Resolving the Origin of Pseudo‐Single Domain Magnetic Behavior

The term “pseudo‐single domain” (PSD) has been used to describe the transitional state in rock magnetism that spans the particle size range between the single domain (SD) and multidomain (MD) states. The particle size range for the stable SD state in the most commonly occurring terrestrial magnetic mineral, magnetite, is so narrow (~20–75 nm) that it is widely considered that much of the paleomagnetic record of interest is carried by PSD rather than stable SD particles. The PSD concept has, thus, become the dominant explanation for the magnetization associated with a major fraction of particles that record paleomagnetic signals throughout geological time. In this paper, we argue that in contrast to the SD and MD states, the term PSD does not describe the relevant physical processes, which have been documented extensively using three‐dimensional micromagnetic modeling and by parallel research in material science and solid‐state physics. We also argue that features attributed to PSD behavior can be explained by nucleation of a single magnetic vortex immediately above the maximum stable SD transition size. With increasing particle size, multiple vortices, antivortices, and domain walls can nucleate, which produce variable cancellation of magnetic moments and a gradual transition into the MD state. Thus, while the term PSD describes a well‐known transitional state, it fails to describe adequately the physics of the relevant processes. We recommend that use of this term should be discontinued in favor of “vortex state,” which spans a range of behaviors associated with magnetic vortices.


Introduction
More than 55 years have passed since Stacey (1961Stacey ( , 1962 coined the term "pseudo-single domain" (PSD) to describe a transitional magnetic state between the stable single domain (SD) and multidomain (MD) states. SD particles are uniformly magnetized; when they occur above the lower threshold size at which they have thermal stability, they can retain a record of the magnetic field in which they were magnetized for durations that exceed the age of the Earth (Néel, 1949(Néel, , 1955. This extraordinary recording capability has made paleomagnetism widely useful in understanding Earth (and solar system) history, particularly through establishment of the geomagnetic polarity timescale and the plate tectonic paradigm. However, the stable SD state occurs over only a narrow range of particle sizes in magnetite (~20-75 nm for equant particles at room temperature, with an upper threshold size of~200 nm for elongated particles with 2:1 axial ratios; Muxworthy & Williams, 2009). In order to minimize the total magnetic energy within and on the surface of a particle, the dominant demagnetization energy drives larger particles to subdivide into magnetic domains, with narrow domain walls between domains. In the MD state, the magnetic moment per unit volume is much lower than for SD particles because the magnetic moments of domains largely cancel each other. MD particles are generally not paleomagnetically stable on geological timescales (e.g., Heider et al., 1988). SD behavior is therefore far more desirable for providing reliable paleomagnetic recording. particle sizes above 20-25 μm (Vlag et al., 1996) or even 100 μm . The particle size range over which this transition occurs depends on the spontaneous magnetization, shape, and the state of internal stress of a particle (e.g., Day et al., 1977;Heider et al., 1987). Manifestations of PSD behavior include gradual variation of the ratio of the saturation remanent magnetization (M rs ) to saturation magnetization (M s ) and coercive force (B c ) across a broad particle size range ( Figure 1) rather than decreasing sharply at the upper threshold size for stable SD behavior. These parameters have intermediate values between ideal SD and MD values.
The importance of the so-called PSD state in paleomagnetism, rock magnetism, and environmental magnetism is difficult to overstate. The fact that stable paleomagnetic behavior is preserved across a much larger particle size range than the narrow SD size range makes paleomagnetism useful for addressing wide-ranging geoscientific problems. The paleomagnetic importance of particles in this larger size range has long been recognized, and the origin of PSD behavior has been much debated for over 50 years (e.g., Banerjee, 1977;Day, 1977;Dunlop, 1977Dunlop, , 1981Dunlop, , 1986Enkin & Dunlop, 1987;Fabian & Hubert, 1999;Fuller, 1984 Day et al. [1977] Worm and Markert [1987] Heider et al. [1987] Argyle and Dunlop [1990] Özdemir and  Grain size ( m)  Figure 1. Illustration of the gradual rather than sharp change in magnetic properties for magnetite with increasing grain size, which has been attributed to "pseudo-single domain" behavior. (a) Plot of M rs /M s versus grain size (after Dunlop (1995), who synthesized data from Argyle & Dunlop (1990); Day et al. (1977), Heider et al. (1987), , and Worm & Markert (1987)) and (b) B c versus grain size (after , who synthesized data from Amin et al. (1987), Argyle & Dunlop (1990), Day et al. (1977), Dunlop (1986), Heider et al. (1987), Levi & Merrill (1978), Özdemir & Banerjee (1982), , Schmidbauer & Schembera (1987), and Worm & Markert (1987)). The "harder" magnetic properties in crushed magnetite are attributed to the higher state of internal stress compared to the hydrothermally grown or glass ceramic crystals.
Journal of Geophysical Research: Solid Earth 10.1002/2017JB014860 & Fuller, 1983;Halgedahl, 1987;Levi & Merrill, 1978;Moskowitz, 1980;Pokhil & Moskowitz, 1997;Stacey, 1961Stacey, , 1962Stacey, , 1963Verhoogen, 1959;Williams & Dunlop, 1995;Xu & Dunlop, 1993). The importance of stable magnetizations in this transitional particle size range makes it unsatisfactory for the origin of PSD behavior to remain uncertain. Below, we provide a historical overview of explanations for PSD behavior, along with key observations that must be explained to account for PSD behavior (section 2). This is followed by presentation of key pieces of evidence that we argue provide a convincing physical explanation for PSD behavior.
In terms of the motivation for this paper, we note that the PSD state was hypothesized at about the same time that two-dimensional (2-D) micromagnetic models became available but long before three-dimensional (3-D) models were applied in a rock magnetic context (Williams & Dunlop, 1989). For example, from micromagnetic considerations, it has long been recognized in materials science and solid-state physics that when particles are larger than the stable SD size, their magnetization no longer reverses in response to an applied field via coherent rotation as envisaged in the Stoner and Wohlfarth (1948) model. Instead, in "soft" ferrimagnetic materials with relatively weak magnetocrystalline anisotropy, a "curling" mode of magnetization reversal occurs in which the magnetization curls around a central axis (Brown, 1957(Brown, , 1958Frei et al., 1957;Luborsky, 1961). This magnetization reversal mode was later referred to as the vortex state when the first 3-D micromagnetic modeling was undertaken (Schabes & Bertram, 1988). A magnetization state known as the "flower" state develops before nucleation of the vortex state, where magnetic moments spread outward at the edges of a particle (Schabes & Bertram, 1988); for the purposes of this paper, we treat the flower state as representing the coarse-grained end of the stable SD state because the magnetization remains essentially uniform at the center of the particle. While the flower state departs from the purely uniform SD state of Néel (1949), such particles have recently been shown to have slightly increased thermal and field stability with respect to true SD particles (Nagy et al., 2017). Once a flower state is no longer stable, a characteristic of the vortex state is that the magnetization reverses by discontinuous jumps at given nucleation fields (Luborsky, 1961). While elements of this reasoning are embedded in the rock magnetic literature, divergent thinking has arisen between the materials science/solid state physics and rock magnetic communities concerning the nature of the SD-MD transition. Soon after the first 3-D micromagnetic models were produced in a rock magnetic context (Williams & Dunlop, 1989), which included recognition of the curling mode of magnetization, specific micromagnetic tests of PSD behavior were conducted Newell & Merrill, 2000;Rave et al., 1998;Williams & Dunlop, 1995). These studies confirmed that the stable magnetizations observed for PSD particles result from nucleation of vortex states. Although this explanation has been accepted by some paleomagnetists (e.g., Tauxe et al., 2002), it has not been accepted universally in the paleomagnetism community despite widespread appreciation of information derived from micromagnetic simulations. For example, Dunlop (2002) referred to the vortex state as exotic and argued that experimental observations of PSD behavior can be explained by mixtures of SD and MD states. Thus, disagreement about the origin of the PSD state has persisted and the term "pseudo-single domain" remains embedded in the literature.
Experimental understanding of the magnetic behavior associated with the PSD state has been problematic. Standard methods that are used to synthesize magnetic particles for fundamental rock magnetic analysis (e.g., hydrothermal and glass-ceramic) usually produce samples with broad particle size distributions, with little or no control of interparticle spacing so that magnetostatic interactions have been a significant complicating factor in resulting magnetic characterizations. This has made it difficult to isolate and study PSD behavior in the bulk samples used in paleomagnetic studies. Electron beam lithography (EBL) has been used to address this problem by producing arrays of identical magnetic particles with controlled spacing (King et al., 1996;Krása et al., 2009). These studies have been helpful in studying PSD behavior; however, production of samples by EBL has also proven to be difficult, with only a few nonideal samples produced, where the samples are often affected by induced stress related to the sample/substrate interface.
Domain state imaging has been a key tool for understanding the magnetic properties of coarser particles in rock magnetism. However, direct imaging of vortex structures in <500 nm particles has been historically difficult for two reasons. First, vortex structures have small external fields. This makes them difficult or impossible (Williams et al., 1992) to image with the dominantly used domain imaging methods like Bitter pattern imaging and magnetic force microscopy that rely on stray surface fields (e.g., de Groot et al., 2014; Journal of Geophysical Research: Solid Earth 10.1002/2017JB014860 Halgedahl, 1991;Halgedahl & Fuller, 1980;Heider et al., 1988;Moskowitz et al., 1988;Pokhil & Moskowitz, 1997;Soffel, 1977). Second, particles in this size range are difficult to resolve with light microscopy, which is used in Bitter pattern imaging. Development and application of advanced transmission electron microscope (TEM) techniques such as off axis electron holography now make it possible to image directly magnetic structures at nanometer scales (Cowley, 1992), although the resulting 2-D images can lead to ambiguity in determining 3-D structures (Almeida et al., 2016). Nevertheless, electron holography has proven to be an exceptionally useful tool in understanding complexities associated with various types of mineral magnetic microstructures (e.g., Almeida et al., 2014;Dunin-Borkowski et al., 1998;Einsle et al., 2016;Harrison et al., 2002). Recently developed 3-D hard X-ray magnetic imaging (Donnelly et al., 2017) has yet to be used in rock magnetism but promises to open new avenues for understanding magnetic structures of relevance to rock magnetism.
In this paper, we combine evidence from first-order reversal curve (FORC) diagrams Roberts et al., 2000), which provide information about all components of magnetization in a sample, with support from numerical micromagnetic simulations and electron holography, to seek to explain the physics of magnetization processes associated with the PSD state. We present observations that confirm that the PSD state has been misdiagnosed routinely in paleomagnetism and rock magnetism and that the key observations that must be explained to account satisfactorily for PSD behavior (see section 2) can be explained by magnetically nonuniform single and multiple vortex states with domain wall nucleation at coarser particle sizes. Dunlop and Özdemir (1997) provide an extensive description of the PSD state to which readers are directed for a comprehensive account. Fabian and Hubert (1999) also provide an excellent concise summary of key concepts associated with development of attempts to explain PSD behavior. We provide a brief overview below not only to trace the major concepts but also to describe the key observations that must be accounted for in any successful explanation of magnetic behavior across the extended particle size range over which PSD behavior occurs.

