WebBook3.4tc

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\doTocEntry\toclikesubsection{}{\csname a:TocLink\endcsname{1}{x1-1000}{QQ2-1-1}{Preface}}{5}\relax 
\doTocEntry\tocsubsection{}{\csname a:TocLink\endcsname{1}{Q1-1-2}{}{Preface}}{v}\relax 
\doTocEntry\toclikechapter{}{\csname a:TocLink\endcsname{2}{x2-2000}{QQ2-2-3}{Contents}}{13}\relax 
\doTocEntry\tocchapter{1}{\csname a:TocLink\endcsname{3}{x3-30001}{QQ2-3-4}{The physics of magnetism}}{5}\relax 
\doTocEntry\tocsection{1.1}{\csname a:TocLink\endcsname{3}{x3-40001.1}{QQ2-3-5}{What is a magnetic field?}}{5}\relax 
\doTocEntry\toclof{1.1}{\csname a:TocLink\endcsname{3}{x3-40011}{}{\ignorespaces  a) Distribution of iron filings on a flat sheet pierced by a wire carrying a current $i$. [From Jiles, 1991.] b) Relationship of magnetic field to current for straight wire. }}{figure}\relax 
\doTocEntry\tocsection{1.2}{\csname a:TocLink\endcsname{3}{x3-50001.2}{QQ2-3-7}{Magnetic moment}}{8}\relax 
\doTocEntry\toclof{1.2}{\csname a:TocLink\endcsname{3}{x3-50022}{}{\ignorespaces a) Iron filings show the magnetic field generated by current flowing in a loop. b) A current loop with current $i$ and area $\pi r^2$ produces a magnetic moment $\mathbf@@ {m}$. c) The magnetic field of loops arranged as a solenoid is the sum of the contribution of the individual loops. [Iron filings pictures from Jiles, 1991.] }}{figure}\relax 
\doTocEntry\tocsection{1.3}{\csname a:TocLink\endcsname{3}{x3-60001.3}{QQ2-3-9}{Magnetic flux}}{12}\relax 
\doTocEntry\toclof{1.3}{\csname a:TocLink\endcsname{3}{x3-60013}{}{\ignorespaces a) A magnetic moment $\mathbf@@ {m}$ makes a vector field $\hbox {\bf  B}$. The lines of flux are represented by the arrows. [Adapted from Tipler, 1999.] b) A magnetic moment $\mathbf@@ {m}$ makes a vector field $\hbox {\bf  B}$ made visible by the iron filings. If this field moves with velocity $\mathbf@@ {v}$, it generates a voltage $V$ in an electrical conductor of length $l$. [Iron filings picture from Jiles, 1991.]}}{figure}\relax 
\doTocEntry\tocsection{1.4}{\csname a:TocLink\endcsname{3}{x3-70001.4}{QQ2-3-11}{Magnetic energy}}{16}\relax 
\doTocEntry\toclof{1.4}{\csname a:TocLink\endcsname{3}{x3-70014}{}{\ignorespaces The magnetic moment $\hbox {\bf  m}$ of, for example, a compass needle, will tend to align itself with a magnetic field $\hbox {\bf  B}$. a) Example of when the field is produced by a current in a wire. b) The aligning energy is the magnetostatic energy which is greatest when the angle $\theta $ between the two vectors of the magnetic moment $\hbox {\bf  m}$ and the magnetic field $\hbox {\bf  B}$ is at a maximum. }}{figure}\relax 
\doTocEntry\tocsection{1.5}{\csname a:TocLink\endcsname{3}{x3-80001.5}{QQ2-3-13}{Magnetization and magnetic susceptibility}}{19}\relax 
\doTocEntry\tocsection{1.6}{\csname a:TocLink\endcsname{3}{x3-90001.6}{QQ2-3-14}{Relationship of $\hbox {\bf  B}$ and $\hbox {\bf  H}$}}{20}\relax 
\doTocEntry\tocsection{1.7}{\csname a:TocLink\endcsname{3}{x3-100001.7}{QQ2-3-15}{A brief tour of magnetic units in the cgs system}}{20}\relax 
\doTocEntry\toclof{1.5}{\csname a:TocLink\endcsname{3}{x3-100015}{}{\ignorespaces The force between two magnetic monopoles $p_1,p_2$ is $p_1p_2\o:over: {r^2}$. \hfill }}{figure}\relax 
\doTocEntry\toclot{1.1}{\csname a:TocLink\endcsname{3}{x3-100041}{}{\ignorespaces Conversion between SI and cgs units.}}{table}\relax 
\doTocEntry\tocsection{1.8}{\csname a:TocLink\endcsname{3}{x3-110001.8}{QQ2-3-18}{The magnetic potential }}{31}\relax 
\doTocEntry\toclof{1.6}{\csname a:TocLink\endcsname{3}{x3-110016}{}{\ignorespaces a) An electric charge produces a field that diverges out from the source. There is a net flux out of the dashed box, quantified by the divergence ($\nabla \cdot \hbox {\bf  E}$), which is is proportional to the magnitude of the sources inside the box. b) There are no isolated magnetic charges, only dipoles. Within any space (e.g., the dashed box) any flux line that goes out must return. The divergence of such a field is zero, i.e., $\nabla \cdot \hbox {\bf  B}= 0$. }}{figure}\relax 
\doTocEntry\toclof{1.7}{\csname a:TocLink\endcsname{3}{x3-110047}{}{\ignorespaces Field $\hbox {\bf  H}$ produced at point P by a magnetic moment $\hbox {\bf  m}$. $\hbox {\bf  H}_r$ and $\hbox {\bf  H}_{\theta }$ are the radial and tangential fields respectively.}}{figure}\relax 
\doTocEntry\tocsection{1.9}{\csname a:TocLink\endcsname{3}{x3-120001.9}{QQ2-3-21}{Origin of the geomagnetic field}}{38}\relax 
\doTocEntry\toclof{1.8}{\csname a:TocLink\endcsname{3}{x3-120018}{}{\ignorespaces Self-exciting disk dynamo. An initial field $B$ is reinforced by dynamo action. When the conducting plate is rotated, electric charge moves perpendicular to the magnetic field setting up an electric potential between the inner conducting rod and the outer rim of the plate. If the conducting plate is connected to a coil wound such that a current produces a magnetic field in the same direction as the initial field, the magnetic field is enhanced. [AfterElsasser, 1958; redrawn from Butler, 1992.]}}{figure}\relax 
\doTocEntry\tocsection{1.10}{\csname a:TocLink\endcsname{3}{x3-130001.10}{QQ2-3-23}{Problems}}{43}\relax 
\doTocEntry\toclof{1.9}{\csname a:TocLink\endcsname{3}{x3-130019}{}{\ignorespaces A magnetic dipole source (${\bf  m}$) located 3480 km from the center of the Earth, below the point of observation at 45$^{\circ }$ latitude. The average radius of the Earth is 6370 km. The field at the point of observation is directed downward (toward the center) with a magnitude of 10 $\mu $T. }}{figure}\relax 
\doTocEntry\tocchapter{2}{\csname a:TocLink\endcsname{4}{x4-140002}{QQ2-4-25}{The geomagnetic field}}{53}\relax 
\doTocEntry\tocsection{2.1}{\csname a:TocLink\endcsname{4}{x4-150002.1}{QQ2-4-26}{Components of magnetic vectors}}{53}\relax 
\doTocEntry\toclof{2.1}{\csname a:TocLink\endcsname{4}{x4-150011}{}{\ignorespaces a) Lines of flux produced by a geocentric axial dipole. b) Lines of flux of the geomagnetic field of 2005. At point P the horizontal component of the field $B_H$, is directed toward the magnetic north. The vertical component $B_V$ is directed down and the field makes an angle $I$ with the horizontal, known as the inclination. c) Components of the geomagnetic field vector $\hbox {\bf  B}$. The angle between the horizontal component (directed toward magnetic north and geographic north is the declination $D$.) [Modified from Ben-Yosef et al., 2008b.] }}{figure}\relax 
\doTocEntry\tocsection{2.2}{\csname a:TocLink\endcsname{4}{x4-160002.2}{QQ2-4-28}{Reference magnetic field}}{58}\relax 
\doTocEntry\toclof{2.2}{\csname a:TocLink\endcsname{4}{x4-160032}{}{\ignorespaces Schmidt polynomials.}}{figure}\relax 
\doTocEntry\toclof{2.3}{\csname a:TocLink\endcsname{4}{x4-160043}{}{\ignorespaces Examples of potential fields (insets) and maps of the associated patterns for global inclinations. Each coefficient is set to 30 $\mu $T. a) Dipole ($g_1^0=30 \mu $T), b) Quadrupole ($g_2^0=30 \mu $T), c) Octupole ($g_3^0=30 \mu $T). }}{figure}\relax 
\doTocEntry\toclot{2.1}{\csname a:TocLink\endcsname{4}{x4-160061}{}{\ignorespaces IGRF, 10$^{th}$ generation (2005) to $l=6$.}}{table}\relax 
\doTocEntry\toclof{2.4}{\csname a:TocLink\endcsname{4}{x4-160074}{}{\ignorespaces  Power at the Earth's surface of the geomagnetic field versus degree for the 2005 IGRF (Table 2.1). }}{figure}\relax 
\doTocEntry\toclof{2.5}{\csname a:TocLink\endcsname{4}{x4-160085}{}{\ignorespaces Maps of geomagnetic field of the IGRF for 2005. a) Intensity (units of $\mu $T), b) inclination, c) potential (units of nT).}}{figure}\relax 
\doTocEntry\tocsection{2.3}{\csname a:TocLink\endcsname{4}{x4-170002.3}{QQ2-4-34}{Geocentric axial dipole (GAD) and other poles}}{76}\relax 
\doTocEntry\toclof{2.6}{\csname a:TocLink\endcsname{4}{x4-170016}{}{\ignorespaces Different poles. The square is the magnetic North Pole, where the magnetic field is straight down $(I = +90^{\circ })$ (82.7$^{\circ }$N, 114.4$^{\circ }$W for the IGRF 2005); the circle is the geomagnetic North Pole, where the axis of the best fitting dipole pierces the surface (9.7$^{\circ }$N, 71.8$^{\circ }$W for the IGRF 2005). The star is the geographic North Pole. [Figure made using Google Earth with seafloor topography of D. Sandwell supplied to Google Earth by D. Staudigel.]}}{figure}\relax 
\doTocEntry\tocsection{2.4}{\csname a:TocLink\endcsname{4}{x4-180002.4}{QQ2-4-36}{Plotting magnetic directional data}}{81}\relax 
\doTocEntry\toclof{2.7}{\csname a:TocLink\endcsname{4}{x4-180017}{}{\ignorespaces  a) Hammer projection of 200 randomly selected locations around the globe. b) Equal area projection of directions of Earth's magnetic field as given by the IGRF evaluated for the year 2005 at locations shown in a). Open (closed) symbols indicate upper (lower) hemisphere. c) Inclinations (I) plotted as a function of site latitude ($\lambda $). The solid line is the inclination expected from the dipole formula (see text). Negative latitudes are south and negative inclinations are up. [Figure redrawn from Tauxe, 1998.]}}{figure}\relax 
\doTocEntry\tocsubsection{2.4.1}{\csname a:TocLink\endcsname{4}{x4-190002.4.1}{QQ2-4-38}{$D', I' $ transformation}}{84}\relax 
\doTocEntry\toclof{2.8}{\csname a:TocLink\endcsname{4}{x4-190028}{}{\ignorespaces a) Vectors evaluated around the globe at 45$^{\circ }$N. Red/green/blue colors reflect the North, East and Down components respectively. b) The unit vectors (assuming unit length) from a). c) Directions from Figure 2.7b transformed using the $D', I'$ transformation.}}{figure}\relax 
\doTocEntry\toclof{2.9}{\csname a:TocLink\endcsname{4}{x4-190039}{}{\ignorespaces Transformation of a vector measured at S into a virtual geomagnetic pole position (VGP) and virtual dipole moment (VDM), using principles of spherical trigonometry and the dipole formula. a) Red dashed line is the magnetic field line observed at S (latitude of $\lambda _s$, longitude of $\phi _s$). This field line is the same as one produced by the VDM at the center of the Earth. The point where the axis of the VDM pierces the Earth's surface is the VGP. b) Observed declination (D) and inclination (converted to $\theta _m$ using the dipole formula (see text) defines angles $D$ and $\theta _m$. $\theta _s$ is the colatitude of the observation site. N is the geographic North Pole (the spin axis of the Earth). The position of the pole at P ($\theta _p,\phi _p$) can be calculated with spherical trigonometry (see text). c) VGP positions converted from directions shown in Figure 2.7b. d) The virtual axial dipole moment giving rise to the observed intensity at S.}}{figure}\relax 
\doTocEntry\tocsubsection{2.4.2}{\csname a:TocLink\endcsname{4}{x4-200002.4.2}{QQ2-4-41}{Virtual geomagnetic poles}}{92}\relax 
\doTocEntry\tocsubsection{2.4.3}{\csname a:TocLink\endcsname{4}{x4-210002.4.3}{QQ2-4-42}{Virtual dipole moment}}{95}\relax 
\doTocEntry\tocsection{2.5}{\csname a:TocLink\endcsname{4}{x4-220002.5}{QQ2-4-43}{Problems}}{96}\relax 
\doTocEntry\tocchapter{3}{\csname a:TocLink\endcsname{5}{x5-230003}{QQ2-5-44}{Induced and remanent magnetism}}{103}\relax 
\doTocEntry\tocsection{3.1}{\csname a:TocLink\endcsname{5}{x5-240003.1}{QQ2-5-45}{Magnetism at the atomic level}}{103}\relax 
\doTocEntry\toclof{3.1}{\csname a:TocLink\endcsname{5}{x5-240011}{}{\ignorespaces Plot of radial distribution and ``dot-density'' for the 1s electron shell.}}{figure}\relax 
\doTocEntry\toclof{3.2}{\csname a:TocLink\endcsname{5}{x5-240042}{}{\ignorespaces Examples of surfaces of equal probability of the first three shells ($l=1,2,3$). Surfaces created with Orbital Viewer. }}{figure}\relax 
\doTocEntry\toclof{3.3}{\csname a:TocLink\endcsname{5}{x5-240063}{}{\ignorespaces Electronic structure of elements from Na to Zn. }}{figure}\relax 
\doTocEntry\tocsection{3.2}{\csname a:TocLink\endcsname{5}{x5-250003.2}{QQ2-5-49}{Induced magnetization}}{115}\relax 
\doTocEntry\toclof{3.4}{\csname a:TocLink\endcsname{5}{x5-250014}{}{\ignorespaces Larmor precession. The orbit of the electron has an angular momentum vector ${\bf  L}$ which creates a magnetic moment. In the presence of a magnetic field $\hbox {\bf  H}$, the moment experiences a torque which causes a change in angular momentum $\Delta L$. The precession of the electronic orbit about $\hbox {\bf  H}$ creates an induced magnetic moment $\Delta m$ in a sense opposite to the applied field $\hbox {\bf  H}$.}}{figure}\relax 
\doTocEntry\tocsubsection{3.2.1}{\csname a:TocLink\endcsname{5}{x5-260003.2.1}{QQ2-5-51}{Orbital contribution and diamagnetism}}{118}\relax 
\doTocEntry\toclof{3.5}{\csname a:TocLink\endcsname{5}{x5-260015}{}{\ignorespaces a) Paramagnetic magnetization (obtained from the Langevin function $\mathcal  {L}(a)$ versus $a= mB/kT$.) b) Paramagnetic magnetization as a function of temperature (Curie Law).}}{figure}\relax 
\doTocEntry\tocsubsection{3.2.2}{\csname a:TocLink\endcsname{5}{x5-270003.2.2}{QQ2-5-53}{Role of electronic spins and paramagnetism}}{122}\relax 
\doTocEntry\toclof{3.6}{\csname a:TocLink\endcsname{5}{x5-270136}{}{\ignorespaces  Exchange energy associated with overlapping orbitals. Example of super-exchange between the $3d$ orbitals of two iron cations through the $2p$ orbitals of the intervening oxygen anion. The two electrons in the $2p$ shells are, by necessity antiparallel. These are shared by the $3d$ shells, hence the two cations have anti-parallel spins. [Figure redrawn from O'Reilly, 1984.]}}{figure}\relax 
\doTocEntry\tocsection{3.3}{\csname a:TocLink\endcsname{5}{x5-280003.3}{QQ2-5-55}{Ferromagnetism}}{129}\relax 
\doTocEntry\toclof{3.7}{\csname a:TocLink\endcsname{5}{x5-280017}{}{\ignorespaces Behavior of magnetization versus temperature of a ferromagnetic substance. Below $T_c$, the magnetization follows Equation 3.9 and is the ferromagnetic magnetization. Above $T_c$ the magnetization follows Equation 3.8 and is the induced magnetization. [Redrawn from Tauxe, 1998.] }}{figure}\relax 
\doTocEntry\toclof{3.8}{\csname a:TocLink\endcsname{5}{x5-280048}{}{\ignorespaces Various data sets for the behavior of $M_s(T)$ for magnetite. }}{figure}\relax 
\doTocEntry\toclof{3.9}{\csname a:TocLink\endcsname{5}{x5-280079}{}{\ignorespaces Types of spin alignment in ferromagnetism {\it  (sensu lato)}: a) ferromagnetism ({\it  sensu stricto}), b) antiferromagnetism, c) spin-canted antiferromagnetism, d) defect anti-ferromagnetism, e) ferrimagnetism. }}{figure}\relax 
\doTocEntry\toclof{3.10}{\csname a:TocLink\endcsname{5}{x5-2800810}{}{\ignorespaces a) Response of a magnetic moment to the torque of an applied field for isolated moments. b) Response of coupled moments to a perturbation. Neighboring spins produce an effect known as ``spin waves''.}}{figure}\relax 
\doTocEntry\tocsection{3.4}{\csname a:TocLink\endcsname{5}{x5-290003.4}{QQ2-5-60}{Problems }}{145}\relax 
\doTocEntry\tocchapter{4}{\csname a:TocLink\endcsname{6}{x6-300004}{QQ2-6-61}{Magnetic anisotropy and domains}}{151}\relax 
\doTocEntry\toclof{4.1}{\csname a:TocLink\endcsname{6}{x6-300011}{}{\ignorespaces a) A magnetite octahedron. [Photo of Lou Perloff in The Photo-Atlas of Minerals.] b) Internal crystal structure. Directions of the body diagonal ([111] direction) and orthogonal to the cubic faces ([001] direction) are shown as arrows. Big red dots are the oxygen anions. The blue dots are iron cations in octahedral coordination and the yellow dots are in tetrahedral coordination. Fe$^{3+}$ sits on the A sites and Fe$^{2+}$ and Fe$^{3+}$ sit on the B sites. c) Magnetocrystalline anisotropy energy as a function of direction within a magnetite crystal at room temperature. The easiest direction to magnetize (the direction with the lowest energy -- note dimples in energy surface) is along the body diagonal (the [111] direction). [Figure from Williams and Dunlop, 1995.] }}{figure}\relax 
\doTocEntry\tocsection{4.1}{\csname a:TocLink\endcsname{6}{x6-310004.1}{QQ2-6-63}{The magnetic energy of particles}}{155}\relax 
\doTocEntry\tocsubsection{4.1.1}{\csname a:TocLink\endcsname{6}{x6-320004.1.1}{QQ2-6-64}{Exchange energy}}{155}\relax 
\doTocEntry\tocsubsection{4.1.2}{\csname a:TocLink\endcsname{6}{x6-330004.1.2}{QQ2-6-65}{Magnetic moments and external fields}}{156}\relax 
\doTocEntry\toclof{4.2}{\csname a:TocLink\endcsname{6}{x6-330022}{}{\ignorespaces Variation of $K_1$ and $K_2$ of magnetite as a function of temperature. Solid lines are data from Syono and Ishikawa (1963). Dashed lines are data from Fletcher and O'Reilly (1974).}}{figure}\relax 
\doTocEntry\tocsubsection{4.1.3}{\csname a:TocLink\endcsname{6}{x6-340004.1.3}{QQ2-6-67}{Magnetocrystalline anisotropy energy}}{160}\relax 
\doTocEntry\toclof{4.3}{\csname a:TocLink\endcsname{6}{x6-340023}{}{\ignorespaces Magnetization curve for magnetite as a function of temperature. The specimen was placed in a very large field, cooled to near absolute zero, then warmed up. The magnetization was measured as it warmed. When it goes through the Verwey transition ($\sim $110 K), a fraction of the magnetization is lost. Data downloaded from ``w5000'' in the ``Rock magnetic Bestiary'' collection at the Institute for Rock Magnetism (\url  {http://www.irm.umn.edu/bestiary}).}}{figure}\relax 
\doTocEntry\tocsubsection{4.1.4}{\csname a:TocLink\endcsname{6}{x6-350004.1.4}{QQ2-6-69}{Magnetostriction - stress anisotropy}}{165}\relax 
\doTocEntry\toclof{4.4}{\csname a:TocLink\endcsname{6}{x6-350014}{}{\ignorespaces a) Internal magnetizations within a ferromagnetic crystal. b) Generation of an identical external field from a series of surface monopoles. c) The internal ``demagnetizing'' field resulting from the surface monopoles. [Redrawn from O'Reilly, 1984]. d) Surface monopoles on a sphere. e) Surface monopoles on an ellipse, with the magnetization parallel to the elongation. f) Demagnetizing field $\hbox {\bf  H}_d$ resulting from magnetization $M$ at angle $\theta $ from $a$ axis in prolate ellipsoid.}}{figure}\relax 
\doTocEntry\tocsubsection{4.1.5}{\csname a:TocLink\endcsname{6}{x6-360004.1.5}{QQ2-6-71}{Magnetostatic (shape) anisotropy}}{169}\relax 
