquisition. Images b, d, f,

and h were obtained after applying identical fields in the opposite directions.

Fig. 27. (a and c) Chemical maps (blue Fe, red Ti) from two regions not shown in Fig. 25. (b and

d) The corresponding magnetic microstructures, in the same format as Fig. 26. (b) Three adjacent

magnetite-rich regions combining to form a single vortex; (d) a small region that is magnetically

antiparallel to its larger neighbors.

107

Fig. 28. Schematic diagrams showing some of the possible magnetization states of three closely

spaced regions of magnetic material.

Fig. 29. Magnetic induction maps of a titanomagnetite inclusion within pyroxene (Feinberg et al.,

2004 and 2005). The inclusion is an intergrowth of elongated magnetite blocks (outlined in white)

separated by lamellae of ulvöspinel. (a) and (b) correspond to saturation remanant states obtained

after tilting the sample to ± 30° in the vertical 2 T of the TEM objective lens, such that the in-plane

component of the applied field was directed along the grey arrows. Note that the magnetic

microstructures in (a) and (b) are the exact reverse of each other, allowing the mean inner potential

to be determined using the method described in Section 5.1.5.

Fig. 30. Magnetic microstructure of ilm70 containing several APDs (Harrison et al. 2005). The

sample edge is indicated by the grey line. Prior to each measurement, the sample was exposed to a

saturating field with an in-plane/out-of-plane component of +1000/+1732 mT, followed by a

smaller field with an in-plane/out-of-plane component of (a) -1.9/+3.3 mT, (b) -10.6/+18.4 mT, (c)

-12.8/+22.2 mT. White arrows indicate the direction of the in-plane component of the applied field.

The hue and intensity of the color indicates the direction and magnitude of the in-plane component

of the magnetic flux in the sample in field-free conditions, as defined by the color wheel on the

right. The blue-purple and green-yellow colors correspond to equal and opposite in-plane

magnetizations in the direction indicated by the black double arrow. The dark bands indicate

regions with weak in-plane magnetization (magnetic domain walls). Dark bands that separate

regions of blue and green color correspond to 180° magnetic and chemical walls (e.g., at regions

labeled ‘1’ and ‘2’, respectively). Dark bands that are surrounded by regions of the same color

correspond to 0° magnetic walls (e.g., at regions labeled ‘3’).

108

Fig. 31. Profiles of the holographic phase shift, φ, across three distinct types of magnetic domain

wall in ilm70 (Harrison et al. 2005). The gradient of each profile is proportional to the in-plane

component of the magnetic flux (Eqn. 15). (a) A free-standing 180° Bloch wall. The solid line is a

least-squares fit to the observed profile, obtained by using Eqn. 28 and yielding a wall width of 19

nm. (b) A 180° ‘chemical’ wall that is coincident with an APB. The dashed line is a fit to the central

portion of the wall, obtained by using Eqn. 28 and yielding a wall width of 7 nm. (c) A 0° magnetic

wall that is coincident with the same APB as in (b). The slope of the phase profile has the same sign

on either side of the wall, indicating that the direction of magnetization is the same. The solid line is

a fit to the profile, obtained using Eqn. 30 and yielding a wall width of 30 nm.

Fig. 32. Discretization of a spherical particle using (a) regular array of 343 cubic elements

(including blanks) and (b) finite element mesh of 60 tetrahedra (image courtesy of W. Williams).

Fig. 33. Calculated domain states occurring in cubic grains of magnetite at room temperature for a

grain with edge length of 120 nm (a) single domain (flower state), (b) single vortex state, and (c)

double vortex state. The [001] axis aligns with the z axis of the cube. It was necessary to constrain

Figure 32c for a 120 nm cube. Reproduced with permission from Muxworthy et al. (2003b).

Fig. 34. Calculated micromagnetic energy density of a magnetite cube as a function of edge length

d for an initial SD configuration at room temperature (Figure 32a). The grain size was gradually

increased until the SD structure collapsed to a vortex structure at d0max = 96 nm. The size was then

gradually decreased until a SD state formed at d0min = 64 nm. Reproduced with permission from

Muxworthy et al. (2003b).

Fig. 35. The magnetization structure of a 1 μm cubic grain of magnetite with lines indicating the

positions of major domain boundaries. The top surface shows a domain wall which is nucleated on

109

a vortex core (labeled a knot in the diagram). The centre of the grain is dominated by domains

aligned towards an easy magnetocrystalline anisotropy axis, and the magnetization of the surface

lies in the plane of the surface to reduce the magnetostatic free pole energy. Reproduced with

permission from Williams and Wright (1998).

Fig. 36. Calculated regions of stability and metastability of the SD state for magnetite. (a) Results

for cuboidal particles. (b) results for particles with morphologies similar to those seen in some

strains of magnetotactic bacteria (c). Displayed is the SD-PDS transition as a function of width over

length of the respective particles. The shaded area delineates microscopically observed

magnetosome shapes (Petersen et al. 1989). The dashed area corresponds to the calculated region

where flower states are metastable. Above this area, the SD state is unstable and cannot persist.

Parts (a) and (b) reproduced with permission from Witt et al. (2005). Part (c) modified from

Bazylinski and Frankel (2004).

Fig. 37. Calculated d0max, d0, and d0min versus temperature for cubic grains of magnetite. Above

d0max only the vortex state is possible, whereas below d0min, only the flower or SD state is possible.

Between d0max and d0min it is possible for the grain to be in either state. Reproduced with permission

from Muxworthy et al. (2003b).

Fig. 38. (a) Schematic of the constrained micromagnetic simulation. A number of cells at the top

have their magnetization constrained to a direction θ1 in the x-y plane, while another set of cells at

the bottom are constrained to a direction θ2 also in the x-y plane. θ1 and θ2 are set to angles between

0 and 360° at regular intervals. The energy is minimized with respect to the magnetization direction

of all the other cells. Energy surfaces for a grain with edge 120 nm and aspect ratio 1.2 at (b) room

temperature and (c) just below Tc. As the grain is asymmetric, there are hard (SDh) and easy (SDe)

110

magnetic directions. Favorable vortex structures are also marked. Reproduced with permission from

Muxworthy et al. (2003b).

Fig. 39. Calculated energy barrier (EB) for magnetization reversal as a function of temperature for

small particles of magnetite; two with d = 80 nm and aspect ratios of 1 and 1.4 (closed and open

circles, respectively), and two for d = 100 nm and aspect ratios of 1 and 1.4 (closed and open

squares, respectively). The two dashed lines at EB = 60 kBT and 25 kBT represent the palaeomagnetic

and laboratory stability criteria, respectively. Reproduced with permission from Muxworthy et al.

(2003b).

Fig. 40. Mrs/Ms versus Hcr/Hc (Day plot) for three different anisotropy assemblages of ideal SD

magnetite grains; uniaxial (closed circles), cubic K1 < 0 (closed squares) and basal plane-uniaxial

(open circles), with a range of interaction spacing 0 ≤ d ≤ 5, where d specifies the distance between

adjacent particles in units of the particle width. For d = 5, particles are well separated and

essentially non interacting. For d = 0 the particles are just touching. Some of the interaction

spacings are marked. The effect of interactions is fairly consistent, so unmarked intermediate points

have intermediate value of d. The anisotropy orientation of the assemblage is random. Reproduced

with permission from Muxworthy et al. (2003a).

Fig. 41. (a) Definition of a first-order reversal curve (FORC). (b) A set of FORCs for a sample of

elongated SD maghemite particles at 20 K (reproduced with permission from Carvallo et al. 2004).

(c) Matrix of Ha and Hb values used to measure magnetization during a typical FORC measurement.

FORC diagrams are usually presented using a rotated set of axes Hu and Hc, covering the area of

Ha-Hb space defined by the pink rectangle. The blue square represents the region of Ha-Hb space

used to fit Eqn. 49 to M(Ha, Hb) about a point P (SF = 2). (d) FORC diagram derived from the

curves in (b) (reproduced with permission from Carvallo et al. 2004).

111

Fig. 42. Magnetization as a function of applied field for particles with (a) irreversible and (b)

reversible hysteresis loops. Irreversible switching occurs at an applied field of ± Hc. (c) In the

presence of a positive interaction field Hu, the irreversible hysteresis loop is shifted to the left, and

switching now occurs at applied fields Ha and Hb. (d) A more general curvilinear hysteresis loop,

which contains both reversible and irreversible magnetization components, can be used to explain

the existence of negative peaks in SD FORC diagrams (Newell 2005).

Fig. 43. Form of the elementary reversible and irreversible magnetization cycles for different

regions of the Ha-Hb plane. The grey area shows the region covered by a remanent Preisach

diagram, the pink area is the region covered by a FORC diagram.

Fig. 44. FORC diagram for a Sony floppy disk sample, showing the reversible ridge at Hc = 0 . In

the contour shading legend above the diagram, Max denotes the value of the FORC distribution at

its ‘‘irreversible’’ peak located at about Hc = 90 mT). A negative region occurs adjacent to the

vertical (Hc = 0) axis at about Hu = 85 mT. Note that the high density of vertical contour lines near

the Hc = 0 axis makes the shading there appear darker than it really is. (b) A horizontal cross section

passing though the irreversible peak at Hb = 5 mT. The ridge at Hc = 0 can also be seen in this plot.

(c) A vertical cross section through the reversible ridge at Hc = 0. Reproduced with permission from

Pike (2003).

Fig. 45. (a) FORC diagram for an SP-hematite-bearing Aptian red-bed sample from the south of

France (b) High-resolution FORC diagram for the lower left-hand portion of the FORC plane for

the same sample. Reproduced with permission from Pike et al. (2001a).

112

Fig. 46. (a) Magnitude of the central ridge FORC signal observed along the Hc axis for a randomly

oriented assemblage of identical uniaxial particles with axial ratio 2, plotted versus Hc/HK, where

HK is the anisotropy field. Note the minimum switching field is 0.5 HK, as predicted by the Stonar-

Wohlfarth model. (b) The continuous part of the FORC function for the same particles as (a). Note

both positive and negative regions separated by the line Hu = -Hc. (c) and (d) Equivalent plots to (a)

and (b) but for a log-normal distribution of axial ratios (average value 2, standard deviation 0.25).

Reproduced with permission after Newell (2005).

Fig. 47. High-resolution FORC diagram for a sediment sample from Lake Ely (Pennsylvania)

containing magnetotactic bacteria. Note the one order of magnitude difference between the

amplitude of the central ridge and the remaining part of the diagram. The color scale is chosen so

that zero is white, negative values are blue, and positive values are yellow to red. Contour lines are

drawn for values specified in the color scale bar. Measurements are not normalized by mass.

Reproduced with permission from Egli et al. (2010).

Fig. 48. FORC diagrams for a series of synthetic PSD magnetite samples with grain sizes of (a) 0.3

μm, (b) 1.7 μm, (c) 7 μm, and (d) 11 μm. Reproduced with permission from Muxworthy and

Dunlop (2002).

Fig. 49. First-order reversal curves and the corresponding distributions for collections of metallic Fe

nanodots with different diameters spanning the SD to SV transition. Families of FORCs, whose

starting points are represented by black dots, are shown in (a-c) for 52, 58, and 67 nm diameter Fe

nanodots, respectively. The corresponding FORC distributions are shown in three-dimensional plots

(d-f) and contour plots (g-i). The characteristic features seen in (h) and (i) are caused by transitions

between SD and SV states in the larger nanodots. The butterfly-shaped feature, consisting of a

positive peak on the Hc axis flanked above and below by negative peaks, is caused by differences in

113

the nucleation and annihilation field for vortices appearing on either side of the particles (Pike and

Fernandez 1999). Reproduced with permission after Dumas et al. (2007).

Fig. 50. FORC diagrams for a series of MD samples. (a) A sample of M80 transformer steel. (b) A 2

mm grain of magnetite, after annealing. (c) The same 2 mm grain of magnetite before annealing. (d)

An unannealed 125 μm magnetite grain. Reproduced with permission from Pike et al. (2001b).

Fig. 51. FORC diagram for a perpendicular recording medium, composed of a perpendicular array

of Ni pillars. Reproduced with permission from Pike et al. (2005).

Fig. 52. FORC diagram of a concentrated sample of magnetotactic bacteria. The FORC distribution

of the MTB sample is bimodal with a broad maximum centered at 42m T and a sharper peak

towards the Hc = 0 axis. The latter is attributed to emerging magnetosomes at the chain ends. (b)

Vertical profile through the high-coercivity peak of the distribution with mean half width field of

6.3 mT at Hc = 41.4 mT. Reproduced with permission from Pan et al. (2005).

114

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Cation-cation distance (Å)

Hematite (Samuelsen and Shirane 1970) Ilmenite (Ishikawa et al. 1985) Magnetite (Uhl and Siberchico 1995)

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L. M. SANDRATSKII et al.: FIRST-PRINCIPLES LSDF STUDY OF WEAK ETC. 451

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0�

Fig. 2. – The total energy as a function of the canting angle ( with spin-orbit coupling,- - - - without spin-orbit coupling).

remains collinear, the canting of the magnetic moments is a direct result of this interaction.Because of the small value of SOC compared with the exchange coupling that is responsiblefor the antiparallel directions of atomic moments, the canting of the moments is rather smalland leads in our calculation, in good agreement with experiment, to a ferromagnetic momentof about 0.002 µB per Fe atom.

The di�erence in the energetics of the system with and without SOC is illustrated schemati-cally in fig. 2, where the total energy as a function of the deviation of the Fe moments from they-axis is graphed. Without SOC the structures with opposite values of the canting angle areevidently equivalent because they can be transformed into each other by a pure spin rotationthrough 180� about the spin direction of the collinear structure. As a result, the collinearstructure possesses a symmetry-determined extremum in the total energy. When, however,the SOC is taken into account, there is no symmetry operation left that could lead to theequivalence of structures with di�erent canting angles. Thus there is no symmetry reason foran extremum in the total energy at the collinear structure. Although curves of the type shownin fig. 2 can be calculated within the LSDF approach using so-called constrained or ‘fixedmoment’ calculations [13], [4], we show in fig. 2 a schematic drawing because the total-energydi�erences from these type of calculations are too small in hematite.