Historical Overview and Key Observations of PSD Behavior
Even before the term PSD was coined, Verhoogen (1959) proposed that stable remanences in large particles could be explained by SD inclusions due to deflected electron spins surrounding local stress concentrations or dislocations within MD grains. Stacey (1961) argued that even if SD regions existed within MD particles, they would be magnetically screened so that MD properties would still dominate. Instead, Stacey (1962) argued that permanent PSD moments could be explained by "Barkhausen discreteness" where changes in remanence occur as a result of Barkhausen jumps of domain walls between positions that are magnetoelastically bound to defects. In this case, domain walls will not annihilate fully the total magnetic moment, and an uncompensated PSD moment was argued to result. PSD moments were not considered to be independent of MD processes in these early models ; later explanations emphasized true SD moments within PSD particles, where SD moments can reverse independently of surrounding domains (Dunlop, 1973(Dunlop, , 1977 or where domain walls fail to renucleate after magnetic saturation is achieved in metastable SD grains (Halgedahl & Fuller, 1980, 1983. The possibility of domain wall moments (referred to as PSARKs), where PSD remanence results from magnetization structures within domain walls (Dunlop, 1977), fails to account for the grain size dependence of the saturation remanence. Domain imbalance in particles with irregular shape (Stacey, 1961) or odd numbers of domains (Shcherbakov, 1978) have been argued to give rise to PSD effects (Dunlop, 1983). However, domain imbalance moments vanish for small particles that still preserve a significant PSD remanence. Surface anisotropy was proposed by Banerjee (1977) to explain high coercivities in PSD particles. However, such surface moments interact strongly with the volume magnetization, which is incompatible with the expected grain size dependence (as is the case for the wall moment hypothesis).
Unconstrained 3-D micromagnetic models provide a powerful means of visualizing inhomogeneous magnetizations at submicron length scales. At fine particle sizes, simple, uniform SD structures are observed. As particle size increases, flower ( Figure 2a) and vortex structures (Figures 2b and 2c) form above the stable SD threshold size (Enkin & Williams, 1994;Fabian et al., 1996;Newell & Merrill, 2000;Rave et al., 1998;Schabes & Bertram, 1988;Williams & Dunlop, 1989, 1995. Coercivity predictions from 3-D micromagnetic models Journal of Geophysical Research: Solid Earth 10.1002/2017JB014860 agree well with experimental data for unstressed cubic magnetite particles, although the saturation remanence is lower than expected (Williams & Dunlop, 1995). Vortex-like structures are the equilibrium or lowest energy state for nonuniformly magnetized particles in 3-D micromagnetic calculations (Williams & Dunlop, 1995). Close agreement between micromagnetic theory and experimental data for PSD particles has led to the suggestion that metastable vortex states, which have high magnetic moment and coercivity, are the dominant magnetization state in small PSD particles, with magnetization reversal controlled by vortex nucleation and propagation (Williams & Dunlop, 1995). In this paper, we provide evidence in support of this conclusion.
In seeking to explain PSD behavior, we bear in mind the comments of Fabian and Hubert (1999) who stated that any explanation needs to account for the universality of PSD remanence in all kinds of magnetic materials across a broad transitional size range, the high intensity of a weak-field thermal remanent magnetization (TRM) above the theoretically predicted SD threshold size, and the coercivity spectrum of a weak-field TRM that is similar to that of a mixture of SD and MD components. A key manifestation of SD-like and MD-like components in PSD magnetite is low-temperature demagnetization (LTD) in magnetite when it is cooled through the isotropic temperature (130 K) and Verwey transition (120 K) (Verwey, 1939). The remanence fraction that survives LTD, or the low-temperature memory, has SD-like behavior during alternating field (AF) demagnetization, whereas that removed during LTD has MD-like behavior (Ozima et al., 1964). Fabian and Hubert (1999) concluded that any explanation for PSD behavior should have a simple and natural explanation that accounts for the above observations. We seek to achieve this with FORC diagrams, micromagnetic simulations, and electron holography to illustrate how single and multiple vortex states, and associated phenomena, can explain PSD behavior.

