On the magnetocrystalline anisotropy of greigite (Fe3S4)

Authors


Abstract

The ferrimagnetic mineral greigite (cubic Fe3S4) is well known as an intracellular biomineralization product in magnetic bacteria and as a widely occurring authigenic mineral in anoxic sediments. Due to the lack of suitable single-crystal specimens, the magnetic anisotropy parameters of greigite have remained poorly constrained, to the point where not even the easy axis of magnetization is known. Here we report on an effort to determine the anisotropy parameters on the basis of ferromagnetic resonance (FMR) powder spectroscopy on hydrothermally synthesized, chemically pure greigite microcrystals dispersed in a nonmagnetic matrix. In terms of easy axis orientations, the FMR data are consistent with <111> or <100>, or less likely, a more general <uv0> type. With a g factor of 2.09, the anisotropy field is about 90 mT and in some samples may reach 125 mT, compared to 30 mT for cubic magnetite. This confirms the dominating role of cubic anisotropy on the magnetic properties of greigite, which we show to be responsible for large SIRM/k values. K1 is in the range −15 … −23 J/m3 (<111>) or +10 … +15 kJ/m3 (<100>), yielding upper limits of 44 or 34 nm for the superparamagnetic grain size, respectively.

1 Introduction

Greigite, the ferrimagnetic iron sulfide with symmetry in the cubic space group Fd3m and composition Fe3S4 (thiospinel analog to magnetite, Fe3O4), was first identified as a mineral in Miocene lake sediments [Skinner et al., 1964]. Owing to a wealth of paleomagnetic and rock magnetic studies since the late 1980s, greigite is now well known as a widespread authigenic mineralization product in anoxic sediments in which bacterial sulfate reduction has occurred [Roberts et al., 2011]. Apart from its significance in sedimentary geochemistry and paleomagnetism, greigite is also an intracellular biomineralization product in sulfate-reducing magnetotactic bacteria, where it forms magnetosome chains, which impart a stable magnetic dipole moment to the cell body [Farina et al., 1990; Heywood et al., 1990, 1991; Mann et al., 1990; Bazylinski et al., 1993a, 1993b, 1995; Pósfai et al., 1998a, 1998b; Kasama et al., 2006; Lins et al., 2007; Lefevre et al., 2012; Wang et al., 2013]. Putative fossil remains of greigite magnetosomes in ∼5 Myr mudstones have also been suggested to carry a primary stable magnetic remanence [Vasiliev et al., 2008].

In the last few years, considerable effort has gone into the experimental determination of the magnetic structure and fundamental magnetic material parameters of greigite [Chang et al., 2008, 2009a]. Earlier attempts were thwarted by impure samples. Fe3S4 has long defied controlled inorganic synthesis [Erd and Evans, 1956] and it was not until 1960 that Fe3S4 crystals of space group Fd3m were finally produced, by hydrothermal precipitation from a solution of sodium sulfide and Mohr salt [Yamaguchi and Katsurai, 1960]. Nevertheless, greigite samples grown with that protocol and modifications thereof [e.g., Uda, 1965; Coey et al., 1970; Dekkers and Schoonen, 1996] contained unrecognized by-products, which in hindsight led to a significant underestimation of the saturation magnetization Ms. Chang et al. [2008], who grew pure microcrystalline samples of greigite according to the synthesis protocol of Tang et al. [2007], obtained Ms = 241 kA/m (241 G) as a new reference value at room temperature, which is significantly larger than earlier reported values of 100 kA/m [Uda, 1965; Dekkers and Schoonen, 1996] and 150 kA/m [Coey et al., 1970], respectively. The higher value is a result of the purity of the samples analyzed by Chang et al. [2008].