\doTocEntry\toclof{4.5}{\csname a:TocLink\endcsname{6}{x6-360055}{}{\ignorespaces Possible non-uniform magnetization configurations that reduce self energy for magnetite with increasing particle widths. The net remanent magnetization reduces with increasingly non-uniform spin configurations. [Data from Tauxe et al., 2002.]}}{figure}\relax 
\doTocEntry\tocsubsection{4.1.6}{\csname a:TocLink\endcsname{6}{x6-370004.1.6}{QQ2-6-73}{Magnetic energy and magnetic stability}}{176}\relax 
\doTocEntry\toclof{4.6}{\csname a:TocLink\endcsname{6}{x6-370026}{}{\ignorespaces A variety of domain structures of a given particle. a) Uniformly magnetized (single domain). [Adapted from Tipler, 1999.] b) Two domains. c) Four domains in a lamellar pattern. d) Essentially two domains with two closure domains.}}{figure}\relax 
\doTocEntry\tocsection{4.2}{\csname a:TocLink\endcsname{6}{x6-380004.2}{QQ2-6-75}{Magnetic domains}}{180}\relax 
\doTocEntry\toclof{4.7}{\csname a:TocLink\endcsname{6}{x6-380017}{}{\ignorespaces Examples of possible domain walls. a) There is a 180$^{\circ }$ switch from one atom to the next. The domain wall is very thin, but the exchange price is very high. b) There is a more gradual switch from one direction to the other [note: each arrow represents several 10's of unit cells]. The exchange energy price is lower, but there are more spins in unfavorable directions from a magnetocrystalline point of view. }}{figure}\relax 
\doTocEntry\toclof{4.8}{\csname a:TocLink\endcsname{6}{x6-380038}{}{\ignorespaces Comparison of ``self'' energy versus the energy of the domain wall in magnetite spheres as a function of particle size.}}{figure}\relax 
\doTocEntry\toclof{4.9}{\csname a:TocLink\endcsname{6}{x6-380049}{}{\ignorespaces a) Bitter patterns from an oriented polished section of magnetite. [Figure from \"Ozdemir et al., 1995]. b) Domains revealed by longitudinal magneto-optical Kerr effect. [Image from Heider and Hoffmann, 1992.] c-e) Magnetic force microscopy technique. [Images from Feinberg et al., 2005.] c) Image of topography of surface of a magnetite inclusion in a non-magnetic matrix. d) Magnetic image from MFM techqnique. e) Interpretation of magnetizations of magnetic domains. }}{figure}\relax 
\doTocEntry\tocsection{4.3}{\csname a:TocLink\endcsname{6}{x6-390004.3}{QQ2-6-79}{Thermal energy}}{189}\relax 
\doTocEntry\toclof{4.10}{\csname a:TocLink\endcsname{6}{x6-3900110}{}{\ignorespaces Relaxation time in magnetite ellipsoids as a function of grain width in nanometers (all length to width ratios of 1.3:1.) }}{figure}\relax 
\doTocEntry\toclof{4.11}{\csname a:TocLink\endcsname{6}{x6-3900411}{}{\ignorespaces  Expected domain states for various sizes and shapes of parallelopipeds of magnetite at room temperature. The parameters $a$ and $b$ are as in Figure 4.4e. Heavy blue (thin green) line is the superparamagnetic threshold assuming a relaxation time of 100s (1 Gyr). Dashed red line marks the SD/MD threshold size. Calculations done using assumptions and parameters described in the text. }}{figure}\relax 
\doTocEntry\tocsection{4.4}{\csname a:TocLink\endcsname{6}{x6-400004.4}{QQ2-6-82}{Putting it all together}}{197}\relax 
\doTocEntry\tocsection{4.5}{\csname a:TocLink\endcsname{6}{x6-410004.5}{QQ2-6-83}{Problems}}{198}\relax 
\doTocEntry\tocchapter{5}{\csname a:TocLink\endcsname{7}{x7-420005}{QQ2-7-84}{Magnetic hysteresis}}{203}\relax 
\doTocEntry\tocsection{5.1}{\csname a:TocLink\endcsname{7}{x7-430005.1}{QQ2-7-85}{The ``flipping'' field}}{203}\relax 
\doTocEntry\toclof{5.1}{\csname a:TocLink\endcsname{7}{x7-430011}{}{\ignorespaces a) Sketch of a magnetic particle with easy axis as shown. In response to a magnetic field $\hbox {\bf  B}$, applied at an angle $\phi $ to the easy axis, the particle moment $\hbox {\bf  m}$ rotates, making an angle $\theta $ with the easy axis. b) Variation of the anisotropy energy density $\epsilon _a = K_u\,\hbox {sin}\,^2\theta $ as a function of $\theta $ for the particle with $\phi =45^{\circ }$ as shown in a). The $\theta $ associated with the minimum energy is indicated by $\epsilon _{min}$. $B$ = 0 mT. c) same as in b) but for $B$ = 30 mT. Also shown the interaction energy density $\epsilon _m=-M_s B\,\hbox {cos}\,(\phi -\theta )$ and the total energy density $\epsilon _t=\epsilon _a+\epsilon _m$.}}{figure}\relax 
\doTocEntry\toclof{5.2}{\csname a:TocLink\endcsname{7}{x7-430032}{}{\ignorespaces a) Variation of the anisotropy energy density $\epsilon _a = K_u\,\hbox {sin}\,^2\theta $, the interaction energy density $\epsilon _m=-M_s B\,\hbox {cos}\,\phi $ and the total energy density $\epsilon _t=\epsilon _a+\epsilon _m$ as a function of $\theta $ for the particle shown in Figure 5.1a. The field was applied with $\phi $ = 180$^{\circ }$ and was 58 mT in magnitude. The $\theta $ associated with the minimum energy is indicated by $\epsilon _{min}$ and is 180$^{\circ }$. b) Variation in first and second derivatives of the energy equation. The flipping condition of both being zero simulaneously is met. c) Same as a) but the field was only 30 mT. d) Same as b but the flipping condition is not met.}}{figure}\relax 
\doTocEntry\toclof{5.3}{\csname a:TocLink\endcsname{7}{x7-430073}{}{\ignorespaces The flipping field $\mu _oH_f$ required to irreversibly switch the magnetization vector from one easy direction to the other in a single domain particle dominated by uniaxial anisotropy. Note that $\phi $ is the angle with the easy axis, but must be the opposite direction from $\textbf  {m}$. }}{figure}\relax 
\doTocEntry\tocsection{5.2}{\csname a:TocLink\endcsname{7}{x7-440005.2}{QQ2-7-89}{Hysteresis loops}}{215}\relax 
\doTocEntry\tocsubsection{5.2.1}{\csname a:TocLink\endcsname{7}{x7-450005.2.1}{QQ2-7-90}{Uniaxial anisotropy}}{215}\relax 
\doTocEntry\toclof{5.4}{\csname a:TocLink\endcsname{7}{x7-450014}{}{\ignorespaces  Moment measured for the particle ($\phi =0^{\circ }$) with applied field starting at 0 mT and increasing in the opposite directions along track #1. When the flipping field $\mu _oH_f$ is reached, the moment switches to the other direction along track #2. The field then switches sign and decreases along track #3 to zero, then increases again to the flipping field. The moment flips and the the field increases along track #4. b) The component of magnetization parallel to +B$_{max}$ versus $B$ for field applied with various angles $\phi $. }}{figure}\relax 
\doTocEntry\toclof{5.5}{\csname a:TocLink\endcsname{7}{x7-450025}{}{\ignorespaces a) Net response of a random assemblage of uniaxial single domain particles. Snap shots of magnetization states (squares labelled 1 to 4) for representative particles are shown in the balloons labelled State 1- 4. The initial demagnetized state is ``State 1''. The initial slope as the field is increased from zero is the low-field susceptibility $\chi _{lf}$. If the field returns to zero after some flipping fields have been exceeded, there is a net  isothermal remanence (IRM). When all the moments are parallel to the applied field (State 2), the magnetization is at saturation $M_s$. When the field is returned to zero, the magnetization is a saturation remanence ($M_r$; State 3). When the field is applied in the opposite direction and has remagnetized half the moments (State 4), the field is the bulk coercive field $\mu _oH_c$. When a field is reached that when reduced to zero leaves zero net remanence, that field is the coercivity of remanence (here labelled $\mu _oH_{cr}'$). b) Curve obtained by subtracting the ascending curve in a) from the descending curve. The field at which half the moments have flipped, leaving a magnetization of one half of saturation is another measure of the coercivity of remanence, here labelled $\mu _oH_{cr}$.}}{figure}\relax 
\doTocEntry\toclof{5.6}{\csname a:TocLink\endcsname{7}{x7-450036}{}{\ignorespaces Heavy green line: initial behavior of demagnetized specimen as applied field ramps up from zero field to a saturating field. The initial slope is the initial or low-field susceptibility $\chi _{lf}$. After saturation is achieved the slope is the high-field susceptibility $\chi _{hf}$ which is the non-ferromagnetic contribution, in this case the paramagnetic susceptibility (because $\chi _{hf}$ is positive.) The dashed blue line is the hysteresis loop after the paramagnetic slope has been subtracted. Saturation magnetization $M_s$ is the maximum value of magnetization after slope correction. Saturation remanence $M_r$ is the value of the magnetization remaining in zero applied field. Coercivity ($\mu _o H_c$) and coercivity of remanence $\mu _oH_{cr}'$ are as in Figure 5.5a. }}{figure}\relax 
\doTocEntry\tocsubsection{5.2.2}{\csname a:TocLink\endcsname{7}{x7-460005.2.2}{QQ2-7-94}{Magnetic susceptibility}}{226}\relax 
\doTocEntry\tocsubsection{5.2.3}{\csname a:TocLink\endcsname{7}{x7-470005.2.3}{QQ2-7-95}{Cubic anisotropy}}{227}\relax 
\doTocEntry\toclof{5.7}{\csname a:TocLink\endcsname{7}{x7-470017}{}{\ignorespaces Heavy lines: theoretical behavior of cubic grains of magnetite. Dashed lines are the responses along particular directions. Light grey lines: hysteresis response for single particles with various orientations with respect to the applied field. [Figure from Tauxe et al., 2002.]}}{figure}\relax 
\doTocEntry\toclof{5.8}{\csname a:TocLink\endcsname{7}{x7-470028}{}{\ignorespaces a) The contribution of SP particles with saturation magnetization $M_s$ and cubic edge length $d$. $\gamma = BM_s d^3/kT$. There is no hysteresis. b) The field at which the magnetization reaches 90\% of the maximum $B_{90}$ is when $ M_s d^3/kT\simeq 10$. [Figure from Tauxe et al., 1996.] }}{figure}\relax 
\doTocEntry\tocsubsection{5.2.4}{\csname a:TocLink\endcsname{7}{x7-480005.2.4}{QQ2-7-98}{Superparamagnetic particles}}{233}\relax 
\doTocEntry\toclof{5.9}{\csname a:TocLink\endcsname{7}{x7-480029}{}{\ignorespaces a) Typical hysteresis loop from a multi-domain assemblage. b) Theoretical behavior for the region in the inset to a). c) Theoretical relationship between $M_r/M_s$ and $H_{cr}/H_c$ for constant $\chi _iH_c/M_s = 0.1$. Heavy red line is the theoretical linear mixing curve of SD/MD end-members. (see text) }}{figure}\relax 
\doTocEntry\toclof{5.10}{\csname a:TocLink\endcsname{7}{x7-4800510}{}{\ignorespaces  Interaction of a domain wall and a void. When the void is within a domain, free poles create a magnetic field which creates a self energy (Chapter 4). When a domain wall intersects the void, the self-energy is reduced. There are no exchange or magnetocrystalline anisotropy energy terms within the void, so the wall energy is reduced.}}{figure}\relax 
\doTocEntry\tocsubsection{5.2.5}{\csname a:TocLink\endcsname{7}{x7-490005.2.5}{QQ2-7-101}{Particles with domain walls}}{242}\relax 
\doTocEntry\toclof{5.11}{\csname a:TocLink\endcsname{7}{x7-4900211}{}{\ignorespaces a) Schematic view of wall energy across a transect of a multi-domain grain. Inset: Placement of domain walls in the demagnetized state. [Domain observations from Halgedahl and Fuller, 1983.] b-g) Schematic view of the magnetization process in MD grain shown in previous figure. b) Demagnetized state, c) in the presence of a saturating field, d) field lowered to +3 mT, e) remanent state, f) backfield of -3 mT, g) resulting loop. Inset shows detail of domain walls moving by small increments called Barkhausen jumps. [Domain wall observations from Halgedahl and Fuller, 1983; schematic loop after O'Reilly, 1984.]}}{figure}\relax 
\doTocEntry\tocsection{5.3}{\csname a:TocLink\endcsname{7}{x7-500005.3}{QQ2-7-103}{Hysteresis of mixtures of SP, SD and MD grains}}{247}\relax 
\doTocEntry\toclof{5.12}{\csname a:TocLink\endcsname{7}{x7-5000112}{}{\ignorespaces a) Dashed line is the descending magnetization curve taken from a saturating field to some field $H_a$. Red line is the first order reversal curve (FORC) from $H_a$ returning to saturation. At any field $H_b>H_a$ there is a value for the magnetization $M(H_a,H_b)$. b) A series of FORCs for a single domain assemblage of particles. At any point there are a set of related ``nearest neighbor'' measurements (circles in inset). A least-squares fit to Equation 5.8 can be determined for each point. c) A contour plot of the FORC density surface for data in b). Specimen is of the Tiva Canyon Tuff, courtesy of the Institute for Rock Magnetism. }}{figure}\relax 
\doTocEntry\toclot{5.1}{\csname a:TocLink\endcsname{7}{x7-500021}{}{\ignorespaces Empirical values for hysteresis parameters measured for single domain (SD) and multi-domain (MD) end-members of Dunlop and Carter-Stiglitz (2006).}}{table}\relax 
\doTocEntry\tocsection{5.4}{\csname a:TocLink\endcsname{7}{x7-510005.4}{QQ2-7-106}{First order reversal curves}}{255}\relax 
\doTocEntry\toclof{5.13}{\csname a:TocLink\endcsname{7}{x7-5100113}{}{\ignorespaces a) A series of FORCs for a ``pseudo-single domain'' specimen. b) FORC diagram for data in a). Specimen is of the Stillwater Layered Intrusion, courtesy of J.S. Gee. }}{figure}\relax 
\doTocEntry\toclof{5.14}{\csname a:TocLink\endcsname{7}{x7-5100314}{}{\ignorespaces a) Illustration of a Zero FORC (ZFORC) whereby the descending loop from saturation is terminated at zero field and the field is then ramped back up to saturation. The transient hysteresis (TH) of Fabian (2003) is the shaded area between the two curves. b) Micromagnetic model of a ZFORC for a 100 nm cube of magnetite. Two snap shots of the internal magnetization on the descending and ascending loops are shown in the insets. [Figure redrawn from Yu and Tauxe, 2005.] }}{figure}\relax 
\doTocEntry\tocsection{5.5}{\csname a:TocLink\endcsname{7}{x7-520005.5}{QQ2-7-109}{Problems}}{262}\relax 
\doTocEntry\toclof{5.15}{\csname a:TocLink\endcsname{7}{x7-5200115}{}{\ignorespaces Various hysteresis plots.}}{figure}\relax 
\doTocEntry\tocchapter{6}{\csname a:TocLink\endcsname{8}{x8-530006}{QQ2-8-111}{Magnetic mineralogy}}{273}\relax 
\doTocEntry\tocsection{6.1}{\csname a:TocLink\endcsname{8}{x8-540006.1}{QQ2-8-112}{Iron-oxides}}{273}\relax 
\doTocEntry\toclof{6.1}{\csname a:TocLink\endcsname{8}{x8-540011}{}{\ignorespaces Phase diagrams for FeTi oxides. The composition is indicated by $x$ or $y$. There is complete solid solution above the solid lines. Exolution begins as the temperature cools below the solid curves. a) Titanomagnetite series. [Redrawn from Nagata, 1961.] b) Titanohematite series. [Redrawn from Robinson et al. 2004.]}}{figure}\relax 
\doTocEntry\toclof{6.2}{\csname a:TocLink\endcsname{8}{x8-540022}{}{\ignorespaces a) Atomic force micrograph image of magnetite inclusion in clinopyroxene. The topographically low areas are ulv\"ospinel while the higher areas are magnetite. b) Magnetic force micrograph of magnetic domains (black and white are oppositely magnetized). The ulv\"ospinel lamellae are essentially non-magnetic and are gray c) Tranmission electron micrograph of ilmenite host with hematite exolution lamellae. Lamellar size gets smaller with proximity to edge. d) Photomicrograph of titanohematite exolution lamellae. Dark bands are Ti-rich (high magnetization, low $T_c$), light grey bands are Ti-poor (low magnetization, high $T_c$). [a and b are from Feinberg et al., 2005, c) from Robinson et al., 2002., d) is modified from S. Haggerty in Butler (1992).]}}{figure}\relax 
\doTocEntry\toclof{6.3}{\csname a:TocLink\endcsname{8}{x8-540033}{}{\ignorespaces Ternary diagram for iron-oxides. The solid lines are solid solution series with increasing titanium concentration ($x$). The dashed lines with arrows indicate the direction of increasing oxidation ($z$). [Figure redrawn from Butler, 1992.] }}{figure}\relax 
\doTocEntry\tocsubsection{6.1.1}{\csname a:TocLink\endcsname{8}{x8-550006.1.1}{QQ2-8-116}{Titanomagnetites Fe$_{3-x}$Ti$_x$O$_4$}}{283}\relax 
\doTocEntry\toclof{6.4}{\csname a:TocLink\endcsname{8}{x8-550014}{}{\ignorespaces Variation of intrinsic parameters with titanium substitution in the titanomagnetite lattice. X is the degree of substitution from 0 (no Ti) to 1 (100\% substitution). a) Variation of the magnetization expressed as Bohr magnetons per unit cell. b) Variation of cell lattice size. c) Variation of Curie temperature. [Data compiled by O'Reilly, 1984.])}}{figure}\relax 
\doTocEntry\toclof{6.5}{\csname a:TocLink\endcsname{8}{x8-550025}{}{\ignorespaces Hematite. a) Photograph of Kidney ore hematite from Michigan by DanielCD. [From commons.wikimedia.org/wiki/File:Hematite.jpg.] b-c) Two views of the crystal structure of hematite. c-axis is perpendicular to the basal plane. [From \url  {http://www.webmineral.com}.] }}{figure}\relax 
\doTocEntry\toclof{6.6}{\csname a:TocLink\endcsname{8}{x8-550036}{}{\ignorespaces Variation of properties with Ti substitution in the titanohematite series. a) Variation of saturation magnetization. b) Variation of N\'eel Temperature. [Modified from Nagata, 1961 and Stacey and Banerjee, 1974.]}}{figure}\relax 
\doTocEntry\tocsubsection{6.1.2}{\csname a:TocLink\endcsname{8}{x8-560006.1.2}{QQ2-8-120}{Hematite-Ilmenite Fe$_{2-y}$Ti$_y$O$_3$}}{292}\relax 
\doTocEntry\toclof{6.7}{\csname a:TocLink\endcsname{8}{x8-560017}{}{\ignorespaces Variation of intrinsic parameters with oxidation in TM60. $z$ is the degree of oxidation. a) Variation of the magnetization. b) Variation of cell lattice size. c) Variation of Curie temperature. [Data compiled by Dunlop and \"Ozdemir, 1997.] inset: A magnetite crystal ($\sim $ 30 $\mu $m) undergoing maghemitization. Because of the change in volume, the crystal begins to crack. [From Gapeyev and Tsel'movich, 1988 in Dunlop and \"Ozdemir, 1997.]}}{figure}\relax 
\doTocEntry\toclof{6.8}{\csname a:TocLink\endcsname{8}{x8-560028}{}{\ignorespaces Effect of maghemitization on Verwey transition. a) Saturation remanence acquired at 10 K observed as it warms up for 37 nm stoichiometric magnetite. b) Same but for partially oxidized magnetite. [Data from \"Ozdemir et al., 1993.] }}{figure}\relax 
\doTocEntry\tocsubsection{6.1.3}{\csname a:TocLink\endcsname{8}{x8-570006.1.3}{QQ2-8-123}{Oxidation of (titano)magnetites to (titano)maghemites}}{298}\relax 
\doTocEntry\toclof{6.9}{\csname a:TocLink\endcsname{8}{x8-570019}{}{\ignorespaces  a) Photograph of goethite. [From en.wikipedia.org/wiki/Image:Goethite3.jpg; photo of Eurico Zimbres.] b) Goethite crystal structure. c) Photograph of greigite. [Photo of William P\'eraud.] d) Greigite crystal structure. e) Photograph of single crystal of pyrrhotite. [Photo of Dan Weinrich.] f) Pyrrhotite crystal structure. [All crystal structure images from \url  {http://www.webminerals.com}.]}}{figure}\relax 