Understanding Magnetic Domain States Using FORC Diagrams
FORC diagrams Roberts et al., 2000) have been used widely to understand microscopic magnetization mechanisms in geological and synthetic samples. A framework for understanding the magnetic behavior of superparamagnetic, magnetostatically interacting, and noninteracting SD, MD, single vortex, and PSD particle systems has been developed through experimental, theoretical, numerical, and micromagnetic approaches. Roberts et al. (2014) provide an up-to-date summary and details concerning the physical meaning and interpretation of FORC diagrams. We provide here a brief outline of the manifestations of SD, MD, and PSD states in FORC diagrams, including observations that suggest vortex nucleation as a credible explanation for PSD behavior. This background is necessary for using FORC diagrams to explain the magnetization mechanisms responsible for PSD behavior.
3.1. The SD, MD, and PSD States Noninteracting uniaxial SD particle systems produce a dominantly horizontal signature on FORC diagrams ( Figure 3a). The magnitude of the FORC distribution along the horizontal axis provides a measure of the  coercivity (switching field) distribution, while the sharp ridge-like distribution indicates a lack of magnetostatic interactions among the uniaxial SD particles (e.g., Dobrota & Stancu, 2013;Egli, 2006;Egli et al., 2010;Muxworthy & Williams, 2005;Newell, 2005;Roberts et al., 2000;Winklhofer & Zimanyi, 2006). As interactions increase, the vertical spread of the FORC distribution increases (Muxworthy et al., 2004;Roberts et al., 2000). By contrast, MD particle systems with weak domain wall pinning give rise to dominantly vertical FORC distributions with low coercivities (Pike et al., 2001;Roberts et al., 2014) (Figure 3b). The weak domain wall pinning regime is approximated by the domain wall pinning model of Néel (1955), where planar domain walls do not interact magnetically with each other (Pike et al., 2001). Experimental results for natural samples indicate that internal stress is important and that it gives rise to FORC distributions that are no longer dominantly vertical but that diverge away from the vertical axis of FORC diagrams to higher coercivities (Pike et al., 2001). In Figures 3c-3f, FORC diagrams are shown for magnetite with particle size from the MD through PSD size range from 76 to 7 μm (Muxworthy & Dunlop, 2002). The large, unstressed 76 μm hydrothermal magnetite sample has a dominantly vertical FORC distribution similar to that for an annealed 2 mm magnetite or a nonannealed 125 μm magnetite reported by Pike et al. (2001). As particle size decreases, the FORC distributions have increasing coercivities with progressively greater divergence from the B i axis. For the finest-grained sample, the peak of the distribution almost closes onto a SD-like peak. Roberts et al. (2000) and  Roberts et al., 2000). The tight distribution of contours along the B i = 0 axis reflects the noninteracting uniaxial SD nature of the particle assemblage, while the distribution along the B c axis reflects its coercivity distribution. (b) Transformer steel sample M-80 (after Pike et al., 2001). The vertical FORC distribution centered near B c = 0 is characteristic of MD particle systems with weak domain wall pinning. Experimental FORC diagrams for a series of synthetic magnetites (H = hydrothermal; W = Wright Industries) with grain size of (c) 76 μm (H), (d) 39 μm (H), (e) 11 μm (W), and (f) 7 μm (W) (after Muxworthy & Dunlop, 2002). With decreasing grain size, FORC distributions progressively diverge away from the vertical distributions expected for MD particle systems ( Figure 3b). We argue in this paper that the divergence is due to coexisting magnetic vortex states. The dashed lines in Figures 3c-3f are 0.05 significance levels for the FORC distributions calculated following Heslop and Roberts (2012).

Journal of Geophysical Research: Solid Earth
10.1002/2017JB014860 Muxworthy and Dunlop (2002) reported that PSD grains have a combination of MD-like moments with both diverging contours and closed SD-like contours. FORC diagrams such as Figure 3f are widely taken to be typical of PSD behavior.

The Single Vortex State
The single vortex magnetic state has been investigated from numerical and experimental perspectives using FORC diagrams. Carvallo et al. (2003) used numerical micromagnetic modeling to simulate FORC diagrams for an elongated magnetite particle just above the stable SD threshold size (100 × 80 × 80 nm). When the magnetization decreases from saturation, it changes progressively from a stable SD state to a flower state to a vortex state, and back to a flower state and stable SD state as it approaches negative saturation. These intermediate magnetization states produce switching branches in simulated FORCs that do not always occur at the same field ( Figure 4a). The different switching branches give rise to multiple positive and negative peaks in a FORC diagram (Figure 4b). This raises questions about whether a randomly oriented assemblage of such particles could give rise to multiple peaks on a FORC diagram that coalesce to form the broader spread observed in geological samples with PSD behavior (e.g., Figure 3f) as suggested by Egli and Winklhofer (2014) and Roberts et al. (2014). Micromagnetic simulation of such realistic assemblages remains computationally prohibitive, but we present results below for a single particle. Before presenting these simulation results, we describe the manifestations of vortex states in FORC diagrams from simple model simulations. Pike and Fernandez (1999) investigated the single vortex state within arrays of cobalt dots (with dimensions of 260 × 450 × 30 nm). They reported FORC diagrams with 3 prominent features associated with single vortex nucleation and annihilation ( Figure 4c), with magnetization switching represented by simplified truncated square hysteresis loops known as hysterons (e.g., Figures 4d and 4e). Hysterons are idealized hysteresis loops where the applied field is parallel to the easy axis of magnetization of uniaxial SD particles. Hysterons are used widely to understand SD particle systems and to illustrate the manifestation of magnetic switching due to coherent rotation of magnetization in FORC diagrams, where the single switching event in a hysteron produces a single point response in a FORC diagram ( Figure 4f) (Roberts et al., 2000(Roberts et al., , 2014. In reality, the magnetization depends on the angle of the applied field to the easy axis of an SD particle. The Stoner and Wohlfarth (1948) model enables such an approximation of SD behavior, with rounded hysteresis loops associated with coherent rotation of magnetization with variable angle of applied field with respect to the easy axis of magnetization (Figures 4d and 4g). Accordingly, the FORC manifestation is not a single point response but is more complicated with weak negative and positive regions and a strong main peak ( Figure 4h) in the same position as for the hysteron approximation ( Figure 4f). Muxworthy et al. (2004), Newell (2005), and Roberts et al. (2014) give detailed explanations of these responses. Pike and Fernandez (1999) simulated vortex nucleation and annihilation using truncated hysterons (Figures 4i and 4k), where the magnetization switches from saturation by forming a vortex at a given nucleation field (B N ) that annihilates at about the same negative applied field (ÀB A ). For the ascending FORC, a vortex nucleates at a given negative field (ÀB N ) and annihilates at about the same positive applied field (B A ). In this representation, two positive peaks occur in the upper and lower half planes of the FORC diagram (Figures 4j and 4l), where the B c and B i coordinates of the peaks depend on the vortex nucleation/annihilation fields with B c = (B A À B N )/2 and B i = ±(B A + B N )/2 (Pike & Fernandez, 1999) (Figures 4i and 4k). Pike and Fernandez (1999) also simulated a weaker "butterfly" feature along the B i = 0 axis that consists of a circular negative region with a superimposed elongated horizontal positive region ( Figure 4c). A vortex will not necessarily annihilate at the same field magnitude at which it nucleated; this possibility becomes more likely if physical irregularities occur within a particle. The butterfly feature along the horizontal axis results from two distinct annihilation fields (Pike & Fernandez, 1999) and will not be present if this condition is not met. Vertical spreading of peaks in the lower and upper half planes of the FORC diagram reflects the distribution of nucleation/annihilation fields. Pike and Fernandez (1999) presented experimental FORC diagrams similar to Figure 4c for arrays of cobalt dots produced by interference lithography.
The observations of Pike and Fernandez (1999) have been confirmed for <100 nm iron nanodots . For 52 nm nanodots, noninteracting SD behavior is observed (Figure 4m) with a negative region near the B i axis (e.g., Figure 4h). As the iron nanodots increase to 58 nm, single vortex behavior appears (Figure 4n), as observed by Pike and Fernandez (1999) (Figure 4c). When the dot size increases to 67 nm, the magnitude of the major peaks and the distance of the peaks from the B i = 0 axis  Micromagnetic results for an elongated single isolated magnetite particle (100 × 80 × 80 nm) that exceeds the stable SD threshold size. (a) Simulated FORCs indicate changes from the stable SD state with decreasing field from positive saturation to a flower state to a vortex state and back with approach to positive saturation. Magnetization switching events at different fields produce (b) a complex FORC diagram with multiple peaks (after Carvallo et al., 2003). (c) Model results (after Pike & Fernandez, 1999) where the offset of the upper and lower peaks from the B i = 0 axis provides a measure of the vortex nucleation/annihilation field. The "butterfly" feature on the right-hand side results from the presence of two distinct annihilation fields (Pike & Fernandez, 1999) and is not always observed in single vortex systems. (d) Illustration of hysteresis loops for uniaxial SD particles with easy axes at different angles with respect to the applied field (from Stoner & Wohlfarth, 1948). (e) Hysteron for an isolated uniaxial SD particle with easy axis parallel to the applied field and (f) its response on a FORC diagram. (g) Hysteresis loop for a Stoner-Wohlfarth particle oriented 25°to the applied field direction and (h) its nonsingular response with strong positive and weak negative regions on a FORC diagram (see Muxworthy et al. (2004), Newell (2005), or Roberts et al. (2014) for details). (i) Split hysteron representation of a magnetic vortex state (following Pike & Fernandez, 1999), where B N is the vortex nucleation field and B A is the annihilation field. (j) FORC diagram for such a split hysteron with positive peaks in both the upper and lower FORC half planes, (k) where the distance of peaks from the B c axis is a measure of the strength of B N and B A , (l) and their distance from the B i axis is a measure of coercivity (hysteron width), as illustrated for a coarser-grained lower coercivity vortex state (Figures 4d-4l are from Zhao et al., 2017). FORC diagrams for iron nanodots with progression from (m) stable SD behavior (for 52 nm nanodots) to (n) single vortex behavior (for 58 nm nanodots) to (o) single vortex behavior with a larger nucleation/annihilation field (for 67 nm nanodots; after . In Figures 4m-4o, the thick black lines denote 0.05 significance levels calculated following Heslop and Roberts (2012).