It is worthwhile to mention an interesting property we obtain for the magnetic moments ofthe O atoms. It is easy to see that there are, due to SOC, two groups of inequivalent O atoms.Therefore, in our calculations O atoms 2 and 3 (fig. 1) carry non-collinear spin moments of�0.002 µB and atom 1 carries a smaller moment of �0.001 µB parallel to the x-axis. Thesemagnetic moments do not compensate and contribute to the weak ferromagnetism of Fe2O3.However, this contribution amounts to only a few per cent of the Fe contribution.

A final interesting observation concerns the calculated non-collinearity of the spin andorbital moments of a given atom. Thus, for example, in the case of Fe the angle betweenthe spin and orbital magnetic moments is about 0.2�, whereas in the case of the O atom, avery large value of 67� is obtained. In general, the following statement holds true: spin andorbital moments of an atom are collinear to one another only in the case when this collinearityfollows from the restrictions imposed by symmetry.

Fig. 2. Calculated hysteresis loops for a 25 As NiFe!O

"particle, with a surface anisotropy of 4 k

#/spin. (a) Particle with no broken bonds

and low roughness (1.8 As RMS), hence no surface spin disorder. (b) Particle broken bond density BBD"0.8 and higher roughness(2.1 As RMS), hence signi"cant surface spin disorder. Reproduced with permission from Ref. [8].

spontaneous magnetization with increasing tem-perature [57,58]:

M(!)"M$[1!B!%&! exp(!!

!/k

#!)],

B"2.61!$(k

#/4!D)%&! (2)

(!$"atomic volume) which reduces to Bloch's

!%&! law when the gap is zero. In itinerant magnets,there can be other low-lying magnetic excitationsknown as Stoner excitations. These are single-elec-tron excitations, as opposed to the collective spinwaves, where an electron from the majority band isexcited into the minority band. Hence, itinerantsystems do not follow Bloch's law as well as localmoment systems. This e!ect has been described in

terms of a temperature dependence in the spin wavesti!ness parameter [58].

In a magnetic nanoparticle, the spin wave spec-trum is quantized due to the "nite size. A roughestimate of the spectrum can be found by assuminga cubic particle with edge length d, in which casethe spin wave energies are given by

E"Dq!""D(n!/d)!, n"1, 2, 3,2 (3)

for an isotropic material. This discrete spectrumresults in a &"nite size gap' in the spin wave spec-trum, given by

!'(

+D!!/d!. (4)

364 R.H. Kodama / Journal of Magnetism and Magnetic Materials 200 (1999) 359}372

(a) (b)

(c) (d)

can also see that for j!0.1, i.e., when the exchange energybecomes comparable with anisotropy and Zeeman energy,

there are more jumps that can be attributed to the switching

of different spherical shells of spins starting from surface

down to the center. This situation is sketched in Fig. 5. For

example, for h!0 one can see that the exchange has a littleinfluence on surface spins, as they are directed almost along

their easy axes; for h!0.64 the surface spins show the samebehavior as in the absence of exchange, but part of core

spins, located near the surface, are deviated from their easy

axes. At the field h!0.8 all these core spins have alreadyswitched.

For j!1!ks , even that there is only one jump, the hys-

teresis loop is not rectangular owing to the fact that the spins

rotate in a noncoherent way, as can be seen in Fig. 6. This is

due to a compromise between anisotropy and exchange en-

ergies, see, for example, the picture for h!0. Moreover,even a small number of neighbors lying in the core produces

a large effect via exchange on the behavior of a surface spin.

For much larger values of j the spins are tightly coupled

and move together, and the corresponding "numerically ob-tained# critical field hc coincides with the "analytical# expres-sion obtained in the limit J!$ , i.e., hc!Nc /N, where Nc is

the number of core spins. This expression for hc has beenobtained by summing over the direction of surface easy axeswhich results in a constant surface energy contribution pro-portional to ks . Hence, due to spherical symmetry, the sur-face anisotropy constant does not enter the final expressionof hc .Now we consider the case of larger values of ks , e.g.,

ks!10, so as to investigate the influence of surface anisot-ropy both in direction and strength. The results are presentedin Fig. 7 "left#.Here, a notable difference with respect to the previous

case, ks!1, is the fact that the core now switches before thesurface and at higher fields. Moreover, there appear morejumps which may be attributed to the switching of variousclusters of surface spins. Both cases show that as the ratioj /ks decreases, the magnetization requires higher fields tosaturate. This is further illustrated by Fig. 7 "right# whereks!10

2! j for a smaller particle.Let us now summarize the ongoing discussion. We ob-

serve that considering a radial distribution for surface anisot-ropy, leads, even in the case of very strong exchange, to animportant quantitative deviation from the classical SWmodel. In particular, the critical field in our model is given

by

FIG. 5. Magnetic structure for j!0.1,ks!1 for the field values h!"4.0,0,0.64,0.8,0.88,4 which correspond to the saturation states anddifferent switching fields shown in Fig. 4. These field values correspond to the pictures when starting from the upper array and moving right,

down left, and then right. Obviously, gray arrows represent core spins and black arrows represent surface spins.

HYSTERETIC PROPERTIES OF A MAGNETIC . . . PHYSICAL REVIEW B 66, 174419 "2002#

174419-5

(a) (b) (c)

(d) (e) (f)

z

H. Kachkachi et al.: Surface e!ects in nanoparticles: application to maghemite !-Fe2O3 683

0 100 200 3000

100

200

300

400

c)b)a)

Ma

gn

etis

ati

on

(em

u/c

m3) H

app= 55 kOe

Temperature (K)Applied field (kOe)

51A (2.7 nm)

36A (4.8 nm)

3D (7.1 nm)

Temperature (K)0 100 200 300

0 20 40 600

100

200

300T(K)

5

10

15

25

50

75

100

125

150

200

250

300

Fig. 1. (a) Magnetisation as a function of the magnetic field of a diluted assembly of !-Fe2O3 nanoparticles with a meandiameter of 2.7 nm. (b) Thermal variation of the magnetisation extracted from a) at a field of 55 kOe. (c) Thermal variation ofthe magnetisation in a field of 55 kOe for three samples with di!erent mean diameters (2.7, 4.8, 7.1 nm).

where J!" (positive or negative) are the exchange couplingconstants between (the !," = A,B) nearest neighborsspanned by the unit vector n; S!

i is the (classical) spinvector of the !th atom at site i; H is the uniform fieldapplied to all spins (of number Nt) in the particle, K > 0is the anisotropy constant and ei the single-site anisotropyaxis (see definition below). A discussion of the core andsurface anisotropy will be presented below. In the sequelthe magnetic field will be set to zero.

To the Dirac-Heisenberg Hamiltonian we add the pair-wise long-range dipolar interactions

Hdip =(gµB)2

2

!

i!=j

(Si · Sj)R2ij ! 3 (Si ·Rij) · (Rij · Sj)

R5ij

(2)

where g is the Lande factor, µB the Bohr magneton andRij the vector connecting any two spins on sites i and jof the particle, Rij " #Rij#.

2.2 Method of simulation

The particle we consider here is a spinel with two dif-ferent iron sites, a tetrahedric Fe3+ site (denoted by A)and an octahedric Fe3+ site (denoted by B). The nearestneighbor exchange interactions are (in units of K) [10,9]:JAB/kB $ !28.1, JBB/kB $ !8.6, and JAA/kB $ !21.0.These coupling constants are used in the Dirac-HeisenbergHamiltonian HDH in order to model the phase transitionfrom the paramagnetic to ferrimagnetic order as the tem-perature is lowered down to zero through Tc $ 906 K.In the spinel structure an atom on site A has 12 nearestneighbors on the sublattice B and 4 on the sublattice A,and an atom on site B has 6 nearest neighbors on A and 6on the B sublattice; the number of B sites is twice that of

sites A. The nominal value of the spin on sites A and B is5/2, and this justifies the use of classical spins. We havealso taken account of 1

3 of lacuna for each two B atomsrandomly distributed in the particle. The nanoparticle wehave studied contains Nt spins ($ 103 ! 105), and its ra-dius is in the range 2-3.5 nm. Our model is based on thehypothesis that the particle is composed of a core of ra-dius containing Nc spins, and a surface shell surroundingit that contains Ns spins, so that Nt = Nc+Ns. Thus vary-ing the size of the particle while maintaining the thicknessof the surface shell constant (% 0.35 nm), is equivalent tovarying the surface to total number of spins, Nst = Ns/Nt,and this allows us to study the e!ect of surfaces of di!er-ent contributions. All spins in the core and on the sur-face are identical, but interact via the, a priori, di!erentcouplings depending on their locus in the whole volume.We will consider both cases of identical interactions, andthat of the general situation with di!erent interactions onthe surface and in the core. Although we treat only thecrystallographically “ideal” surface, we do allow for per-turbations in the exchange constants on the surface. Thisis meant to take into account, though in a somewhat phe-nomenological way, the possible defects on the surface,and the possible interactions between the particles andthe matrix in which they are embedded. In [9] it was as-sumed that the pairwise exchange interactions are of thesame magnitude for the core and surface atoms, but therewas postulated the existence of a fraction of missing bondson the surface. On the other hand, we consider that theexchange interactions between the core and surface spinsare the same as those inside the core. We also stress thatwe are only concerned with non interacting particles, sowe ignore the e!ect of interparticle interactions on theexchange couplings at the surface of the particle.

In our simulations we start with a regular box(X & Y & Z) of spins with the spinel structure having

686 The European Physical Journal B

Surface

Core

Nt = 2009, N

st = 46%

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

!core

Core

Surface

Nt = 3766, N

st = 41%

Ma

gn

etis

ati

on

0.2 0.4 0.6 0.8 1.0

Nst = 26%

Nst = 41%

Nst = 46%

Nst = 53%

!core

0.2

0.4

0.6

0.8

1.0

d)

b)

c)

a)

Surface

Core

Nt = 909, N

st = 53%

Fig. 3. (a)-(c) Thermal variation of the surface and core magnetisation (per site) (Fig. 3d) and mean magnetisation as obtainedfrom the Monte Carlo simulations of an ellipsoidal nanoparticle. The anisotropy constants are given in the text; the exchangeinteractions on the surface are taken to be 1/10 times those in the core.

the surface is, of course, enhanced by the single-site sur-face anisotropy which tends to orientate the spins normalto the surface. In Figure 3d we see that the more impor-tant is the surface contribution the more enhanced andrapid is the raising of the mean magnetisation at low tem-peratures, and this behavior bears some resemblance toFigure 1c.

In Figure 4 we plot the core magnetisation of an el-lipsoidal nanoparticle with Nt = 909, 3766, 6330 andthe magnetisation of the isotropic system with the spinelstructure and periodic boundary conditions1 as functionsof the reduced temperature !PBC ! T/TPBC

c , and Nst =53%, 41%, 26%. Comparing the di!erent curves, it is seenthat both the critical temperature and the value of themagnetisation are dramatically reduced in the core of theparticle. The reduction of the critical temperature is obvi-ously due to the finite-size and surface e!ects [11]. Thereis a size-dependent reduction of the critical temperatureby up to 50% for the smallest particle. The same resulthas been found by Hendriksen et al. [18] for small clus-ters of various structures (bcc, fcc, and disordered) usingspin-wave theory. As to the magnetisation, the reductionshows that the core of the particle does not exhibit thesame magnetic properties as the bulk material, and asdiscussed before, it is influenced by the misaligned spinson the surface.

1 This system is a perfectly ferrimagnetic material with peri-odic spinel structure and without vacancies, though such ma-terial does not exist in reality since all spinels present somedegree of vacancy. This system will be referred to in the sequelas the PBC system.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0 MPBC

Mcore

(Nt = 909)

Mcore

(Nt = 3766)

Mcore

(Nt = 6330)

Ma

gn

eti

sati

on

!PBC

Fig. 4. Thermal variation of the magnetisation of the PBCsystem with Nt = 403, and the core magnetisation for Nt =909, 3766, 6330 with Nst = 53%, 41%, 26%, respectively, asfunctions of !PBC (see text). The exchange interactions on thesurface are taken equal to 1/10 times those in the core.

In Figure 4 we can also see that the higher Nt the lowerthe magnetisation in the critical region and the higherthe temperature at which the magnetisation approacheszero, and this is consistent with the fact that M " 1/

#Nt

at high temperatures, as discussed earlier. However, theincrease of the critical temperature with Nt is not asclear-cut as it could be expected, and this can be under-stood by noting that the disordered surface (Js = Jc/10,small coordination numbers, and single-site anisotropy)strongly influences the magnetic order in the core through

a b

AF R3c + PM R3

PM R3c

PM R3

PM R3c + PM R3

AF R3c

2000

1900

1800

1700

1600

1500

1400

1300

1200

1100

1000

900

800

700

600

500

400

300

200

100

0

Tem

per

ature

(K

)

100806040200

Mol% FeTiO3

(a) C

atio

ns

(b) C

om

positio

n(c

) Spin

s(d

) Magnetiz

atio

n

R3c

R3 R3c

R3

a

b

c

d

e

f

g

h

25 K

400 K

Annealed at 1100 KAnnealed at 850 K

-1.0

-0.5

0.0

0.5

1.0

Ord

er p

aram

eter

403020100

1.0

0.8

0.6

0.4

0.2

0.0

Composition

403020100

-2

-1

0

1

2

Spin

403020100

-1.0

-0.5

0.0

0.5

1.0

Ord

er p

aram

eter

403020100

1.0

0.8

0.6

0.4

0.2

0.0

Composition

403020100

-2

-1

0

1

2

Spin

403020100

25 K

APB APB

APBAPB

APD

APD

APD APD

-2

-1

0

1

2

Spin

403020100

Layer

375 K

-2

-1

0

1

2

Spin

403020100

Layer

850 K 1100 K

1.5

1.0

0.5

0.0

8006004002000

Temperature (K)

-0.10

-0.05

0.00

0.05

0.10

Spin

600400200

Temperature (K)

2.0

1.5

1.0

0.5

0.0

-0.5

Subla

ttic

e sp

in

800700600500400300200100

Temperature (K)

0.10

0.08

0.06

0.04

0.02

0.00

Spin

800600400200

Temperature (K)

0

π

2π

ϕ

x

Coherent Electron Beam

* * * * * * * * * * * * * * * * * * * * * * * *

0

π

2π

ϕ

x

Coherent Electron Beam

* * * * * * * * * * * * * * * *

********

* **

+ + + + + + + + +

+ + + + + + + + +

+ + + + + + + + +

+ + + + + + + + +

+ + + + + + + + +

- - - - - - - - -

- - - - - - - - -

- - - - - - - - -

- - - - - - - - -

- - - - - - - - -

-

-

-

-

-

-

-

-

-

-

+

+

+

+

+

+

+

+

+

+

* *

*

(a)