The two sets of magnetic material parameters related to anisotropy remain to be robustly determined experimentally: the magnetocrystalline anisotropy constants K1, K2, and the magnetoelastic parameters λ100, λ111. Both sets of parameters influence derived magnetic parameters such as coercivity, susceptibility, blocking volume, domain size, and domain-wall width. The single-most limiting obstacle for experimental determination of these parameters has been and continues to be the lack of suitable single crystals that can be oriented and measured along well-defined crystallographic directions. The largest greigite crystals obtained by Chang et al. [2008] had a grain size of <50 µm, which is too small for an accurate anisotropy to be determined from measurements of magnetization curves [e.g., McKeehan, 1937; Kakol and Honig, 1989], torque [e.g., Kouvel and Graham, 1957; Aubert, 1968; Pecora, 1989], or single-crystal ferromagnetic resonance (FMR) absorption [Bickford, 1950; Bonstrom et al., 1961].

Recently, an estimate of K1 was presented in Roberts et al. [2011] on the basis of single-domain hysteresis properties of (polycrystalline) authigenic greigite nodules that are dominated by cubic magnetocrystalline anisotropy as indicated by remanence ratios greater than 0.5, typically 0.5–0.6 [Roberts, 1995], but occasionally ∼0.75 [Sagnotti and Winkler, 1999]. In contrast, available samples of synthetic single-domain greigite contain a significant fraction of particles that are too small to be magnetically stable at room temperature as indicated by remanence ratios of 0.3 [Chang et al., 2009a]. Therefore, no reliable estimates of K1 for pure synthetic greigite samples exist so far.

Due to lack of availability of ideal single-crystal samples, we here resort to fitting of FMR spectra of synthetic greigite powders. This method has been successfully applied to determine K1 of polycrystalline Al-Ni-ferrites [Schlömann and Zeender, 1958], of iron dispersed in silicon [Griscom et al., 1979], and more recently to simultaneously determine shaped-induced uniaxial anisotropy and cubic K1 of whole-cell samples of magnetite-producing magnetotactic bacteria [Charilaou et al., 2011, 2012]. This approach works well on sufficiently diluted systems that show sharp features in the FMR derivative spectrum (see Figures 1c and 2c), but may yield less accurate results on concentrated magnetic particle systems, where magnetic (dipolar) interactions cause excessive intrinsic line broadening, which obscures the sharp features [Valstyn et al., 1962; Kopp et al., 2006]. This appears to be the case in published FMR powder spectra for pure greigite samples [Chang et al., 2012]. It is even more puzzling that FMR spectra for samples containing authigenic greigite nodules differ significantly from those for synthetic greigite in terms of skewness and effective g value [Chang et al., 2012]. FMR spectra for synthetic greigite samples [Chang et al., 2012] are asymmetric in the sense that the low-field branch of the absorption curve is steeper and narrower than the high-field branch (i.e., positive skewness), which suggests a negative value of K1 [Valstyn et al., 1962; Griscom, 1974] (cf. Figure 1d). In contrast, FMR spectra for the natural samples have negative skewness and therefore appear to be consistent with positive K1 [Chang et al., 2012] (cf. Figure 2b), provided the sample consists exclusively of greigite. However, in these natural samples, greigite is not the only magnetic mineral; aggregates of pyrite and hexagonal pyrrhotite surround the greigite particles in the studied iron sulfide nodules [van Dongen et al., 2007].

Figure 1.

Computed resonance fields and FMR powder spectra for cubic magnetocrystalline anisotropy for the case K1 < 0 with inline image, inline image, g = 2, and inline image (i.e., inline image), compared with results obtained from a first-order approximation. (a) Angular dependence of resonance field, inline image. The solid white contour connects all inline image pairs with resonance fields that are equal in strength to inline image. The dashed white contour connects all isotropic events defined by inline image = B0. (b) Probability density function of resonance fields, inline image, obtained by binning the values from a). The dashed line is an approximative pdf, inline image resulting from evaluating Schlömann's [1958] first-order expansion. (c) Simulated FMR absorption derivative spectra, computed by convolving inline image with a Gaussian, equation (11), for three values of intrinsic line width ΔB (red: 10 mT; green: 20 mT; blue: 30 mT). (d) Simulated FMR absorbtion spectra based on equation (10); same color code as in (c). (e) and (f), as in c) and d), but with inline image instead of inline image.