Journal of Geophysical Research: Solid Earth
10.1002/2017JB014860 (i.e., the nucleation/annihilation field) increase ( Figure 4o) and a secondary "butterfly" feature appears. These results illustrate clearly the effects of vortex nucleation/annihilation on FORC diagrams for samples with precisely controlled particle size.

The Vortex State and PSD Behavior
Despite the simplicity of the numerical model used by Pike and Fernandez (1999) to represent the single vortex state, it replicates key aspects of experimental results for synthetic samples with tightly controlled magnetic particle size . The observation that the vertical separation between the upper and lower peaks of the FORC distribution depends on the magnitude of vortex nucleation/annihilation fields raises questions about whether a broader distribution of single vortex nucleation/annihilation fields associated with broader particle size distributions could give rise to the FORC diagrams observed for geological samples with PSD behavior (e.g., Figure 3f). We explore this possibility using several examples in Figure 5.
FORC diagrams for geological samples routinely provide evidence of PSD behavior as in Figure 3f and do not often contain evidence of single vortex states as illustrated in Figures 4c, 4n, and 4o. An exception is shown in Figure 5a in which single vortex behavior is clear in a conventional FORC diagram (from Zhao et al., 2017). The sample in question is an extracellular magnetite (Li, 2012) produced by thermophilic iron-reducing bacteria (Thermobacter spp. strain TOR39), which occur as hexagonal plates (Figures 5d-5f). Importantly, the plates have a relatively constant thickness and a relatively narrow particle size distribution across several hundred nanometers, which is within the size range expected for PSD behavior (Muxworthy & Williams, 2009). It is the relatively narrow particle size distribution of this sample that gives rise to the clear single vortex FORC distribution in Figure 5a, although magnetostatic interactions associated with packing of the magnetite particles will have produced additional vertical spreading of the FORC distribution.
As demonstrated by Zhao et al. (2017), using a new FORC measurement type that includes transientmagnetization-free measurements, we can isolate the effect of transient magnetization processes due to self-demagnetization in coarse-grained magnetic particles. The transient magnetization is the magnetization difference between the upper and lower branches of a minor hysteresis loop between the saturating field and zero field (Fabian, 2003). When the magnetization decreases from saturation, it is controlled by both field-driven magnetization changes and transient processes. The transient magnetization will decay to zero when the applied field returns to zero and will stay at zero even if the field increases (Fabian, 2003). Thus, when the magnetization is measured from zero-field to saturation (referred to as the zero-FORC by Yu & Tauxe, 2005), only field-driven (i.e., transient-free) magnetization reversal occurs. By measuring a series of zero FORCs within a conventional set of FORC measurements following the procedure of Zhao et al. (2017), it is possible to determine a FORC distribution for the transient-free magnetization component ( Figure 5b). Then, by subtracting the transient-free FORC (tfFORC) distribution from the conventional FORC distribution, it is possible to obtain the FORC distribution due to the transient magnetization component (Figure 5c). Thus, the transient FORC (tFORC) diagram (Zhao et al., 2017) provides a measure of the distributions of nucleation and annihilation fields of magnetic vortices and domains. In the tFORC diagram for the Thermobacter magnetite in Figure 5c, the peaks above and below the B i = 0 axis reflect the distribution of nucleation and annihilation fields of magnetic vortices associated with the particle size distribution illustrated in Figures 5d-5f. Interpretation in terms of vortex nucleation and annihilation follows naturally from the discussion above about the manifestations of single vortex behavior in FORC diagrams (Figure 4). These peaks are evident in the conventional FORC diagram for the Thermobacter magnetite (Figure 5a), but they are not evident in conventional FORC diagrams for typical geological samples that contain "PSD" magnetic particles. Transient FORC diagrams are, therefore, a powerful tool for assessing the origin of "PSD" behavior in natural samples.
A more typical PSD FORC diagram is shown in Figure 5g for an andesite sample from Mount Ruapehu, New Zealand (Ingham et al., 2017). This conventional FORC diagram is not obviously representative of single vortex behavior (compare Figures 3f and 5g with Figures 4n and 4o). However, the characteristic vortex nucleation and annihilation field distributions above and below the B i = 0 axis become evident in a corresponding tFORC diagram (Figure 5i). The vortex signature is not evident in conventional FORC diagrams because they include contributions from many magnetization processes, including remanent, induced, and transient magnetization components (Zhao et al., 2017). In contrast, tFORC diagrams contain contributions Journal of Geophysical Research: Solid Earth 10.1002/2017JB014860 from only the transient magnetization processes described by Fabian (2003), which provides clear evidence of magnetic vortex states without being masked by coexisting magnetization processes (Zhao et al., 2017). Routine determination of tFORC diagrams (Zhao et al., 2017) should lead to more frequent diagnosis of PSD behavior as due to vortex states. In promoting tFORC diagrams, we understand that the potential exists for false diagnosis of the vortex state. For example, when a magnetofossil chain, produced originally by magnetotactic bacteria, collapses, the magnetic flux associated with individual particles can link to create a super vortex-like structure that would be manifest as a vortex signature in a FORC diagram (Egli & Journal of Geophysical Research: Solid Earth 10.1002/2017JB014860 Winklhofer, 2014). The extent of vortex state magnetizations in geological materials can be assessed by use of samples that do not contain magnetofossils, as will be shown in forthcoming work.
Published demonstrations of single vortex states in conventional FORC diagrams for geological samples remain nonexistent to our knowledge. Lappe et al. (2011Lappe et al. ( , 2013 reported single vortex FORC features for synthetic dusty olivine samples. A noninteracting SD component may obscure some of the features observed in the FORC model results of Pike and Fernandez (1999), but the lower part of the "butterfly" feature is preserved, along with the main positive FORC features expected for single vortex behavior (these features were originally attributed to PSD behavior (Lappe et al., 2011) but were later attributed to single vortex behavior (Lappe et al., 2013)). Both stable SD and single vortex states are observed by electron holography (Lappe et al., 2011) in particles that span a continuous grain size distribution. This provides confirmation that single vortex nucleation and annihilation can explain important aspects of PSD behavior.