(b)

z

z

+

Field Emission Gun

Sample

Lorentz Lens

Back Focal Plane

S1 S2

Biprism

Hologram

a b

c d

e f

30

°3

0 °

BK

=2KM

S

B<

0.52BK

B�

0.52BK

a b c

d e f

g h i

j k l

100 nm

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Phase shift (rad)

100 nm

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Phase shift (rad)

100 nm

403020100

Thickness (nm)

100 nm

120

110

100

9080706050

MIP (V)

(a)(b)

(c)(d)

-20

-10 0 10 20

Mx or My (x 106µB)

12080

400

Radius (nm

)

My

Mx

My

50 nm 50 nm

(111)

(111)

(101)(010)

(101)

[111]

[111] [111]

a b

c d

e

-0.6

-0.4

-0.2

0.0

0.2

0.4

Phase shift (rad)

25

02

00

15

01

00

50

0

Dis

tance (n

m)

Room

tem

pera

ture

B =

0.6

± 0

.12

T

0.4

0.2

0.0

-0.2

-0.4

Phase shift (rad)

30

02

50

20

01

50

10

05

00

Dis

tance (n

m)

90

KB =

0.4

6 ±

0.0

9 T

1

2

4

6

810

2

4

6

8100

2

4

6

81000

Part

icle

Length

(nm

)

1.00.80.60.40.20.0

Axial Ratio (width/length)

4x109

y

100 s

Multi Domain

Superparamagnetic

Single Domain

Two Domain

Single Domain

Vortex

2.2 High-resolution electron microscopy

Information about the relative orientations

of the magnetosomes in a single chain can

be obtained from either zone-axis selected

area electron diffraction (SAED) patterns

or high-resolution (HR) images of

individual crystals. Figure 3 shows a

bright-field image of a double chain of

magnetite crystals. By inspection, and with

reference to previous literature on

magnetite magnetosomes [3], apart from

the crystals at the ends of the chains, [111]

is nearly parallel to the chain axis in all of

the magnetosomes, as indicated by the line

of white arrows.

Figures 4 and 5 show zone-axis HR

images and SAED patterns acquired from

crystals 4 and 7, respectively. One of the

tilt axes of the double-tilt specimen holder

(y) was approximately parallel to the chain

axis, while the other (x) was perpendicular

to it. The small tilts about x that were

required to achieve zone axis orientations

in crystals 3-7 from a zone axis orientation

in crystal 2 suggest that their [111]

directions are approximately parallel to

one another (and to the chain axis), the

largest difference being 4.5º between

crystals 3 and 7. In contrast, much larger

tilt angles were required about y to achieve

zone axis orientations. The measured

crystal orientations (relative to crystal 2)

are plotted on a stereogram in figure 6,

which highlights this difference. Assuming

that sample preparation has not altered the

relative orientations of the crystals, the

chain is therefore analogous to beads on a

string that are allowed to rotate freely.

Biological control over the orientations of

the crystals appears to be stricter in setting

[111] parallel to the chain axis than in

constraining their orientation about this

direction.

From a magnetic perspective, the

alignment of the crystals ensures that their

magnetocrystalline easy axes are closely

parallel to the chain axis at room

temperature. However, this relationship

no longer holds below the Verwey

transition, as discussed below.

Figure 3. Bright-field image of a double

chain of magnetite magnetosomes, acquired at

400 kV using a JEOL 4000EX TEM. The

orientations of the crystals marked 1-7 are

referred to in the text and in subsequent

figures. The white arrows are approximately

parallel to [111] in each crystal.

3.3 Study of magnetotactic bacteria

3.3.3 Electron tomography

High angle annular dark field (HAADF)

electron tomography allows the three-

dimensional morphology of a sample to

be deduced from a series of two-

dimensional images taken at di�erent

tilt angles, typically from plus 70� to

minus 70� (17).

Electron tomography has its origins

in the biological sciences where conven-

tional bright-field micrographs were used

to reconstruct three dimensional mor-

phologies. For denser material speci-

mens, such as the magnetosomes in bac-

teria, bright-field imaging is unsuitable

as the contrast of the images is not mono-

tonically dependent on the thickness; ef-

fects from di�raction or Fresnel fringes

can cause problems. HAADF imaging

is therefore used in a scanning transmis-

sion electron microscope (STEM) where

the intensity is proportional to thick-

ness squared, the contrast is strong and

there are little or no di�raction e�ects.

Reconstruction of the 3D morphol-

ogy from 2D micrographs is done us-

ing specialised software, the resulting

model containing information not only

on the shape, but also on the variation

in density, and even chemical composi-

tion of the structure if energy filtered

TEM is used.

Figure 3.6: Reconstruction of the 3D

morpohology of the double magneto-

some chain shown in figure 3.3, from

electron tomography. Tomography by

R K K Chong.

17

sample [5,6]. The approach relies on being able to reverse the direction of magnetisation in the

sample exactly. For chains of crystals this condition is likely to be met, and can be checked by

repeating the same experiment several times. The approach also relies on diffraction contrast in the

crystals being identical in each pair of images. Artefacts arising when this is not the case can usually

be identified visually.

2.4. Off-axis electron holography of magnetite chains

Off-axis electron holograms of bacterial magnetite chains were recorded in magnetic-field-free

conditions, both at room temperature and with the sample cooled using a double tilt liquid nitrogen

cold holder. A thermocouple indicated that the cold holder nominally cooled the sample to 116 K,

which is in the vicinity of the Verwey Transition (119 K), so we can be confident of being below the

isotropic point of magnetite (130 K). However it was not possible to acquire diffraction patterns of the

crystals to assess the true temperature of the crystals under the conditions used for electron

holography. The present experiment can therefore be regarded as a preliminary study of the effect of

temperature on magnetic microstructure in biogenic magnetite crystals.

Figure 8 shows a representative magnetic induction map of two double chains of magnetite crystals

determined from holograms acquired at room temperature with the chains magnetised parallel and

antiparallel to their length. The contours are highly constrained to be parallel to each other within the

crystals and to follow the chain axis, although they deviate when breaks in the chain occur. Each

chain is seen to behave relatively independently, however there is some transfer of magnetic flux

between adjacent chains, as well as between the two pairs of chains in the figure.

Figure 8. Magnetic

phase contours

measured using

electron holography

from two pairs of

bacterial magnetite

chains at 293 K,

after magnetising

the sample parallel

and antiparallel to

the direction of the

white arrow. The

colours, which were

determined from the

local gradient of the

phase image, show

the direction of the

magnetic induction

according to the

colour wheel shown

below. The contour

spacing is 0.25

radians.

a

b

300 n

m

ac

1000 m

Tb

1000 m

T

1.9

mT

+

12.8

mT

+

1000 m

T

10.6

mT

+

800 n

m

1

1

2

23

2

12

12

3

3

-10

-9

-8

-7

-6

-5

-4

Magneti

c p

hase

shift

(rad)

200150100500

Position (nm)

2w = 30 nm

0° wall

-8

-7

-6

-5

-4

Magneti

c p

hase

shift

(rad)

200150100500

2w = 7 nm

180° chemical wall

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

Magneti

c p

hase

shift

(rad)

150100500

2w = 19 nm

Free-standing 180° walla

b

c

Fig. 2 Harrison et al.

three-dimensional micromagnetic algorithms [e.g., Williamsand Dunlop, 1989, 1995; Fabian et al., 1996]. Thesestudies have shown that the domain state of PSD grainsjust above the SD/MD threshold size is not a two domainstructure with a 180! domain wall, as postulated by formerPSD theories, but a vortex structure (Figure 1b). For slightlylarger PSD grains, Fabian et al. [1996] have shown that adouble-vortex (DV) (effectively three-domain) structure islikely (Figure 1c). In addition these models have shown thatSD grains just below the SD/MD threshold display ‘‘flower-ing’’ of the domain structure near the edge of the grains(Figure 1a).[6] In this paper we examine the stability of PSD magnet-

ite remanence structures like those shown in Figure 1 as afunction of temperature up to near the Curie temperature(!580!C) using a three-dimensional micromagnetic modelwith a conjugate-gradient (CG) minimization algorithm.There have been several previous papers which haveexamined various aspects of SD and PSD remanencestability as a function of temperature using micromagnetics[Dunlop et al., 1994; Thomson et al., 1994; Winklhofer etal., 1997]. The results of Dunlop et al. [1994] were ground-breaking, but they only used a one-dimensional model.However, Thomson et al. [1994] and Winklhofer et al.[1997] both incorporated simulated annealing (SA) intotheir three-dimensional models which greatly increasescomputational time. Consequently, both studies concen-trated on smaller SD/flower structures which they couldaccurately model, however, the larger PSD structures alsoconsidered were modeled using insufficient resolutions[Rave et al., 1998]. Winklhofer et al. [1997] realized thisand tested their SA minimizations with CG solutionsdetermined at higher resolutions. The comparison wasfavorable.[7] With the rapid improvement in computing resources it

has now become feasible to model such PSD structuresusing the correct minimum resolution. In this paper weconsider both changes in grain size (30–300 nm) anddifferences in shape (cubic and elongated grains up to anaxial ratio q of 1.4). In addition to examining the SD-vortextransition using constrained and unconstrained models as in

previous papers, we examine for the first time the stabilityof DV structures (Figure 1c).

2. Discrete Micromagnetic Model

[8] The basic algorithm used to calculate the results inthis paper was fully described by Wright et al. [1997]. Themodel subdivides a grain into a number of finite elementsubcubes. Each sub-cube represents the averaged magnet-ization direction of many hundreds of atomic magneticdipole moments. All the subcubes have equal magneticmagnitude, but their magnetization can vary in direction.The domain structure was calculated by minimizing thetotal magnetic energy Etot, which is the sum of the exchangeenergy Eex, the magnetostatic energy Ed and the anisotropyEanis [Williams and Dunlop, 1989; Wright et al., 1997]. Thedomain state of a grain is calculated by minimizing Etot bythe CG method with a fast Fourier transform (FFT) to givethe local energy minimum (LEM) [Fabian et al., 1996;Wright et al., 1997]. The calculation of the energy terms andthe implementation of the FFT are exactly the same as in thework of Wright et al. [1997].[9] It was not necessary to include magnetostrictive

anisotropy in the model [Fabian and Heider, 1996] becausefor magnetite grains <5000 nm in size, its contribution isinsignificant over the temperature range considered in thispaper [Muxworthy and Williams, 1999]. The structures inthis study were calculated for stress-free samples, i.e., nodislocations and no external stress, making the contributionfrom the magnetoelastic anisotropy zero.[10] In the model Eex / the exchange constant A, Ed /

the spontaneous magnetization Ms and Eanis / the firstcubic magnetocrystalline anisotropy K1. The thermal behav-ior of A, Ms and K1 was taken from Heider and Williams[1988], Pauthenet and Bochirol [1951], and Fletcher andO’Reilly [1974], respectively.[11] To accurately model domain structures it is necessary

to have a minimum model resolution of two cells perexchange length (exchange length =

!!!!!!!!!!!

A=Kd

p

, where Kd =m0Ms

2/2 and m0 is the permeability of free space [Rave et al.,1998]). This minimum resolution was used at all times in

Figure 1. Domain states occurring in cubic grains of magnetite at room temperature for a grain withedge length of 120 nm (a) single domain (flower state), (b) single vortex state, and (c) double vortex state.In this paper the term ‘‘SD state’’ refers not just to homogeneous magnetization structures as in Neeltheory but also to nonuniform domain structures as shown in Figure 1a which are basically SD-like with adegree of flowering toward the edges of the grain. The [001] axis aligns with the z axis of the cube. It wasnecessary to constrain Figure 1c for a 120 nm cube.

EPM 18 - 2 MUXWORTHY ET AL.: STABILITY OF PSD MAGNETIC

this study. This meant that the models were significantlylarger than in previous studies, e.g., for the largest grain thatWinklhofer et al. [1997] modeled, i.e. 120 nm, they used aresolution of 5 ! 5 ! 5, whereas the resolution used in thisstudy for a 120 nm grain was 17 ! 17 ! 17.[12] The increase in resolution meant that it was imprac-

tical to incorporate SA in the model, and the minimizationwas based on the CG algorithm. The SA method generallyfinds lower energy states than CG algorithms. However, thedifference has been shown not to be significant [Thomson,1993]. Nevertheless, the higher energy estimates from theCG algorithm are likely to lead to slightly higher energybarrier estimates between LEM states in the constrainedmodel calculations (section 4). Therefore these resultsshould be treated as upper energy barrier estimates.[13] The effect of applying external fields similar to the

strength of the earth’s field was found to be negligible forboth the constrained and unconstrained models. Winklhoferet al. [1997] drew similar conclusions.

3. Unconstrained Models

[14] There are several methods of determining the possi-ble and favorable domain structure as a function of temper-ature. Here the unconstrained method of Fabian et al.[1996] and Williams and Wright [1998] is described. In thisapproach a very small grain, say "20 nm, with an initial SDstructure is gradually increased in size until the domainstructure collapses to a vortex structure at do

max (Figure 2).The grain size is then decreased until the vortex structurebecomes SD at do

min (Figure 2). domin and do

max are inter-

preted as the lower and upper bounds where both SD andvortex structures can co-exist.[15] Previous studies have only made these calculations at

room temperature [e.g., Fabian et al., 1996; Winklhofer etal., 1997; Williams and Wright, 1998]. The room temper-ature curve (Figure 2) is in rough agreement with Williamsand Wright [1998], with a transition from a SD (flower) tovortex state. In contrast, Fabian et al. [1996] found that theSD state collapsed to a DV structure, not a vortex state. Thisdifference in findings raises questions about the the exis-tence of the DV state in ideal magnetite cubes in this narrowgrain size range. Clearly, for certain grain sizes, grainshapes and mineralogy, DV states will be favorable [Raveet al., 1998; Williams and Wright, 1998]. However, do theyoccur in magnetite in this grain size range? Initially, thetransition from SD to DV state was thought to be due toincomplete minimization, but recent calculations suggestthat it may be due to the degree of numerical precision inthe model; stable DV states occur when the numericalprecision of the model is high. Either the high precisioncalculations introduce artificial LEM states or reducedprecision calculations simply ‘‘step-over’’ the energy bar-riers (K. Fabian, personal communication, 2003).[16] do

min and domax were determined for each temperature

and are plotted as a function of temperature in Figure 3. dois the average of do

min and domax. As the temperature

increases from room temperature domin and do

max initiallydiverge. However, above "300!C, the stability range for SDand vortex co-existence is seen to narrow. On approach tothe Curie temperature Tc, do

max increases sharply to "200nm just below Tc, and the grain size range of co-existenceincreases.[17] Compared to the 1-D micromagnetic model for a

grain with q = 1.5 [Dunlop et al., 1994], it is seen that therange where vortex states and SD states can co-exist ismuch narrower, especially at room temperature. Fromhysteresis data do was estimated to be more or less inde-

Figure 2. Energy density of a magnetite cube as a functionof edge length d for an initial SD configuration at roomtemperature (Figure 1a). The grain size was graduallyincreased until the SD structure collapsed to a vortexstructure at d0

max = 96 nm. The size was then graduallydecreased until a SD state formed at d0

min = 64 nm. Tomaximize computer efficiency the resolution was increased/decreased with each increase/decrease in size, and thedomain structure rescaled between each pair of calculations.