Figure 2.

Computed resonance fields and FMR powder spectra for cubic magnetocrystalline anisotropy for the case K2 > 0 with inline image, inline image, g = 2, and inline image (i.e., inline image). (a) Probability density function of resonance fields, inline image, and approximative inline image (dashed). (b) Simulated FMR absorption spectra from inline image based on equation (9), for three values of the intrinsic line width ΔB (red: 10 mT; green: 20 mT; blue: 30 mT). (c) Simulated FMR derivative spectra based on equation (10) (same color code as in Figure 2c).

2 Theory

Following the pioneering FMR work of Kittel [1947, 1948], the theory of FMR absorption for crystals with cubic anisotropy was developed in the 1950s [Van Vleck, 1950; Bagguley, 1955; Standley and Stevens, 1956; Artman, 1956, 1957; Schlömann, 1958]. The general FMR condition was independently derived by Suhl [1955] and Smit and Beljiers [1955] as:

display math(1a)

where h is the Planck constant inline image, v is the microwave frequency (in Hz), g is the spectroscopic splitting factor, μB is the Bohr magneton inline image, Ms is the saturation magnetization (in A/m), and inline image is the total magnetic Helmholtz free energy density (at constant volume) with respect to the azimuth ϕ and polar angle θ of the magnetization vector. The second derivatives in (1a) are to be taken at equilibrium:

display math(1b)

The total free energy density for a homogeneously magnetized isometric particle is the sum of external field energy and magnetocrystalline energy,

display math(2)

where

display math(2a)
display math(2b)

and inline image is the external direct current (DC) magnetic field H expressed as vacuum flux density, ϕB, θB are the azimuth and polar angle of H, and K1, K2, and K3 are the phenomenological first, second, and third-order cubic magnetocrystalline anisotropy constants. The αi are the Cartesian components of the unit magnetization vector given by (ϕ, θ), which like the ϕB, θB are defined with respect to the [001] axis of the crystal.

Before presenting details of the FMR spectra calculation for samples with randomly oriented crystallographic axes, it is most instructive to consider the special case when the external magnetic field is parallel to any of the major cubic symmetry axes. In this case, the equilibrium conditions in equation (1b) are fulfilled for inline image and exact analytical expressions for the resonance field can be obtained in closed form by solving equation (1a) at inline image, which yields

display math(3a)
display math(3b)
display math(3c)

where inline image is the “isotropic” resonance field and

display math(4)

The expressions given by Griscom et al. [1979] for inline image and inline image, respectively, differ from those given here ((3b) and (3c)) as far as the k3 terms are concerned because the expansion of the cubic anisotropy energy used by Griscom is different to the one we have in equation (2b) (see also Appendix, section A1a in supporting information). The resonance fields (equation (3)) are related to the anisotropy fields, which from equation (2) with inline image are obtained as

display math(5a)
display math(5b)
display math(5c)

and define the theoretical switching fields when the external field is parallel to a given crystallographic axis. Thus, inline image and inline image.

Except for some exotic combinations of anisotropy constants (see Appendix, section A1b and Figures A1a and A1b), inline image defines the maximum in the powder FMR absorption spectrum [Bagguley, 1955], while inline image and inline image (equations (3a) and (3c)) produce the first and last resonance events, separated by a distance inline image Provided that the g factor is known, the three unknowns K1, K2, K3 can in principle be determined from equations (3a), (3b), (3c) by identifying inline image in a measured FMR absorption spectrum for a chemically pure sample consisting of well-diluted (non-interacting) isometric SD particles with randomly oriented crystallographic axes. The SD condition is realized in multidomain (MD) particles once the applied field exceeds the saturating field Bs, defined by the onset of approach-to-saturation behavior. For B < Bs, additional resonances due to MD effects appear (low-field losses), but these are secondary in magnitude [cf. Artman, 1956]. More importantly, however, whether the resonance events at inline image can be identified in practice depends on the line width associated with the individual resonance events.