MD Behavior Contrasted With Coarse Vortex State Behavior
A conventional FORC diagram for a 120 μm natural magnetite sample is shown in Figure 5j. The divergent contours are characteristic of MD behavior (Pike et al., 2001). In a tFORC diagram (Figure 5l), the upper and lower parts of the distribution have a "winged" structure (Zhao et al., 2017) where the peaks of the distribution of domain nucleation/annihilation fields lie near the origin of the FORC diagram and the distribution intersects the B c = 0 axis. This behavior contrasts with the lobate tFORC distributions in Figures 5c and 5i, where the lobes are indicative of vortex nucleation/annihilation fields. In Figure 5c, FORC contours close around the lobes, whereas in Figure 5l, the MD contours diverge away from the origin. These contrasting features could be useful for distinguishing the extent of coexisting single vortex and domain walls. For example, the lobate features in Figure 5i close around each other, but the outer parts of the tFORC distribution also diverge away from the lobes, particularly in the lower FORC half plane. This could be evidence of combined vortex/domain wall behavior within the same particles, which is suggested by micromagnetic simulations (e.g., Figures 2c and 2d). The transition from the coarsest part of the vortex state to the MD state is discussed further below.

Micromagnetic Simulation of the Single Vortex State
We now present results of numerical simulations to further illustrate how vortex states can explain PSD behavior. To approximate magnetic behavior in the magnetite plates illustrated in Figures 5d-5f, we simulated hysteresis behavior micromagnetically for a circular magnetite disk with 240 nm diameter and 40 nm height (Figures 6a and 6b) and easy axis of magnetization in the plane of the disk. Calculations were made using the Micromagnetic Earth Related Rapid Interpreted Language Laboratory (MERRILL) code, in which finite element/ boundary element methods are used to solve for the magnetic scalar potential inside a particle to calculate the demagnetizing energy; to sum the exchange, cubic anisotropy, magnetostatic, and demagnetizing energies; and to minimize the total free magnetic energy. Energy minimization was performed using a conjugate gradient method. Material parameters used are for magnetite at 20°C. Two sets of calculations were made. First, the angular dependence of hysteresis was calculated, where an external field was applied through a range of angles from parallel to the easy axis of magnetization (0°) to perpendicular to the plate (90°) at 5°increments (Figure 6b). Hysteresis loops are presented at 15°increments in Figure 6d. This simulated disk is in the single vortex magnetic state at 20°C. The vortex core lies at the center of the particle (Figure 6c) so that all magnetic moments in the plane of the disk are balanced and the net remanent magnetization will lie along the direction of the vortex core. The vortex state persists with increasing angle of applied field with respect to the plate. Figure 6f is an average of simulated loops for the disk at 5°intervals, which was calculated by accounting for geometrical weighting over the surface of a sphere for different disk orientations with respect to the applied field. Hysteresis loops with such shapes are rarely observed in nature, but this is an average of simulations at only 19 angles; natural samples will contain many millions of particles and will, therefore, give rise to smooth, rounded bulk hysteresis loops.
Natural samples also contain magnetic particles with variable size, which was simulated in a second set of calculations for disks with 40 nm thickness and long dimension varying from 20 to 320 nm at 20 nm increments (Figure 6e). For diameters of 20-100 nm, the particle is in the SD state. A vortex state nucleates at 120 nm and persists to the maximum simulated diameter of 320 nm. The average of 16 loops for plate Journal of Geophysical Research: Solid Earth 10.1002/2017JB014860 diameters between 20 and 320 nm is shown in Figure 6g. It is usually assumed that magnetic particles in paleomagnetic samples are distributed randomly with respect to the applied field. Particle sizes are also likely to range between tens of nanometers and tens of microns. Extrapolating from these limited simulations, natural samples are expected to contain magnetic particles that range from having weak remanence/low coercivity to high remanence/high coercivity, with particles in the vortex state expected to have variable remanence and coercivity (Figures 6f and 6g). When such particles are present in geological samples, natural particle size distributions will give rise to broad-ranging coercivity and vortex nucleation/ annihilation fields. Thus, our simple micromagnetic simulations further raise the question of whether typical PSD FORC diagrams (e.g., Figure 3f) can be explained by vortex state particles with broad particle size distributions. We explore this possibility with micromagnetic simulations of FORCs below. (b) Side-on view of the disk with relationship between applied field angle and the disk. (c) Micromagnetic simulation of the remanence state of the disk when the field is applied parallel to its easy axis (0°). Results of micromagnetic simulations of hysteresis loops for the disk for (d) variable applied field angle (calculated at 5°increments, but shown at 15°increments) and (e) variable disk diameter (from 20 nm to 320 nm). Hysteresis loops for the sum of the (f) 19 angles and (g) 16 disk diameters illustrate the smoothing that occurs in real samples with mixed particle sizes and easy axis orientations. For the millions of particles in a typical paleomagnetic sample, smooth loops will result and steps due to vortex nucleation/ annihilation will not be evident. The parameters used are for magnetite at 20°C as implemented in MERRILL.   Figure 4c). For the 90°case, a SD peak is evident.