\doTocEntry\tocsection{6.2}{\csname a:TocLink\endcsname{8}{x8-580006.2}{QQ2-8-125}{Iron-oxyhydroxides and iron-sulfides}}{301}\relax 
\doTocEntry\toclof{6.10}{\csname a:TocLink\endcsname{8}{x8-5800110}{}{\ignorespaces a) Low-temperature transition in monoclinic pyrhotite. [Data from Snowball and Torrii, 1999.] Thermomagnetic curves for b) monoclinic c) hexagonal and d) mixture of b) and c) pyrrhotite. [Data from Dekkers, 1988.]}}{figure}\relax 
\doTocEntry\tocsection{6.3}{\csname a:TocLink\endcsname{8}{x8-590006.3}{QQ2-8-127}{FeTi oxides in igneous rocks}}{304}\relax 
\doTocEntry\toclof{6.11}{\csname a:TocLink\endcsname{8}{x8-5900111}{}{\ignorespaces Occurrence of FeTi oxides in igneous rocks. [Data from Frost and Lindsley, 1991.]}}{figure}\relax 
\doTocEntry\toclof{6.12}{\csname a:TocLink\endcsname{8}{x8-5900212}{}{\ignorespaces Photomicrographs of bacterial magnetites produced by magnetotactic bacteria. a) Intact magnetosome in living bacterium. [False color image from H. Vali in Maher and Thompson, 1999.] b) Chains recovered from ODP Site 1006D in the Bahamas [From M. Hounslow in Maher and Thompson, 1999.]}}{figure}\relax 
\doTocEntry\tocsection{6.4}{\csname a:TocLink\endcsname{8}{x8-600006.4}{QQ2-8-130}{Magnetic mineralogy of soils and sediments}}{310}\relax 
\doTocEntry\toclot{}{\csname a:TocLink\endcsname{8}{x8-60002}{}{\numberline {6.1}{Physical properties of magnetic minerals.}}}{311}\relax 
\doTocEntry\toclof{6.13}{\csname a:TocLink\endcsname{8}{x8-6000313}{}{\ignorespaces Curie Temperature curves for two samples, A and B. [Figure redrawn from Butler, 1992.]}}{figure}\relax 
\doTocEntry\tocsection{6.5}{\csname a:TocLink\endcsname{8}{x8-610006.5}{QQ2-8-133}{Problems}}{316}\relax 
\doTocEntry\toclof{6.14}{\csname a:TocLink\endcsname{8}{x8-6100114}{}{\ignorespaces a) Thermomagnetic run of mineral whereby magnetization (normalized by the initial value) is measured as a function of temperature. The red line is the heating curve and the blue line is the cooling curve. b) Electron microprobe data from FeTi oxides (dots in yellow field) plotted on TiO$_2$-FeO-Fe$_2$O$_3$. ternary diagram. [Figure redrawn from Butler, 1992.]}}{figure}\relax 
\doTocEntry\tocchapter{7}{\csname a:TocLink\endcsname{9}{x9-620007}{QQ2-9-135}{How rocks get and stay magnetized}}{323}\relax 
\doTocEntry\tocsection{7.1}{\csname a:TocLink\endcsname{9}{x9-630007.1}{QQ2-9-136}{The concept of dynamic equilibrium}}{323}\relax 
\doTocEntry\toclof{7.1}{\csname a:TocLink\endcsname{9}{x9-630011}{}{\ignorespaces Illustration of dynamic equilibrium. If conditions on either side of the fence are equally pleasant, an equal number of sheep will be on either side of the fence, despite the fact that sheep are constantly jumping over the fence. If one side is preferable (sunny rather than rainy), there will tend to be more sheep on the nicer side. [Drawing by Genevieve Tauxe modified from animation available at:  \url  {http://magician.ucsd.edu/Lab_tour/movs/equilibrium.mov}.]}}{figure}\relax 
\doTocEntry\toclof{7.2}{\csname a:TocLink\endcsname{9}{x9-630022}{}{\ignorespaces  a) Magnetic relaxation in an assemblage of single domain ferromagnetic grains. The initial magnetization $M_o$ decays to $1\o:over: e$ of its original strength in time $\tau $. b) Relaxation times of single domain grains on a plot of grain volume, $v$, against an anisotropy energy constant ($K$), for a given temperature. Grains with short relaxation times plot toward the lower left and are in equilibrium with the magnetic field (they are superparamagnetic). Grains with long relaxation times plot toward the upper right; their moments are blocked, preserving the magnetization for geologically significant times. Inset shows the effect of temperature on the relaxation time curves which move toward the right and up with increasing temperature, changing ``blocked'' remanences to unblocked ones. }}{figure}\relax 
\doTocEntry\tocsection{7.2}{\csname a:TocLink\endcsname{9}{x9-640007.2}{QQ2-9-139}{Essential N\'eel theory}}{330}\relax 
\doTocEntry\toclof{7.3}{\csname a:TocLink\endcsname{9}{x9-640033}{}{\ignorespaces Lines of equal blocking energy in plot of grain volume, $v$, against the anisotropy energy density, $K$. Lines of equal blocking energy (product $Kv$) are also lines of equal relaxation time, $\tau $, at a given temperature (here assumed to be room temperature). Contours are for a hypothetical population of magnetic grains. Grains with short $\tau $ plot toward the lower left. Grains with long $\tau $ plot toward the upper right; superparamagnetic grains with $\tau < 100$s plot to the left or below the ``superparamagnetic line'' when $\tau \simeq $ 100s . Stable single domain grains with $\tau > $100s plot above or to right of superparamagnetic line. }}{figure}\relax 
\doTocEntry\tocsection{7.3}{\csname a:TocLink\endcsname{9}{x9-650007.3}{QQ2-9-141}{Viscous remanent magnetization}}{335}\relax 
\doTocEntry\toclof{7.4}{\csname a:TocLink\endcsname{9}{x9-650024}{}{\ignorespaces Magnetization versus time for a) Saturation remanence placed in zero field. b) Zero initial magnetization placed in a field. c) Magnetization placed in an antiparallel field. }}{figure}\relax 
\doTocEntry\toclof{7.5}{\csname a:TocLink\endcsname{9}{x9-650055}{}{\ignorespaces Migration of the relaxation times of a population of magnetic grains from a) low anisotropy energy at high temperature to b) high anisotropy energy at lower temperatures and the resulting change in relaxation times. The relaxation time curves also migrate up and to the right with lower thermal energy. Any particle initially to the right or above the superparamagnetic line would acquire a TRM its anisotropy energy density migrated across the line by cooling. Note that the anisotropy energy density ($K$ from Chapter 4) itself is a function of temperature through its dependence on magnetization, so a given population of grains will change with changing temperature, migrating to the left with higher temperature as magnetization goes down . }}{figure}\relax 
\doTocEntry\toclof{7.6}{\csname a:TocLink\endcsname{9}{x9-650066}{}{\ignorespaces Variation of relaxation time versus temperature for magnetite ellipsoids of different widths (all with length to width ratios of 1.3:1). }}{figure}\relax 
\doTocEntry\tocsection{7.4}{\csname a:TocLink\endcsname{9}{x9-660007.4}{QQ2-9-145}{Thermal remanent magnetization}}{347}\relax 
\doTocEntry\toclof{7.7}{\csname a:TocLink\endcsname{9}{x9-660017}{}{\ignorespaces a) Picture of lava flow courtesy of Daniel Staudigel. b) While the lava is still well above the Curie temperature, crystals start to form, but are non-magnetic. c) Below the Curie temperature but above the blocking temperature, certain minerals become magnetic, but their moments continually flip among the easy axes with a statistical preference for the applied magnetic field. As the lava cools down, the moments become fixed, preserving a thermal remanence. [b) and c) modified from animation of Genevieve Tauxe available at:  \url  {http://magician.ucsd.edu/Lab_tour/movs/TRM.mov}.] [Figure from Tauxe and Yamazaki, 2007.] }}{figure}\relax 
\doTocEntry\toclof{7.8}{\csname a:TocLink\endcsname{9}{x9-660038}{}{\ignorespaces Relationship of TRM with respect to the applied field for different assemblages of magnetite grains. a) Theoretical calculations of TRM acquisition for different assemblages of randomly oriented non-interacting single domain ellipsoids of magnetite. b) Experimentally determined TRM acquisition in three natural specimens. [Redrawn from Selkin et al., 2007.]}}{figure}\relax 
\doTocEntry\toclof{7.9}{\csname a:TocLink\endcsname{9}{x9-660049}{}{\ignorespaces Distribution of blocking temperatures of a typical basaltic specimen. The solid line labeled TRM indicates the amount of TRM remaining after step heating to increasingly higher temperature. The colored blocks labeled PTRM shows the amount of TRM blocked within corresponding temperature intervals. }}{figure}\relax 
\doTocEntry\toclof{7.10}{\csname a:TocLink\endcsname{9}{x9-6600510}{}{\ignorespaces Dependence of intensity of TRM on particle diameter of magnetite. Magnetite particles were dispersed in a non-magnetic matrix; the intensity of TRM is determined per unit volume of magnetite and normalized to the maximum TRM observed to allow comparison between experiments that used varying concentrations of dispersed magnetite; the magnetizing field was 100 $\mu $T. [Data compiled by Dunlop and \"Ozdemir, 1997.] }}{figure}\relax 
\doTocEntry\toclof{7.11}{\csname a:TocLink\endcsname{9}{x9-6600611}{}{\ignorespaces Migration of the blocking energy of grains by increasing volume. The relaxation times of a population of magnetic grains from a) short relaxation times when the particles are small to b) long relaxation times when the grains have grown through their blocking volumes. }}{figure}\relax 
\doTocEntry\tocsection{7.5}{\csname a:TocLink\endcsname{9}{x9-670007.5}{QQ2-9-151}{Chemical remanent magnetization}}{366}\relax 
\doTocEntry\toclof{7.12}{\csname a:TocLink\endcsname{9}{x9-6700112}{}{\ignorespaces Grain growth CRM. a) Red beds of the Chinji Formation, Siwaliks, Pakistan. The red soil horizons have a CRM carried by pigmentary hematite. b) Initial state of non-magnetic matrix. c) Formation of superparamagnetic minerals with a statistical alignment with the ambient magnetic field (shown in blue). }}{figure}\relax 
\doTocEntry\toclof{7.13}{\csname a:TocLink\endcsname{9}{x9-6700313}{}{\ignorespaces Depositional remanence versus applied field for redeposited glacial varves. $B_o$ was the field in the lab. [Data from Johnson et al., 1948; figure from Tauxe, 1993.]}}{figure}\relax 
\doTocEntry\tocsection{7.6}{\csname a:TocLink\endcsname{9}{x9-680007.6}{QQ2-9-154}{Detrital remanent magnetization}}{373}\relax 
\doTocEntry\tocsubsection{7.6.1}{\csname a:TocLink\endcsname{9}{x9-690007.6.1}{QQ2-9-155}{Physical alignment of magnetic moments in viscous fluids}}{373}\relax 
\doTocEntry\toclof{7.14}{\csname a:TocLink\endcsname{9}{x9-6900414}{}{\ignorespaces a) Schematic drawing of traditional view of the journey of magnetic particles from the water column to burial in a non-flocculating (freshwater) environment. Magnetic particles are black. b) View of depositional remanence in a flocculating (marine) environment. [Figure from Tauxe and Yamazaki, 2007.] }}{figure}\relax 
\doTocEntry\toclof{7.15}{\csname a:TocLink\endcsname{9}{x9-6900515}{}{\ignorespaces a) Numerical simulations of Brownian remanent magnetization (BRM) for various sizes of magnetite. b) BRM simulated for distribution of particle sizes of magnetite shown in inset. [Figure from Tauxe and Yamazaki, 2007.]}}{figure}\relax 
\doTocEntry\tocsubsubsection{}{\csname a:TocLink\endcsname{9}{x9-700007.6.1}{QQ2-9-158}{Non-flocculating environments}}{381}\relax 
\doTocEntry\toclof{7.16}{\csname a:TocLink\endcsname{9}{x9-7000316}{}{\ignorespaces a) Results of numerical experiments of the flocculation model using the parameters: $l=0.2$ m and the viscosity of water. $M/M_o$ is the DRM expressed as a fraction of saturation, holding $\mathaccent "7016\relax m$ constant and varying $B$. For a given field strength, particles are either at saturation or randomly oriented, except for within a very narrow size range. b) Same as a) but plotted versus applied field ($B$). [Figures from Tauxe et al., 2006.] }}{figure}\relax 
\doTocEntry\tocsubsubsection{}{\csname a:TocLink\endcsname{9}{x9-710007.6.1}{QQ2-9-160}{Flocculating environments}}{386}\relax 
\doTocEntry\toclof{7.17}{\csname a:TocLink\endcsname{9}{x9-7100117}{}{\ignorespaces  Results of settling experiments as a function of field ($B$) in a flocculating environment. The assumed mean and standard deviations of truncated log-normal distributions for floc radii are shown in the legends and are indicated using the different line styles in the figure. [Figure from Tauxe and Yamazaki, 2007 after Tauxe et al. 2006.]}}{figure}\relax 
\doTocEntry\toclof{7.18}{\csname a:TocLink\endcsname{9}{x9-7100218}{}{\ignorespaces Applied field inclination versus remanent inclination for redeposited river sediments. Best fit line is with $f=0.55$. [Data from Tauxe and Kent, 1984.] }}{figure}\relax 
\doTocEntry\tocsubsection{7.6.2}{\csname a:TocLink\endcsname{9}{x9-720007.6.2}{QQ2-9-163}{Post-depositional processes}}{392}\relax 
\doTocEntry\tocsubsection{7.6.3}{\csname a:TocLink\endcsname{9}{x9-730007.6.3}{QQ2-9-164}{Inclination Error}}{392}\relax 
\doTocEntry\toclof{7.19}{\csname a:TocLink\endcsname{9}{x9-7300119}{}{\ignorespaces Outcrop photo showing sampling locations and charred stump of tree that was hit by lightning in foreground. b) Impulse field required to reproduce the NRM intensity as an IRM, plotted as a function of distance from the tree shown in a). Dashed line is best-fit to the data assuming that the tree at the center of the photo was the site of a remagnetizing line current (lightning bolt) of 300,000 Amps. [Figures from Tauxe et al., 2003.]}}{figure}\relax 
\doTocEntry\tocsection{7.7}{\csname a:TocLink\endcsname{9}{x9-740007.7}{QQ2-9-166}{Isothermal remanent magnetization}}{396}\relax 
\doTocEntry\toclof{7.20}{\csname a:TocLink\endcsname{9}{x9-7400120}{}{\ignorespaces Acquisition of IRM by exposure to large magnetic fields. After saturation, the remanence remaining is $M_r$. One can then turn the sample around and applied smaller fields in the opposite direction to determine the field necessary to reduce the net remanence to zero. Also shown are two methods of estimating coercivity of remanence ($H_{cr}''$ and $H_{cr}'''$; see Appendix C for summary). }}{figure}\relax 
\doTocEntry\tocsection{7.8}{\csname a:TocLink\endcsname{9}{x9-750007.8}{QQ2-9-168}{Thermo-viscous remanent magnetization}}{399}\relax 
\doTocEntry\toclof{7.21}{\csname a:TocLink\endcsname{9}{x9-7500421}{}{\ignorespaces Theoretical nomogram relating relaxation time and blocking temperature. a) magnetite and b) hematite. }}{figure}\relax 
\doTocEntry\tocsection{7.9}{\csname a:TocLink\endcsname{9}{x9-760007.9}{QQ2-9-170}{Natural remanent magnetization}}{404}\relax 
\doTocEntry\toclof{7.22}{\csname a:TocLink\endcsname{9}{x9-7600122}{}{\ignorespaces Acquisition of ARM in alternating magnetic field. A total ARM is acquired if the DC field is switched on throughout the experiment (red dashed line) and a partial ARM (pARM) is acquired if the field is switched on only for part of the experiment (blue dash-dot line). }}{figure}\relax 
\doTocEntry\tocsection{7.10}{\csname a:TocLink\endcsname{9}{x9-770007.10}{QQ2-9-172}{Artificial remanences }}{407}\relax 
\doTocEntry\tocsection{7.11}{\csname a:TocLink\endcsname{9}{x9-780007.11}{QQ2-9-173}{Problems}}{408}\relax 
\doTocEntry\tocchapter{8}{\csname a:TocLink\endcsname{10}{x10-790008}{QQ2-10-174}{Applied rock (environmental) magnetism}}{413}\relax 
\doTocEntry\tocsection{8.1}{\csname a:TocLink\endcsname{10}{x10-800008.1}{QQ2-10-175}{Images}}{413}\relax 
\doTocEntry\toclof{8.1}{\csname a:TocLink\endcsname{10}{x10-800011}{}{\ignorespaces Images of various magnetic phases. a) 300 $\mu $m titanomagnetite grain of igneous origin showing high temperature exsolution lamellae [Photo from R. Reynolds in Maher and Thompson, 1999.]. b) Detrital and aeolian (titano)magnetites from Chinese Loess. [Photo from Maher and Thompson, 1999.] c) Hematite rosettes on a smectite surface. [Photo from Reynolds et al., 1985.] d) Backscatter SEM image of fly-ash spherule. [Photo of J. Matzka, in Maher and Thompson, 1999.] The bright grains are iron rich particles embedded in a silicate matrix. e) Silicate spherule with dendrites of Fe-rich material of cosmic origin, showing characteristic pitting of the surface. [Photo from M. Hounslow in Maher and Thompson, 1999.] }}{figure}\relax 
\doTocEntry\toclot{8.1}{\csname a:TocLink\endcsname{10}{x10-800021}{}{\ignorespaces Summary of environmental magnetic parameters.}}{table}\relax 
\doTocEntry\tocsection{8.2}{\csname a:TocLink\endcsname{10}{x10-810008.2}{QQ2-10-178}{Critical temperatures}}{420}\relax 
\doTocEntry\toclof{8.2}{\csname a:TocLink\endcsname{10}{x10-810012}{}{\ignorespaces a) Translation curie balance in the Scripps Laboratory. b) Schematic drawing of the key elements of a) (top view). }}{figure}\relax 
\doTocEntry\toclof{8.3}{\csname a:TocLink\endcsname{10}{x10-810023}{}{\ignorespaces a) $M_s-T$ data for magnetite. Inset illustrates intersecting tangent method of Curie temperature estimation. b) Data from a) differentiated once. c) Data from a) differentiated twice. Peak shows temperature of maximum curvature, interpreted as the Curie temperature for this specimen. }}{figure}\relax 
\doTocEntry\tocsection{8.3}{\csname a:TocLink\endcsname{10}{x10-820008.3}{QQ2-10-181}{Magnetic susceptibility}}{426}\relax 
\doTocEntry\toclof{8.4}{\csname a:TocLink\endcsname{10}{x10-820014}{}{\ignorespaces Measuring magnetic susceptibility. a) An alternating current applied in the coil on the right induces a current in the left-hand coil. This induces a magnetization in the specimen shown in b), which in turn offsets the current in the coil to the right. The offset is proportional to the magnetic susceptibility of the specimen. [Modified from Genevieve Tauxe animation at: \url  {http://magician.ucsd.edu/Lab_tour/movs/isosuscp.mov}.] }}{figure}\relax 
\doTocEntry\tocsubsection{8.3.1}{\csname a:TocLink\endcsname{10}{x10-830008.3.1}{QQ2-10-183}{Measurement of magnetic susceptibility}}{430}\relax 
\doTocEntry\toclof{8.5}{\csname a:TocLink\endcsname{10}{x10-830015}{}{\ignorespaces a) Schematic drawings of paramagnetic (solid line) and diamagnetic (dashed line) magnetic susceptibility as a function of temperature. b) Behavior of ferromagnetic susceptibility (solid line) as the material approaches its Curie temperature ($M_s-T$ data shown as dashed line). }}{figure}\relax 
\doTocEntry\tocsubsection{8.3.2}{\csname a:TocLink\endcsname{10}{x10-840008.3.2}{QQ2-10-185}{Temperature dependence}}{434}\relax 