Figure 3. d0max, d0, and d0

min versus temperature for cubicgrains (q = 1). Above d0

max only the vortex state is possible,whereas below d0

min, only the flower or SD state is possible.Between d0

max and d0min it is possible for the grain to be in

either state.

MUXWORTHY ET AL.: STABILITY OF PSD MAGNETIC EPM 18 - 3

optim

izemagnetic

momentandcoerciv

ity.Their

use

formagneto

taxis

requires

anoften

substan

tialchain

momentto

produce

amagnetic

torquewhich

counter-

actsthevisco

usdrag

ofthebacterial

bodyandkeep

sit

aligned

with

theextern

alfield

.If

several

chain

sare

presen

t,they

may

alsoact

asamagnetic

skeleto

nwhich

stabilizes

thebacterial

shape[39].

Theopti-

mality

request

holdsbest

ifthemagneto

somes

adopt

an–at

leastmeta-stab

le–SD

stateinsid

ethechain

.Tests

ofthis

assumptio

ncommonly

use

thestab

ilitydiag

ramofButler

andBanerjee

[40],which

isbased

onaone-d

imensio

nal

domain

wall

model

forfin

itegrain

s[41].A

refined

versio

nofthisdiag

ramhas

been

obtain

edusin

ganumerical

three-d

imensio

nal

micro

-magnetic

model

forrectan

gular

magnetite

particles

[2].Theleft

panelofFig.9showstheresu

ltsofthese

computatio

nsin

compariso

nwith

aspect

ratiosof

micro

scopically

observ

edmagneto

somes.

Apparen

tly,afractio

nofthebacterial

magneto

somes

liesoutsid

ethereg

ionwhere

theSD

stateismeta-stab

le.Acco

rding

tothe

model

calculatio

ns,

itwould

assumeavortex

magnetizatio

nstate

ifnototherw

isestab

ilized.Indeed

,inhomogeneous

magnetizatio

nstru

ctures

havebeen

observ

edin

artificialmagnet-

izationstates

oflarg

emagneto

somes

[42].Additio

nal

stabilizatio

narises

bymagneto

staticinteractio

nwith

inthemagneto

somechain

.Thecorresp

ondinginterac-

tionfield

dependscritically

onrelativ

epositio

nand

spacin

gbetw

eenthemagneto

somes.

Both

param

eterschangewith

movem

entandgrowth

ofthebacteria

which

alsomay

impose

consid

erable

bendingofthe

chain

[43].Durin

gcell

divisio

n,this

interactio

ncan

even

break

downcompletely

which

would

then

leadto

irreversib

ledem

agnetizatio

niftheSD

structu

reis

intrin

sicallyunstab

lewith

inthenon-in

teractingmag-

neto

some.

Wetherefo

rehypothesize

that

alsomagne-

tosomes

with

inchain

sdosupport

ameta-stab

leSD

statewith

outmagneto

staticstab

ilization.Thisim

plies

that

thecharacteristic

magneto

someshapeand

the

corresp

ondingorien

tationofthecubicaniso

tropyaxes

should

significan

tlystab

ilizetheSD

statein

large

magneto

somes.

Wetested

this

assumptio

nbymodel-

lingaparticle

geometry

which

istypical

formagneto

-somes

[24].

Thegeneral

shapeand

orien

tation

ofthecubic

aniso

tropyaxes

used

inourmodellin

gare

sketch

edin

Fig.4.Particle

length

lvaries

alongthecen

tral[111

]-axisandthedefin

itionofthewidth-to

-length

ratioqis

exten

ded

tonon-rectan

gular

shap

esby

setting

q!

!!!!!!!!!!

V=l 3

p

,which

forrectan

gular

particles

ofsize

w"w"lcorrectly

yield

sw/l.

Theenerg

eticallyoptim

alPSD-states

ofmagneto

-somes

atlarg

ergrain

sizesdependonthevalu

eofq.

Foramagneto

somewith

q=1,a

vortex

inthexz-p

lane

ofFig.4req

uires

lessenerg

ythan

avortex

inthexy-

plan

e.In

contrast,

inmagneto

somes

with

q=0.9,0.8,

Fig.9.Regionsofstab

ilityandmeta-stab

ilityoftheSDstate

forrectan

gular

particles

(left)orcharacteristic

magneto

somes

(right).

Disp

layed

is

theSD-PDStran

sitionas

afunctio

nofwidth

over

length

oftheresp

ectiveparticles.

Theshaded

areadelin

eatesmicro

scopically

observ

ed

magneto

someshapes,

Petersen

etal.

[9].Thedash

edarea

corresp

ondsto

themicro

magnetically

calculated

regionwhere

flower

statesare

meta-

stable.

Abovethisarea,

theSD

stateisunstab

leandcan

notpersist.

A.Witt

etal./Earth

andPlaneta

ryScien

ceLetters

233(2005)311–324

320

this study. This meant that the models were significantlylarger than in previous studies, e.g., for the largest grain thatWinklhofer et al. [1997] modeled, i.e. 120 nm, they used aresolution of 5 ! 5 ! 5, whereas the resolution used in thisstudy for a 120 nm grain was 17 ! 17 ! 17.[12] The increase in resolution meant that it was imprac-

tical to incorporate SA in the model, and the minimizationwas based on the CG algorithm. The SA method generallyfinds lower energy states than CG algorithms. However, thedifference has been shown not to be significant [Thomson,1993]. Nevertheless, the higher energy estimates from theCG algorithm are likely to lead to slightly higher energybarrier estimates between LEM states in the constrainedmodel calculations (section 4). Therefore these resultsshould be treated as upper energy barrier estimates.[13] The effect of applying external fields similar to the

strength of the earth’s field was found to be negligible forboth the constrained and unconstrained models. Winklhoferet al. [1997] drew similar conclusions.

3. Unconstrained Models

[14] There are several methods of determining the possi-ble and favorable domain structure as a function of temper-ature. Here the unconstrained method of Fabian et al.[1996] and Williams and Wright [1998] is described. In thisapproach a very small grain, say "20 nm, with an initial SDstructure is gradually increased in size until the domainstructure collapses to a vortex structure at do

max (Figure 2).The grain size is then decreased until the vortex structurebecomes SD at do

min (Figure 2). domin and do

max are inter-

preted as the lower and upper bounds where both SD andvortex structures can co-exist.[15] Previous studies have only made these calculations at

room temperature [e.g., Fabian et al., 1996; Winklhofer etal., 1997; Williams and Wright, 1998]. The room temper-ature curve (Figure 2) is in rough agreement with Williamsand Wright [1998], with a transition from a SD (flower) tovortex state. In contrast, Fabian et al. [1996] found that theSD state collapsed to a DV structure, not a vortex state. Thisdifference in findings raises questions about the the exis-tence of the DV state in ideal magnetite cubes in this narrowgrain size range. Clearly, for certain grain sizes, grainshapes and mineralogy, DV states will be favorable [Raveet al., 1998; Williams and Wright, 1998]. However, do theyoccur in magnetite in this grain size range? Initially, thetransition from SD to DV state was thought to be due toincomplete minimization, but recent calculations suggestthat it may be due to the degree of numerical precision inthe model; stable DV states occur when the numericalprecision of the model is high. Either the high precisioncalculations introduce artificial LEM states or reducedprecision calculations simply ‘‘step-over’’ the energy bar-riers (K. Fabian, personal communication, 2003).[16] do

min and domax were determined for each temperature

and are plotted as a function of temperature in Figure 3. dois the average of do

min and domax. As the temperature

increases from room temperature domin and do

max initiallydiverge. However, above "300!C, the stability range for SDand vortex co-existence is seen to narrow. On approach tothe Curie temperature Tc, do

max increases sharply to "200nm just below Tc, and the grain size range of co-existenceincreases.[17] Compared to the 1-D micromagnetic model for a

grain with q = 1.5 [Dunlop et al., 1994], it is seen that therange where vortex states and SD states can co-exist ismuch narrower, especially at room temperature. Fromhysteresis data do was estimated to be more or less inde-

Figure 2. Energy density of a magnetite cube as a functionof edge length d for an initial SD configuration at roomtemperature (Figure 1a). The grain size was graduallyincreased until the SD structure collapsed to a vortexstructure at d0

max = 96 nm. The size was then graduallydecreased until a SD state formed at d0

min = 64 nm. Tomaximize computer efficiency the resolution was increased/decreased with each increase/decrease in size, and thedomain structure rescaled between each pair of calculations.

Figure 3. d0max, d0, and d0

min versus temperature for cubicgrains (q = 1). Above d0

max only the vortex state is possible,whereas below d0

min, only the flower or SD state is possible.Between d0

max and d0min it is possible for the grain to be in

either state.

MUXWORTHY ET AL.: STABILITY OF PSD MAGNETIC EPM 18 - 3

pendentoftem

peratu

reuntil

450!C

,where

itincreases

rapidly

[Dunlop,1987].

4.

Constra

ined

High-Tem

pera

ture

Models

[18]

To

determ

inethestab

ilityofan

LEM

state,it

isnecessary

tocalcu

latetheenerg

ybarriers

(EB )

which

trapit.

This

isdonebyconstrain

ingdomain

structu

resofagrain

into

interm

ediate

non-LEM

states.[ 19]

Constrain

edmodels

ofdomain

structu

rewere

calcu-

latedusin

gasim

ilarproced

ure

tothat

firstdescrib

edby

Enkin

andWillia

ms[1994].In

this

approach

,anumber

of

cellsare

constrain

edto

setangles,

andthen

thetotal

energ

yis

minim

izedwith

respect

totheother

unconstrain

edcells.

This

techniqueallo

wsnon-LEM

magnetic

statesto

be

produced

sothat

transitio

npath

sbetw

eenLEM

statescan

beexam

ined.Twosets

ofconstrain

edcells

atopposite

sides

ofthemodel

grain

arerotated

through360!

atsomestep

interv

al(Figure

4).

From

these

two

degrees

offreed

om

energ

ysurfaces

canbeplotted

,fro

mwhich

theenerg

ybarriers

betw

eenLEM

statesare

determ

ined

[Enkin

and

Willia

ms,1994].

[ 20]

Prev

iousconstrain

edmodels

haveonly

consid

eredSD-vortex

transitio

ns[Enkin

andWillia

ms,1

994;W

inklh

ofer

etal.,

1997;Muxw

orth

yandWillia

ms,1999].In

thispaper

wealso

consid

erSD-D

Vandvortex

-DV

transitio

ns.

[ 21]

Forvery

small

grain

snear

d0in

size,consid

ering

only

SD-vortex

transitio

nsis

reasonable

becau

seas

afirst

approxim

ationthere

areonly

twoLEM

domain

states,the

SD

stateandthevortex

state.In

fact,justas

SD

statehave

differen

tdegrees

offlo

werin

g,there

aredifferen

ttypes

of

vortex

state,e.g

.,Rave

etal.[1998]foundsev

endifferen

tvortex

statesin

uniax

ialmaterials.

However,

inthis

study

wegroupall

these

vortex

statesinto

onecateg

ory,

aswe

consid

erthestu

dy

ofsubtle

differen

cesbetw

eenvortex

statesin

magnetite

outsid

ethesco

peofthispaper.

[ 22]

Asthegrain

sizebeco

mes

larger

thenumber

of

possib

leLEM

statesincreases.

Thenextmost

realisticLEM

domain

stateto

constrain

ingrain

slarger

than

d0 maxis

theDV

state(Figure

1c).

Toproduce

DV

structu

resitis

necessary

toconstrain

theedges

ofthemodels,

notthe

surfaces

asin

SD-vortex

constrain

edmodels

(Figure

4),

i.e.,thefourcorners

ofthemiddlelay

erofcells

lyingin

the

zplan

ewere

constrain

ed.Toobtain

SD-D

Vtran

sitions,the

constrain

edcells

inthefourcornergroupswere

splitin

totwo

pairs

facingeach

other

across

thediag

onal.

Each

pair

was

then

rotated

separately

through180!.

Toproduce

DV-vortex

transitio

npath

s,twosets

ofconstrain

edcells

separated

bythe

edgeofthecubewere

keptfix

edandwere

anti-p

arallel,while

theother

twosets

were

rotated

independently

through180!.

Asthenumber

ofpossib

leLEM

statesbeco

mes

greater,

the

number

ofpossib

letran

sitionpath

sbetw

eenLEM

statesincreases.

Itis

possib

letherefo

rethat

inconstrain

ingSD-

DV

andDV-vortex

transitio

ns,

sometran

sitionpath

swith

lower

energ

ybarriers

areoverlo

oked,e.g

.,an

SD

toDV

transitio

nwith

aninterm

ediate

vortex

state.Therefo

rethe

energ

ybarriers

determ

ined

forDV-SD

andDV-vortex

tran-

sitionscould

possib

lybeoverestim

ates.Thisproblem

isnot

uniqueto

constrain

edCGminim

izations,butalso

applies

toconstrain

edSAcalcu

lations.

[ 23]

Energ

y-su

rfaceplots

were

determ

ined

asafunctio

nofgrain

size,tem

peratu

reandshape(Figure

5).Asthegrain

sizeincreased

,themodel

resolutio

nwas

increased

,e.g

.,for

a100nm

cubic

grain

thereso

lutio

nwas

14!