Journal of Geophysical
Journal of Geophysical Research: Solid Earth 10.1002/2017JB014860 (Figures 7b,7d,7f,7h,7j,7l,and 7n), where the main secondary features in addition to the main peaks are positive and negative distributions that extend from the main peaks back to the B i axis at 45°. Pike and Fernandez (1999) explained these features as due to reversible magnetizations associated with curved portions of FORCs that are analogous to those in the Stoner and Wohlfarth (1948) model (Figures 4g and  4h). Importantly, the main peaks in the FORC diagrams indicate that both the coercivity and strength of vortex nucleation/annihilation fields vary with applied field angle with respect to the easy axis of magnetization. These micromagnetic results, thus, provide a direct link between the physics of vortex nucleation/annihilation and expected FORC responses (Pike & Fernandez, 1999). Even for so few simulations, when results for 19 orientations at 5°increments are averaged (Figure 7p), it becomes evident that averaging over a larger particle size range could get close to providing an overall bulk PSD-like response (e.g., Figure 3f), as suggested by Egli and Winklhofer (2014) and Roberts et al. (2014). However, it is likely that simulation of single vortex states alone will be insufficient and that multivortex features, as discussed below, will need to be included to achieve realistic simulations of bulk PSD-like responses. Such a micromagnetic simulation is computationally expensive and has yet to be undertaken.
Overall, our micromagnetic results validate the simple calculations of Pike and Fernandez (1999) where vortex nucleation/annihilation fields produce FORC distributions that are dominated by a simple two-peak geometry. Our results are consistent with the thesis that the fine-grained end of bulk PSD behavior is represented by the magnetic single vortex state.   Heslop and Roberts (2012). (c-h) Magnetic induction maps reconstructed from room temperature electron holograms (in-plane fields were applied parallel to the black arrow in each image to induce the remanent magnetization represented in each induction map). The contour spacing is 0.53 rad for all induction maps; magnetization directions are indicated with arrows (depicted in the color wheels). The widest (~300 nm) magnetite lath 6 occurs in either a nearly uniformly magnetized state (Figures 9e and 9f) or with a series of multiple stable vortices (Figures 9c, 9d, 9g, and 9h) depending on applied field. (i and j) Micromagnetic simulations conducted with a finite difference algorithm with regular grid points indicate that magnetization switching occurs by nucleation of a vortex that sweeps down the lath. Continuous magnetic flux lines between laths indicate strong magnetic interactions.

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Curie temperature of magnetite (Almeida et al., , 2016Einsle et al., 2016;Muxworthy et al., 2014;Nagy et al., 2017), which suggests that they are capable of recording paleomagnetic information over long geological periods. This confirms the paleomagnetic recording fidelity of PSD particles that has been postulated for over 55 years.

The Multivortex State
In addition to the single vortex state (Figure 8), multivortex states are evident in electron holographic images (Figures 9c, 9d, 9g, and 9h). These images and micromagnetic simulations (Figure 9j) are for magnetite laths with widths of 100-150 nm (numbered 1-7 in Figure 9a) that are separated by titanohematite lamellae (Church, 2010). The nearly pure magnetite laths and paramagnetic titanohematite lamellae (Fe 1.16 Ti 0.84 O 3 ) were produced synthetically by oxy-exsolution of an original titanomagnetite (Fe 2.4 Ti 0.6 O 4 ). A conventional FORC diagram for this sample is shown in Figure 9b and is typical of bulk PSD behavior. In addition, the FORC diagram has a superimposed "wishbone" structure that has been observed in strongly interacting Ni nanopillar arrays, where the positive peak is shifted above B i = 0 due to a mean (negative) demagnetizing field produced by interparticle interactions (Pike et al., 2005). Magnetite lath 6 ( Figure 9a) is the widest (~300 nm), and either occurs in a nearly uniformly magnetized state (Figures 9e, 9f, and 9i) or with a series of stable vortices (Figures 9c, 9d, 9g, 9h, and 9j) depending on the applied field. Micromagnetic simulations indicate that magnetization switching occurs by nucleation of a vortex that sweeps down the length of the lath. Also, continuous magnetic flux lines between laths indicate that this sample is affected strongly by magnetic interactions, which is also evident in the positive upward shift of the main peak in the FORC diagram.
Magnetic superstates due to magnetic interactions among such exsolution features have been well documented (Feinberg et al., 2006;Harrison et al., 2002). Regardless, this illustration demonstrates that in larger particles in the hundreds of nm size range, such as magnetite lath 6, complex internal multivortex features occur, which indicates that the single vortex state does not transition directly into the MD state. At these still relatively small particle sizes, multivortex behavior becomes an important part of the spectrum of magnetic behavior associated with the PSD state (Einsle et al., 2016). Simulation of bulk PSD FORC diagrams, therefore, require consideration of multivortex behavior. Multivortex magnetic structures are also well known in materials science (e.g., Donnelly et al., 2017;Harada et al., 1996;Ivanov et al., 2016).

Other Magnetic Structures Associated With the Vortex State
As particle size increases, further phenomena must be considered to understand magnetization structures within magnetic materials in the vortex state. A major limitation in understanding the roles of relevant magnetic features has been the inability to undertake 3-D imaging of magnetization structures. The recent work of Donnelly et al. (2017) opens the possibility of imaging the 3-D magnetization state in micron-scale particles of interest for understanding the magnetic vortex state in rock magnetism. Donnelly et al. (2017) documented various magnetic structures that will need to be considered to develop a detailed understanding of the vortex state. For example, they demonstrated the existence of vortices with clockwise and anticlockwise vorticity. The magnetization is almost uniform between such structures with opposite vorticity, with antivortices occurring in magnetically inhomogeneous regions between vortices with the same vorticity. Sequences of vortex and antivortex structures give rise to cross tie walls that extend through a magnetic material and separate regions with oppositely polarized magnetizations. A magnetic singularity, known as a Bloch point, can form when a vortex core intersects a domain wall. Magnetic order is destroyed at such singularities, which occur over a radius of order of the exchange length. When this range of magnetic structures can be resolved, the 3-D magnetization configuration is complex for particles that span the vortex state. While many new things will be learned with application of such new techniques, the observations of Donnelly et al. (2017) strengthen our case for referring to the PSD state as the vortex state because the complex magnetization structures that they documented are associated with magnetic vortices and their interactions within a magnetic material.

Discussion
We have provided evidence that the PSD state can be explained by the magnetic vortex state based on FORC diagrams, micromagnetic simulations, and electron holography. However, key aspects of PSD behavior need further explanation, including the nature of the transitions from the stable SD to the single vortex state and Journal of Geophysical Research: Solid Earth 10.1002/2017JB014860 from the multivortex to the MD state, along with issues mentioned by Fabian and Hubert (1999) as indicated in section 2. We provide such explanations below.

Transition From the SD State to the Single Vortex State
Results from micromagnetic modeling, numerical simulation, experimental analysis of nanoparticle arrays with controlled particle size and interparticle spacing, and natural geological/bacterial/synthetic samples with narrow particle size ranges all point to an immediate transition from the stable SD state (including a flower state) to a single vortex state. This transition is sharp (Williams & Dunlop, 1989, 1995. For some materials such as equant iron particles, which are important in meteorites, there is no stable SD state at room temperature and particles pass directly from the superparamagnetic to the vortex or MD states (Butler & Banerjee, 1975;Muxworthy & Williams, 2015). How, then, is the sharpness of the stable SD to single vortex transition reconciled with the gradual magnetic property variations (e.g., M rs , M rs /M s , and B c ; Figure 1) observed through the PSD size range? We explore this subject in section 5.2.

Why Do M rs and B c Vary Continuously With Particle Size in the Vortex State?
Particle size variations have frustrated domain state diagnosis in natural and synthetic samples, but they are also the likely cause of gradual magnetic property variations through the PSD size range. When the magnetic responses of particles with variable sizes and shapes are summed (typically millions of particles in a paleomagnetic sample), sharp transitions due to individual vortex state particles will be smoothed (smoothing of abrupt magnetization switching is even evident in the limited averaging in Figure 6g). Thus, even if all magnetic particles in a sample occur in a vortex state, vortex nucleation/annihilation field distributions will cause gradually varying bulk M rs , M rs /M s , and B c values with changes in average magnetic particle size. Randomized easy axis orientations in natural samples ( Figure 6) will also contribute to smooth rather than abrupt magnetic property changes through the PSD size range.
Use of tFORC diagrams (Zhao et al., 2017) enables isolation of the effects of vortex and domain wall nucleation/annihilation. When assessed using tFORC diagrams, typical PSD systems manifest magnetic behavior associated with the vortex state ( Figure 5i). Distributed particle sizes are also evident in the coercivity and nucleation/annihilation field distributions in Figures 5c and 5i. Even for the Thermobacter sample that appears to have obvious single vortex behavior in a conventional FORC diagram (Figure 5a), a significant stable SD component is present in the tfFORC diagram ( Figure 5b). This is also the case for a typical PSD sample (Figures 5g and 5h). Thus, widely documented SD-like moments in PSD samples could partially be due to stable SD particles. Our ability to produce samples with narrow particle size and shape distributions is limited, which makes it difficult to rule out the presence of stable SD behavior. Documentation of widespread SD nanoparticles as inclusions within host silicates in the geological record provides support for this possibility (Chang et al., 2016;Evans et al., 1968). FORC unmixing (Channell et al., 2016;Lascu et al., 2015) now makes it possible, along with the additional FORC measurements of Zhao et al. (2017), to discriminate SD, vortex, and MD moments in natural samples. Regardless of the possible presence of SD grains, particle size and shape variations are widespread in natural samples and are likely to be responsible for smooth magnetic property variations throughout the PSD size range.