2.1 Intrinsic Line Width

In a single resonance event, that is, in a single ideal crystal that satisfies the resonance condition in equation (1) for a given orientation of the external magnetic field, the magnetization will precess about the DC field vector H. It will have precession frequency γB/2π equal to the driving frequency v of the microwave magnetic field hrf that is applied perpendicular to H, where γ is the gyromagnetic ratio of the electron, with inline image for g = 2 [Kittel, 1948]. The amplitude of the precessional motion is damped by relaxation processes, which lead to a finite line width of the resonance event. The lower limit of the intrinsic line width ΔBint is determined by the relaxation time τ associated with the interchange of energy between the absorbing mechanism and its surroundings, for example, by way of spin-wave excitation [Miles, 1954]. The lower limit of ΔBint is given by:

display math(6)

(for a derivation of equation (6) and for a definition of various τ values, see Lax and Button [1962, pp. 169 and 231]). Fe2+ in cubic ferrites, no matter whether it is octahedrally or tetrahedrally coordinated, has a short τ and hence causes a broad line width [White, 1959]. Fe2+ is the source of anisotropy. That is, it couples to the crystal lattice, and thereby can transfer energy from a uniform precession mode to the lattice, either directly or by spin waves [White, 1959]. In general, the intrinsic line width in ferrites increases not only with microwave input power [Bloembergen and Wang, 1954], but also with frequency [Tannenwald, 1955; Mastrogiacomo et al., 2010], which is why X-band measurements have higher spectral resolution than K or Q band measurements, provided the sample can be saturated in X-band fields. Depending on the type of ferrite, the intrinsic line width may increase with increasing temperature (as for ordered magnetite [Bickford, 1950]) or it may decrease (as for Mn ferrite) [Tannenwald, 1955].

The intrinsic line shape depends on the details of the relaxation process and is usually a Lorentzian. In contrast, asymmetric Dysonian lineshapes are expected for electrically conducting samples that are thicker than the skin depth [cf. Poole 1983, chapter 12].

In a cubic crystal the intrinsic line width is anisotropic and is given by:

display math(7)

[Suhl, 1955], where α is the Landau-Lifshitz loss parameter. Angular variations of the line width can be pronounced in thin film samples [Goryonov et al., 1995], but otherwise they are usually small, i.e., 10–20% variation relative to the “isotropic” line width [Tannenwald, 1955; Callen and Pittelli, 1960; Vittoria et al., 1967]. The total line width of a single resonance event may have a number of external contributions arising from, for example, impurities, crystal defects, uncompensated surface spins, and dipolar interactions, all of which tend to broaden the absorption line [Vittoria et al., 1967; Goryonov et al., 1995; Schmool et al., 2007]. In metals, exchange-conductivity coupling due to the skin-effect limitation of microwave penetration produces additional broadening [Vittoria et al., 1967].

2.2 Simulation of FMR Spectra

FMR spectra for polycrystalline or powdered samples with randomly oriented crystallographic axes can be forward modeled by determining inline image for all possible crystal orientations relative to a fixed field direction [Standley and Stevens, 1956; Tsay et al., 1971; Griscom, 1980], or, equivalently, for all possible field orientations relative to a fixed crystallographic direction [e.g., Charilaou et al., 2011]. By selecting the crystallographic directions of highest symmetry, i.e., one of the <100> directions, say [001], the cubic symmetry can be taken advantage of to restrict the numerical procedure to a 1/16th of the full solid angle by selecting the upper (or lower) hemisphere of an eighth wedge given by the inline image domain S16 = [0, 45°]×[0, 1]. The spectrum is the superposition of all individual resonance events inline image in S16:

display math(8)

where inline image is the imaginary or absorptive part of the complex susceptibility (if multiplied with the saturation magnetic moment per sample volume), L is the line shape function and inline image is the total line width for resonance event inline image, including intrinsic and external contributions. If the anisotropy of the line width is negligible, then it is more convenient to employ the histogramming approach [Taylor and Bray, 1970; Griscom, 1974] where the inline image determined for a large number of orientations over the sphere are binned on a magnetic field scale to obtain a one-dimensional probability density function (pdf) of resonance fields,

display math(9)

which can then be directly convolved with the line shape function L to obtain the FMR absorption spectrum as:

display math(10)

Likewise, the derivative absorption spectrum is obtained by:

display math(11)

By using dL/dB and not dpdf/dB in equation (11), the derivative spectrum remains finite at the three singular points inline image, where the field derivative of inline image diverges.

Schlömann [1958] derived analytically the behavior of inline image in the vicinity of the three singular cases inline image and constructed the full inline image semiempirically by interpolation. Here we simulate FMR spectra with a histogramming technique. For each prescribed pair of (K1, K2), inline image is computed for about 2000 different pairs of inline image according to equations (1) and (2) with high numerical precision and without making simplifying approximations. inline image obtained this way varies so smoothly (Figure 1a) that it can be interpolated on a finer grid to obtain a sufficiently dense set of data points that upon binning produces a smooth, yet accurate, histogram function (Figure 1b). Numerical evaluation of the convolution integrals (equations (10) and (11)) has been performed by means of digital Fourier transformation, e.g.,

display math(12)

where inline image and inline image represent the forward and inverse Fourier transformations, respectively. For L, we select a Gaussian line shape, centered on the resonance field:

display math(13)

where 2ΔB is the full width at half-maximum of inline image, or equivalently, the peak-to-trough distance of inline image From our modeling, we found a Gaussian be in better agreement with the experimental spectra than a Lorenztian, which would give too broad tails.

The forward-modeling results shown in Figures 1a–1d are for inline image and inline image, which yields inline image, and inline image. The three singular points, inline image, occur at inline image, which agree within 0.1 mT with the analytical predictions from equations (3a), (3b), (3c). In contrast, the first-order approximation given by Schlömann [1958] and used by Kopp et al. [2006] for modeling FMR spectra of magnetosome chains yields {241.6, 289.9 and 499.4} mT More severely, the shape of the approximative histogram is distorted (dashed line in Figure 1b) compared to the accurate one (solid line in Figure 1b), which then will also yield qualitatively different FMR spectra (compare Figure 1e with Figure 1c and Figure 1f with Figure 1d).

Next, we explore the effects of the intrinsic linewidth ΔB on the FMR spectra. With increasing ΔB, the absorption maximum is shifted toward the low-field end of the spectrum and increases the apparent or effective g value geff, defined by inline image, where Bmax is the position of the maximum in the FMR absorption spectrum. The simulated FMR absorption derivative spectrum (Figure 1c) does not isolate the resonance events at inline image and inline image once inline image (blue curve in Figure 1c, inline image). Nevertheless, for K1 < 0, all absorption spectra have positive skew, with a tail on the high-field side, whereas for K1 > 0, the skew is negative, with a tail on the low-field side (see Figure 2, inline image, i.e., k1 = 0.3).