14!

14and

fora300nm

grain

thereso

lutio

nwas

44!

44!

44.Asthe

largergrain

sizesreq

uired

more

CPU

time,less

variatio

nin

temperatu

reandshapecould

beexam

ined

forthese

grain

s.Elongated

grain

swere

consid

eredas

cuboidsofsquare

cross-sectio

nandelo

ngated

alongoneaxis.

Thismean

tthat

themodel

sizehad

tobeincreased

,e.g

.,forthe300nm

modelwith

alongaxis/sh

ortaxisratio

q=1.4,thegrid

sizewas

62

!44

!44.Thenumber

ofconstrain

edcells

increased

with

resolutio

nbutwas

keptto

"2–5%

ofthe

total

number

ofcells.

Durin

gtheenerg

y-su

rfaceplotcalcu

-latio

ns,domain

structu

reswere

visu

allycheck

edforsm

ooth

consisten

tbehaviorwith

noabruptchanges.

[ 24]

Theminim

um

energ

ybarrier

betw

eenLEM

stateswas

determ

ined

byconsid

eringsad

dle-p

ointsbetw

eenmeta-

stable

states(Figure

5).In

particu

lar,tran

sitionpath

sfro

mvortex

andDV

stateswere

difficu

ltto

determ

ine.

Inthese

cases,theenerg

ybarrier

was

defin

edas

theenerg

yneed

edto

rotate

themomentby90!.

4.1.

SD-Vortex

Transitio

ns

[ 25]

FortheSD-vortex

transitio

n,twosets

ofspinswere

constrain

edonopposite

sides

ofthegrain

inan

identical

proced

ure

tothat

ofEnkin

andWillia

ms[1994].

Typical

energ

y-surface

plotsare

shownin

Figure

6foragrain

with

d=120nm

andq=1.2.Atroom

temperatu

rethevortex

stateis

themost

favorab

leLEM

state(Figure

6a).

The

energ

ybarrier

(EB )

betw

eentwosuch

identical

stateswas

Figure

4.

Schem

aticoftheconstrain

edSD-vortex

micro

-magnetic

model.

Anumber

ofcells

atthetophavetheir

magnetizatio

nconstrain

edto

adirectio

nq1in

thex-y

plan

e,while

another

setofcells

atthebotto

mare

constrain

edto

adirectio

nq2also

inthex-y

plan

e.Theenerg

yisminim

izedwith

respect

tothemagnetizatio

ndirectio

nofall

theother

cells.q1andq2are

settoangles

betw

een0!

to360!

atinterv

alspacin

gsof15!,

30!

or45!

dependingonmodelreso

lutio

n.

Thetotal

number

ofconstrain

edcells

varies

with

model

resolutio

nbutwas

keptbetw

een2and5%

ofthetotal.

EPM

18-4

MUXWORTHY

ETAL.:STABILITY

OFPSD

MAGNETIC

determ

ined

tobe590kT.At567!C

theSD

stateisthemost

favorab

leLEM

stateandtheenerg

ybarrier

betw

eentwo

identical

SD

stateswas

calculated

tobe6.1

kT.

[ 26]

From

such

energ

yplots,

energ

ybarriers

(EB )

were

determ

ined

asafunctio

noftem

peratu

re(Figure

7).In

cluded

onFigure

7are

theBoltzm

annenerg

iesfortworelax

ation

times;

alab

orato

ryrelax

ationtim

e!1sgivingEB"

25kT

andageological

relaxatio

ntim

e!1billio

nyrs

with

EB"

60kT.Athightem

peratu

resnear

Tc ,EB<25kT

forall

grain

sizesandshapes.H

owever,

asthetem

peratu

redecreases,

EB

increases

sharp

ly.Therate

ofincrease

isgreatest

forlarger

grain

s,andat

room

temperatu

rethelargest

grain

(300nm)

has

thelargest

EB .

Itis

atfirst

sightsurprisin

gthat

grain

swith

SD

stateshavelower

energ

ybarriers

than

grain

swith

vortex

states.However,

this

isdueto

acombinatio

nof

effects.[ 27]

Firstly

thecompetin

gmagnetic

energ

iesincrease

with

grain

volume.

Forexam

ple,

inafirst

approxim

ation

forsim

ple

magnetic

structu

res,Eexand

Eanis

increase

linearly

with

grain

volume,

while

themagneto

staticenerg

yEdincreases

asthesquare

ofgrain

volume.

Themagneto

-static

energ

y’sstro

nggrain

sizedependence

causes

both

Etot

andEBto

increase

sharp

lywith

grain

size.[ 28]

Seco

ndly

theconfig

uratio

nal

aniso

tropydisp

laysa

grain

sizedependency.

Theconfig

uratio

nal

aniso

tropyis

aterm

coined

todescrib

etheenerg

ybarrier

associated

with

interm

ediate

statesin

atran

sitionpath

.Tem

porarily

ignorin

gmagneto

crystallin

eaniso

tropy,consid

eraSD-lik

eorflo

wer

statein

acubic

grain

(Figure

1a).

Theenerg

yofaSD

statealig

ned

along‘‘x’’

or‘‘y’’

areequivalen

tdueto

symmetry

;

Figure

5.

SD-vortex

energ

ysurface

(contourmap

of(E

tot–minimum

Etot )/kT

fordifferen

tconstrain

edthree-d

imensio

nalmagnetic

structu

res)foragrain

with

edge120nm

andq=1.4at567!C

.Asthegrain

isasy

mmetric,

there

arefav

orab

le(easy

)andunfav

orab

le(hard

)SD

magnetic

states.Unfav

orab

levortex

structu

resare

alsomark

ed.A

possib

letran

sitionpath

overasad

dlepointishighlig

hted

.Thetwoanglesq

1

andq2refer

totheangles

ofthetwosets

ofconstrain

edspins(Figure

4).Themodel

resolutio

nused

was

24#

17#

17.

Figure

6.

SD-vortex

energ

ysurfaces

foragrain

with

edge

120nm

andq=1.2

at(a)

room

temperatu

reand(b)just

belo

wTc .Asthegrain

isasy

mmetric

there

arehard

(SDh )

and

easy(SDe )

magnetic

directio

ns.

Favorab

levortex

structu

resare

alsomark

ed.Themodel

resolutio

nused

was

21#

17#

17.

MUXWORTHY

ETAL.:STABILITY

OFPSD

MAGNETIC

EPM

18-5

the degree of flowering will be identical. For a SD to rotatecoherently from the x direction to the y direction or viceversa it will have to pass through an intermediate state. Thedegree of flowering varies depending on the direction of themagnetization with respect to the cube faces. Intermediatestates have less flowering due to geometry considerationsgiving rise to an effective energy barrier. If no floweringoccurs, i.e., an ideal SD grain, then for cubic grains with nomagnetocrystalline anisotropy there would be no energybarrier for this rotation. However, in magnetite flowering isin reality common. Since the degree of flowering increasesas the grain size increases, the energy barrier along thetransition path increases. This effect occurs for other typesof transitions, e.g., between vortex states. Configurational

anisotropy will always exist in cubic structures, but willoften be masked by magnetocrystalline anisotropy or otheranisotropy created by applied fields. Only a sphere will haveno configurational anisotropy.[29] Generally, the results agree well with those of

Winklhofer et al. [1997] who modeled grains up to 120 nmusing simulated annealing. The agreement is good even forthe larger grains where Winklhofer et al. [1997] used aresolution of only 5 ! 5 ! 5.

4.2. SD-DV and Vortex-DV Transitions

[30] SD-DV and vortex-DV transitions were determinedfor grains in the range 140–200 nm. Below 140 nm, theDV state is not an LEM state [Fabian et al., 1996]. BothSD and DV states were found to be LEM states in this sizerange and produced energy surface plots similar to those inFigure 6. The DV-vortex energy surfaces are less easilyinterpreted (Figure 8). Generally the vortex state was theabsolute energy minimum, and the DV state had a muchhigher energy. The DV state was often located near veryshallow LEM states, which are not thought to be stable (themodel does not include thermal fluctuations which wouldmake them even less stable). This implies that even if thegrain is in a stable LEM state in a SD-DV energy plotdiagram, in an unconstrained system the DV state wouldactually minimize to a vortex state, in effect resulting in aSD-vortex energy surface plot (Figures 5 and 6). Becausethe DV state was not a significant LEM state, no values forEB were determined.

5. Blocking Temperatures and Relaxation Times

[31] From plots of EB versus temperature (Figure 7),blocking temperature diagrams as a function of grain sizewere determined (Figure 9). Also shown in Figure 9 are theexperimental data of Dunlop [1973]. As the grain sizeincreases, the blocking temperature increases, reflectingthe increase in EB with grain size (Figure 7). As q increases

Figure 7. Energy barrier (EB) as a function of temperaturefor a selection of small particles of magnetite; two with d =80 nm (q = 1 and 1.4) and two for d = 100 nm (q = 1 and1.4). The two dashed lines at EB = 60 kT and 25 kT representthe palaeomagnetic and laboratory stability criteria.

Figure 8. DV-vortex energy surface for a grain with edge160 nm and q = 1.0 just below Tc. The regions for DVstructures and vortex structures are highlighted. Intermedi-ate structures are positioned in between these two domainstates. The model resolution used was 23 ! 23 ! 23. q1 andq2 refer to the angles of the two sets of constrained spins, seetext for explanation.

Figure 9. Calculated blocking temperatures as a functionof grain size for different aspect ratios (q = 1 (cubic), 1.2and 1.4). Open symbols represent blocking temperatures incooling under laboratory conditions, while solid symbolsindicate cooling over geological timescales. The shadedareas represent experimentally obtained blocking tempera-tures from three magnetite samples of near cubic shape[Dunlop, 1973].

EPM 18 - 6 MUXWORTHY ET AL.: STABILITY OF PSD MAGNETIC

and 0.866 respectively [Kneller, 1969; Tauxe et al., 2002].The trigonal anisotropy MRS/MS ratio agreed within twodecimal places with theory, i.e., 0.65, and the b-uniaxialwas 0.50 as expected from theory [Dunlop, 1971].

3.2. Anisotropy Control

[17] The effect of interactions is partially controlled by thetype of anisotropy [Kneller, 1969]. If the anisotropy isrestricted to the basal plane, then the effect of interactionsis greatly reduced (Figure 3). This is point emphasized byconsidering dMRS

, which for nonplanar anisotropies is greaterthan dMRS

for planar anisotropies (Table 1). As the order ofanisotropy increases dMRS

decreases, e.g., dMRSfor uniaxial

< dMRSfor cubic (8 fold). For all three nonplanar anisotropy

regimes, dHC> dMRS

(Table 1), which suggests that HC ismore sensitive to interactions than MRS/MS. However,MRS/MS is clearly more strongly affected by the interactions.For example, for the cubic anisotropy with K1 > 0, MRS/MS

! 0 for d < 0.3, and equals 0.83 for the noninteractingregime, however,HC increases from only!4 mT to!16 mTover the same range.

3.3. Day Plots for Uniform SD Grains

[18] It is of interest to the paleomagnetist to plot simu-lated ‘‘Day plots’’, i.e., MRS/MS versus HCR/HC [Day et al.,1977], with the effect of grain interaction spacing depicted(Figure 4). As the two planar and the two cubic anisotropiesshow similar behavior, only the b-uniaxial and cubicanisotropy with K1 < 0 are plotted. The effect of decreasingd decreases MRS/MS and increases HCR/HC, causing thehysteresis parameters plot position to move from the

Figure 3. (a) MRS/MS, (b) HC and (c) HCR versus spacingd for five different anisotropy assemblages of ideal SDgrains; uniaxial, cubic K1 > 1, cubic K1 < 1, b-uniaxial andtrigonal. The anisotropy orientation of each assemblage israndom.

Table 1. Estimates for dMRSand dHC

for Ideal SD Grainsa

Anisotropy dMRSdHC

Nonplanar Anisotropyuniaxial 1.2 2.0

cubic (K1 > 1) 2.0 2.0cubic (K1 < 1) 2.5 2.0

Basal Plane Anisotropyb-uniaxial 0.6 . . .trigonal 0.6 . . .

adMRSis the value of d in between d = 0 and 5, where MRS/MS becomes

independent of interaction spacing (Figure 3a), similarly for HC and dHC

(Figure 3b).

Figure 4. MRS/MS versus HCR/HC (Day plot) for threedifferent anisotropy assemblages of ideal SD grains;uniaxial, cubic K1 < 1 and b-uniaxial, with a range ofinteraction spacing; 0 " d " 5. Some of the interactionspacings are marked. The effect of interactions is fairlyconsistent, so unmarked intermediate points have inter-mediate value of d. The anisotropy orientation of theassemblage is random.

EPM 4 - 4 MUXWORTHY ET AL.: MAGNETOSTATIC INTERACTIONS AND HYSTERESIS

Ha

Hb

M (H

a , Hb)

M (Am2)

H (mT)

accentuated

inthethree

lowest

contours,

thecen

traldistri-

butio

nbein

gclo

seto

circular.

The!Hulobeis

flanked

atlow

Hcby

asm

allnegativ

epeak

,which

persists

atall

temperatu

resupto

700K

(428!C

,Figure

3d)andseem

sto

bereal.

Thepattern

isrem

iniscen

tofthedistrib

utio

nfound

theoretically

byPike

etal.[1999]usin

gamovingPreisach

model

with

both

local

interactio

nfield

s[Neel,

1954]anda

mean

interactio

nfield

proportio

nal

tothenet

sample

mag-

netizatio

n.Thismodel

ismore

physically

realisticthan

the

local

interactio

nfield

model.

Hejd

aand

Zelin

ka[1990]

showed

that

most

oftheasy

mmetry

seenin

theclassical

Preisach

interp

retationcan

beacco

unted

forbythemoving

Preisach

model.

However,

inPikeet

al.’sdistrib

utio

nthe

negativ

epeak

liesim

mediately

belo

wthemain

positiv

epeak

andforces

contoursinward

,whereas

ournegativ

epeak

isoffset

diag

onally,

andthemain

peak

contours

‘‘spill’’

aroundit.

Thusitisuncertain

wheth

erthepersisten

tpattern

inourFORC

distrib

utio

nsis

evidence

ofmean

-fieldinter-

actionsornot.