What Controls the Remanence Direction in Vortex State Particles?
Is it reasonable to ask what processes control the remanence direction for single vortex particles? Einsle et al. (2016) conducted the most extensive study of this question by characterizing magnetic domain states within iron particles in dusty olivine from a chondritic meteorite using focused ion beam nanotomography, electron tomography, and finite element micromagnetic models. They found that the remanence vector is controlled primarily by the magnetization of the vortex core and the sense of rotation of bulk spins. Slight irregularities in particle shape, including different sizes of corresponding crystal surface facets, can give rise to net remanences due to additional bulk spins in these regions. Such an imbalance between bulk spins in the single vortex state is illustrated in the experimental electron hologram in Figure 8b, where the asymmetric particle shape gives rise to a much larger volume of spins on the right compared to the left of the image. This produces a net in-plane component of the total moment in addition to the out-of-plane moment due to the vortex core. Depending on the details of a particle, the total moment can be vortex core dominated, bulk dominated, or some combination of these two. Thus, some aspects of the early domain imbalance moment explanation for PSD behavior (Dunlop, 1983;Shcherbakov, 1978;Stacey, 1961)  to developing a more detailed understanding of the full range of magnetic behaviors in the vortex state.
Overall, Einsle et al. (2016) concluded that particles in the single vortex state with high coercivity and large volume can have long-term paleomagnetic stability and can be reliable paleomagnetic recorders.

Transition From the Multivortex State to the MD State
There has been much debate as to where PSD behavior stops and true MD behavior begins (e.g., Heider et al., 1996;Vlag et al., 1996) even though it is acknowledged to correspond to the point at which the magnetization is controlled by the response of domains and domain walls to the internal demagnetizing field (Dunlop, 2002;Néel, 1955). In the terms discussed here, this transition corresponds to that between the multivortex and MD states. Three-dimensional micromagnetic model results reveal that as particle volume increases, vortex cores and domain walls coexist (e.g., Figures 2c and 2d) (Muxworthy, 2013). Einsle et al. (2016) found that in particles several hundred nanometers across, the magnetic structure is dominated by the single vortex state, which gives way to multiple vortex and domain wall structures in larger particles with~1 μm long axes. Vortex cores in these larger particles develop "winged" structures that protrude along directions in which domain walls nucleate (Figures 2c and 2d). Some multivortex scenarios simulated by Einsle et al. (2016) are similar to electron holograms in Figure 9 and to the complex magnetization structures documented by Donnelly et al. (2017). The presence of both multiple vortex cores and domain walls in larger particles suggests that the vortex to MD transition is gradual. Important transient magnetization processes that are not discussed in detail here will also contribute to the bulk magnetization in particles through this size range. For example, pinning and partial switching of vortex core or domain wall positions are possible and will be responsible for transient magnetizations. Such processes are part of the spectrum associated with magnetic vortices, so we expect that vortex and domain wall nucleation/annihilation represent the fundamental first-order physical processes.

Can Coarse Particles With Multiple Vortex States Be Discriminated From MD Particles?
Vortex nucleation/annihilation fields and coercivity decrease with increasing particle size (illustrated schematically in Figures 4i and 4k). The gradual nature of the transition between the vortex and MD states is particularly evident in the orientations of elongated upper and lower lobes of tFORC distributions. These lobes have decreasing nucleation/annihilation fields with increasing particle size and merge to B i = 0, coercivity decreases progressively, and the FORC distribution shifts to the left and merges onto the B i axis (compare Figures 5c, 5i, and 5l). Novosad et al. (2001) also observed that vortex nucleation fields decrease with increasing particle size. These observations make it reasonable to ask at which point does a particle change from a single vortex state to a multiple vortex state to the MD state? This question is not easy to answer, but we discuss it below.
When concentric contours define a nucleation/annihilation field distribution in a tFORC diagram, particles must be present in the single vortex state no matter how near the B c and B i axes the distribution lies. But such tFORC distributions also partially intersect the B c axis, which indicates that MD particles are also present. This interpreted dominance of single vortex states is consistent with numerical hysteresis loop simulations for particles with multiple vortex cores, which are more MD-like than vortex-like (e.g., particle 233 of Einsle et al. (2016)). This interpretation is also consistent with micromagnetic simulations of equidimensional magnetite in Figure 2, which indicate a large single vortex core in an 80 nm particle, while at 300 nm, the vortex core remains and domain walls have also nucleated at particular angles with respect to the vortex core. Overall, the transition from the vortex state to the true MD state appears to be gradual because of the coexistence of vortex cores and domain walls within the same particle (e.g., Figures 2c and 2d). The gradual nature of this transition makes it difficult to diagnose. The existence of new microscale tools for imaging magnetic structures (e.g., Donnelly et al., 2017) means that the important vortex state particle size range can now be subjected to the detailed 3-D characterizations needed to develop a meaningful understanding of magnetization structures in such particles. This will be important in moving away from a phenomenological understanding of the PSD state and toward a detailed physical understanding of the relevant magnetic structures.

Low-Temperature Behavior of PSD Magnetite
Fabian and Hubert (1999) concluded that any explanation for PSD behavior should include the presence of SD-like and MD-like coercivity components in LTD results. When cycled through the Verwey transition (120 K) for magnetite (Verwey, 1939), where the crystal lattice of magnetite changes from a cubic to a Journal of Geophysical Research: Solid Earth 10.1002/2017JB014860 monoclinic structure, the magnetocrystalline anisotropy increases by an order of magnitude, which increases the SD to single vortex transition size from~80 nm at room temperature to~140 nm at~100 K (Muxworthy & Williams, 1999). Smirnov (2006Smirnov ( , 2007 presented low-temperature FORC results for polycrystalline synthetic PSD and MD magnetite samples (similar results were published independently by Carvallo and Muxworthy (2006)). From room temperature down to near 120 K, FORC diagrams are characteristic of those reported for PSD magnetite at room temperature (compare Figures 10e and 10f with Figure 3f). Below the Verwey transition temperature, the central peak in the PSD FORC distribution starts to split (Figure 10d) into separate peaks with concentric contours in the upper and lower parts of the FORC diagrams (Figures 10a-10c). With decreasing temperature, the peaks separate progressively (Figures 10a-10d). These low-temperature FORC diagrams resemble those associated with single vortex systems discussed above and reported by Brandt et al. (2013), , , Dumas et al. (2009Dumas et al. ( , 2012, Pike and Fernandez (1999), Winklhofer et al. (2008), and Zhao et al. (2017). Smirnov (2006) recognized this possibility but dismissed it because the upper and lower peaks have significantly different magnitudes and because FORC diagrams for sample P1 lack the "butterfly" feature ( Figure 4c) recognized by Pike and Fernandez (1999). However, this asymmetry can be explained by different nucleation/annihilation fields (Brandt et al., 2013;Dumas et al., 2012). Likewise, a "butterfly" feature will only be present when there are two distinct annihilation fields (Pike & Fernandez, 1999), which does not appear to be the case with the samples of Smirnov (2006).
At face value, the low-temperature FORC results of Smirnov (2006) for PSD magnetite are consistent with the presence of single vortex states. However, magnetic domain state interpretation for magnetite below the Verwey transition is complicated by the formation of crystallographic twin domains that control the local magnetocrystalline anisotropy. The positions of magnetic domain walls are often pinned strongly at twin Figure 10. Temperature-dependent FORC diagrams for PSD magnetite above and below the Verwey transition temperature. FORC diagrams (smoothing factor (SF) = 2 for all) measured after zero field cooling during warming from 20 to 140 K at (a) 20 K, (b) 80 K, (c) 95 K, (d) 105 K, (e) 110 K, and (f) 140 K (from Smirnov, 2006). Above the Verwey transition temperature, FORC diagrams are typical of PSD particle systems (e.g., Figure 3f). Below the transition, the FORC distributions split into upper and lower peaks, which resemble those for single vortex systems (Figure 4), but without a "butterfly" feature. The thick black lines represent 0.05 significance levels calculated following Heslop and Roberts (2012).

