Andrew J. Newell

The effect of remanence anisotropy on paleointensity estimates: a case study from the Archean Stillwater Complex

Paleomagnetism of Archean rocks potentially provides information about the early development of the Earth and of the geodynamo. Precambrian layered intrusive rocks are good candidates for paleomagnetic studies: such complexes are commonly relatively unaltered and may contain some single-domain magnetite ‘armored’ by silicate mineral grains. However, layered intrusives often have a strong petrofabric that may result in a strong remanence anisotropy. Magnetic anisotropy can have particularly disastrous consequences for paleointensity experiments if the anisotropy is unrecognized and if its effects remain uncorrected. Here we examine the magnetic anisotropy of an anorthosite sample with a well-developed magmatic foliation. The effect of the sample’s remanence fabric on paleointensity determinations is significant: paleointensities estimated by the method of Thellier and Thellier range from 17 to 55 μT for specimens magnetized in a field of 25 μT. We describe a technique based on the remanence anisotropy tensor to correct paleointensity estimates for the effects of magnetic fabric and use it to estimate a paleointensity for the Stillwater Complex (MT, USA) of ∼32 μT (adjusted for the effects of slow cooling).

A GENERALIZATION OF THE DEMAGNETIZING TENSOR FOR NONUNIFORM MAGNETIZATION

The demagnetizing tensor for ferromagnets is generalized to include interactions between uniformly magnetized bodies. This “mutual” demagnetizing tensor is symmetric, has a trace of zero, and has other simple geometric properties. The tensor is then used to develop an expression for the macroscopic magnetic field in non-uniformly magnetized bodies of arbitrary shape. Finally, the theory is applied to a block model of magnetization and explicit formulae for the tensor components are given.

Single‐domain critical sizes for coercivity and remanence

It is usually assumed that magnetic parameters such as coercivity and saturation remanence are single-domain (SD) over the same size range. In reality, there is a different SD size range for each parameter. We define critical sizes LSDcoerc for coercivity and LSDrem for remanence. In general, LSDcoerc ≤ LSDrem. Up to L = LSDrem, the saturation remanent state is single-domain. If a sufficiently large reverse field is applied, a conventional SD state would reverse by uniform rotation. However, the mode of reversal is nonuniform if the grain size is between LSDcoerc and LSDrem, so in this size range the SD state is less stable. To calculate the critical sizes, we use rigorous nucleation theory and obtain analytical expressions. The analytical form allows us to explore the effect of grain shape, stress, crystallographic orientation and titanium content in titanomagnetites. We adapt the theory to cubic anisotropy with K1 < 0, which allows us to apply the expressions to titanomagnetites. We find that the size range for SD coercivity is always small. The size range for SD remanence can vary enormously depending on the anisotropy. If the easy axes are oriented favorably, the SD state can occur in large x = 0.6 titanomagnetite grains. Ensembles of magnetite grains with aspect ratios greater than 5 have SD-like remanence but low coercivity. However, most synthetic magnetite grains are nearly equant, and the predicted size range for SD remanence is small to nonexistent. This, rather than grain interactions, may be the reason they have properties such as saturation remanence that do not agree well with standard SD theory.

A two‐dimensional micromagnetic model of magnetizations and fields in magnetite

A two-dimensional micromagnetic model is used to obtain magnetic domain structures for cubic particles of magnetite with sides 1 μm or less. We show that complex submicron structures evolve continuously with increasing particle size into recognizable classical domain structures. Two such sequences of equilibrium states are seen. The lowest energy state for particle sizes above 0.06 μm is the “vortex” state (already seen in three-dimensional models). Another state is seen in larger particles which has angles between walls of 90°, 125°, and 145°, agreeing with domain observations of magnetite. These are usually called 71°, 109°, and 180° walls, based on the assumption that the domain moments are in the easy ([111]) directions, but we find that the domains are actually magnetized in the [100] and [110] directions. Because both states eventually become single domain in sufficiently small particles, there is no critical single domain size in a conventional sense, but there is a rapid decrease in normalized remanence over the size range 0.06 – 0.18 μm. Bloch-like and Neel-like walls are seen, but strong surface fields are only found over the Bloch-like walls. Depending on the geometry of the particle observed, one may see only a partial expression of closure domains in domain observations.

Nucleation and stability of ferromagnetic states

Each ferromagnetic state, such as the single-domain (SD) state, has critical fields and grain sizes at which it becomes unstable. To determine the properties of these critical points, a three-dimensional numerical micromagnetic model is combined with nucleation theory. Isothermal hysteresis and grain growth are simulated for cuboids with no internal stress or magnetocrystalline anisotropy. Most jumps in hysteresis loops are turning points at which the susceptibility goes to infinity. The SD state becomes unstable at a pitchfork bifurcation, which has a jump in susceptibility but a continuous change in magnetic moment. This is a generalization of curling mode nucleation. As the grain size increases, there is an increasing gap in field between the curling mode nucleation and the first irreversible jump in magnetization. A similar gap is often seen experimentally between the formation of a small spike domain and the appearance of a full size body domain. For the first time in a micromagnetic simulation, minor branches are traced from the main hysteresis loop. When they occur, the main loop becomes wasp waisted. At any given grain size the lowest-energy state has SD-like stability in response to changes in magnetic field. A high-stability component of remanence is commonly observed in pseudo-single-domain grains. It has previously been assumed that the high stability must be due to SD-like regions in larger grains, but the micromagnetic simulations demonstrate that SD-like stability does not require a SD-like mechanism.