\doTocEntry\toclof{8.6}{\csname a:TocLink\endcsname{10}{x10-840016}{}{\ignorespaces a) Magnetic susceptibility as a function of frequency. The decrease in frequency dependence of susceptibility with increasing frequency is caused by the superparamagnetic particles in the specimen. b) Plot showing temperature and frequency dependence of the same specimen as in a). [Data from Tiva Canyon Tuff, Carter-Stiglitz et al. 2006.] }}{figure}\relax 
\doTocEntry\tocsubsection{8.3.3}{\csname a:TocLink\endcsname{10}{x10-850008.3.3}{QQ2-10-187}{Frequency dependence}}{437}\relax 
\doTocEntry\toclof{8.7}{\csname a:TocLink\endcsname{10}{x10-850017}{}{\ignorespaces Map of magnetic susceptibility as a function of distance from the road. [Data from Hoffmann et al., 1999; Figure of M. Knab.]}}{figure}\relax 
\doTocEntry\tocsubsection{8.3.4}{\csname a:TocLink\endcsname{10}{x10-860008.3.4}{QQ2-10-189}{Outcrop measurements}}{440}\relax 
\doTocEntry\tocsection{8.4}{\csname a:TocLink\endcsname{10}{x10-870008.4}{QQ2-10-190}{Magnetization}}{440}\relax 
\doTocEntry\toclof{8.8}{\csname a:TocLink\endcsname{10}{x10-870018}{}{\ignorespaces a) IRM acquisition (solid lines) versus progressive demagnetization of IRM with alternating fields (dashed lines) for two specimens. Circles are the Lambert plagioclase (non-interacting uniaxial single domain magnetite particles) and squares are chiton teeth (interacting magnetite particles). The field at which the demagnetization and acquisition curves cross (the cross-over point $R_x$) is sensitive to particle interaction. [Data of Cisowski, 1981.] b) ARM acquisition as a function of DC bias field for two specimens with different concentrations of magnetite. The squares are for a low concentration of 2.6 x 10$^{-4}$ volume percent magnetite while the circles are for a high concentration of 2.33 volume percent. [Data of Sugiura, 1979.] }}{figure}\relax 
\doTocEntry\tocsubsection{8.4.1}{\csname a:TocLink\endcsname{10}{x10-880008.4.1}{QQ2-10-192}{Magnetic interactions: IRM and ARM techniques}}{443}\relax 
\doTocEntry\toclof{8.9}{\csname a:TocLink\endcsname{10}{x10-880019}{}{\ignorespaces Theoretical curve for the acquisition of IRM with two magnetic components with different coercivity spectra (see insert). The acquisition curve can be differentiated to get the heavy solid line in the insert and then decomposed into the different components assuming some distribution of coercivity (in this case log-normal). The main plot is a ``linear acquisition plot'' (LAP) and the heavy solid line in the inset is a ``gradient of acquisition plot'' (GAP) in the terminology of Kruiver et al. (2001). $H_{1/2}$ and $DP$ are the fields required to magnetize half the population and the ``dispersion parameter'' of Robertson and France (1994) respectively. Note that $H_{1/2}$ is a measure of $H_{cr} (H'''_{cr}$ in Table C.1) if there is only one population of coercivities. }}{figure}\relax 
\doTocEntry\tocsubsection{8.4.2}{\csname a:TocLink\endcsname{10}{x10-890008.4.2}{QQ2-10-194}{IRM ``unmixing''}}{446}\relax 
\doTocEntry\tocsubsection{8.4.3}{\csname a:TocLink\endcsname{10}{x10-900008.4.3}{QQ2-10-195}{Combining thermal and isothermal information for rock magnetic characterization}}{446}\relax 
\doTocEntry\toclof{8.10}{\csname a:TocLink\endcsname{10}{x10-9000110}{}{\ignorespaces a) Acquisition of IRM ($M_r$). After applying a field of 2 T, the specimen was subjected to two additional IRMs: 0.4 T and 0.12 T along orthogonal axes. b) Thermal demagnetization of a 3-axis IRM. Each component is plotted separately. [Figure from Tauxe, 1998.] }}{figure}\relax 
\doTocEntry\tocsection{8.5}{\csname a:TocLink\endcsname{10}{x10-910008.5}{QQ2-10-197}{Hysteresis parameters}}{450}\relax 
\doTocEntry\tocsubsection{8.5.1}{\csname a:TocLink\endcsname{10}{x10-920008.5.1}{QQ2-10-198}{The building blocks of hysteresis loops}}{451}\relax 
\doTocEntry\toclof{8.11}{\csname a:TocLink\endcsname{10}{x10-9200111}{}{\ignorespaces Hysteresis loops of end-member behaviors: a) diamagnetic, b) paramagnetic, c) superparamagnetic (data for submarine basaltic glass), d) uniaxial, single domain, e) magnetocrystalline, single domain, f) ``pseudo-single domain''. Hysteresis behavior of various mixtures: g) magnetite, and hematite, h) SD/SP magnetite (data from Tauxe et al. 1996), i) another example of SD/SP magnetite with a finer grained SP distribution. [Figures redrawn from Tauxe, 1998.]}}{figure}\relax 
\doTocEntry\toclof{8.12}{\csname a:TocLink\endcsname{10}{x10-9200212}{}{\ignorespaces  a-d) Hysteresis curves, e-h: $\Delta M$ curves and i-l) $d\Delta M/dH$ curves. Columns from the left to right: hematite, SD magnetite, hematite plus magnetite, and SD plus SP magnetite. [Redrawn from Tauxe, 1998.]}}{figure}\relax 
\doTocEntry\tocsubsection{8.5.2}{\csname a:TocLink\endcsname{10}{x10-930008.5.2}{QQ2-10-201}{Hysteresis behavior of mixtures}}{457}\relax 
\doTocEntry\tocsection{8.6}{\csname a:TocLink\endcsname{10}{x10-940008.6}{QQ2-10-202}{Trends in parameters with grain size}}{457}\relax 
\doTocEntry\toclof{8.13}{\csname a:TocLink\endcsname{10}{x10-9400113}{}{\ignorespaces  Grain size dependence in hysteresis parameters. Crushed grains (red) indicated by ``C'', glass ceramic grains (blue) indicated by GC; hydrothermal grains (green) indicated by ``H''. a) Variation of coercivity ($\mu _oH_c$). b) Variation of $M_r/M_s$. c) Variation of coercivity of remanence $\mu _oH_{cr}$. [Data compiled by Hunt et al., 1995.] d) Variation of susceptibility with grain size. [Data compiled by Heider et al., 1996.] e) Variation in $\chi _{ARM}$ with grain size. [Data compiled by Dunlop and Argyle, 1997.] }}{figure}\relax 
\doTocEntry\toclof{8.14}{\csname a:TocLink\endcsname{10}{x10-9400214}{}{\ignorespaces Plots of hysteresis parameters from a collection of related specimens. a) Plot of $M_r/M_s$ versus $H_{cr}/H_c$. Inset shows typical loop from which the ratios were derived. b) Plot of $M_r/M_s$ versus $\mu _oH_c$. [Data from Ben-Yosef et al., 2008.]}}{figure}\relax 
\doTocEntry\tocsection{8.7}{\csname a:TocLink\endcsname{10}{x10-950008.7}{QQ2-10-205}{Ratios}}{464}\relax 
\doTocEntry\tocsection{8.8}{\csname a:TocLink\endcsname{10}{x10-960008.8}{QQ2-10-206}{Applications of rock magnetism}}{466}\relax 
\doTocEntry\tocsubsection{8.8.1}{\csname a:TocLink\endcsname{10}{x10-970008.8.1}{QQ2-10-207}{Paleoclimatic information from lake sediments}}{466}\relax 
\doTocEntry\toclof{8.15}{\csname a:TocLink\endcsname{10}{x10-9700115}{}{\ignorespaces Plot of ARM versus magnetic susceptibility for a core from Minnesota. The different slopes are correlated with major climatic and anthropogenic events during the Holocene. [Redrawn from Banerjee et al., 1981.]}}{figure}\relax 
\doTocEntry\toclof{8.16}{\csname a:TocLink\endcsname{10}{x10-9700216}{}{\ignorespaces Rock magnetic and trace element data from Buck Lake [Data downloaded from \url  {http://pubs.usgs.gov/of/1995/of95-673/of95-673.html} and interpreted as in Rosenbaum et al., 1996].}}{figure}\relax 
\doTocEntry\toclof{8.17}{\csname a:TocLink\endcsname{10}{x10-9700317}{}{\ignorespaces Biplots of various trace elements and rock magnetic parameters. Solid lines are best-fit lines. Dashed lines are theoretical lines with no Fe-loss. Open symbols were excluded from best-fit line. Note that many data are off the plot. a) Zr against Ti. b) Fe against Ti. c) HIRM (hematite component) against Ti (proxy for detrital input). d) $\chi $ (magnetite component) against Ti. [Figures re-drawn from Rosenbaum et al. (1996) using data in Figure 8.15.]}}{figure}\relax 
\doTocEntry\tocsubsection{8.8.2}{\csname a:TocLink\endcsname{10}{x10-980008.8.2}{QQ2-10-211}{Paramagnetic contributions to magnetic susceptibility}}{475}\relax 
\doTocEntry\toclof{8.18}{\csname a:TocLink\endcsname{10}{x10-9800118}{}{\ignorespaces a) Frequency dependence of magnetic susceptibility ($\chi _{fd}$) versus age for NP21, a pelagic clay core. b) Low-frequency magnetic susceptiblity ($\chi _{l}$) versus the difference between the low and high-frequency magnetic susceptibilities ($\chi _l-\chi _h$) for core NP21. The value at the intersection of a linear regression line with the $\chi _l$ axis is interpreted as the frequency-independent fraction. c) Ratio of ARM to $\chi $ versus age for the uncorrected (U: open symbols) and corrected (C: solid symbols) data using the paramagnetic fraction of the susceptibility for core NP21. [Data of Yamazaki and Ioka, 1997.]}}{figure}\relax 
\doTocEntry\toclof{8.19}{\csname a:TocLink\endcsname{10}{x10-9800219}{}{\ignorespaces  a) Calculated grain distribution for the mixture of two Tiva Canyon Tuff specimens with different mean grain sizes and aspect ratios (contour interval = f$_{max}$/10). b) Calculated back-field spectra. [Redrawn from Figure 18 in Jackson et al. 2006.]}}{figure}\relax 
\doTocEntry\tocsubsection{8.8.3}{\csname a:TocLink\endcsname{10}{x10-990008.8.3}{QQ2-10-214}{Separation of two superparamagnetic particle size distributions}}{482}\relax 
\doTocEntry\toclof{8.20}{\csname a:TocLink\endcsname{10}{x10-9900120}{}{\ignorespaces  a) Reconstructed grain distribution (contour interval of fmax/10) and b) best fit back-field spectra for a paleosol specimen. The RMS misfit is $<5$\%. [Redrawn from Figure 21 in Jackson et al. 2006.]}}{figure}\relax 
\doTocEntry\tocsubsection{8.8.4}{\csname a:TocLink\endcsname{10}{x10-1000008.8.4}{QQ2-10-216}{Identification of biogenic magnetite in natural samples}}{486}\relax 
\doTocEntry\toclof{8.21}{\csname a:TocLink\endcsname{10}{x10-10000221}{}{\ignorespaces Low-temperature FC (dashed line) and ZFC (solid line) demagnetization curves for selected water depths corresponding to a) above the oxic-anoxic interface (OAI;2.7 m), b) bottom of OAI (3.5 m) , and c) below the OAI (4.0 m). [Redrawn from Figure 9 in Moskowitz et al. (2008).]}}{figure}\relax 
\doTocEntry\toclof{8.22}{\csname a:TocLink\endcsname{10}{x10-10000322}{}{\ignorespaces $\delta _{FC}/\delta _{ZFC}$ ratios as a function of water depth. Shaded zone is the location of the OAI based on chemical profiles. sr: short-rod shaped magnetotactic bacteria. Values of $\delta _{FC}/\delta _{ZFC} > 2.0$ are characteristic of MMB and MRP bacteria that have magnetite magnetosomes organized in chains. [Redrawn from Figure 10b in Moskowitz et al., 2008.]}}{figure}\relax 
\doTocEntry\tocsection{8.9}{\csname a:TocLink\endcsname{10}{x10-1010008.9}{QQ2-10-219}{Concluding remarks}}{493}\relax 
\doTocEntry\tocsection{8.10}{\csname a:TocLink\endcsname{10}{x10-1020008.10}{QQ2-10-220}{Problems}}{494}\relax 
\doTocEntry\toclot{8.2}{\csname a:TocLink\endcsname{10}{x10-1020012}{}{\ignorespaces Data for beach project.}}{table}\relax 
\doTocEntry\tocchapter{9}{\csname a:TocLink\endcsname{11}{x11-1030009}{QQ2-11-222}{ Getting a paleomagnetic direction}}{503}\relax 
\doTocEntry\tocsection{9.1}{\csname a:TocLink\endcsname{11}{x11-1040009.1}{QQ2-11-223}{Paleomagnetic sampling}}{503}\relax 
\doTocEntry\toclof{9.1}{\csname a:TocLink\endcsname{11}{x11-1040011}{}{\ignorespaces Sampling technique with a water-cooled drill. [Photos of Daniel Staudigel.] a) Drill the sample. b) Insert a non-magnetic slotted tube with an adjustable platform around the sample. Rotate the slot to the upper side of the sample. Note the azimuth and plunge of the drill direction (into the outcrop) with a sun and/or magnetic compass and inclinometer. Mark the sample through the slot with a brass or copper wire. c) Extract the sample. d) Make a permanent arrow on the upper side of the sample in the direction of drilling and label the sample with the sample name. Make a note of the name and orientation of the arrow in a field notebook. }}{figure}\relax 
\doTocEntry\toclof{9.2}{\csname a:TocLink\endcsname{11}{x11-1040022}{}{\ignorespaces Hand sampling technique for soft sediment: a) Dig down to fresh material. b) Rasp off a flat surface. c) Mark the strike and dip on the sample. d) Extract the sample and label it.}}{figure}\relax 
\doTocEntry\tocsubsection{9.1.1}{\csname a:TocLink\endcsname{11}{x11-1050009.1.1}{QQ2-11-226}{Types of samples}}{510}\relax 
\doTocEntry\toclof{9.3}{\csname a:TocLink\endcsname{11}{x11-1050053}{}{\ignorespaces Sampling of a sediment core. A plastic cube with a hole in it to let the air escape is pressed into the split surface of a core. The orientation arrow points ``up core''. After extraction, a label with the sample name is put on. [Photo from Kurt Schwehr.] }}{figure}\relax 
\doTocEntry\toclof{9.4}{\csname a:TocLink\endcsname{11}{x11-1050084}{}{\ignorespaces Orientation system for sample collected by portable core drill. a) Schematic representation of core sample in situ. The $Z$ axis points into outcrop; the $X $axis is perpendicular to $Z$ and is in the vertical plane; the $Y$ axis in the horizontal plane and is positive to the right of $X$. b) Orientation angles for core samples. The angles measured are the hade of the $Z$ axis (angle of $Z$ from vertical) and geographic azimuth of the horizontal projection of the +$X$ axis measured clockwise from geographic north. }}{figure}\relax 
\doTocEntry\toclof{9.5}{\csname a:TocLink\endcsname{11}{x11-1050095}{}{\ignorespaces a) Pomeroy orientation device in use as a sun compass. b) Schematic of the principles of sun compass orientation.}}{figure}\relax 
\doTocEntry\tocsubsection{9.1.2}{\csname a:TocLink\endcsname{11}{x11-1060009.1.2}{QQ2-11-230}{Orientation in the field}}{518}\relax 
\doTocEntry\toclof{9.6}{\csname a:TocLink\endcsname{11}{x11-1060016}{}{\ignorespaces Back-sighting technique using a Pomeroy orientation device and two Brunton Compasses. One is used with the Pomeroy to measure the direction of drill and the other is used to check for deflection caused by local magnetic anomalies.}}{figure}\relax 
\doTocEntry\toclof{9.7}{\csname a:TocLink\endcsname{11}{x11-1060027}{}{\ignorespaces  Differential GPS system for orienting paleomagnetic samples in polar regions. Photo taken during sampling trip to the foothills of the Royal Society Ranges in Antarctica, Jan. 2004. }}{figure}\relax 
\doTocEntry\toclof{9.8}{\csname a:TocLink\endcsname{11}{x11-1060038}{}{\ignorespaces Various types of possible specimen shapes and orientation conventions. a) A one inch slice from a drilled core. b) A cubic specimen of sediment sanded from a hand sample. c) A specimen (also sample) from a piston core.}}{figure}\relax 
\doTocEntry\tocsubsection{9.1.3}{\csname a:TocLink\endcsname{11}{x11-1070009.1.3}{QQ2-11-234}{A note on terminology}}{528}\relax 
\doTocEntry\tocsection{9.2}{\csname a:TocLink\endcsname{11}{x11-1080009.2}{QQ2-11-235}{Measurement of magnetic remanence}}{528}\relax 
\doTocEntry\tocsection{9.3}{\csname a:TocLink\endcsname{11}{x11-1090009.3}{QQ2-11-236}{Changing coordinate systems}}{529}\relax 
\doTocEntry\toclof{9.9}{\csname a:TocLink\endcsname{11}{x11-1090019}{}{\ignorespaces a) Specimen coordinates with $X_1$ being along the ``lab arrow''. A magnetic moment $m$ was measured relative to the specimen coordinate system with components $x_1, x_2, x_3$. The orientation of the lab arrow with respect to geographic coordinates ($X'_1 = N$) is specified by the azimuth and plunge ($Az, Pl$) of the lab arrow. }}{figure}\relax 
\doTocEntry\toclof{9.10}{\csname a:TocLink\endcsname{11}{x11-10900210}{}{\ignorespaces Principle of progressive demagnetization. Specimens with two components of magnetization (shown by heavy arrows on the right hand side), with discrete coercivities (plotted as histograms to the left). The original ``NRM'' is the sum of the two magnetic components and is shown as the + in the diagrams to the right. Successive demagnetization steps (numbered) remove the component with coercivities lower than the peak field, and the NRM vector changes as a result. a) The two distributions of coercivity are completely separate. b) The two distributions partially overlap resulting in simultaneous removal of both components. c) The two distributions completely overlap. d) One distribution envelopes the other. [Figure redrawn from Tauxe, 1998.] }}{figure}\relax 
\doTocEntry\toclof{9.11}{\csname a:TocLink\endcsname{11}{x11-10900311}{}{\ignorespaces a) Solid (open) symbols are horizontal (vertical) projections respectively. Peak alternating fields for each demagnetizing step (in mT) are indicated. Inset is equal area plot of the same data. Solid (open) symbols are projections onto the lower (upper) hemisphere. b) Intensity as a function of demagnetization step. Data from a). The median destructive field (mdf of Chapter 8) also shown. c) Specimen with two components with overlapping stabilities. Inset as in a). Best fit great circle is shown as the curve through the data (dashed portion is upper hemisphere projection). d) Data from specimen showing evidence of GRM (see Chapter 7). During demagnetization, the vector grows perpendicular the last demagnetization direction (-Y). Deviation ANGle, DANG also shown. }}{figure}\relax 
\doTocEntry\tocsection{9.4}{\csname a:TocLink\endcsname{11}{x11-1100009.4}{QQ2-11-240}{Demagnetization techniques}}{542}\relax 
\doTocEntry\tocsection{9.5}{\csname a:TocLink\endcsname{11}{x11-1110009.5}{QQ2-11-241}{Estimating directions from demagnetization data}}{544}\relax 
\doTocEntry\tocsection{9.6}{\csname a:TocLink\endcsname{11}{x11-1120009.6}{QQ2-11-242}{Vector difference sum}}{547}\relax 
\doTocEntry\toclof{9.12}{\csname a:TocLink\endcsname{11}{x11-11200112}{}{\ignorespaces Sampling units with different bedding attitudes in the ``fold test''. a) Example of folded beds. [Photo from G. Dupont-Nivet.] b) Hypothetical paleomagnetic directions are shown on equal area projections before and after adjusting for bedding tilt. Top pair represents the case in which the grouping of paleomagnetic directions is improved after adjusting for tilt which would argue for a pre-tilt acquisition of remanence. Lower pair represents a post-tilt acquisition of remanence in which the grouping is worse after restoring beds to the horizontal position. }}{figure}\relax 
\doTocEntry\tocsection{9.7}{\csname a:TocLink\endcsname{11}{x11-1130009.7}{QQ2-11-244}{Best-fit lines and planes}}{550}\relax 