[ 15]

FORC

distrib

utio

nsdeterm

ined

atandaboveroom

temperatu

reappear

inFigure

3.TheHcscale

has

been

expanded

30%

relativeto

theHuscale

inFigures

3a–3d(th

escales

areidentical

alongthetwoaxes

inFigures

3eand3f

andin

Figure

2).AsTincreases

from

150!C

to428!C

,the

tailing

ofthe

micro

coerciv

itydistrib

utio

nto

high

Hc

beco

mes

more

mark

ed.Thehigh-H

clobenow

inclu

des

thecen

tralcontours,

notjusttheouter

ones,

andtheoverall

asymmetry

inf(H

c )ismoreaccen

tuated

.Beginningwith

the

312!C

data,

thepeak

ofthespectru

mmoves

tolower

Hc ,

eventually

merg

ing

with

theHuaxis

at580!C

.Athigh

temperatu

res,more

smoothing

isnecessary

becau

sethe

magnetizatio

nis

much

weak

er.Figures

3g–3ishow

the

effectofsm

oothingonthese

more

noisy

data.

[16]

Inanumber

ofother

FORCstu

dies

thediag

onallin

ein

thenegativ

eHureg

ionstartin

gfro

mtheorig

inofthe

FORCdiag

ramoften

showsaparticu

larpattern

.Forexam

-ple,

alternatin

gpositiv

eandnegativ

epeak

salo

ngthis

line

aresometim

esobserv

ed,in

both

modeled

andmeasu

redFORC

diag

rams[e.g

.,Carva

lloet

al.,

2003].In

modelin

g,

thelin

eisdueto

thepertu

rbatio

nsintro

duced

inthemodel

totest

thestab

ilityandwhich

cause

theBark

hausen

jumps

tooccu

rat

differen

tfield

son

eachrev

ersalcurve.

At

temperatu

resless

than

428!C

the!Hulobe,which

roughly

follo

wsthislin

e,can

besuspected

tobean

artifactcreated

byprocessin

gandmay

notbephysically

realistic.[ 17]

Theshift

ofthepeak

toward

lower

Hcis

somew

hat

similar

tothetren

dseen

byMuxw

orth

yandDunlop[2002]

over

asim

ilarT

range

fortheir

finest

(0.3

mm)PSD

magnetite

and

was

interp

retedby

them

asmark

ing

aprogressiv

echangefro

mSD-lik

eto

MD-lik

ebehavior.

However,

whereas

their

contours

spread

along

the+Hu

and

!Huaxes

inasymmetrical

fashion

athigh

T(a

characteristic

MDpattern

),oursrem

ainhighly

asymmetrical

even

at580!C

.There

isnospread

ingat

allalo

ngthe+Hu

axis.

Inthe!Hudirectio

n,thecontours

areactu

allycom-

pressed

,while

abruptbendsin

these

contours

defin

ea

largelysep

arated!Hulobeexten

dingatleast

3tim

esfarth

erthan

themain

distrib

utio

n.Asthe!Hulobemoves

with

the

main

spectru

mto

lower

Hcat

553!C

and

580!C

,the

flankingnegativ

epeak

isannihilated

.

5.

Analysis

andDiscu

ssion

[18]

Ozdem

ir[1990]foundthatM

rs /Msofthismaghem

iterem

ained

close

to0.5

forT

"500!C

,reach

ed0.45

by

Figure

1.

First-o

rder

reversal

curves

at20K.

Figure

2.

Low-tem

peratu

reFORC

diag

rams:(a)

300K,(b)200K,(c)

100K,(d)20K

(SF=2).

B04105

CARVALLO

ETAL.:FORC

DIA

GRAMSOFELONGATED

SIN

GLE-D

OMAIN

GRAIN

S

3of8

B04105

Hb

Ha

Hc Hu

accentuated

inthethree

lowest

contours,

thecen

traldistri-

butio

nbein

gclo

seto

circular.

The!Hulobeis

flanked

atlow

Hcby

asm

allnegativ

epeak

,which

persists

atall

temperatu

resupto

700K

(428!C

,Figure

3d)andseem

sto

bereal.

Thepattern

isrem

iniscen

tofthedistrib

utio

nfound

theoretically

byPike

etal.[1999]usin

gamovingPreisach

model

with

both

local

interactio

nfield

s[Neel,

1954]anda

mean

interactio

nfield

proportio

nal

tothenet

sample

mag-

netizatio

n.Thismodel

ismore

physically

realisticthan

the

local

interactio

nfield

model.

Hejd

aand

Zelin

ka[1990]

showed

that

most

oftheasy

mmetry

seenin

theclassical

Preisach

interp

retationcan

beacco

unted

forbythemoving

Preisach

model.

However,

inPikeet

al.’sdistrib

utio

nthe

negativ

epeak

liesim

mediately

belo

wthemain

positiv

epeak

andforces

contoursinward

,whereas

ournegativ

epeak

isoffset

diag

onally,

andthemain

peak

contours

‘‘spill’’

aroundit.

Thusitisuncertain

wheth

erthepersisten

tpattern

inourFORC

distrib

utio

nsis

evidence

ofmean

-fieldinter-

actionsornot.

[ 15]

FORC

distrib

utio

nsdeterm

ined

atandaboveroom

temperatu

reappear

inFigure

3.TheHcscale

has

been

expanded

30%

relativeto

theHuscale

inFigures

3a–3d(th

escales

areidentical

alongthetwoaxes

inFigures

3eand3f

andin

Figure

2).AsTincreases

from

150!C

to428!C

,the

tailing

ofthe

micro

coerciv

itydistrib

utio

nto

high

Hc

beco

mes

more

mark

ed.Thehigh-H

clobenow

inclu

des

thecen

tralcontours,

notjusttheouter

ones,

andtheoverall

asymmetry

inf(H

c )ismoreaccen

tuated

.Beginningwith

the

312!C

data,

thepeak

ofthespectru

mmoves

tolower

Hc ,

eventually

merg

ing

with

theHuaxis

at580!C

.Athigh

temperatu

res,more

smoothing

isnecessary

becau

sethe

magnetizatio

nis

much

weak

er.Figures

3g–3ishow

the

effectofsm

oothingonthese

more

noisy

data.

[16]

Inanumber

ofother

FORCstu

dies

thediag

onallin

ein

thenegativ

eHureg

ionstartin

gfro

mtheorig

inofthe

FORCdiag

ramoften

showsaparticu

larpattern

.Forexam

-ple,

alternatin

gpositiv

eandnegativ

epeak

salo

ngthis

line

aresometim

esobserv

ed,in

both

modeled

andmeasu

redFORC

diag

rams[e.g

.,Carva

lloet

al.,

2003].In

modelin

g,

thelin

eisdueto

thepertu

rbatio

nsintro

duced

inthemodel

totest

thestab

ilityandwhich

cause

theBark

hausen

jumps

tooccu

rat

differen

tfield

son

eachrev

ersalcurve.

At

temperatu

resless

than

428!C

the!Hulobe,which

roughly

follo

wsthislin

e,can

besuspected

tobean

artifactcreated

byprocessin

gandmay

notbephysically

realistic.[ 17]

Theshift

ofthepeak

toward

lower

Hcis

somew

hat

similar

tothetren

dseen

byMuxw

orth

yandDunlop[2002]

over

asim

ilarT

range

fortheir

finest

(0.3

mm)PSD

magnetite

and

was

interp

retedby

them

asmark

ing

aprogressiv

echangefro

mSD-lik

eto

MD-lik

ebehavior.

However,

whereas

their

contours

spread

along

the+Hu

and

!Huaxes

inasymmetrical

fashion

athigh

T(a

characteristic

MDpattern

),oursrem

ainhighly

asymmetrical

even

at580!C

.There

isnospread

ingat

allalo

ngthe+Hu

axis.

Inthe!Hudirectio

n,thecontours

areactu

allycom-

pressed

,while

abruptbendsin

these

contours

defin

ea

largelysep

arated!Hulobeexten

dingatleast

3tim

esfarth

erthan

themain

distrib

utio

n.Asthe!Hulobemoves

with

the

main

spectru

mto

lower

Hcat

553!C

and

580!C

,the

flankingnegativ

epeak

isannihilated

.

5.

Analysis

andDiscu

ssion

[18]

Ozdem

ir[1990]foundthatM

rs /Msofthismaghem

iterem

ained

close

to0.5

forT

"500!C

,reach

ed0.45

by

Figure

1.

First-o

rder

reversal

curves

at20K.

Figure

2.

Low-tem

peratu

reFORC

diag

rams:(a)

300K,(b)200K,(c)

100K,(d)20K

(SF=2).

B04105

CARVALLO

ETAL.:FORC

DIA

GRAMSOFELONGATED

SIN

GLE-D

OMAIN

GRAIN

S

3of8

B04105

accentuated

inthethree

lowest

contours,

thecen

traldistri-

butio

nbein

gclo

seto

circular.

The!Hulobeis

flanked

atlow

Hcby

asm

allnegativ

epeak

,which

persists

atall

temperatu

resupto

700K

(428!C

,Figure

3d)andseem

sto

bereal.

Thepattern

isrem

iniscen

tofthedistrib

utio

nfound

theoretically

byPike

etal.[1999]usin

gamovingPreisach

model

with

both

local

interactio

nfield

s[Neel,

1954]anda

mean

interactio

nfield

proportio

nal

tothenet

sample

mag-

netizatio

n.Thismodel

ismore

physically

realisticthan

the

local

interactio

nfield

model.

Hejd

aand

Zelin

ka[1990]

showed

that

most

oftheasy

mmetry

seenin

theclassical

Preisach

interp

retationcan

beacco

unted

forbythemoving

Preisach

model.

However,

inPikeet

al.’sdistrib

utio

nthe

negativ

epeak

liesim

mediately

belo

wthemain

positiv

epeak

andforces

contoursinward

,whereas

ournegativ

epeak

isoffset

diag

onally,

andthemain

peak

contours

‘‘spill’’

aroundit.

Thusitisuncertain

wheth

erthepersisten

tpattern

inourFORC

distrib

utio

nsis

evidence

ofmean

-fieldinter-

actionsornot.

[ 15]

FORC

distrib

utio

nsdeterm

ined

atandaboveroom

temperatu

reappear

inFigure

3.TheHcscale

has

been

expanded

30%

relativeto

theHuscale

inFigures

3a–3d(th

escales

areidentical

alongthetwoaxes

inFigures

3eand3f

andin

Figure

2).AsTincreases

from

150!C

to428!C

,the

tailing

ofthe

micro

coerciv

itydistrib

utio

nto

high

Hc

beco

mes

more

mark

ed.Thehigh-H

clobenow

inclu

des

thecen

tralcontours,

notjusttheouter

ones,

andtheoverall

asymmetry

inf(H

c )ismoreaccen

tuated

.Beginningwith

the

312!C

data,

thepeak

ofthespectru

mmoves

tolower

Hc ,

eventually

merg

ing

with

theHuaxis

at580!C

.Athigh

temperatu

res,more

smoothing

isnecessary

becau

sethe

magnetizatio

nis

much

weak

er.Figures

3g–3ishow

the

effectofsm

oothingonthese

more

noisy

data.

[16]

Inanumber

ofother

FORCstu

dies

thediag

onallin

ein

thenegativ

eHureg

ionstartin

gfro

mtheorig

inofthe

FORCdiag

ramoften

showsaparticu

larpattern

.Forexam

-ple,

alternatin

gpositiv

eandnegativ

epeak

salo

ngthis

line

aresometim

esobserv

ed,in

both

modeled

andmeasu

redFORC

diag

rams[e.g

.,Carva

lloet

al.,

2003].In

modelin

g,

thelin

eisdueto

thepertu

rbatio

nsintro

duced

inthemodel

totest

thestab

ilityandwhich

cause

theBark

hausen

jumps

tooccu

rat

differen

tfield

son

eachrev

ersalcurve.

At

temperatu

resless

than

428!C

the!Hulobe,which

roughly

follo

wsthislin

e,can

besuspected

tobean

artifactcreated

byprocessin

gandmay

notbephysically

realistic.[ 17]

Theshift

ofthepeak

toward

lower

Hcis

somew

hat

similar

tothetren

dseen

byMuxw

orth

yandDunlop[2002]

over

asim

ilarT

range

fortheir

finest

(0.3

mm)PSD

magnetite

and

was

interp

retedby

them

asmark

ing

aprogressiv

echangefro

mSD-lik

eto

MD-lik

ebehavior.

However,

whereas

their

contours

spread

along

the+Hu

and

!Huaxes

inasymmetrical

fashion

athigh

T(a

characteristic

MDpattern

),oursrem

ainhighly

asymmetrical

even

at580!C

.There

isnospread

ingat

allalo

ngthe+Hu

axis.

Inthe!Hudirectio

n,thecontours

areactu

allycom-

pressed

,while

abruptbendsin

these

contours

defin

ea

largelysep

arated!Hulobeexten

dingatleast

3tim

esfarth

erthan

themain

distrib

utio

n.Asthe!Hulobemoves

with

the

main

spectru

mto

lower

Hcat

553!C

and

580!C

,the

flankingnegativ

epeak

isannihilated

.

5.

Analysis

andDiscu

ssion

[18]

Ozdem

ir[1990]foundthatM

rs /Msofthismaghem

iterem

ained

close

to0.5

forT

"500!C

,reach

ed0.45

by

Figure

1.

First-o

rder

reversal

curves

at20K.

Figure

2.

Low-tem

peratu

reFORC

diag

rams:(a)

300K,(b)200K,(c)

100K,(d)20K

(SF=2).

B04105

CARVALLO

ETAL.:FORC

DIA

GRAMSOFELONGATED

SIN

GLE-D

OMAIN

GRAIN

S

3of8

B04105

(a)

(b)

(c)

(d)

Hsat

P

+MS

-MS

H+Hc-Hc

+MS

-MS

HbHaHu

+MS

-MS

H

+MS

-MS

H+Hc-Hc

(a) (b)

(c) (d)

+Hc-Hc

a b

b

a b

Hu

Ha

Hb

ab

ab

ab Preisach space

FORC space

Hc

a

a b

!"Hc!0,Hb#!$"Hc#1

2 ! lim

H!Hr

"

%M "H ,Hr#

%H "#Hr!H

b

.