3 Material and Experimental Methods

We used greigite powder SYN702 (mean grain size of about 8 µm, which is comparable to SYN706 of Chang et al. [2007, 2009a]). The greigite powder had been synthesized earlier [Chang et al., 2008] and kept in a sealed glass tube to prevent oxidation. FMR samples were obtained by dispersing the greigite powder in a nonmagnetic, electrically insulating embedding medium (CC HIGH TEMPerature cement kit, Omega Engineering Inc., Stamford, CT, USA). First, a mortar was prepared by adding the liquid binder (an aqueous sodium silicate solution) to the cement powder (a mixture of zirconium silicate, sodium silico fluoride, and silica). The greigite powder was then mixed with the mortar by stirring. Samples were then filled into SiO2 EPR tubes (707-SQ-250M, Wilmad Lab Glass) to set for about 24 hours before measurement. No sulfide smell was released during the procedure, which indicates that the greigite did not decompose. For background control measurements, some EPR tubes were not filled with greigite, but otherwise prepared in the same way as the greigite samples.

FMR spectra were measured on a JEOL X-band (∼3.3 cm) EPR spectrometer (JES-FA 200) with a cylindrical cavity resonating in TE011 mode at v = 9.07 GHz. The spectrometer was calibrated with free radical standards. Microwave input power was limited to 1 mW to avoid additional line broadening and nonlinear effects. To amplify weak absorption signals, the DC magnetic field is modulated with a low amplitude, low frequency alternating current (AC) magnetic field (0.4 mT, 100 kHz). Phase sensitive measurement of the AC amplification with a lock-in amplifier yielded the derivative of the absorption spectrum with respect to the applied static field. The DC field was swept from 0 to 800 mT at a rate of about 1 T/min and the spectra were sampled with 65,536 equidistant data points.

4 Results

4.1 Experimental FMR Spectra

A typical experimental X-band FMR derivative absorption spectrum for greigite embedded in ceramic cement (generic sample name SYN702cc) and recorded at room temperature is shown in Figure 3a. FMR parameters from the absorption spectrum are inline image compared to inline image, inline image (X-band) or inline image (Q-band) for synthetic greigite samples dispersed in eicosane [Chang et al., 2012]. As with the published spectra for synthetic greigite [Chang et al., 2012], the ones recorded here have positive skewness and appear to reflect excessive linewidth broadening that obscures diagnostic cubic FMR features as expected from Figures 1c and 2c (blue lines). If we assign the peak of the spectrum as low field peak BLF (here 207.5 mT), which due to line width broadening may be shifted by some δB > 0 from its “intrinsic position” (see Table S1 for δB simulations), our first rough estimates for the anisotropy field are inline image for inline image or inline image for inline image For <111> easy axes, these anisotropy fields translate into inline image (g = 2) and inline image (g = 2.1). For <100> easy axes, we have inline image (g = 2) or inline image (g = 2.1). More precise values were obtained from spectral fitting (below).

4.2 Results From Spectral Fitting

Technical details of the fitting procedure are described in Appendix 2. In short, an anisotropy model (with parameters K1, K2) was fitted to a recorded spectrum by minimizing the squared difference between recorded spectrum and forward computed spectrum on the basis of equations (1), (2), (9), and (11). The model parameters K1, K2 and linewidth parameter ΔB were adjusted iteratively according to a downhill-simplex algorithm. Since the g factor for greigite is not known, we performed the procedure for a range of prescribed g factors ranging from 2.0 to 2.15 in steps of 0.01 (Tables A1 and A2). In the target region, we refined the solution by allowing for a finite K3. The low field part (B < 60 mT) of the spectrum was not taken into consideration because of inhomogeneous magnetization behavior in the MD particles to which the saturation-based FMR theory does not apply. In earlier hysteresis measurements, the onset of approach to saturation (reversible magnetization rotations) was about 150 mT, where the difference between the upper and lower hysteresis branch was as low as 0.5% [Chang et al., 2008].