\doTocEntry\toclof{9.13}{\csname a:TocLink\endcsname{11}{x11-11300213}{}{\ignorespaces The paleomagnetic conglomerate test. a) The target lithology was involved in a catastrophic event leading to incorporation into a conglomerate bed. Samples are taken from individual clasts. The directions of samples from the target lithology are shown in b) indicating that it is relatively homogeneously magnetized. c) Directions from the conglomerate clasts are also homogeneously magnetized; the magnetization must post-date formation of the conglomerate. In a positive conglomerate test d), the magnetization vectors of samples from the conglomerate clasts are random. }}{figure}\relax 
\doTocEntry\toclof{9.14}{\csname a:TocLink\endcsname{11}{x11-11300414}{}{\ignorespaces The baked contact test. In a positive test, zones baked by the intrusion are remagnetized and have directions that grade from that of the intrusion to that of the host rock. If all the material is homogeneously magnetized, then the age of the intrusion places an upper bound on the age of magnetization.}}{figure}\relax 
\doTocEntry\tocsection{9.8}{\csname a:TocLink\endcsname{11}{x11-1140009.8}{QQ2-11-247}{Field strategies}}{557}\relax 
\doTocEntry\tocsection{9.9}{\csname a:TocLink\endcsname{11}{x11-1150009.9}{QQ2-11-248}{Problems}}{558}\relax 
\doTocEntry\toclof{9.15}{\csname a:TocLink\endcsname{11}{x11-11500115}{}{\ignorespaces Paleomagnetic site NS034. a) Photo of the ``red'' team. b) Photo showing sample holes with labels. The picture was taken in an easterly direction (see look direction in notebook page.) c) Page from the notebook. }}{figure}\relax 
\doTocEntry\tocchapter{10}{\csname a:TocLink\endcsname{12}{x12-11600010}{QQ2-12-250}{Paleointensity}}{573}\relax 
\doTocEntry\toclof{10.1}{\csname a:TocLink\endcsname{12}{x12-1160021}{}{\ignorespaces Principles of paleointensity estimation. The remanent magnetization is assumed linear with the magnetic field. If the slope $\nu $ can be determined through laboratory proxy measurements ($M_{lab}/B_{lab}$), then the NRM of a given specimen, $M_{NRM},$ can be mapped to an estimate of the ancient magnetic field $B_{anc}$.}}{figure}\relax 
\doTocEntry\tocsection{10.1}{\csname a:TocLink\endcsname{12}{x12-11700010.1}{QQ2-12-252}{Paleointensity with TRMs}}{578}\relax 
\doTocEntry\tocsubsection{10.1.1}{\csname a:TocLink\endcsname{12}{x12-11800010.1.1}{QQ2-12-253}{Stepwise heating family of experiments}}{579}\relax 
\doTocEntry\toclof{10.2}{\csname a:TocLink\endcsname{12}{x12-1180012}{}{\ignorespaces Example of thermal normalization experiment of K\"onigsberger (1938). A specimen is heated to given temperature and cooled in a field of +0.4 Oe (40 $\mu $T) (e.g., step labeled #4). Then the specimen is heated to same temperature and cooled in field of -0.4 Oe (e.g., step #5). The two curves can be decomposed to give $M_{nrm}$ and $M_{lab}$, the ratio of which was termed $Q_{nt}$ by K\"onigsberger. Note that $J_{rn}$ and $J_{rt}$ are $M_{NRM}$ and $M_{pTRM}$ respectively here. [Figure redrawn from K\"onigsberger (1938) by Tauxe and Yamazaki, 2007.]}}{figure}\relax 
\doTocEntry\toclof{10.3}{\csname a:TocLink\endcsname{12}{x12-1180023}{}{\ignorespaces Illustration of step-wise heating method for determining absolute paleointensity. a) Thermal demagnetization of NRM shown as filled circles and the laboratory acquired pTRM shown as open symbols. b) Plot of NRM component remaining versus pTRM gained at each temperature step. Triangles are the second in-field heating step (pTRM check step) at a given temperature. The difference, e.g., $\delta _{300}$, is an indication of possible alteration during the heating experiment. }}{figure}\relax 
\doTocEntry\toclof{10.4}{\csname a:TocLink\endcsname{12}{x12-1180034}{}{\ignorespaces  a) Stepwise thermal demagnetization of pTRMs imparted by applying a small DC field during cooling from 370 to 350$^{\circ }$C in magnetite specimens of known grain size. Between 50 and 90\% of the remanence unblocks at temperatures below (a low temperature pTRM tail) or above (a high temperature pTRM tail) the pTRM blocking temperature range. The failure of reciprocity is most extreme for the largest grain sizes. b) Step-wise heating paleointensity experiments on specimens with a laboratory TRM. Heavy red line is theoretical SD behavior. All specimens give results that sag below the ideal SD line, an expression of the pTRM tails exhibited by some of the same specimens in a). [Data of Dunlop and \"Ozdemir, 2001.] }}{figure}\relax 
\doTocEntry\toclof{10.5}{\csname a:TocLink\endcsname{12}{x12-1180045}{}{\ignorespaces Schematic diagram of the IZZI experimental protocol. [Figure from Ben-Yosef et al., 2008.] }}{figure}\relax 
\doTocEntry\toclof{10.6}{\csname a:TocLink\endcsname{12}{x12-1180056}{}{\ignorespaces Example of results from an IZZI paleointensity experiment. a) NRM remaining after demagnetization in zero field (blue circles) and pTRM gained after heating and cooling in the laboratory field (red squares). Both remanences were normalized by the initial NRM. b) Arai plot of data in a). Open (closed) symbols are the IZ (ZI) steps. Triangles are pTRM check steps and blue squares are the pTRM tail check steps. The zig-zag behavior is characteristic of the effect of pTRM tails. }}{figure}\relax 
\doTocEntry\toclof{10.7}{\csname a:TocLink\endcsname{12}{x12-1180067}{}{\ignorespaces Data from an experiment with an anisotropic specimen given a total TRM in different orientations with respect to the laboratory field. The relative TRM magnitudes are plotted as squares and a best fit model intensity based on the TRM anisotropy tensor is shown as the solid line. [Redrawn from Selkin et al., 2000.]}}{figure}\relax 
\doTocEntry\toclof{10.8}{\csname a:TocLink\endcsname{12}{x12-1180078}{}{\ignorespaces Ratio of estimated field intensity $B_{est}$ to actual ancient field intensity $B_{anc}$ versus the ratio of cooling rates at the blocking temperature using the method of Halgedahl et al. (1980) but the variation of $M_s(T)$ in Chapter 3 ($\gamma =0.38$). Laboratory blocking temperatures are shown as examples. [Figure courtesy of R. Mitra.] }}{figure}\relax 
\doTocEntry\tocsubsection{10.1.2}{\csname a:TocLink\endcsname{12}{x12-11900010.1.2}{QQ2-12-261}{Reducing the effect of heating}}{603}\relax 
\doTocEntry\tocsubsubsection{}{\csname a:TocLink\endcsname{12}{x12-12000010.1.2}{QQ2-12-262}{Controlled atomospheres}}{603}\relax 
\doTocEntry\tocsubsubsection{}{\csname a:TocLink\endcsname{12}{x12-12100010.1.2}{QQ2-12-263}{Perpendicular field method}}{603}\relax 
\doTocEntry\tocsubsubsection{}{\csname a:TocLink\endcsname{12}{x12-12200010.1.2}{QQ2-12-264}{Multi-specimen techniques}}{604}\relax 
\doTocEntry\tocsubsubsection{}{\csname a:TocLink\endcsname{12}{x12-12300010.1.2}{QQ2-12-265}{Shaw family of experiments}}{605}\relax 
\doTocEntry\toclof{10.9}{\csname a:TocLink\endcsname{12}{x12-1230019}{}{\ignorespaces  Shaw family of methods (see text). a) Plot of pairs of NRM and the first TRM for each AF demagnetization step. b) Plot of pairs of the first ARM and the second ARM for each AF demagnetization step. c) Plot of pairs of NRM and TRM adjusted by the ratio of ARM1/ARM2 for that AF step from b) (TRM1*). d) same as a) but for the first and second TRMs. e) same as a) but for the second and third ARMs. f) Same as c) but for first and second TRM where TRM2* is adjusted using ARM2/ARM3 ratio from e). [Data of Yamamoto et al., 2003; figure from Tauxe and Yamazaki, 2007.]}}{figure}\relax 
\doTocEntry\tocsubsubsection{}{\csname a:TocLink\endcsname{12}{x12-12400010.1.2}{QQ2-12-267}{Use of microwaves for thermal excitation}}{608}\relax 
\doTocEntry\tocsubsubsection{}{\csname a:TocLink\endcsname{12}{x12-12500010.1.2}{QQ2-12-268}{Using materials resistant to alteration}}{610}\relax 
\doTocEntry\tocsubsubsection{}{\csname a:TocLink\endcsname{12}{x12-12600010.1.2}{QQ2-12-269}{Use of IRM normalization}}{610}\relax 
\doTocEntry\tocsubsection{10.1.3}{\csname a:TocLink\endcsname{12}{x12-12700010.1.3}{QQ2-12-270}{Quality assurance and data selection}}{611}\relax 
\doTocEntry\tocsection{10.2}{\csname a:TocLink\endcsname{12}{x12-12800010.2}{QQ2-12-271}{Paleointensity with DRMs}}{612}\relax 
\doTocEntry\toclof{10.10}{\csname a:TocLink\endcsname{12}{x12-12800110}{}{\ignorespaces Principles of relative paleointensity. The original DRM is plotted as open symbols. It is a function not only of the applied field, but also of the magnetic activity $[a_m]$ of the specimen. When normalized by $[a_m]$ (dots), the DRM is a linear function of applied field $B$. [Redrawn from Tauxe, 1993.] }}{figure}\relax 
\doTocEntry\tocsection{10.3}{\csname a:TocLink\endcsname{12}{x12-12900010.3}{QQ2-12-273}{Problems}}{617}\relax 
\doTocEntry\tocchapter{11}{\csname a:TocLink\endcsname{13}{x13-13000011}{QQ2-13-274}{Fisher statistics}}{623}\relax 
\doTocEntry\tocsection{11.1}{\csname a:TocLink\endcsname{13}{x13-13100011.1}{QQ2-13-275}{The normal distribution}}{623}\relax 
\doTocEntry\toclof{11.1}{\csname a:TocLink\endcsname{13}{x13-1310011}{}{\ignorespaces a) The Gaussian probability density function (normal distribution, Equation 11.1). The proportion of observations within an interval $dz$ centered on $ z$ is $f(z)dz$. b) Histogram of 1000 measurements of bed thickness in a sedimentary formation. Also shown is the smooth curve of a normal distribution with a mean of 10 and a standard deviation of 3. c) Histogram of the means from 100 repeated sets of 1000 measurements from the same sedimentary formation. The distribution of the means is much tighter. d) Histogram of the variances ($s^2$) from the same set of experiments as in c). The distribution of variances is not bell shaped; it is $\chi ^2$. }}{figure}\relax 
\doTocEntry\tocsection{11.2}{\csname a:TocLink\endcsname{13}{x13-13200011.2}{QQ2-13-277}{Statistics of vectors}}{633}\relax 
\doTocEntry\toclof{11.2}{\csname a:TocLink\endcsname{13}{x13-1320152}{}{\ignorespaces Hypothetical data sets drawn from Fisher distributions with vertical true directions with $\kappa $ = 5 (a-c), $\kappa $ = 10 (d-f), $\kappa $ = 50 (g-i). Estimated $\mathaccent "7016\relax D, \mathaccent "7016\relax I, \kappa , \alpha _{95}$ shown in insets. }}{figure}\relax 
\doTocEntry\tocsubsection{11.2.1}{\csname a:TocLink\endcsname{13}{x13-13300011.2.1}{QQ2-13-279}{Estimation of Fisher statistics}}{636}\relax 
\doTocEntry\toclof{11.3}{\csname a:TocLink\endcsname{13}{x13-1330023}{}{\ignorespaces a) Probability of finding a direction within an angular area, $dA$ centered at an angle $\alpha $ from the true mean. b) Probability of finding a direction at angle $\alpha $ away from the true mean direction.}}{figure}\relax 
\doTocEntry\toclof{11.4}{\csname a:TocLink\endcsname{13}{x13-1330074}{}{\ignorespaces Vector addition of eight unit vectors ($m_i$) to yield resultant vector $R$. [Figure redrawn from Butler, 1992.] }}{figure}\relax 
\doTocEntry\tocsubsection{11.2.2}{\csname a:TocLink\endcsname{13}{x13-13400011.2.2}{QQ2-13-282}{Some illustrations}}{649}\relax 
\doTocEntry\toclof{11.5}{\csname a:TocLink\endcsname{13}{x13-1340075}{}{\ignorespaces  Dependence of estimated angular standard deviation, CSD and $\delta $, and confidence limit, $\alpha _{95}$, on the number of directions in a data set. An increasing number of directions were selected from a Fisherian sample of directions with angular standard deviation $S$ = 15$^{\circ }$ ($\kappa $ = 29.2), shown by the horizontal line. }}{figure}\relax 
\doTocEntry\tocsection{11.3}{\csname a:TocLink\endcsname{13}{x13-13500011.3}{QQ2-13-284}{Significance Tests}}{654}\relax 
\doTocEntry\tocsubsection{11.3.1}{\csname a:TocLink\endcsname{13}{x13-13600011.3.1}{QQ2-13-285}{Watson's test for randomness}}{656}\relax 
\doTocEntry\tocsubsection{11.3.2}{\csname a:TocLink\endcsname{13}{x13-13700011.3.2}{QQ2-13-286}{Comparison of precision }}{657}\relax 
\doTocEntry\toclof{11.6}{\csname a:TocLink\endcsname{13}{x13-1370026}{}{\ignorespaces a) Equal area projections of declinations and inclinations of two hypothetical data sets. b) Fisher means and circles of confidence from the data sets in a). c) Distribution of $V_w$ for simulated Fisher distributions with the same $N$ and $\kappa $ as the two shown in a). The dashed line is the upper bound for the smallest 95\% of the $V_w$s calculated for the simulations ($V_{crit}$). The solid vertical line is the $V_w$ calculated for the two data sets. According to this test, the two data sets do not have a common mean, despite their overlapping confidence ellipses.}}{figure}\relax 
\doTocEntry\tocsubsection{11.3.3}{\csname a:TocLink\endcsname{13}{x13-13800011.3.3}{QQ2-13-288}{Comparing known and estimated directions}}{662}\relax 
\doTocEntry\toclof{11.7}{\csname a:TocLink\endcsname{13}{x13-1380017}{}{\ignorespaces  Examples of demagnetization data from a site whose mean is partially constrained by a great circle. The best-fit great circle and six directed lines allow a mean (diamond) and associated $\alpha _{95}$ to be calculated using the method of McFadden and McElhinny (1988). Demagnetization data for two of the directed lines are shown at the top of the diagram while those for the great circle are shown at the bottom. [Data from Tauxe et al., 2003.] }}{figure}\relax 
\doTocEntry\tocsubsection{11.3.4}{\csname a:TocLink\endcsname{13}{x13-13900011.3.4}{QQ2-13-290}{Comparing two estimated directions}}{665}\relax 
\doTocEntry\toclof{11.8}{\csname a:TocLink\endcsname{13}{x13-1390028}{}{\ignorespaces Directions drawn from a Fisher distribution with a near vertical true mean direction. The Fisher mean direction from the sample is shown by the triangle. The Gaussian average inclination ($<I>= 70^{\circ }$) is shallower than the Fisher mean $I_F = 75^{\circ }$. The estimated inclination using the maximum likelihood estimate of McFadden and Reid (1982) ($I_{MF}=73^{\circ }$ is closer to the Fisher mean than the Gaussian average).}}{figure}\relax 
\doTocEntry\tocsubsection{11.3.5}{\csname a:TocLink\endcsname{13}{x13-14000011.3.5}{QQ2-13-292}{Combining directions and great circles}}{670}\relax 
\doTocEntry\tocsection{11.4}{\csname a:TocLink\endcsname{13}{x13-14100011.4}{QQ2-13-293}{Inclination only data}}{670}\relax 
\doTocEntry\toclof{11.9}{\csname a:TocLink\endcsname{13}{x13-1410019}{}{\ignorespaces  Transformation of coordinates from a) geographic to b) ``data'' coordinates. The direction of the principal eigenvector $\hbox {\bf  V}_1$ is shown by the triangle in both plots. [Figure redrawn from Tauxe, 1998.]}}{figure}\relax 
\doTocEntry\toclof{11.10}{\csname a:TocLink\endcsname{13}{x13-14100210}{}{\ignorespaces  a) Declinations and b) co-inclinations ($\alpha $) from Figure 11.9. Also shown are behaviors expected for $D$ and $I$ from a Fisher distribution, i.e., declinations are uniformly distributed while co-inclinations are exponentially distributed. [Figure from Tauxe, 1998.]}}{figure}\relax 
\doTocEntry\tocsection{11.5}{\csname a:TocLink\endcsname{13}{x13-14200011.5}{QQ2-13-296}{Is a given data set Fisher distributed?}}{677}\relax 
\doTocEntry\toclof{11.11}{\csname a:TocLink\endcsname{13}{x13-14200111}{}{\ignorespaces  a) Quantile-quantile plot of declinations (in data coordinates) from Figure 11.9 plotted against an assumed uniform distribution. b) Same for inclinations plotted against an assumed exponential distribution. The data are Fisher distributed. [Figure from Tauxe, 1998.]}}{figure}\relax 
\doTocEntry\tocsection{11.6}{\csname a:TocLink\endcsname{13}{x13-14300011.6}{QQ2-13-298}{Problems}}{681}\relax 
\doTocEntry\tocchapter{12}{\csname a:TocLink\endcsname{14}{x14-14400012}{QQ2-14-299}{Beyond Fisher statistics}}{689}\relax 
\doTocEntry\toclof{12.1}{\csname a:TocLink\endcsname{14}{x14-1440011}{}{\ignorespaces a) VGPs from geomagnetic vectors evaluated from the statistical field model of Tauxe and Kent (2004) at 30$^{\circ }$N (site of observation shown as square). The geographic pole is shown as a triangle. A set of VGP positions at 60$^{\circ }$ N are shown as the black ring. b) Directions observed at the site of observation [square in a)] converted from black ring of VGPs in a) which correspond to the VGP positions at 60$^{\circ }$N. These directions have been projected along expected direction at site of observation (triangle). Note that a circularly symmetric ring about the geographic pole gives an asymmetric distribution of directions with a shallow bias. [Figures from Tauxe and Kent, 2004.]}}{figure}\relax 
\doTocEntry\tocsection{12.1}{\csname a:TocLink\endcsname{14}{x14-14500012.1}{QQ2-14-301}{Non-Fisherian parametric approaches}}{692}\relax 
\doTocEntry\tocsubsection{12.1.1}{\csname a:TocLink\endcsname{14}{x14-14600012.1.1}{QQ2-14-302}{The Kent distribution}}{693}\relax 
\doTocEntry\toclof{12.2}{\csname a:TocLink\endcsname{14}{x14-1460022}{}{\ignorespaces  a) An example of data obtained from a hypothetical equatorial sampling site plotted with the Fisher circle of confidence. The data have been transposed such that the expected direction (0, 0) is at the center of the diagram and ``up'' is at the top. b) Same data but with the Kent 95\% confidence ellipse. c) Data from a) with some directions transposed to the antipode; directions plotted with the Bingham 95\% confidence ellipse. }}{figure}\relax 
\doTocEntry\tocsubsection{12.1.2}{\csname a:TocLink\endcsname{14}{x14-14700012.1.2}{QQ2-14-304}{The Bingham distribution}}{697}\relax 
\doTocEntry\tocsubsection{12.1.3}{\csname a:TocLink\endcsname{14}{x14-14800012.1.3}{QQ2-14-305}{The Bingham-LeGoff approximation}}{698}\relax 
\doTocEntry\toclof{12.3}{\csname a:TocLink\endcsname{14}{x14-1480013}{}{\ignorespaces A bi-gaussian set of vectors suitable for treatment using the method of Love and Constable (2003). }}{figure}\relax 
\doTocEntry\tocsubsection{12.1.4}{\csname a:TocLink\endcsname{14}{x14-14900012.1.4}{QQ2-14-307}{The Bi-Gaussian distribution}}{701}\relax 
\doTocEntry\toclof{12.4}{\csname a:TocLink\endcsname{14}{x14-1490014}{}{\ignorespaces a) Hypothetical non-Fisherian data set. Normal and reversed polarity data that are not symmetrically distributed. Filled (open) circles plot on the lower (upper) hemisphere. b) Equal area projection of 500 bootstrapped means for pseudo-samples drawn from the data shown in a). c) Same as a) but with the bootstrapped confidence ellipses shown. }}{figure}\relax 
\doTocEntry\tocsection{12.2}{\csname a:TocLink\endcsname{14}{x14-15000012.2}{QQ2-14-309}{The simple (na\"ive) bootstrap}}{704}\relax 