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boundaries , and no lower size limit is known for which twin domains will not form. Furthermore, these twin walls have been shown through TEM observations to move within applied external magnetic fields at temperatures close to the Verwey transition temperature (Kasama et al., 2010), although twin walls are static below the transition . Using detailed TEM and electron holographic observations, Kasama et al. (2013) demonstrated that vortex structures above the Verwey transition are converted into vortex-like structures that are defined by twin walls in the monoclinic phase below the transition. Bryson et al. (2013) simulated micromagnetically a field-induced "freezing-in" of a vortex core by twin wall formation below the transition. "Vortex nucleation" below the transition is due to nucleation and movement of 180°domain walls that lock into place when the magnetization aligns with twin walls. Vortex "annihilation" then occurs when magnetic walls depin from twin walls. The simulations of Bryson et al. (2013, Figure 7) provide strong evidence that vortex states, including those associated with twin-induced closure domain structures, can explain the low-temperature hysteresis behavior discussed by Smirnov (2006Smirnov ( , 2007. Based on the above evidence, we conclude that the low-temperature FORC diagrams in Figure 10 reflect the dominance of the single vortex state (Figures 4, 5, and 7) in the studied sample.  observed that vortex states in Fe dots persist to low temperatures, that they are thermally activated, and that vortex nucleation/annihilation fields increase with decreasing temperatures. Magnetite seems to behave similarly ( Figure 10). The results of Carvallo and Muxworthy (2006) and Smirnov (2006Smirnov ( , 2007, coupled with the experimental and numerical insights of Kasama et al. (2013) and Bryson et al. (2013), could, therefore, provide the beginning of an understanding of the temperature dependence of the vortex state in magnetite where magnetic vortices act as a template for the location of twin walls below the Verwey transition.
Returning to the issue of LTD in PSD magnetite, Muxworthy and McClelland (2000) assessed the causes of this phenomenon. They concluded that LTD has two main contributions due to (1) kinematic domain state reorganization and (2) the change from cubic to monoclinic anisotropy, where some domains move and occupy new positions that lead to a lower net remanence after warming in zero field. Their key finding is that internal stress, rather than particle size, controls the recovery mechanism so that AF demagnetization is unsatisfactory for distinguishing between MD-like and SD-like behaviors (Muxworthy & McClelland, 2000). The two contributions to LTD proposed by Muxworthy and McClelland (2000) are explained better by TEM and electron holographic observations and micromagnetic simulations that provide strong evidence that the locations of magnetic vortices above the Verwey transition are a template for the placement of twin walls below the transition, so that their interactions can explain the preservation and recovery of remanence after cycling through the Verwey transition (Bryson et al., 2013;Kasama et al., 2013). The above evidence, which combines bulk magnetic measurements with nanoscale experimental observations and micromagnetic simulations, provides hope that we are close to explaining key aspects of PSD behavior that have remained elusive for decades.

Universality of PSD Behavior
Finally, Fabian and Hubert (1999) stated that explanations for the PSD state must account for the universality of PSD remanence in all kinds of magnetic materials across a broad transitional size range. From the above treatment, based on various observations, as particle size increases above the stable SD threshold size and flower state in any magnetic material (e.g., magnetite, Fe, Co, and Ni), the vortex state becomes the next stable domain state. The vortex state, therefore, provides an explanation for the universality of PSD behavior. Like the terms "single domain" and "multidomain," the term "vortex state" provides a description of the physics of the dominant magnetization process, which "pseudo-single domain" does not. Thus, we recommend abandonment of PSD in favor of "vortex state." The vortex state spans a complex and wide range of phenomena associated with magnetic vortices, including single and multiple vortices, antivortices, cross tie walls, domain walls, and Bloch points. The term "vortex state" is, therefore, a catchall term like PSD. However, it at least provides a meaningful description of the relevant physics, whereas the term PSD is misleading at a fundamental level because the origin of magnetic stability in PSD particles is not due to SD-like magnetic moments.

Conclusions
Since being proposed by Stacey (1961Stacey ( , 1962, the term "pseudo-single domain" has come to be the prevalent term used to describe the transitional state between the single and multidomain states in paleomagnetism Journal of Geophysical Research: Solid Earth 10.1002/2017JB014860 and rock magnetism. We present evidence from FORC diagrams, micromagnetic simulations, and electron holography to make the case that the PSD state is better described by the vortex state. As particle size increases above the range over which uniform magnetizations occur, flower ( Figure 2a) and vortex structures (Figures 2b and 2c) nucleate. Vortex-like structures, which are observed in all types of magnetic materials (e.g., Coey, 2010), are the equilibrium or lowest energy state for nonuniformly magnetized particles in 3-D micromagnetic calculations (Williams & Dunlop, 1995). The transition from the stable SD/flower state to the vortex state is sharp, yet magnetic properties through the PSD size range vary gradually. This observation is best reconciled by the fact that materials studied in paleomagnetism and rock magnetism almost always have magnetic particle size and shape distributions, which result in gradual variation of bulk magnetic properties due to the summed responses of millions of magnetic particles. As vortex state particles increase in size, more complex magnetization structures occur, where domain walls nucleate alongside single (Figures 2c and  2d) or multiple vortex cores, along with less well documented magnetic structures such as antivortices, cross tie walls, and Bloch points. Thus, single vortex states explain the fine end of the PSD particle size range, while domain walls along with single/multiple vortex cores control magnetization structures at the coarser end of the PSD size range. When genuine MD behavior occurs, magnetizations will be dominated by domain wall pinning (Néel, 1955;Pike et al., 2001). The persistence of weak but stable remanences and the gradual decline in remanence with increasing particle size through the PSD size range is likely explained by the imbalanced nature of magnetic moments in complex magnetic structures within magnetic particles in the vortex state (Einsle et al., 2016). Available evidence suggests that these structures can record paleomagnetically meaningful information over long geological periods (Almeida et al., , 2016Nagy et al., 2017).
Many details of complex magnetization structures across the critical vortex state size range remain to be well understood, but the tools now exist to enable their systematic documentation and understanding. We propose that the term "vortex state" better describes the physics of the relevant magnetization processes and that the term "pseudo-single domain" should be abandoned.