Possible anisotropy models, inline image obtained by spectral fitting of the X-band spectra of the greigite crystals embedded in ceramic-cement are summarized in Table 1 (for more diagnostics, see Table A2), along with anisotropy fields (equation (5)) and energy barriers ΔFSD (per unit volume for an SD particle), which represent the difference in anisotropy energy density (at zero external field) between easy and intermediate axis, where Fan for the major axes is given by:

display math(14)
Table 1. Overview of Anisotropy Models inline image Consistent With the Experimental X-Band Spectrum (Figure 3a) of the Ceramic-Cement Embedded Particles From Greigite Powder SYN702, Recorded at 9.07 GHza
ModelK1 (kJ/m3)K2/K1K3/K1gΔB (mT)sBan (mT)ΔFSD (kJ/m3)
  1. a

    Prescribed values are in italics, g is the g-factor, ΔB is the Gaussian linewidth (equation (13)) and s is a misfit parameter, which was minimized in the fitting procedure. Ban is the resulting anisotropy field for each model (equations (5a) and (5c)) and ΔFSD is the free energy difference between intermediate and easy axis (equation (14)).

<111>a−15.7+0.3102.08162.40.20895.81.49
<111>b−15.3+0.3002.09063.40.19493.21.45
<111>c−13.9+0.33+0.372.165.90.17285.21.20
<100>a+10.1+4.2302.0865.70.19284.12.53
<100>b+11.0+4.17+0.102.07965.00.19184.12.60
<1r0>a−18.7−6.78−3.232.0866.70.18960.30.90
<1r0>b−53.6−3.95−1.361.967.30.33465.91.52
Figure 3.

(a) Typical experimental X-band FMR derivative spectrum of synthetic greigite powder SYN702 embedded in ceramic cement (black solid curve). Dotted: Simulated spectra for the two models from Table 1 (orange: <111>c, blue <100>b) and residuals. (b) Residuals between data and fitted curves for models <111>c (orange), <100>b (blue), <1r0>a (green), and <1r0>b (magenta). (c) inline image each model from Table 1. The inline image for <111>b and <100>a are very similar to that for <111>a and <100>b, respectively.

An interesting outcome is that the two best-fitting anisotropy models for <100> easy axis and <111> easy axis orientation yield anisotropy fields of similar magnitude (85 mT). By comparison, for magnetite at room temperature, inline image (with inline image and inline image) [Bickford, 1950]. Significantly, any <100> easy axes model requires a K2/K1 ratio of about 4 to explain the skewness of the experimental spectra. We also tried to impose an <110> easy axis model on the data by setting the constraints inline image and K1 < 0, but found no numerically stable solution, indicated by the fact that the minimization procedure was persistently pushing K2 toward inline image, obviously trying to leave the unfavorable region in model space to converge on a solution with significantly smaller K2 when K1 < 0. Inclusion of a finite third-order anisotropy constant in the fitting procedure does not greatly improve the fits obtained for the <111> or <100> models without K3 (see Table 1). Importantly, the best fitting g factor for all these <111> and <100> models, with or without K3, is found to lie in the range 2.08 … 2.10. From independent measurements, we obtain inline image, where μtot is the total magnetic moment per formula unit (3.12 μB from Ms) [Chang et al., 2008] and inline image [Chang et al., 2009] is the “spin-only” magnetic moment obtained from neutron diffraction.

The last row in Table 1 shows the importance of the g factor in obtaining anisotropy models from FMR data. Here we prescribed an unrealistically low value of 1.9 for the g factor and obtained a model with large negative K1, which nevertheless fits the experimental spectrum (see red points in Figure 3a). Without a relatively large contribution from K3, no fits were found with such low g values. The best fit solution for g=1.9 represents an anisotropy model with easy axes along nonmajor axes (here <1r0>, with r between 0 and 1, see also Appendix section A1b and Figure A1b). A <1r0> solution matching the data was also found for g=2.08. The <1r0> model solutions are characterized by lower anisotropy fields compared to the <111> or <100> easy axis solutions. Anyway, we consider all these <1r0> solutions as contrived, resulting from forcing a model with large contribution from K3 on the data. Nonmajor easy axes orientations are a common feature in the cubic rare-earth Lave compounds (CeFe2, LuFe2, HoCo2) [Atzmony and Dariel, 1974, 1976], where K3 is of the same order as K1, but we know of no cubic rare-earth free transition metal compound where K3 is important.