\doTocEntry\toclof{12.5}{\csname a:TocLink\endcsname{14}{x14-1500015}{}{\ignorespaces Test for common mean with two directional data sets. a) Equal-area projections of two simulated Fisherian data sets (triangles and circles) each with $\kappa $ of 20. b) Means and $\alpha _{95}$s of data sets shown in a). }}{figure}\relax 
\doTocEntry\tocsection{12.3}{\csname a:TocLink\endcsname{14}{x14-15100012.3}{QQ2-14-311}{The parametric bootstrap}}{708}\relax 
\doTocEntry\toclof{12.6}{\csname a:TocLink\endcsname{14}{x14-1510016}{}{\ignorespaces  Cumulative distributions of Cartesian components of the bootstrapped means from 500 pseudo-samples from data shown in Figure 12.5. a) $X$ components. b) $ Y$, and c) $Z$. Also shown are the bounds for each data set that include 95\% of the components. The confidence intervals for the different data sets overlap for $X$ and $Z$ but not for $Y$. }}{figure}\relax 
\doTocEntry\tocsection{12.4}{\csname a:TocLink\endcsname{14}{x14-15200012.4}{QQ2-14-313}{When are two data sets distinct?}}{711}\relax 
\doTocEntry\tocsection{12.5}{\csname a:TocLink\endcsname{14}{x14-15300012.5}{QQ2-14-314}{Application to the ``reversals test''}}{711}\relax 
\doTocEntry\toclof{12.7}{\csname a:TocLink\endcsname{14}{x14-1530017}{}{\ignorespaces Cumulative distributions of Cartesian coordinates of means of pseudo-samples drawn from the data shown in Figure 12.4a. The reverse polarity mode has been flipped to its antipode. The intervals containing 95\% of each set of components are also drawn (vertical lines). Because the confidence bounds from the two data sets overlap in all three components, the means of the reverse and normal modes cannot be distinguished at the 95\% level of confidence; they pass the bootstrap reversals test.}}{figure}\relax 
\doTocEntry\tocsection{12.6}{\csname a:TocLink\endcsname{14}{x14-15400012.6}{QQ2-14-316}{Application to the ``fold test''}}{714}\relax 
\doTocEntry\toclof{12.8}{\csname a:TocLink\endcsname{14}{x14-1540018}{}{\ignorespaces  a) Equal area projection of a set of directions in geographic coordinates. The data were drawn from the same distribution of directions that gave rise to the VGPs shown in Figure 12.1a. They have been rotated about strike on two simulated limbs of the fold, one to the northeast and one to the southwest, resulting in a streaked (girdle) distribution. The original polarity of many data points is ambiguous. b) Data from a) after back tilting by 100\% of the original tilt. Polarities are more readily identifiable. c) Red dashed lines: trends of the largest eigenvalues ($\tau _1$s) of the orientation matrices from representative pseudo-samples drawn from a) as they evolve during untilting. The directions are adjusted for tilt incrementally from -10\% to 110\%. The largest value of ($\tau _1$ occurs near 100\% in all of the pseudo-samples sets. The cumulative distribution is of 500 maxima of $\tau _1$ and the bounds that enclose 95\% of them. These data ``pass'' the bootstrap fold test.}}{figure}\relax 
\doTocEntry\tocsection{12.7}{\csname a:TocLink\endcsname{14}{x14-15500012.7}{QQ2-14-318}{Problems}}{718}\relax 
\doTocEntry\tocchapter{13}{\csname a:TocLink\endcsname{15}{x15-15600013}{QQ2-15-319}{Paleomagnetic tensors}}{725}\relax 
\doTocEntry\toclof{13.1}{\csname a:TocLink\endcsname{15}{x15-1560011}{}{\ignorespaces Definition of specimen coordinate system. b) Six measurement scheme for determining the anisotropy ellipsoid. c) Position of the specimen in the magnetic smagusceptibility meter. }}{figure}\relax 
\doTocEntry\tocsection{13.1}{\csname a:TocLink\endcsname{15}{x15-15700013.1}{QQ2-15-321}{Anisotropy of magnetic susceptibility}}{728}\relax 
\doTocEntry\toclof{13.2}{\csname a:TocLink\endcsname{15}{x15-1570092}{}{\ignorespaces a) Arbitrary coordinate system of a specimen. b) The magnitude ellipsoid of AMS. Its coordinate system is defined by the eigenvectors $\hbox {\bf  V}_i$. The lengths along the eigenvectors of the ellipsoid surface are related to the eigenvalues $\tau _i$ (see text). }}{figure}\relax 
\doTocEntry\tocsection{13.2}{\csname a:TocLink\endcsname{15}{x15-15800013.2}{QQ2-15-323}{Hext Statistics}}{738}\relax 
\doTocEntry\toclof{13.3}{\csname a:TocLink\endcsname{15}{x15-1580043}{}{\ignorespaces Relationship of the uncertainty ellipses (calculated by Hext statistics for AMS data) to the principal axes. The major and minor semi-axes of the uncertainty ellipses are oriented along the axes defined by the eigenvectors. [Figure from Tauxe, 1998.] }}{figure}\relax 
\doTocEntry\tocsubsection{13.2.1}{\csname a:TocLink\endcsname{15}{x15-15900013.2.1}{QQ2-15-325}{Hext confidence ellipses}}{744}\relax 
\doTocEntry\tocsubsection{13.2.2}{\csname a:TocLink\endcsname{15}{x15-16000013.2.2}{QQ2-15-326}{Hext $F$ statistics for significance of eigenvalue ratios}}{746}\relax 
\doTocEntry\tocsection{13.3}{\csname a:TocLink\endcsname{15}{x15-16100013.3}{QQ2-15-327}{Limitations of Hext statistics}}{747}\relax 
\doTocEntry\tocsection{13.4}{\csname a:TocLink\endcsname{15}{x15-16200013.4}{QQ2-15-328}{Bootstrap confidence ellipses}}{748}\relax 
\doTocEntry\toclof{13.4}{\csname a:TocLink\endcsname{15}{x15-1620014}{}{\ignorespaces  a) Lower hemisphere projection of directions of $\hbox {\bf  V}_1$ (squares), $\hbox {\bf  V}_2$ (triangles), and $\hbox {\bf  V}_3$ (circles) from the margin of a volcanic dike. Open symbols are the Hext means. Thin blue lines are the Hext 95\% confidence ellipses (dashed portion are on the upper hemisphere). b) Equal area projection of principal eigenvectors ($\hbox {\bf  V}_1$) of 500 pseudo-samples drawn from the data in a). c) Same as b) for the major eigenvectors ($\hbox {\bf  V}_2$). d) Same as b) for the minor eigenvectors ($\hbox {\bf  V}_3$). [Data from Tauxe et al., 1998.]}}{figure}\relax 
\doTocEntry\toclof{13.5}{\csname a:TocLink\endcsname{15}{x15-1620025}{}{\ignorespaces a) AMS data from Cretaceous carbonate limestones in Italy (the Scaglia Bianca Formation) in tilt adjusted coordinates. a) Lower hemisphere projections of the principal $\hbox {\bf  V}_1$ (squares), major $\hbox {\bf  V}_2$ (triangles), and minor $\hbox {\bf  V}_3$ (circles) eigenvectors. b) Bootstrapped eigenvectors from pseudo-samples of the data in a). c) Cumulative distribution of the $v_{31}$ with bounds containing 95\% of the components plotted as dashed lines. The zero value expected from a vertical direction is shown as a vertical solid line. d) Same as c) but for the $v_{32}$ components. [Data from Cronin et al., 2001.] }}{figure}\relax 
\doTocEntry\tocsection{13.5}{\csname a:TocLink\endcsname{15}{x15-16300013.5}{QQ2-15-331}{Comparing mean eigenvectors with other axes}}{755}\relax 
\doTocEntry\toclof{13.6}{\csname a:TocLink\endcsname{15}{x15-1630016}{}{\ignorespaces Principles of AMS for interpretation of flow directions in dikes. [Figure from Tauxe, 1998 after Knight and Walker, 1988; ]}}{figure}\relax 
\doTocEntry\toclof{13.7}{\csname a:TocLink\endcsname{15}{x15-1630027}{}{\ignorespaces Characteristics of AMS data from sediments deposited in a) quiet water, b) moderate water flow, and c) flow that is sufficient to entrain particles. [Figure adapted from Tauxe, 1998.] }}{figure}\relax 
\doTocEntry\toclof{13.8}{\csname a:TocLink\endcsname{15}{x15-1630038}{}{\ignorespaces Determination of the shape of AMS data using the bootstrap. Conventions as in Figure 13.4 a-d) Selected data sets plotted as eigenvector directions from individual specimens. e-h) Bootstrapped eigenvectors from a-d) respectively. i-l) Cumulative distributions of the bootstrapped eigenvalues associated with the eigenvectors plotted in e-h). The bounds containing 95\% of each eigenvalue are shown as vertical dashed dot line for $\tau _3$, dashed for $\tau _2$ and solid for $\tau _1$. }}{figure}\relax 
\doTocEntry\tocsection{13.6}{\csname a:TocLink\endcsname{15}{x15-16400013.6}{QQ2-15-335}{Shape}}{764}\relax 
\doTocEntry\toclof{13.9}{\csname a:TocLink\endcsname{15}{x15-1640019}{}{\ignorespaces Properties of various AMS diagrams: a) Flinn, b) Ramsay and c) Jelinek. [Figure from Tauxe, 1998.] }}{figure}\relax 
\doTocEntry\toclof{13.10}{\csname a:TocLink\endcsname{15}{x15-16400210}{}{\ignorespaces Properties of the Ternary diagram: a) There are three axes with limits of $\tau _1,\tau _2, \tau _3$. Because of the constraint that $\tau _1>\tau _2>\tau _3$, only the shaded region is allowed. This is bounded at the top by a sphere when all three eigenvalues are equal, to the bottom left by a disk and to the bottom right by a needle. Geological materials generally have a low percentage of anisotropy and plot close to the sphere. This region is enlarged in b) which illustrates how the ternary projection can be plotted as $E'$ versus $R$ and how shape (oblate, prolate, sphere) and percent anisotropy appear on the diagram. [Figure from Tauxe, 1998.] }}{figure}\relax 
\doTocEntry\toclot{13.1}{\csname a:TocLink\endcsname{15}{x15-1640031}{}{\ignorespaces Assorted anisotropy statistics.}}{table}\relax 
\doTocEntry\tocsection{13.7}{\csname a:TocLink\endcsname{15}{x15-16500013.7}{QQ2-15-339}{Anisotropy of magnetic remanence}}{774}\relax 
\doTocEntry\tocsubsection{13.7.1}{\csname a:TocLink\endcsname{15}{x15-16600013.7.1}{QQ2-15-340}{Anisotropy of ARM and TRM}}{774}\relax 
\doTocEntry\tocsubsection{13.7.2}{\csname a:TocLink\endcsname{15}{x15-16700013.7.2}{QQ2-15-341}{Anisotropy of DRM}}{776}\relax 
\doTocEntry\tocsection{13.8}{\csname a:TocLink\endcsname{15}{x15-16800013.8}{QQ2-15-342}{Problems}}{778}\relax 
\doTocEntry\tocchapter{14}{\csname a:TocLink\endcsname{16}{x16-16900014}{QQ2-16-343}{The ancient geomagnetic field}}{787}\relax 
\doTocEntry\toclof{14.1}{\csname a:TocLink\endcsname{16}{x16-1690051}{}{\ignorespaces a) Radiocarbon calibration data from from Cariaco ODP Leg 165, Holes 1002D and 1002E (blue circles), plotted versus calendar age assigned by correlation of detailed paleoclimate records to the Greenland Ice Core GISP2. The thin black line is high-resolution radiocarbon calibration data from tree rings joined at 12 cal. ka B.P. to the varve counting chronology. Red squares are paired $^{14}$C-U/Th dates from corals. Light gray shading represents the uncertainties in the Cariaco calibration. The radiocarbon dates are too young, falling well below the dotted line of 1:1 correlation. b) Compilation of data interpreted as production rate changes in radiocarbon ($\Delta ^{14}$C) versus calender age. (symbols same as in a). c) Predicted variation of $\Delta ^{14}$C from the geomagnetic field intensity variations from sediments of the north Atlantic (Laj et al., 2002) using the model of Masarik and Beer (1999). [Figure modified from Hughen et al., 2004.]}}{figure}\relax 
\doTocEntry\toclof{14.2}{\csname a:TocLink\endcsname{16}{x16-1690142}{}{\ignorespaces a) A reconstruction (Wang , 1948) of the south pointing spoon ({\it  shao}) used by the Chinese in the first century CE. [Photo of Stan Sherer.] ] b) Measurements of magnetic declination made in China from 720 CE to 1829. [Data quoted in Smith and Needham, 1967.] }}{figure}\relax 
\doTocEntry\tocsection{14.1}{\csname a:TocLink\endcsname{16}{x16-17000014.1}{QQ2-16-346}{Historical measurements}}{794}\relax 
\doTocEntry\toclof{14.3}{\csname a:TocLink\endcsname{16}{x16-1700013}{}{\ignorespaces Chart of magnetic declination of Halley. Shown in blue is the line of zero variation from the 2005 IGRF. [Figure modified from Cook, 2001.]}}{figure}\relax 
\doTocEntry\toclof{14.4}{\csname a:TocLink\endcsname{16}{x16-1700024}{}{\ignorespaces Maps of the strength of the radial magnetic field at the core mantle boundary from the {\bf  GUFM1} secular variation model of Jackson et al., (2000). a) For 1600 CE. b) For 1990. c) Field strength in San Diego, CA evaluated from the {\bf  GUFM1} model.}}{figure}\relax 
\doTocEntry\toclof{14.5}{\csname a:TocLink\endcsname{16}{x16-1700035}{}{\ignorespaces Inclinations evaluated at 100 year intervals from the PSVMOD1.0 of Constable et al. (2000) for selected records. These are plotted from East to West. Maxima and minima are noted. Westward drift would imply that these correlated features would ``rise'' to the right.}}{figure}\relax 
\doTocEntry\tocsection{14.2}{\csname a:TocLink\endcsname{16}{x16-17100014.2}{QQ2-16-350}{Archaeo- and paleomagnetic records}}{804}\relax 
\doTocEntry\tocsubsection{14.2.1}{\csname a:TocLink\endcsname{16}{x16-17200014.2.1}{QQ2-16-351}{Pioneers in paleomagnetism}}{805}\relax 
\doTocEntry\toclof{14.6}{\csname a:TocLink\endcsname{16}{x16-1720016}{}{\ignorespaces  Paleosecular variation of the magnetic field ($D$ and $I$) observed in the Wilson Creek section north of Mono Lake. The inclination expected from a geocentric axial dipole is shown as a dashed line. The declination is expected to be zero. The so-called ``Mono Lake'' excursion is marked. The data are from Lund et al. (1988) and represent some 23 kyr of time. }}{figure}\relax 
\doTocEntry\tocsubsection{14.2.2}{\csname a:TocLink\endcsname{16}{x16-17300014.2.2}{QQ2-16-353}{The last seven millenia}}{808}\relax 
\doTocEntry\tocsubsection{14.2.3}{\csname a:TocLink\endcsname{16}{x16-17400014.2.3}{QQ2-16-354}{Westward drift}}{809}\relax 
\doTocEntry\toclof{14.7}{\csname a:TocLink\endcsname{16}{x16-1740017}{}{\ignorespaces Stack of relative paleointensity records from deep sea sediments. [Figure modified from Guyodo and Valet, 1999.] }}{figure}\relax 
\doTocEntry\tocsubsection{14.2.4}{\csname a:TocLink\endcsname{16}{x16-17500014.2.4}{QQ2-16-356}{The more distant past}}{812}\relax 
\doTocEntry\toclof{14.8}{\csname a:TocLink\endcsname{16}{x16-1750058}{}{\ignorespaces Relative paleointensity records spanning the last 100 kyr with independent age control based on $\delta ^{18}$O. The solid red bars indicate intensity lows that are possibly related to the ``Laschamp excursion'' and the blue bars are a later paleointensity low, referred to as the ``Mono Lake excursion''. [Figure from Tauxe and Yamazaki, 2007.] }}{figure}\relax 
\doTocEntry\tocsection{14.3}{\csname a:TocLink\endcsname{16}{x16-17600014.3}{QQ2-16-358}{Time series of paleomagnetic data}}{816}\relax 
\doTocEntry\tocsubsection{14.3.1}{\csname a:TocLink\endcsname{16}{x16-17700014.3.1}{QQ2-16-359}{Excursions}}{816}\relax 
\doTocEntry\toclof{14.9}{\csname a:TocLink\endcsname{16}{x16-1770019}{}{\ignorespaces Directional data from ODP Site 919. Declination ($D$) and inclination ($I$) data from continuous core (``u-channel'') measurements (dark/green closed symbols connected by line), deconvolved u-channel data (closed gray/blue symbols connected by dashed line) and data from 1cc discrete samples (open/red squares without connecting line). [Figure redrawm from Channell (2006).] }}{figure}\relax 
\doTocEntry\toclof{14.10}{\csname a:TocLink\endcsname{16}{x16-17700210}{}{\ignorespaces  a) The lower Jaramillo geomagnetic polarity reversal as recorded in deep sea sediments from core RC14-14. Inclinations and declinations expected from a normal and reverse GAD field are shown as dashed lines. [Data from Clement and Kent, 1984]. b) Record of polarity transition recorded at Steens Mountain. [Data from Camps et al., 1999.] }}{figure}\relax 
\doTocEntry\toclof{14.11}{\csname a:TocLink\endcsname{16}{x16-17700311}{}{\ignorespaces  VDM versus VGP latitude from data in the PINT06 database compiled by Tauxe and Yamazaki (2007). The red triangles are from double heating experiments with pTRM checks (see Chapter 10). b) Plot of transitional VGPs (blue dots) from the TRANS data base (McElhinny and Lock, 1996). No selection criteria were applied. c) Shear wave velocity SB448 model of Masters et al. (2000) evaluated at 2770 km (core mantle boundary region). There is a fast (cold) ring around the Pacific, presumably from the influence of subducted slabs. }}{figure}\relax 
\doTocEntry\tocsubsection{14.3.2}{\csname a:TocLink\endcsname{16}{x16-17800014.3.2}{QQ2-16-363}{Reversals}}{827}\relax 
\doTocEntry\toclof{14.12}{\csname a:TocLink\endcsname{16}{x16-17800112}{}{\ignorespaces Barcode: The Geomagnetic Polarity Time Scale (GPTS) for the last 160 Ma (Berggren et al., 1995; Gradstein et al., 1995). Line traces the reversal frequency (number of reversals in a four million year interval) estimated by Constable (2003).}}{figure}\relax 
\doTocEntry\tocsection{14.4}{\csname a:TocLink\endcsname{16}{x16-17900014.4}{QQ2-16-365}{Geomagnetic polarity time scale -- a first look}}{830}\relax 
\doTocEntry\toclof{14.13}{\csname a:TocLink\endcsname{16}{x16-17900113}{}{\ignorespaces Time averaged intensity of the geomagnetic field. [Model of Hatakeyama and Kono. 2002.] }}{figure}\relax 
\doTocEntry\tocsection{14.5}{\csname a:TocLink\endcsname{16}{x16-18000014.5}{QQ2-16-367}{The time averaged field}}{833}\relax 
\doTocEntry\toclof{14.14}{\csname a:TocLink\endcsname{16}{x16-18000114}{}{\ignorespaces  a) Paleointensity versus latitude of the Pint06 database (grey crosses) (see Tauxe and Yamazaki, 2007) and paleointensity estimates from Lawrence et al. (2009) for data with ages less than 5 Ma, $d\sigma B \le $ 15 $\mu $T, and $N_{site} \ge $ 2. Mean paleointensity results (diamonds) are calculated for 15$^{\circ }$ latitude bins and errors are shown as 2$\sigma $. The black line is the longitudinal-averaged intensity for today's field. The vertical dashed line is the surface expression of the edge of the tangent cylinder. Southern hemisphere data have been flipped to the Northern hemisphere. The black line represents the mean intensity for today's field as defined by the 2005 IGRF model coefficients, while the red dashed line represents the intensity associated with a geocentric axial dipole with a dipole term of 30 $\mu $T. b) Illustration of outer core flow regimes. The tangent cylinder is denoted by the blue cylinder tangential to the red sphere (inner core). [Figures redrawn from Lawrence et al., 2009.] }}{figure}\relax 
\doTocEntry\toclof{14.15}{\csname a:TocLink\endcsname{16}{x16-18000215}{}{\ignorespaces Summary of data in the PINT06 compilation of Tauxe and Yamazaki (2007) meeting minimum acceptance criteria for last 200 Ma. Blue dots are submarine basaltic glass data. Red diamonds are single crystal results. Triangles are all other data meeting the same consistency criteria ($\sigma < $5\% of mean or $<$5$\mu $T); At the bottom is the Geomagnetic Polarity Time Scale showing the Cretaceous Normal Superchron (CNS) and the M-sequence of magnetic anomalies. [Figure from Tauxe and Yamazaki, 2007.]}}{figure}\relax 