"13#

The derivative in Eq. "13# is just the reversible magnetizationon the descending major hysteresis loop at applied field

Hb . It should be noted that for H#Hr , or equivalently Hc

#0, the FORC distribution is equal to zero.

III. DEMONSTRATION

We next demonstrate the application of these extended

FORC datasets to experimental data with a sample of a Sony

high-density floppy disk magnetic medium. The exact com-

position of this medium is proprietary, but the magnetic com-

ponent consists of fine &-Fe2O3 single-domain particles. Themagnetization of the data has been normalized so that Ms

!1. The FORC diagram for this sample is shown in Fig.

3"a#, in the 'Hc ,Hb( coordinates. In the contour shadinglegend, Max denotes the value of the FORC distribution at

its ‘‘irreversible’’ peak "located at roughly Hc!90 mT). The! distribution goes to zero at the upper, bottom, and right

hand boundaries of the FORC diagram. The shading at these

boundaries corresponds to !)0 and shadings lighter thanthis represent negative regions of ! , as indicated in the con-tour shading.

The FORC diagram in Fig. 3"a# shows a sharply peakedridge on the Hc!0 axis. This ridge is just the $ function inEq. "13#, although it has been smoothed somewhat by thelocal polynomial fit described earlier. If the resolution of the

dataset were increased, this ridge would approach a $ func-tion. Since this ridge is due to the presence of reversible

magnetization, we will refer to it as the ‘‘reversible’’ ridge.

We should note that the high density of vertical contour lines

near the Hc!0 axis in Fig. 3"a# makes the shading of thereversible ridge appear somewhat darker than it really is. The

horizontal cross section at Hb!$5 mT in Fig. 3"b# gives abetter measure of the magnitude of this ridge.

The FORC diagram in Fig. 3 also shows two somewhat

surprising features: If the system has a reversible magnetiza-

tion of the form Mrev(H), then Mrev(H) should be an odd

function of H and therefore the ridge should be a symmetric

function of Hb . But the vertical cross section though the

reversible ridge "at Hc!0) in Fig. 3"c# shows that the weightof the ridge as a function of Hb is nonsymmetric about Hb

!0. A second surprising feature is a negative region in Fig.3"a# adjacent to the vertical axis in the vicinity of Hb

!$85 mT. To help us interpret these two features of theFORC diagram in Fig. 3, we next look at a simple model.

Let us begin our modeling work by defining the ‘‘square’’

FIG. 2. On a field plot, each FORC is plotted on a horizontal

line with vertical position equal to Hr . Each data point on a FORC

appears at a horizontal coordinate equal to the applied field H at that

data point. Our datasets make up a square grid on a field plot. An

actual dataset would include thousands of data points. The FORC

distribution at a point P is obtained with a local polynomial fit on a

5%5 square grid centered at P, as indicated above.

FIG. 3. "a# FORC diagram for Sony floppy disk sample, show-

ing the reversible ridge at Hc!0. In the contour shading legendabove the diagram, Max denotes the value of the FORC distribution

at its ‘‘irreversible’’ peak "located at about Hc!90 mT). A negativeregion occurs adjacent to the vertical (Hc!0) axis at about Hb!$85 mT. Note that the high density of vertical contour lines nearthe Hc!0 axis makes the shading there appear darker than it reallyis. "b# A horizontal cross section passing though the irreversible

peak at Hb!$5 mT. The ridge at Hc!0 can also be seen in thisplot. "c# A vertical cross section through the reversible ridge at Hc

!0.

FIRST-ORDER REVERSAL-CURVE DIAGRAMS AND . . . PHYSICAL REVIEW B 68, 104424 "2003#

104424-3

Hu

(m

T)

Hu (mT)

Hu

Chantrell 1992)

*E :;

;:

( ) ! koVMsHk

21+

H

Hk

! "2

, with "Hk < H < Hk ,

(2)

and where the moment will have a negative orientation forH<xHk, and a positive orientation for H>Hk.The Neel–Arrhenius law states that the magnetic moment

of a ferromagnetic single-domain particle will pass over anenergy barrier DE at the rate foexp[–DE/(kBT)], where kB is theBoltzmann constant, T is the absolute temperature and fo(sometimes referred to as an attempt frequency) is estimated tolie between 108 and 1013 (Xiao et al. 1986;Moskowitz et al. 1997).Let us introduce the dimensionless applied field hwH/Hk andC(h)wexp[xmoVMsHk(1+h)2/2kBT]. For x1<h<1, the rateof change of M will be

_M ! "#1$M%fo!&h' $ #1"M%fo!&"h'

! fo#"Mf!&h' $ !&"h'g$ f!&"h' " !&h'g% : (3)

With an initial magnetization Mo, eq. (3) has the solution

M#t% !Mo exp#"fotf!&h' " !&"h'g%

$ &1" exp#"fotf!&h' " !&"h'g%' !&"h' " !&h'!&h' $ !&"h'

# $: (4)

Let us next index the points in a set of FORC data by{i, j}, where i denotes the position of a FORC within a setof FORCs, and j denotes the position of a data point on anindividual FORC, and where {i, j=1} is the reversal pointon the ith FORC. Let hi, j and Mi, j denote the applied fieldand magnetization at the {i, j} data point. We then havehi, j= h1,1+ [(ix 1)x ( jx 1)]FS, where FS is the field stepbetween successive measurements. FS and h1,1 now determinethe entire data set.In the following model, the system will spend a time tm at

each point on a FORC, where tm is the measurement time. Themagnetization at the end of this time will be taken as themagnetization at the corresponding data point. We will treatthe applied field as if it instantaneously jumps from one fieldvalue to the next on a FORC. However, we need to considerexplicitly that some amount of time is necessary to ramp the

Hu

Hc

0 100 200 300 400 500

-60

0

60

Hu

H c

Figure 4. (a) FORC diagram for the red-bed sample 90-VAU-42 (tm=0.7 s). (b) High-resolution FORC diagram for the lower left-hand portion ofthe FORC plane for the red-bed sample (tm=1 s; SF=3).

FORC diagrams and thermal relaxation effects 725

# 2001 RAS, GJI 145, 721–730

[36] The coefficients A and B are plotted in Figure 6.A has an asymptote at Hc = 0.5HK. This occursbecause the switching fields are concentrated nearha = !1/2, and the derivative qs0(h) approachesinfinity at h = !1/2 (Figure 3b). A also has anegative region between 0.806HK and HK. Thisfield range corresponds to angles between 86.5!and 90!. This is only part of the range of angles(76.72! to 90!) for which the jump at hs is negative.The narrowing of the range occurs because the sameswitching fields are possessed by particles in acomplementary range of small angles [0!, 22.3!].In these particles the jumps are downward andlarger in magnitude than the upward jumps. How-ever, on a sphere small angles of q occupy less solidangle. Thus downward jumps dominate until theratio between solid angles offsets the ratio of jumpsizes. The negative swing in A is too small to see inFigure 6a.

[37] The component B contains the derivative Dh fand therefore represents differences between slopesof the upper and lower curves of each particle.Since the functions sinqa, cosqa and jqa0 j are alwayspositive, B and Dhf have opposite signs. Near theHu axis, as Hb approaches Ha the slope of the uppercurve approaches infinity while the slope of thelower curve stays finite. Thus B approaches nega-tive infinity. Near the Hc axis the roles are reversedand B approaches positive infinity. This asymptoteis at Hb = !Ha because the jumps in Stoner-Wohlfarth particles are symmetrically placed aboutH = 0. Other kinds of ferromagnets do not have thissymmetry.

[38] B has an asymptote at Ha = !HK/2 for thesame reason that A does. This is a line at a 45!angle with respect to the Hu and Hc axes. Awayfrom this line B has a downward trend.

[39] In summary, the FORC function is concen-trated around three lines that meet at 135! angles.One line (Hc = 0) is potentially significant beyondthe Stoner-Wohlfarth model (section 4.1), one(Hu = 0) reflects a symmetry of Stoner-Wohlfarthhysteresis, and one (Ha = !HK/2) comes from theangular dependence of the switching field in theStoner-Wohlfarth model. The greatest concentra-tion is around the two meeting points at (Hc, Hu) =(0, HK/2) and (Hc, Hu) = (HK/2, 0). The spreadabout these peaks along the ridges gives rise to the‘‘boomerang’’ shape noted by Muxworthy et al.[2004].

[40] As Figure 3a shows, Dh f is antisymmetricabout hb = 0 for all particle orientations. ThereforeB is also antisymmetric about Hb = 0 or Hu = !Hc,as can be seen in Figure 6b. By contrast, A isentirely on one side of this axis.

3.3. Average Over Random Orientations(Anisotropic)

[41] Suppose that some easy axis directions aremore probable than others. In general this anisot-ropy can be described by a probability distributionof the form n(q, f; K), where K is a vector ofdistribution parameters. Since each easy axis hastwo directions, an integral of the FORC functionover any half sphere should give the same averageFORC function. This condition is satisfied if thedistribution is bimodal: n(p ! q, f; K) = n(q, f; K).One such distribution is the Bingham distribution[Mardia, 1972]. In general, an integral over theBingham distribution is messy. To give someinsight into the effect of anisotropy I use a simple

Figure 6. The FORC function (21) for an isotropicsample of identical particles with aspect ratio q = 2. Ithas a continuous part in the region Hu < 0 and a deltafunction along the Hc axis. (a) The delta functionmultiplier A, representing jumps in magnetization.(b) The continuous part B, representing changes in theslope of the magnetization curve. B is zero outside of thecolored region. A narrower color scale is used than inFigure 1b.

GeochemistryGeophysicsGeosystems G3G3 newell: forc 10.1029/2004GC000877

7 of 14

[47] The FORC function is obtained by integratingthe function for identical particles (9) over N:

m Ha;Hb;S! " # 1

Ms

Z 1

0

~mHa

NMs;Hb

NMs

! "

r N ;S! " dNN2

: !26"

[48] The integrand is only nonzero for N1 $ N $N2, where N1 = jHaj/Ms and N2 = Min(1/2, 2jHaj/Ms) (see Appendix A). The new coefficients are

A Ha;S! " #Z N2

N1

aHa

NMs

! "

r N ;S! " dNN

!27a"

B Ha;Hb;S! " #Z N2

N1

bHa

NMs;Hb

NMs

! "

r N ;S! " dNN2

: !27b"

[49] The theory in this article applies only toparticles with uniaxial anisotropy, but many geo-logically interesting materials such as magnetitehave a cubic magnetocrystalline anisotropy. Fortu-nately, the cubic anisotropy can be neglected if theparticles are elongated. For such particles it is bestto start with a pdf for the particle aspect ratio q anduse it to derive the pdf for the demagnetizing factor(Appendix A). The integral is evaluated usingadaptive Gauss/Lobatto quadrature [Gander andGautschi, 2000] with a relative accuracy of 10%6

or better.

[50] Many of the properties of a system of identicalparticles are still true of a system with a shapedistribution. For example, in a set of identicalparticles, B(Ha, %Hb) = %B(Ha, Hb). In (27) thisimplies that B(Ha, %Hb; S) = %B(Ha, Hb; S). Thusthe antisymmetry of B about Hu = %Hc ispreserved. Also, the positive and negative peakshave the same relative weights as in section 3.3.

[51] The arguments are plotted in Figure 8 for alognormal distribution of aspect ratios with mean!q = 2, corresponding to a demagnetizing factorN0(!q) = 0.24. Equation (27b) predicts that theshape distribution will spread the FORC functionequally in all directions, and this is consistent withFigure 8. One result of this spreading is that theslanted ridges are replaced by humps that areroughly symmetric about the peaks.

4. Discussion

[52] This model makes two kinds of predictionsabout FORC functions of uniaxial SD particles.Some predictions follow directly from the proper-ties of the single-particle hysteresis loops. Theseinclude the negative region near the Hu axis, the

positive region near the Hc axis, the delta functionon the Hc axis, and the equal distances of thesepeaks from the origin. Also, the FORC function isidentically zero for Hu > 0. These predictions arerobust because they do not depend on the distribu-tions of particle orientations or shapes. The otherkind of prediction does depend on these distribu-tions. The distribution of particle orientations haslittle effect on the shapes of the positive andnegative peaks, but it has a strong effect on therelative size of these peaks. The distribution ofparticle shapes affects the shapes of the peaksequally, with a realistic distribution tending tosmear them out and remove the ridge betweenthe peaks.

[53] In this section I discuss two robust predictionsand their significance. The first, a negative peaknear the Hu axis, is one of the surprises to come outof plots of experimental FORC functions. I clarifyits physical significance in section 4.1. The secondis that the function is identically zero for Hu > 0.

Figure 8. The components of the FORC function foran isotropic sample with a lognormal distribution ofaspect ratios (q = 2 and s = 0.25). (a) A. The functionfor identical particles with aspect ratio q is shown as adashed line. (b) B. The same color scale is used as inFigure 1b.

GeochemistryGeophysicsGeosystems G3G3 newell: forc 10.1029/2004GC000877

9 of 14

(c)

(d)

Figure 1 on a sample from Lake Ely (Pennsylvania)[Kim et al., 2005].

3.2. Selecting Suitable FORCMeasurement Parameters

[15] Suitable measurement parameters must bechosen to correctly resolve the FORC signaturesof UNISD particles. The FORC acquisition pro-cedure is automated by the Micromag

1

softwarethat controls the VSM or the AGM. At the start ofthe experiment, the user is prompted to input: thesaturating field Hs, the Hu range (given by ‘‘Hb1’’and ‘‘Hb2’’), the Hc range (given by ‘‘Hc1’’ and‘‘Hc2’’), the averaging time, the field incrementdH, the number N of FORC curves to be mea-sured, and other parameters that are not discussedhere. Best choice of the measurement parametersdepends on the sample. Given the low concentra-tion of magnetic minerals in typical sediments, it isimportant to select the smallest possible measure-ment range. Typical averaging times are between

0.2 and 1 s. Increasing the averaging time helps toreduce measurement noise, except for the noisederiving from the electromagnets, but it alsoincreases instrumental drift effects. Therefore,averaging multiple FORC runs is more effectivethan increasing the averaging time in case of partic-ularly weak samples. Care should be taken to avoiddrift artifacts, which are particularly pronouncedduring the first 20 min of instrument operation andare not completely removable by data processing.