Transdomain thermoremanent magnetization

Transdomain thermoremanent magnetization (TRM) is produced when thermally activated transitions between different domain structures become blocked during cooling. This paper investigates transdomain remanence in the pseudo-single-domain size range ( 6000kT for SD ↔ 2D transitions. Transdomain viscous remanent magnetization (VRM) will not occur, even on geological timescales, by a one-dimensional excitation such as edge nucleation of a domain wall. Transdomain blocking temperatures, at which energy barriers fall to 25kT–60kT, are ≥553°C for SD ↔ 2D and ≥574°C for 2D ↔ 3D transitions. There are two separate blocking temperatures, e.g. TB1 for SD ↔ 2D and TB2 for 2D ↔ SD transitions. Usually, only the higher of the two has practical significance because the favored (lower energy) state is already 100% populated at this temperature. Our theory is the first to make quantitative predictions of transition paths, relaxation times, and blocking temperatures for transdomain TRM. It is also quite robust. Relaxing the one-dimensional constraint and introducing crystal defects would make it easier to nucleate domains, but energy barriers and blocking temperatures would not be reduced greatly. Our principal conclusion, that only the lowest energy state at blocking is significantly populated, is a fundamental consequence of the Boltzmann statistics of equilibrium states and is unaffected by the details of transitions between states. Grain interactions may be responsible for the multiplicity of states observed in large titanomagnetite grains following replicate TRM experiments.

Size dependence of hysteresis properties of small pseudo‐single‐domain grains

The Day plot (M rs /M s versus H cr /H c ) is widely used by paleomagnetists to estimate the size of ferromagnetic grains and classify them as single-domain (SD), pseudo-single-domain (PSD,) or multidomain (MD). How reliable is this plot? To find out, a numerical micromagnetic model is used to calculate hysteresis loops as a function of grain size for two grain shapes (cube and cuboid with X = 1.5Y = 1.4Z). Magnetocrystalline anisotropy is ignored. The average M rs /M s is calculated for a collection of randomly oriented grains: In the elongated grain it drops from 0.4 to 0.06 over a negligible size range, almost missing the usual PSD range altogether. Other hysteresis parameters, H c , H cr , and X 0 , can only be calculated for a grain at a time. This is done for two magnetic field directions (close to the longest axis and close to the shortest axis). The single-grain values of H cr /H c depend strongly on field direction, but it is clear that the average jumps rapidly from SD to MD values. There are large, rapid fluctuations in H cr and X 0 associated with changes in remanent states. However, these fluctuations may not be apparent when averaged over a broad size range. This may explain why H cr and X 0 depend weakly on grain size in real samples. In tile size range studied (L < 0.25 μm), hysteresis parameters do not represent a typical grain size. Instead, they depend strongly on the size distribution.

The curling nucleation mode in a ferromagnetic cube

A ferromagnetic sphere is uniformly magnetized in a large field. As the field decreases, there is a critical field (the nucleation field) at which the magnetization becomes nonuniform. The initial deviation from uniform magnetization (the nucleation mode) is either coherent rotation or curling. By contrast, a ferromagnetic cube is never uniformly magnetized, and theorists have disagreed on whether nucleation theory for a sphere can be generalized to a cube. A three-dimensional numerical model of a cube is used to show there is a bifurcation at which a curl appears in the magnetization. This bifurcation is a generalized curling-mode nucleation. The nucleation field for a small cube fits the equation for a sphere of the same volume with a reduced demagnetizing factor. The nucleation mode is well approximated by the curling mode for the sphere. The magnetization changes continuously, so the nucleation field is not equal to the switching field or coercivity. When the cube size is twice the single-domain criti...

Micromagnetic modeling of two‐dimensional domain structures in magnetite

Micromagnetic studies of two-dimensional domain structures in various local energy minimum states were carried out for 1- and 5-μm cubes of magnetite. Domain structures obtained for the cubes are relatively simple compared to ones from previous two-and three-dimensional calculations for smaller magnetite grains (Newell et al., 1993a; Williams and Dunlop, 1989, 1990). With an initially assumed one-dimensional lamellar structure, the final two-dimensional structure reveals the formation of closure domains at the grain surfaces. The magnetization directions in closure domains are found to be determined largely by the magnetostatic energy, rather than the magnetocrystalline anisotropy as normally expected. The number of body domains that gives the lowest-energy state for the 1-μm magnetite cube is 2, while the number for the 5-μm cube is 4. These are much smaller equilibrium numbers of domains than the 6 and 13 domains obtained from a one-dimensional model by Moon and Merrill (1985) and the 4 and 10 domains predicted from a quasi-two-dimensional analysis by Ye and Merrill (1991) for 1-and 5-μm cubes, respectively. The numbers of domains predicted are comparable to the numbers observed in magnetite grains of similar sizes.

A re‐evaluation of magnetocrystalline anisotropy and magnetostriction constants

The standard theory of magnetocrystalline anisotropy for a rigid ferromagnetic body, with cubic symmetry, makes use of a “zero strain” anisotropy constant K1. If the effect of magnetostriction is taken into account, this constant must be replaced by a “zero stress” anisotropy constant K′1 which differs from K1 by a term involving magnetostriction constants and elastic constants. There are also zero strain and zero stress versions of higher order constants (K2, etc) for cubic symmetry, as well as for other symmetries. The constant K′1 also appears in dynamic behavior of ferromagnets driven by an applied field, unless the field changes too rapidly for the system to remain in equilibrium. It is predicted that ferromagnetic resonance (FMR) experiments give a direct measurement of K1.