\doTocEntry\tocsection{14.6}{\csname a:TocLink\endcsname{16}{x16-18100014.6}{QQ2-16-370}{Long term changes in paleointensity}}{841}\relax 
\doTocEntry\toclof{14.16}{\csname a:TocLink\endcsname{16}{x16-18100116}{}{\ignorespaces a) Paleomagnetic directions from the PSVRL database (see McElhinny and McFadden, 1997) compiled for latitude band 0-5$^{\circ }$ (N\&S). Antipodes of reverse directions are used. The expected direction is at the star at the center of the equal area projection. Directions in the upper (lower) half are above (below) those expected and those to the right (left) are right-handed (left-handed). The red ellipse illustrates the elongation $E$ of the directional data where $E$ is the ratio of the eigenvalues along the maximum and minimum axes (here vertical and E-W respectively). b) Same as a) but for 25-35$^{\circ }$ (N\&S) latitude band. c) Same as a) but for 55-65$^{\circ }$ (N\&S) latitude band. [Figures redrawn from Tauxe and Kent, 2004.] }}{figure}\relax 
\doTocEntry\toclof{14.17}{\csname a:TocLink\endcsname{16}{x16-18100217}{}{\ignorespaces a) Illustration of a normal distribution with varying standard deviations. b) Variation of standard deviation $\sigma $ as a function of spherical harmonic degree $l$ in the CP88 model. }}{figure}\relax 
\doTocEntry\tocsection{14.7}{\csname a:TocLink\endcsname{16}{x16-18200014.7}{QQ2-16-373}{Statistical models of paleosecular variation}}{847}\relax 
\doTocEntry\toclof{14.18}{\csname a:TocLink\endcsname{16}{x16-18200318}{}{\ignorespaces a) Variation of the standard deviation $\sigma _l$ as a function of harmonic degree $l$ for asymmetric and symmetric terms for the statistical field model TK03.GAD. All terms have zero mean except the axial dipole term. b) Estimated behavior of $S'$ from the data compilation of McElhinny and McFadden (1997) (circles). Blue line is the predicted variation of $S'$ from the TK03.GAD model of Tauxe and Kent (2004). c) 1000 vector endpoints from realizations of model TK03.GAD at 30$^{\circ }$N. d) Elongation versus inclination predicted from the TK03.GAD model. Compilation of data from LIPs of Tauxe et al. (2008). Crossed open circles are data from large igneous provinces back through time. Yemeni traps: Riisager et al. (2005). Deccan traps: Vandamme et al. (1991), Vandamme and Courtillot (1992). Faroe Island basalts: Riisager et al. (2002). Kerguelen: Plenier et al. (2002). [Figures from Tauxe and Kent, 2004; Tauxe 2005; Tauxe et al., 2008.]}}{figure}\relax 
\doTocEntry\tocsection{14.8}{\csname a:TocLink\endcsname{16}{x16-18300014.8}{QQ2-16-375}{Problems}}{857}\relax 
\doTocEntry\tocchapter{15}{\csname a:TocLink\endcsname{17}{x17-18400015}{QQ2-17-376}{The GPTS and magnetostratigraphy}}{865}\relax 
\doTocEntry\tocsection{15.1}{\csname a:TocLink\endcsname{17}{x17-18500015.1}{QQ2-17-377}{Early efforts in defining the GPTS}}{865}\relax 
\doTocEntry\toclof{15.1}{\csname a:TocLink\endcsname{17}{x17-1850011}{}{\ignorespaces  Magnetic polarities from volcanic units plotted against age as determined by the potassium-argon method. The first three long intervals were named after famous geomagnetists. [Figure redrawn from Cox et al., 1964].}}{figure}\relax 
\doTocEntry\toclof{15.2}{\csname a:TocLink\endcsname{17}{x17-1850022}{}{\ignorespaces Map of the pattern of magnetic anomalies off northwestern North America. [Figure from Mason and Raff, 1961.]}}{figure}\relax 
\doTocEntry\toclof{15.3}{\csname a:TocLink\endcsname{17}{x17-1850033}{}{\ignorespaces A profile of bathymetry (bottom panel) and magnetic anomalies (labelled ``profile'') obtained from the East Pacific Rise (Eltanin 19 profile, also known as ``the magic profile''.) The magnetic anomaly profile, flipped east-to-west is replotted above (labelled ``profile backwards''). Assuming a magnetization of a 500 m thick section of oceanic crust (black and white pattern above), a model for the predicted anomalies could be generated (labelled ``model''). Above is the inferred time scale. The position of the Gauss/Gilbert boundary is marked by stars. [Adapted from Pitman and Heirtzler, 1966.]}}{figure}\relax 
\doTocEntry\toclof{15.4}{\csname a:TocLink\endcsname{17}{x17-1850044}{}{\ignorespaces a) Inclinations from core V16-134 plotted against depth. b) The GPTS as it was known in 1966. c) Faunal zones of the southern ocean identified within the core. [Data from Opdyke et al., 1966.]}}{figure}\relax 
\doTocEntry\toclof{15.5}{\csname a:TocLink\endcsname{17}{x17-1850055}{}{\ignorespaces Declinations from deep-sea piston core RC12-65 from the equatorial Pacific Ocean (using an arbitrary zero line because the cores were not oriented). The epoch system of magnetostratigrahic nomenclature was extended back to Epoch 11 in this core and to Epoch 19 in companion cores. [Figure redrawn from Opdyke et al., 1974].}}{figure}\relax 
\doTocEntry\toclof{15.6}{\csname a:TocLink\endcsname{17}{x17-1850066}{}{\ignorespaces Illustration of the ``astrochronology'' dating method. The sequence of polarity intervals and climatically induced sapropel layers is correlated to the GPTS (left) and the orbital cycles (right). The numerical ages from the orbital cycles can then be transferred to the GPTS. [Adapted from Hilgen, 1991.]}}{figure}\relax 
\doTocEntry\tocsubsection{15.1.1}{\csname a:TocLink\endcsname{17}{x17-18600015.1.1}{QQ2-17-384}{The addition of biostratigraphy}}{885}\relax 
\doTocEntry\toclof{15.7}{\csname a:TocLink\endcsname{17}{x17-1860017}{}{\ignorespaces  Left: Lithostratigraphic and magnetostratigraphic pattern derived from overlapping drill cores into the Newark Basin. Right: Interpretation for the GPTS based on astrochronology and correlation to the Geological Time Scale. [Adapted from Kent et al., 1995 and Kent and Olsen, 1999.]}}{figure}\relax 
\doTocEntry\toclof{15.8}{\csname a:TocLink\endcsname{17}{x17-1860028}{}{\ignorespaces The Neogene of the Geological Time Scale. [Figure created using TSCreator software from \url  {http://chronos.org} based on time scale of Lourens et al., 2004.]}}{figure}\relax 
\doTocEntry\tocsubsection{15.1.2}{\csname a:TocLink\endcsname{17}{x17-18700015.1.2}{QQ2-17-387}{Astrochronology}}{895}\relax 
\doTocEntry\tocsubsection{15.1.3}{\csname a:TocLink\endcsname{17}{x17-18800015.1.3}{QQ2-17-388}{ A note on terminology}}{895}\relax 
\doTocEntry\tocsection{15.2}{\csname a:TocLink\endcsname{17}{x17-18900015.2}{QQ2-17-389}{Current status of the geological time scale}}{897}\relax 
\doTocEntry\toclof{15.9}{\csname a:TocLink\endcsname{17}{x17-1890019}{}{\ignorespaces Plot of distance from the ridge crest in the South Atlantic versus age using the GPTS of Gradstein et al., (2004). The differential of this curve gives the inferred instantaneous spreading rate.}}{figure}\relax 
\doTocEntry\tocsection{15.3}{\csname a:TocLink\endcsname{17}{x17-19000015.3}{QQ2-17-391}{Applications}}{900}\relax 
\doTocEntry\tocsubsection{15.3.1}{\csname a:TocLink\endcsname{17}{x17-19100015.3.1}{QQ2-17-392}{Dating geological sequences}}{900}\relax 
\doTocEntry\toclof{15.10}{\csname a:TocLink\endcsname{17}{x17-19101510}{}{\ignorespaces Application of magnetostratigraphic techniques for delineating isochronous horizons in a series of stratigraphic sections. The polarities of sampling sites are shown by open (reverse) and solid (normal) symbols. The light shading indicates silts, while the darker shaded units (labelled A-C) represent sand bodies, which were not suitable for paleomagnetic analysis in this example. The inferred isochrons (horizons that separate polarity zones) are shown as heavy dashed lines. [Figure modified from Behrensmeyer and Tauxe, 1982.]}}{figure}\relax 
\doTocEntry\tocsubsection{15.3.2}{\csname a:TocLink\endcsname{17}{x17-19200015.3.2}{QQ2-17-394}{Measuring rates}}{905}\relax 
\doTocEntry\tocsubsection{15.3.3}{\csname a:TocLink\endcsname{17}{x17-19300015.3.3}{QQ2-17-395}{Tracing of magnetic isochrons}}{905}\relax 
\doTocEntry\toclot{}{\csname a:TocLink\endcsname{17}{x17-193002}{}{\numberline {15.1}{Geomagnetic polarity time scale}}}{906}\relax 
\doTocEntry\tocsection{15.4}{\csname a:TocLink\endcsname{17}{x17-19400015.4}{QQ2-17-397}{Problems}}{908}\relax 
\doTocEntry\tocchapter{16}{\csname a:TocLink\endcsname{18}{x18-19500016}{QQ2-18-398}{Tectonic applications of paleomagnetism}}{913}\relax 
\doTocEntry\toclof{16.1}{\csname a:TocLink\endcsname{18}{x18-1950011}{}{\ignorespaces  a) A moving continent will retain a record of changing paleomagnetic directions through time that reflect the changing orientations and distances to the pole (which is held fixed). The resulting path of observed pole positions is called an ``apparent polar wander path'' or APWP because in this case the pole is actually fixed and only appears to move when viewed from the continental frame of reference. b) On the other hand, if a continent is held fixed, the same changing paleomagnetic directions reflect the wandering of the pole itself. This is called ``true polar wander'' or TPW.}}{figure}\relax 
\doTocEntry\tocsection{16.1}{\csname a:TocLink\endcsname{18}{x18-19600016.1}{QQ2-18-400}{Essentials of plate tectonic theory}}{916}\relax 
\doTocEntry\toclof{16.2}{\csname a:TocLink\endcsname{18}{x18-1960022}{}{\ignorespaces a) Some of the major lithospheric plates. b) Motion of North America with respect to Europe around the Euler pole shown as a blue square. Projection is such that current Euler pole North America (NAM) with respect to Europe (EUR) is at the ``North pole''. Lines of co-latitude are the angular distance from the Euler pole, $\theta $. Velocities of NAM with respect to EUR at two points with different $\theta $ are shown as black arrows. }}{figure}\relax 
\doTocEntry\toclof{16.3}{\csname a:TocLink\endcsname{18}{x18-1960033}{}{\ignorespaces a) Finite rotation of North America from one frame of reference to another. Finite rotation pole is located at $\lambda _f,\phi _f$ and the finite rotation is $\Omega $. b) Estimating a finite rotation of a continental fragment from a paleomagnetic pole.}}{figure}\relax 
\doTocEntry\toclof{16.4}{\csname a:TocLink\endcsname{18}{x18-1960044}{}{\ignorespaces Polarity and paleolongitude can be ambiguous from paleomagnetic data alone. All three positions of the continental fragment (a,b,c) could be reconstructions of the same observed direction. a) and b) differ with assumed polarity. b) and c) differ with assumed longitude.}}{figure}\relax 
\doTocEntry\tocsection{16.2}{\csname a:TocLink\endcsname{18}{x18-19700016.2}{QQ2-18-404}{Poles and apparent polar wander}}{927}\relax 
\doTocEntry\toclof{16.5}{\csname a:TocLink\endcsname{18}{x18-1970015}{}{\ignorespaces Paleomagnetic poles from Australia for the last 200 Ma from GPMDB. a) No selection criteria. b) The selection criteria of BC02.}}{figure}\relax 
\doTocEntry\toclof{16.6}{\csname a:TocLink\endcsname{18}{x18-1970166}{}{\ignorespaces Examples of how to construct an APWP. a) Discrete window. b) Key pole approach. c) Moving window (Besse and Courtillot, 2002) versus spline (Torsvik et al., 2008). }}{figure}\relax 
\doTocEntry\toclof{16.7}{\csname a:TocLink\endcsname{18}{x18-1970257}{}{\ignorespaces a) Paleomagnetic Euler pole method for determining APWPs. A continent is rotating about a fixed Euler pole (green triangle). As the continent moves, rocks record paleomagnetic directions reflecting the position of the spin axis at that particular age. b) When viewed in the present coordinate system and converted to paleomagnetic poles, these will fall on the small circle APWP track. c) PEP analysis for Jurassic APWP for North America of May and Butler (1986). Poles are interpreted to lie along small circle tracks (J1/J2) separated by a cusp (J2 cusp) located at the LM pole. The J1 and J2 tracks are small circles about their respective Euler poles, shown as blue triangles. }}{figure}\relax 
\doTocEntry\toclof{16.8}{\csname a:TocLink\endcsname{18}{x18-1970328}{}{\ignorespaces  Master path approach: Maps of continental reconstructions for a) present, b) 50, c) 100, and d) 200 Ma. e) Poles and APWP for various continents for the last 200 million years, evaluated at five million year intervals. [Reconstructions using finite rotation poles of Torsvik et al. 2008 (see Appendix\nobreakspace  {}\o:ref {app:polerot}).] Paleomagnetic poles from the synthetic APWP constructed by Besse and Courtillot, (2002) exported to the different continents. }}{figure}\relax 
\doTocEntry\toclof{16.9}{\csname a:TocLink\endcsname{18}{x18-1970339}{}{\ignorespaces Sampling sites are marked by triangles. Inclinations from the sites can be used to calculate the paleomagnetic colatitude of the site using the dipole formula (see Chapter 2) which defines a small circle along which the paleomagnetic pole must lie. The intersection of three such small circles uniquely defines the position of the paleomagnetic pole. }}{figure}\relax 
\doTocEntry\toclot{16.1}{\csname a:TocLink\endcsname{18}{x18-1970341}{}{\ignorespaces Demagnetization Codes (DC) summarized by McElhinny and McFadden (2000).}}{table}\relax 
\doTocEntry\toclof{16.10}{\csname a:TocLink\endcsname{18}{x18-19703510}{}{\ignorespaces The South African APWP for the Phanerozoic. a) South African poles only (Table 1 of Torsvik and van der Voo, 2002). b) Smoothed APWP spline path using master path approach for Gondwana in South African coordinates. }}{figure}\relax 
\doTocEntry\tocsection{16.3}{\csname a:TocLink\endcsname{18}{x18-19800016.3}{QQ2-18-412}{The Gondwana APWP}}{935}\relax 
\doTocEntry\toclof{16.11}{\csname a:TocLink\endcsname{18}{x18-19800111}{}{\ignorespaces a) Set of possible geomagnetic field directions plotted in equal area projection. Lower (upper) hemisphere directions are solid (open) symbols. b) Directions recorded by the sediment using the flattening function. [Figure modified from Tauxe et al., 2008] }}{figure}\relax 
\doTocEntry\tocsection{16.4}{\csname a:TocLink\endcsname{18}{x18-19900016.4}{QQ2-18-414}{Inclination shallowing and GAD}}{937}\relax 
\doTocEntry\toclof{16.12}{\csname a:TocLink\endcsname{18}{x18-19900112}{}{\ignorespaces a) Paleomagnetic directions of Oligo-Miocene redbeds from Asia in equal area projection (stratigraphic coordinates). [Redrawn from Tauxe and Kent, 2004; data from Gilder et al., 2001.]. b) Plot of elongation versus inclination for the data (heavy red line) and for the TK03.GAD model (dashed green line). Also shown are results from 20 bootstrapped datasets (yellow). The crossing points represents the inclination/elongation pair most consistent with the TK03.GAD model. Elongation direction is shown as a dash-dotted (purple) line and ranges from E-W at low inclination to more N-S at steeper inclinations. c) Cumulative distribution of crossing points from 5000 bootstrapped datasets. The inclination of the whole data set (64.4$^{\circ }$) is consistent with that predicted from the {{\it  } Besse and Courtillot } (2002) European APWP. The 95\% confidence bounds on this estimate are 55.6-71.2$^{\circ }$. }}{figure}\relax 
\doTocEntry\toclof{16.13}{\csname a:TocLink\endcsname{18}{x18-19900213}{}{\ignorespaces a) Pangea A reconstruction (``Bullard fit''; Smith and Hallam, 1970), Bullard et al., 1965). b) Pangea A-2 reconstruction (van der Voo and French, 1974). c) Pangea B reconstruction (Morel and Irving, 1981). Note: a) and b) are reconstructions to fit the continental margins and do not take into account paleolatitudes. }}{figure}\relax 
\doTocEntry\toclot{16.2}{\csname a:TocLink\endcsname{18}{x18-1990032}{}{\ignorespaces Rotation poles for various versions of Pangea.}}{table}\relax 
\doTocEntry\toclot{16.3}{\csname a:TocLink\endcsname{18}{x18-1990043}{}{\ignorespaces  Paleomagnetic data for Kimmeridgian. }}{table}\relax 
\doTocEntry\toclof{16.14}{\csname a:TocLink\endcsname{18}{x18-19900514}{}{\ignorespaces Using paleomagnetic poles as a test for reconstructions. The continental outlines are rotated according to the same finite rotation poles for each reconstruction (in light grey). a) Poles for the period 180-200 Ma Pangea from the Besse and Courtillot (2002) and Torsvik et al. (2008) compilations rotated to the Pangea A reconstruction of Bullard (1965) and Smith and Hallam (1970). b) Poles for the Permian ($\sim $250-300 Ma) from Torsvik et al. (2008) compilation shown in Pangea A reconstruction. c) Same as b) but for Pangea A-2 reconstruction of van der Voo and French (1974). d) same as b) but for Pangea B reconstruction (Morel and Irving, 1981). e) Same as d) but just the lower Permian poles. }}{figure}\relax 
\doTocEntry\tocsection{16.5}{\csname a:TocLink\endcsname{18}{x18-20000016.5}{QQ2-18-420}{Paleomagnetism and plate reconstructions}}{944}\relax 
\doTocEntry\tocsection{16.6}{\csname a:TocLink\endcsname{18}{x18-20100016.6}{QQ2-18-421}{Discordant poles and displaced terranes}}{945}\relax 
\doTocEntry\tocsection{16.7}{\csname a:TocLink\endcsname{18}{x18-20200016.7}{QQ2-18-422}{Inclination only data and APWPs}}{946}\relax 
\doTocEntry\toclof{16.15}{\csname a:TocLink\endcsname{18}{x18-20200115}{}{\ignorespaces Circles are ``reliable'' mean poles from cratonic North America. (Data as listed in van der Voo, 1990). So-called ``discordant poles'' from western North America are plotted as triangles. [Data from van der Voo, 1981.]}}{figure}\relax 
\doTocEntry\toclof{16.16}{\csname a:TocLink\endcsname{18}{x18-20200216}{}{\ignorespaces If local rotations are suspected for a given region, the inclination information can be converted to the equivalent paleo-colatitude small circles (green solid lines) on which the paleopole must lie. Site locations from `mobile regions' are shown as open circles. LB: Lebanon, SP: Spain; IT: Italy; CH: Chile; NA: Morrison Formation on the Colorado Plateau of North America. Small circles (solid green lines) are the paleomagnetic colatitudes ($\theta $ in Table 16.3) from inclination data. The dashed line is the paleolatitude from uncorrected inclination data of the Morrison Formation. Shaded ellipse indicates region of overlap among all small circles. Fully oriented poles are shown as purple triangles. Numbers are the GPMDB reference numbers followed by the age in Ma. See Table 16.3. All poles and observation sites have been rotated into South African coordinates for 155 Ma (see Appendix\nobreakspace  {}\o:ref {app:polerot}) as have the continents. Pole number 268 is the Canelo Hills Volcanics from Arizona. If this region rotated about a vertical axis, the pole would lie along the solid blue line. Blue stars are the predicted poles of Besse and Courtillot (2002) in South African coordinates. }}{figure}\relax 