[16] The most critical parameters are the Hc and Hu

ranges, which determine the FORC space coveredby the measurement, and the field increment dH(Figure 2). If Hc is the largest switching field ofinterest, and if the FORC diagram is expected toextend by Hu above the Hc axis, sufficient FORCspace coverage with minimum amount of measure-ments is obtained by choosing Hc1 = 0, Hc2 ! Hc,Hb1 = "Hc " Hu, and Hb2 = Hu. The last twoimportant measurement parameters are the fieldincrement dH and the number N of FORC curves.

Figure 1. High-resolution FORC diagram for a sediment sample from Lake Ely (Pennsylvania). Note the one orderof magnitude difference between the amplitude of the central ridge and the remaining part of the diagram. The colorscale is chosen so that zero is white, negative values are blue, and positive values are yellow to red. Contour lines aredrawn for values specified in the color scale bar. Measurements are not normalized by mass.

GeochemistryGeophysicsGeosystems G3G3 egli et al.: detection of noninteracting sd particles 10.1029/2009GC002916

5 of 22

factor(SF

)and

isgiven

by(2SF

+1) 2.

For

exam-

plefor

SF=3,

thesm

oothingis

performed

acrossa7U

7array

ofdata

points.The

magnetization

atthese

pointsis

then¢tted

with

apolynom

i-al

surfaceof

theform

:a1 +

a2 H

a +a3 H

2a +a4 H

b +a5 H

2b +a6 H

a Hb ,

where

thevalue

3a6represents

b(Ha ,H

b )at

P.Taking

thesecond

derivativein

Eq.1

magni¢es

thenoise

thatisinevitably

presentin

themagnetization

measurem

ents.Therefore,

FORC

diagramsproduced

with

SF=1

containsgreater

noise.This

canbe

reducedby

increasingthe

sizeof

SF;how

ever,thecost

ofincreasing

SFisthat

¢nescale

featuresdisappear.In

addition,incalculating

theFORC

distribution,no

pointsare

determined

inthe

regionbetw

eenthe

HUaxis

and2U

SFU

FS(FS=¢eld

spacingduring

theFORC

measurem

ent)and

itis

necessaryto

make

anex-

trapolationof

theFORC

surfaceonto

theH

Uaxis.

IncreasingSF

increasesthe

errorin

thisex-

trapolation.The

FORC

distributionof

anassem

blageof

non-interactingSD

particlesis

narrowly

con¢nedto

thecentral

horizontalaxis

[1,8].Magnetostatic

interactionsbetw

eenSD

grainscauses

verticalspread

ofthe

contoursabout

thepeak,w

hilether-

mal

relaxationof

¢neSD

particlesshifts

theFORC

distributionto

lower

coercivities[8,9].

Incontrast

MD

FORC

distributionshave

nocentral

peak,and

thecontours

tendto

spreadbroadly

parallelto

theH

U=0axis

[1,10].

5.Room

-temperature

resultson

PSD

andMD

magnetite

Room

-temperature

FORC

diagramsare

shown

forthe

fourWright

samples

inFig.3,and

forthe

threehydrotherm

allygrow

nsam

plesin

Fig.

4.The

FORC

distributionschange

markedly

with

grainsize.

Samples

W(0.3

Wm)and

W(1.7

Wm)

displaydistinct

closed-contourpeaks

between

25and

50mT

inthe

FORC

distribution,while

thepeak

ofthe

FORC

distributionlie

nearthe

originfor

thelarger

samples.

The

cross-sectionof

theFORC

distributionalong

theH

Caxis

isplotted

inFig.

5.According

tothe

Preisach^Ne ¤el

theory,this

plotis

thecoercivity

distributiong(H

C ).This

distributionis

seento

evolvecontinuously

with

grainsize

(Figs.

3^5).The

hydrothermally

grown

samples

(Fig.

4)display

more

MD-like

FORC

diagramsthan

theWright

samples

(Fig.

3)for

samples

with

similar

grainsizes,

e.g.W(7

Wm)

andH(7.5

Wm).T

hisre£ects

di¡erencesin

internalstress

anddislocation

densities.Pro¢les

ofthe

FORC

distributionsin

theH

Udirection

graduallybecom

ebroader

and£atter

with

increasinggrain

size.The

behaviorof

thelarger

grainsis

consistentwith

observationson

MD

grains[10].

This

changere£ects

thedi¡eren-

cesbetw

eenPSD

(grainscontaining

onlya

fewless

mobile

walls)

andMD

(grainscontaining

many

mobile

walls).

Inaddition

allthe

FORC

Fig.

3.Room

-temperature

FORC

diagramsfor

thefour

Wright

samples.

Scalingfactors:

(a)SF

=2,

(b)SF

=3,

(c)SF

=2and

(d)SF

=2.

EPSL

639225-9-02

A.R.Muxw

orthy,D.J.

Dunlop

/Earth

andPlanetary

Science

Letters

203(2002)

369^382373

ployed to study details of the magnetization reversal. Aftersaturation, the magnetization M is measured starting from areversal field HR back to positive saturation, tracing out aFORC. A family of FORC’s is measured at different HR, withequal field spacing, thus filling the interior of the major hys-teresis loop !Figs. 2"a#–2"c#$. The FORC distribution is de-fined as a mixed second order derivative17–21

!"HR,H# % !12

!2M"HR,H#!HR!H

, "1#

which eliminates the purely reversible components of themagnetization. Thus any nonzero ! corresponds to irrevers-ible switching processes.19–21 The FORC distribution is plot-ted against "H ,HR# coordinates on a contour map or a three-dimensional "3D# plot. For example, along each FORC inFig 4"a# with a specific reversal field HR, the magnetizationM is measured with increasing applied field H; the corre-sponding FORC distribution ! in Fig. 4"b# is represented bya horizontal line scan at that HR along H. Alternatively ! canbe plotted in coordinates of "HC ,HB#, where HC is the localcoercive field and HB is the local interaction or bias field.This transformation is accomplished by a simple rotation ofthe coordinate system defined by: HB= "H+HR# /2 and HC= "H!HR# /2. Both coordinate systems are discussed in thispaper.

III. RESULTS

Families of the FORC’s for the 52, 58, and 67 nm nan-odots are shown in Figs. 2"a#–2"c#. The major hysteresisloops, delineated by the outer boundaries of the FORC’s,exhibit only subtle differences. The 52 nm nanodots show a

FIG. 1. Scanning electron micrograph of the 67 nm diameternanodot sample. Inset is a histogram showing the distribution ofnanodot sizes.

FIG. 2. First-order reversal curves and the corresponding distributions. Families of FORC’s, whose starting points are represented byblack dots, are shown in "a#–"c# for the 52, 58, and 67 nm Fe nanodots, respectively. The corresponding FORC distributions are shown inthree-dimensional plots "d#–"f# and contour plots "g#–"i#.

DUMAS et al. PHYSICAL REVIEW B 75, 134405 "2007#

134405-2

C.R.Pike

etal./P

hysics

oftheEarth

andPlaneta

ryInterio

rs126(2001)11–25

17

Fig.6.FORCdiagram

s(SF

=3)for(a)

asam

pleofM80tran

sformersteel;

(b)a2mmgrain

ofmagnetite,

afterannealin

g;(c)

thesam

e

2mmgrain

ofmagnetite

beforeannealin

g;and(d)a125

!mmagnetite

grain

(noannealin

g)fromsam

pleHM4(see

Hartstra,

1982).

contourpattern

shavealso

been

observ

edinPreisach

diagram

sfornatural

MDsam

ples

(e.g.Mullinsand

Tite,

1973;Ivanovetal.,

1981;IvanovandSholpo,

1982;Zelin

kaetal.,

1987;HejdaandZelin

ka,1990;

Dunlopetal.,

1990;Fabian

andvonDobeneck,1997).

Roberts

etal.(2000)showed,empirically,

thatnatural

sampleslocated

furtherintheMDdirectio

nonaDay

plot(Dayetal.,

1977)haveFORCdistrib

utionswith

largerdegrees

ofdivergence.

Fig.7.FORCdiagram

(SF

=4)foranassem

blageofMDparticles

insam

pleODP887B-2H-6-70fromtheNorth

Pacifi

cOcean

(see

Roberts

etal.,

1995).

Themeasu

redFORC

diagram

forourM80

transformersteel

sample(Fig.6(a))

isconsisten

twith

theanalytical

resultofBerto

ttietal.(1999a)forDW

pinningwith

aWLprocess

(Fig.4(b)).Thatis,the

diagram

consists

ofvertical

contours,with

aFORC

distrib

utionfunctio

nthatdecreases

with

increasin

g

Hc .Thisresu

ltindicates

thatthesim

pleclassical

model,alth

oughitisbased

onadubiousphysical

pic-

ture,somehowcaptures

thephysics

ofthehysteresis

mechanism

sinthissam

ple.TheFORCdistrib

ution

forthe2mm-sized

magnetite

grain

afterannealin

g

(Fig.6(b))also

hasvertical

contoursconsisten

twith

thoseofthetran

sformersteel

andtheresu

ltofBerto

tti

etal.(1999a).However,theFORCdiagram

forthe

2mmgrain

beforeannealin

g(Fig.6(c))

isinconsisten

t

withtheresu

ltofBerto

ttietal.(1999a)andtheFORC

diagram

forthe125

!mmagnetite

(Fig.6(d))isin-

termediate

between

theresu

ltsfortheannealed

and

unannealed

2mmmagnetite

sample.

Theinconsis-

tencybetween

theresu

ltsfortheannealed

andunan-

nealed

samplesimplies

thatstress

mightberesp

onsible

forthedeviatio

n.Exactly

howstress

givesrise

tothis

pattern

isunknown.Wesuggest

thatintheannealed

state,thepinningsites

arehomogeneouslydistrib

uted

throughoutthesam

pleinamannerthatisconsisten

t

witharandomprocess.

Intheunannealed

state,how-

ever,with

stresspresen

t,“pinning”mightoccuron

to the substrate. The magnetization was normalized by the

saturation magnetization. The FORC distribution generated

from this data is shown in Fig. 5!b". Again, Max denotes thevalue of ! at the “irreversible peak,” which is located in thecase at about Hc=23 mT, Hb=20 mT. A large reversible

ridge can be seen on the Hc=0 axis. The most prominent

feature of the FORC distribution, aside from the reversible

ridge, is a two branch “wishbone” structure. The vertical

cross-section through this ridge is shown in Fig. 5!c". Twonegative “valleys” can be also be seen in Fig. 5!b": one athigh coercivity just below the Hb=0 axis, and another adja-

cent to the reversible ridge.

In the following sections we develop a qualitative under-

standing of the physical mechanisms which give rise to the

features of this measured FORC signature.

IV. MEAN FIELD MODELING

We next show that the wishbone structure of the FORC

signature in Fig. 5!b" can be qualitatively accounted for us-ing a interacting hysteron model with a negative !antiparal-lel" mean field and distributed coercivities. Let us begin witha collection of N square and symmetric !zero bias" hysterons.The state of the ith hysteron is denoted by si, which can take

values of ±1. The pillars in this array do not have a perfectly

uniform shape and size; they also likely contain a high den-

sity of defects, vacancies, and inhomogeneities. Therefore a

distribution of coercivities is expected. The coercivity of the

ith hysteron is denoted by hic. The distribution of coercivities

is denoted by f!hc". The total normalized magnetization ofthe system is given by

M =# si/N . !5"

Since the magnetization of the nickel pillars is dominantly

oriented perpendicular to the plane of the substrate, then the

dipolar interaction is antiparallel to the direction of the mag-

netization. In this section we will represent this interaction

by a mean interaction field written as

Hint = ! JM , !6"

where J is the magnitude of the total interaction field seen by

the hysterons in the saturated state. The total field is the sum

of the externally applied field H and Hint. In our algorithm

for calculating FORCs, the applied field H was initially

given a large value and the si were all set to +1. Then H is

lowered in small “field steps”. To obtain robust numerical

results with interacting systems, the size of the field steps

should be much smaller than the width of the coercivity dis-

tribution. Note that a “field step” is distinct from a “field

FIG. 4. Scanning electron micrograph of pillar sample.

FIG. 5. !a" The first-order reversal curve !FORC" data for nickelpillar sample. To make it easier for the eye to resolve individual

curves, only 70 of the 140 measured FORCs are shown. !b" TheFORC distribution generated from this data. Max denotes the value

of the distribution at the “irreversible” peak located at about Hc

=23 mT, Hb=20 mT. On the Hc=0 vertical axis is a sharply peak

ridge due to reversible magnetization.!c" The vertical cross sectionthrough the “reversible” ridge at Hc=0 as a function of Hb.

PIKE et al. PHYSICAL REVIEW B 71, 134407 !2005"

134407-4

Hu

100 and 70 K, and stays nearly constant down to 5 K,where the remanence gained is nearly 4% compared tothe SIRM300K. Upon warming curves (5Y300 K),

the remanence shows reversible behavior until 70–80K with respect to the cooling curve, and partiallyincreases below 100–130 K, then gradually decreases

0 0.5 1 1.5 2x 10-10

10 20 30 40 50 60 70 80 90

-50

-40

-30

-20

-10

0

10

20

30

Hc [mT]

Hc [mT]

Hb

[m

T]

Hb

[m

T]

0 0.2 0.4 0.6 0.8 1-60

-50

-40

-30

-20

-10

0

10

20

30

40

0 2 4 6 8 10 12x 10-8

10 20 30 40 50 60 70

-40

-30

-20

-10

0

10

20

-10

0 0.2 0.4 0.6 0.8 1-50

-40

-30

-20

0

10

20

30(b)

Hb1/2 = 6.3 mT

Hb1/2 = 8.3 mT

(a)

Fig. 4. FORC diagram of MTB sample P3 (a) and single-domain magnetite powder sample (b), both derived with a smoothing factor of 2. The

FORC distribution of the MTB sample (a) is bimodal with a broad maximum centered at 42 mT and a sharper peak towards the Hc=0 axis. The

latter is attributed to nascent magnetosomes as they typically occur at the chain ends. Insets in (a) and (b): Vertical profile through the high-

coercivity peak of the distribution rendering a measure of the characteristic interaction strength, Hb1/2, defined as the value of the interaction

field where the FORC distribution has reduced to half of its maximum value. Mean half-width field Hb1/2=6.3 mT at Hc=41.4 mT (a), mean

Hb1/2=8.3 mT at Hc=17.8 mT (b).

Y. Pan et al. / Earth and Planetary Science Letters 237 (2005) 311–325318

Hu