\doTocEntry\tocsection{16.8}{\csname a:TocLink\endcsname{18}{x18-20300016.8}{QQ2-18-425}{Concluding remarks}}{950}\relax 
\doTocEntry\tocsection{16.9}{\csname a:TocLink\endcsname{18}{x18-20400016.9}{QQ2-18-426}{Problems}}{951}\relax 
\doTocEntry\tocchapter{}{\csname a:TocLink\endcsname{18}{Q1-18-427}{}{Appendices}}{956}\relax 
\doTocEntry\tocappendix{A}{\csname a:TocLink\endcsname{19}{x19-205000A}{QQ2-19-428}{Definitions, derivations and tricks}}{957}\relax 
\doTocEntry\tocsection{A.1}{\csname a:TocLink\endcsname{19}{x19-206000A.1}{QQ2-19-429}{Definitions}}{957}\relax 
\doTocEntry\toclot{}{\csname a:TocLink\endcsname{19}{x19-206002}{}{\numberline {A.1}{Acronyms}}}{957}\relax 
\doTocEntry\toclot{}{\csname a:TocLink\endcsname{19}{x19-206004}{}{\numberline {A.2}{ Physical Parameters and Constants}}}{959}\relax 
\doTocEntry\toclot{}{\csname a:TocLink\endcsname{19}{x19-206006}{}{\numberline {A.3}{ Statistics}}}{960}\relax 
\doTocEntry\tocsection{A.2}{\csname a:TocLink\endcsname{19}{x19-207000A.2}{QQ2-19-433}{Derivations}}{961}\relax 
\doTocEntry\tocsubsection{A.2.1}{\csname a:TocLink\endcsname{19}{x19-208000A.2.1}{QQ2-19-434}{Langevin function for a paramagnetic substance}}{962}\relax 
\doTocEntry\tocsubsection{A.2.2}{\csname a:TocLink\endcsname{19}{x19-209000A.2.2}{QQ2-19-435}{Superparamagnetism}}{965}\relax 
\doTocEntry\tocsection{A.3}{\csname a:TocLink\endcsname{19}{x19-210000A.3}{QQ2-19-436}{Useful tricks}}{967}\relax 
\doTocEntry\tocsubsection{A.3.1}{\csname a:TocLink\endcsname{19}{x19-211000A.3.1}{QQ2-19-437}{Spherical trigonometry}}{968}\relax 
\doTocEntry\toclof{A.1}{\csname a:TocLink\endcsname{19}{x19-2110011}{}{\ignorespaces Rules of spherical trigonometry. $a,b,c$ are all great circle tracks on a sphere which form a triangle with apices $A,B,C$. The lengths of $a,b,c$ on a unit sphere are equal to the angles subtended by radii that intersect the globe at the apices, as shown in the inset. $\alpha ,\beta ,\gamma $ are the angles between the great circles. }}{figure}\relax 
\doTocEntry\tocsubsection{A.3.2}{\csname a:TocLink\endcsname{19}{x19-212000A.3.2}{QQ2-19-439}{Vector addition}}{969}\relax 
\doTocEntry\toclof{A.2}{\csname a:TocLink\endcsname{19}{x19-2120012}{}{\ignorespaces Vectors $\hbox {\bf  A}$ and $\hbox {\bf  B}$, their components A$_{x,y}$, B$_{x,y}$ and the angles between them and the $X$ axis, $\alpha $ and $\beta $. The angle between the two vectors is $\alpha $ -$\beta $ = $\Delta $. Unit vectors in the directions of the axes are $\mathaccent "705E\relax x$ and $\mathaccent "705E\relax y$ respectively. }}{figure}\relax 
\doTocEntry\tocsubsection{A.3.3}{\csname a:TocLink\endcsname{19}{x19-213000A.3.3}{QQ2-19-441}{Vector subtraction}}{971}\relax 
\doTocEntry\tocsubsection{A.3.4}{\csname a:TocLink\endcsname{19}{x19-214000A.3.4}{QQ2-19-442}{Vector multiplication}}{971}\relax 
\doTocEntry\toclof{A.3}{\csname a:TocLink\endcsname{19}{x19-2140013}{}{\ignorespaces Illustration of cross product of vectors $A$ and $B$ separated by angle $\theta $ to get the orthogonal vector $C$. }}{figure}\relax 
\doTocEntry\tocsubsection{A.3.5}{\csname a:TocLink\endcsname{19}{x19-215000A.3.5}{QQ2-19-444}{Tricks with tensors}}{972}\relax 
\doTocEntry\tocsubsubsection{}{\csname a:TocLink\endcsname{19}{x19-216000A.3.5}{QQ2-19-445}{Direction cosines}}{973}\relax 
\doTocEntry\toclof{A.4}{\csname a:TocLink\endcsname{19}{x19-2160014}{}{\ignorespaces Definition of direction cosines in two dimensions. a) Definition of vector in one set of coordinates, $x_1, x_2$. b) Definition of angles relating $X$ axes to $X'$. }}{figure}\relax 
\doTocEntry\tocsubsubsection{}{\csname a:TocLink\endcsname{19}{x19-217000A.3.5}{QQ2-19-447}{Changing coordinate systems}}{974}\relax 
\doTocEntry\toclof{A.5}{\csname a:TocLink\endcsname{19}{x19-2170025}{}{\ignorespaces  a) Sample coordinate system. b) Trigonometric relations between two cartesian coordinate systems, $\hbox {\bf  X}_i$ and $\hbox {\bf  X}'_i$. $\lambda ,\phi ,\psi $ are all known and the angles between the various axes can be calculated using spherical trigonometry. For example, the angle $\alpha $ between $\hbox {\bf  X}_1$ and $\hbox {\bf  X}_1'$ forms one side of the triangle shown by dash-dot lines. Thus, $\,\hbox {cos}\,\alpha = \,\hbox {cos}\,\lambda \,\hbox {cos}\,\phi + \,\hbox {sin}\,\lambda \,\hbox {sin}\,\phi \,\hbox {cos}\,\psi $. [Figure from Tauxe, 1998.]}}{figure}\relax 
\doTocEntry\tocsubsubsection{}{\csname a:TocLink\endcsname{19}{x19-218000A.3.5}{QQ2-19-449}{Method for rotating points on a globe using finite rotation poles}}{978}\relax 
\doTocEntry\toclot{}{\csname a:TocLink\endcsname{19}{x19-218008}{}{\numberline {A.4}{Finite Rotations for Gondwanda continents }}}{981}\relax 
\doTocEntry\toclot{}{\csname a:TocLink\endcsname{19}{x19-218010}{}{\numberline {A.5}{Finite Rotations for laurentian continents }}}{982}\relax 
\doTocEntry\toclot{A.6}{\csname a:TocLink\endcsname{19}{x19-2180116}{}{\ignorespaces Finite Rotations for South Africa }}{table}\relax 
\doTocEntry\tocsubsubsection{}{\csname a:TocLink\endcsname{19}{x19-219000A.3.5}{QQ2-19-453}{The orientation tensor and eigenvectors}}{985}\relax 
\doTocEntry\tocsubsection{A.3.6}{\csname a:TocLink\endcsname{19}{x19-220000A.3.6}{QQ2-19-454}{Upside down triangles, $\nabla $}}{989}\relax 
\doTocEntry\tocsubsubsection{}{\csname a:TocLink\endcsname{19}{x19-221000A.3.6}{QQ2-19-455}{Gradient}}{989}\relax 
\doTocEntry\toclof{A.6}{\csname a:TocLink\endcsname{19}{x19-2210016}{}{\ignorespaces Illustration of the relationship between a vector field (direction and magnitude of steepest slope at every point, e.g., red arrows) and the scalar field (height) of a ski slope.}}{figure}\relax 
\doTocEntry\toclof{A.7}{\csname a:TocLink\endcsname{19}{x19-2210027}{}{\ignorespaces Example of a vector field with a non-zero divergence.}}{figure}\relax 
\doTocEntry\tocsubsubsection{}{\csname a:TocLink\endcsname{19}{x19-222000A.3.6}{QQ2-19-458}{Divergence}}{991}\relax 
\doTocEntry\toclof{A.8}{\csname a:TocLink\endcsname{19}{x19-2220018}{}{\ignorespaces Example of a vector field with zero divergence.}}{figure}\relax 
\doTocEntry\toclof{A.9}{\csname a:TocLink\endcsname{19}{x19-2220029}{}{\ignorespaces Example of a vector field with non-zero curl.}}{figure}\relax 
\doTocEntry\tocsubsubsection{}{\csname a:TocLink\endcsname{19}{x19-223000A.3.6}{QQ2-19-461}{Curl}}{994}\relax 
\doTocEntry\tocsubsection{A.3.7}{\csname a:TocLink\endcsname{19}{x19-224000A.3.7}{QQ2-19-462}{The statistical bootstrap}}{996}\relax 
\doTocEntry\toclof{A.10}{\csname a:TocLink\endcsname{19}{x19-22400110}{}{\ignorespaces  Bootstrapping applied to a normal distribution. a) 500 data points are drawn from a Gaussian distribution with mean of 10 and a standard deviation of 2. b) Q-Q plot of data in a). The 95\% confidence interval for the mean is given by Gauss statistics as $\pm $ 0.17. 10,000 new (para) data sets are generated by randomly drawing $N$ data points from the original data set shown in a). c) A histogram of the means from all the para-data sets. 95\% of the means fall within the interval 10.06$^{+0.16}_{-0.16}$, hence the bootstrap confidence interval is similar to that calculated with Gaussian statistics. [Figure from Tauxe, 1998.]}}{figure}\relax 
\doTocEntry\toclof{A.11}{\csname a:TocLink\endcsname{19}{x19-22400211}{}{\ignorespaces Calculation of the azimuth of the shadow direction ($\beta '$) relative to true North, using a sun compass. L is the site location (at $\lambda _L,\phi _L$), S is the position on the Earth where the sun is directly overhead ($\lambda _S,\phi _S$). [Figure from Tauxe, 1998.]}}{figure}\relax 
\doTocEntry\tocsubsection{A.3.8}{\csname a:TocLink\endcsname{19}{x19-225000A.3.8}{QQ2-19-465}{Directions using a sun compass}}{999}\relax 
\doTocEntry\tocappendix{B}{\csname a:TocLink\endcsname{20}{x20-226000B}{QQ2-20-466}{Plots useful in paleomagnetism}}{1003}\relax 
\doTocEntry\tocsection{B.1}{\csname a:TocLink\endcsname{20}{x20-227000B.1}{QQ2-20-467}{Equal area projections}}{1003}\relax 
\doTocEntry\tocsubsection{B.1.1}{\csname a:TocLink\endcsname{20}{x20-228000B.1.1}{QQ2-20-468}{Calculation of an equal area projection}}{1003}\relax 
\doTocEntry\toclof{B.1}{\csname a:TocLink\endcsname{20}{x20-2280011}{}{\ignorespaces Construction of an equal area projection for a point P corresponding to a $D$ of 40$^{\circ }$ and an $I$ of 35$^{\circ }$. [Figure from Tauxe, 1998.]}}{figure}\relax 
\doTocEntry\toclof{B.2}{\csname a:TocLink\endcsname{20}{x20-2280032}{}{\ignorespaces Schmidt (equal area) net.}}{figure}\relax 
\doTocEntry\toclof{B.3}{\csname a:TocLink\endcsname{20}{x20-2280043}{}{\ignorespaces How to use an equal area net (see text). }}{figure}\relax 
\doTocEntry\tocsubsection{B.1.2}{\csname a:TocLink\endcsname{20}{x20-229000B.1.2}{QQ2-20-472}{Plotting directions}}{1006}\relax 
\doTocEntry\tocsubsection{B.1.3}{\csname a:TocLink\endcsname{20}{x20-230000B.1.3}{QQ2-20-473}{Bedding-tilt corrections}}{1006}\relax 
\doTocEntry\toclof{B.4}{\csname a:TocLink\endcsname{20}{x20-2300034}{}{\ignorespaces Example of structural corrections to NRM directions. The bedding attitude is specified by dip and dip direction (squares on the equal-area projections); the azimuth of the strike is 90$^{\circ }$ anti-clockwise from the dip direction; the rotation required to restore the bedding to horizontal is clockwise (as viewed along the strike line) by the dip angle and is shown by the rotation symbol; the {\it  in situ} NRM direction is at the tail of the arrow, and the structurally corrected NRM direction is at the head of the arrow.}}{figure}\relax 
\doTocEntry\tocsubsection{B.1.4}{\csname a:TocLink\endcsname{20}{x20-231000B.1.4}{QQ2-20-475}{Reading ternary diagrams}}{1009}\relax 
\doTocEntry\toclof{B.5}{\csname a:TocLink\endcsname{20}{x20-2310015}{}{\ignorespaces How to read a ternary diagram. The three apices are components A,B,C. A composition is plotted as the star. a) Shows the percentage of component A (60\%). b) Shows the percentage of component B (15\%) and c) shows the percentage of component C (25\%). }}{figure}\relax 
\doTocEntry\tocsubsection{B.1.5}{\csname a:TocLink\endcsname{20}{x20-232000B.1.5}{QQ2-20-477}{Quantile-Quantile plots}}{1010}\relax 
\doTocEntry\toclof{B.6}{\csname a:TocLink\endcsname{20}{x20-2320016}{}{\ignorespaces  a) Illustration of how the sorted data $\zeta _i$ divide the density curve into areas $A_i$ with an average area of $1/(N+1)$. b) The values of $z_i$ which divide the density function into equal areas $a_i=1/(N+1)$. c) Q-Q plot of $z$ and $\zeta $. [Figure from Tauxe, 1998.] }}{figure}\relax 
\doTocEntry\tocsubsubsection{}{\csname a:TocLink\endcsname{20}{x20-233000B.1.5}{QQ2-20-479}{Q-Q plots for Fisher distributions}}{1011}\relax 
\doTocEntry\tocsubsubsection{}{\csname a:TocLink\endcsname{20}{x20-234000B.1.5}{QQ2-20-480}{Q-Q plots for normal distributions}}{1015}\relax 
\doTocEntry\tocappendix{C}{\csname a:TocLink\endcsname{21}{x21-235000C}{QQ2-21-481}{Paleomagnetic statistics and parameter estimation}}{1019}\relax 
\doTocEntry\tocsection{C.1}{\csname a:TocLink\endcsname{21}{x21-236000C.1}{QQ2-21-482}{Hysteresis Parameters}}{1019}\relax 
\doTocEntry\toclot{C.1}{\csname a:TocLink\endcsname{21}{x21-2360051}{}{\ignorespaces Summary of hysteresis parameters.}}{table}\relax 
\doTocEntry\toclof{C.1}{\csname a:TocLink\endcsname{21}{x21-2360061}{}{\ignorespaces Typical hysteresis experiment. a) Raw data are solid red line. Data are processed (see text) by closing the ascending and descending loops, subtracting the high field slope ($\chi _{hf}$) and adjusting such that the y-intercepts are equal (that for the descending loop is labeled $M_r$). Processed data are dotted blue line. Coercivity ($\mu _oH_{c}$) is the applied field for which a saturation magnetization ($M_s$) is reduced to zero. b) Difference between processed ascending and descending loops is the $\Delta M$ curve (solid blue line). Back-field IRM data shown normalized by saturation remanence ($M_r$) -- dashed green line. Two methods of estimating coercivity of remanence shown (see text).}}{figure}\relax 
\doTocEntry\tocsection{C.2}{\csname a:TocLink\endcsname{21}{x21-237000C.2}{QQ2-21-485}{Directional statistics}}{1022}\relax 
\doTocEntry\toclot{C.2}{\csname a:TocLink\endcsname{21}{x21-2370012}{}{\ignorespaces Critical values of $R_o$ for a random distribution [Watson, 1956.]}}{table}\relax 
\doTocEntry\tocsubsection{C.2.1}{\csname a:TocLink\endcsname{21}{x21-238000C.2.1}{QQ2-21-487}{Calculation of Watson's $V_w$}}{1023}\relax 
\doTocEntry\tocsubsection{C.2.2}{\csname a:TocLink\endcsname{21}{x21-239000C.2.2}{QQ2-21-488}{Combining lines and planes}}{1025}\relax 
\doTocEntry\toclot{C.3}{\csname a:TocLink\endcsname{21}{x21-2390113}{}{\ignorespaces Maximum likelihood estimators of $k_1, k_2$ in the Bingham distribution for given eigenvalues $\omega _1, \omega _2$. Data from Mardia and Zemroch (1977). Upper (lower) number is $k_1(k_2)$}}{table}\relax 
\doTocEntry\tocsubsection{C.2.3}{\csname a:TocLink\endcsname{21}{x21-240000C.2.3}{QQ2-21-490}{Inclination only calculation}}{1027}\relax 
\doTocEntry\tocsubsection{C.2.4}{\csname a:TocLink\endcsname{21}{x21-241000C.2.4}{QQ2-21-491}{Kent 95\% confidence ellipse}}{1029}\relax 
\doTocEntry\tocsubsection{C.2.5}{\csname a:TocLink\endcsname{21}{x21-242000C.2.5}{QQ2-21-492}{Bingham 95\% confidence parameters}}{1031}\relax 
\doTocEntry\tocsection{C.3}{\csname a:TocLink\endcsname{21}{x21-243000C.3}{QQ2-21-493}{Paleointensity statistics}}{1034}\relax 
\doTocEntry\toclof{C.2}{\csname a:TocLink\endcsname{21}{x21-2430012}{}{\ignorespaces Illustration of paleointensity parameters. Arai plots: The magnitude of the NRM remaining after each step is plotted versus the pTRM gained at each temperature step. Closed symbols are zero-field first followed by in-field steps (ZI) while open symbols are in-field first followed by zero field (IZ). Triangles are pTRM checks and squares are pTRM tail checks. Horizontal dashed lines are the vector difference sum (VDS) of the NRM steps. Vector endpoint plots: Insets are the x,y (solid symbols) and x,z (open symbols) projections of the (unoriented) natural remanence (zero field steps) as it evolves from the initial state (plus signs) to the demagnetized state. The laboratory field was applied along -Z. Diamonds indicate bounding steps for calculations. a) The $f_{vds}$ is the fraction of the component used of the total VDS. The difference between the pTRM check and the original measurement at each step is $\delta T_i$. The inset shows the deviation angle (DANG) that a component of NRM makes with the origin. The maximum angle of deviation MAD is calculated from the scatter of the points about the best-fit line (solid green line). b) Data exhibit Òzig-zag behaviorÓ diagnostic for significant difference between blocking and unblocking temperatures. The Zig-zag for slopes compares slopes calculated between ZI and IZ steps ($b_{zi}$) with those connecting IZ and ZI steps $b_{iz}$). The difference between the pTRM tail check and the original measurement at each step is $\Delta T_i$. c) $\beta $ reflects the scatter ($\delta _x, \delta _y$) about the best-fit slope (solid green line). The Zig-zag for directions compares those calculated between ZI and IZ steps ($D_{zi}$) with those connecting IZ and ZI steps $D_{iz}$). [Figures from Ben-Yosef et al., 2008.] }}{figure}\relax 
\doTocEntry\tocappendix{D}{\csname a:TocLink\endcsname{22}{x22-244000D}{QQ2-22-495}{Anisotropy in paleomagnetism}}{1043}\relax 
\doTocEntry\tocsection{D.1}{\csname a:TocLink\endcsname{22}{x22-245000D.1}{QQ2-22-496}{The 15 measurement protocol}}{1043}\relax 
\doTocEntry\toclof{D.2}{\csname a:TocLink\endcsname{22}{x22-2450032}{}{\ignorespaces  The 15 position scheme of Jelinek (1976) for measuring the AMS of a sample. [Figure from Tauxe, 1998.] }}{figure}\relax 
\doTocEntry\tocsection{D.2}{\csname a:TocLink\endcsname{22}{x22-246000D.2}{QQ2-22-498}{The spinning protocol}}{1046}\relax 
\doTocEntry\toclof{D.3}{\csname a:TocLink\endcsname{22}{x22-2460013}{}{\ignorespaces Specimen orientations for the three spins used with spinning magnetic susceptibility meters. The heavy gray arrows show the axes of rotation; one oriented toward the user for Position 1 (a) and 2 (b) away from the user for Position 3 (c). The orientation of the specimen coordinate system in space is specified by the azimuth and plunge of either the arrow along the core length (+x$_3$ axis, black) or the +x$_1$ axis (red arrow on core top). d) orientation of applied field (coil axis) relative to specimen coordinates in Position 3. [Figure from Gee et al. 2008.]}}{figure}\relax 
\doTocEntry\toclof{D.4}{\csname a:TocLink\endcsname{22}{x22-2460024}{}{\ignorespaces Processing steps for data spin protocol. a) From a single spin with eight revolutions. Raw data with peaks (red dots) identified by peak-finding algorithm and best fit linear trend. Data are detrended using peaks. b) Data from detrended individual revolutions and best fit 2-D model. c) Original (zero-mean) deviatoric susceptibility data from three spins. The best fit 2-D model for each spin provides an estimate of two elements of the deviatoric susceptibility tensor (square, $\chi _{11}$; hexagon, $\chi _{22}$; circle, $\chi _{33}$). Thick bars indicate the calculated offsets for Positions 1 and 2. d) Crossover adjustment for data from three positions. Original (zero-mean) deviatoric susceptibility data from three positions are scaled to absolute values (right-hand scale) using a bulk measurement in spin Position 3, and adjusted to minimize cross over error. [Figure modified from Gee et al. 2008.]}}{figure}\relax 
\doTocEntry\tocsection{D.3}{\csname a:TocLink\endcsname{22}{x22-247000D.3}{QQ2-22-501}{Correction of inclination error with AARM}}{1048}\relax 
\doTocEntry\toclikechapter{}{\csname a:TocLink\endcsname{23}{x23-248000D.3}{QQ2-23-502}{Bibliography}}{1053}\relax