4.3 Comparison With FMR Data for Other Greigite Powders

All the anisotropy models listed in Table 1 do fit the data very well, as seen in the residuals (Figure 3b), which demonstrate the equivalence of the <100> and <111> models under the forgiving linewdith broadening. For these, even the underlying resonance field distributions (Figure 3c) are roughly similar. On the basis of the misfit parameter s, some models appear to be better matches than others, which is deceptive however, because such numerical diagnostics are related to precision, not trueness. It is clear that more independent data are needed to obtain realistic error bars in terms of uncertainties. Therefore, we also performed fits of X and Q-band spectra on different greigite powder samples published earlier [Chang et al., 2012]. For one of the samples (SYN519), both X and Q-band spectra were available, which gave us the opportunity to predict the Q-band spectrum for a best-fit X-band spectrum and vice versa. These efforts are documented in Table A3 and Figure A2. We obtained only a few model solutions that explained both X and Q-band spectra (<111>dS519 and in <111>eS519 in Figure A2). However, the observation that these solutions emerged with a g factor of 1.9 to us suggests that the parameters values do not reflect intrinsic properties, but rather apparent ones due to us forcing an ideal model on the data, which may reflect nonideal sample behavior (interaction field bias, shape effects) or unrecognized instrumental artifacts such as drift or nonlinear effects. The different embedding media could play a role, too. The mechanically strong ceramic cement used for SYN702 is likely to promote constant volume (strain free) conditions so that the anisotropy parameters are related to the Helmholtz free energy (equation (2)). In contrast, the eicosane used for dispersing SYN519 is elastically softer than greigite, thereby allowing the greigite crystals to deform by magnetostriction (stress free condition, i.e., Gibbs free energy). According to Ye et al. [1994], FMR is considered to “feel” the strain-free anisotropy because the time scale of the resonance events is too short for the lattice to deform to its equilibrium value. Thus, for the stress-free crystals in eicosane, the resonance condition (equation (1a)) would reflect strain-free conditions, while equation (1b) now applies to stress-free conditions, with the result that all resonance events would be shifted relative to the strain-free equilibrium conditions, except those due to <100,110,111>, for which inline image. For all other events, the angle of the equilibrium magnetization relative to the field changes when going from strain-free to stress-free conditions. However, these differences are relevant only if greigite has large magnetostriction constants.

Given these uncertainties, it appears reasonable to use the approximative method outlined in section 'Experimental FMR Spectra' to determine the upper bound for K1 > 0 and lower bound for K1 < 0, respectively, for more conservative anisotropy models where the low-field FMR peak is related to either a <100> or <111> easy axis. For BLF, we used the low-field peak values from the derivative spectra, referred to as Bp1 in Chang et al. [2012, Table 1]. As shown in Table 2 here, the boundary values for K1 thus derived for the Q-band spectra of Chang et al. [2012] are, by and large, in good agreement with those for the X-band spectrum of SYN702, unless a value of the g factor closer to 2.0 is assumed, in which case the Q-band derived bounds soar in magnitude. In other words, the uncertainty in the intrinsic g factor translates into a larger uncertainty in estimates for K1 from Q-band spectra than from X-band spectra. For, the resonance fields are nearly 4 times higher in Q-band compared to X-band FMR, so that the difference in the isotropic resonance field for g = 2.04 and g = 2.10 is 9 mT in X-band (9.07 GHz), but 34 mT in Q-band (34.0 GHz), which explains the large uncertainty for the estimated anisotropy fields from Q-band. We therefore conclude that our X-band estimates are more precise than those that could be obtained from Q-band.

Table 2. Estimated Anisotropy Parameters for Various Greigite Samplesa
Samplev [Ghz]g inline image [mT] inline image [kJ/m3] inline image [kJ/m3]
  1. a

    The symbols with a tilde denote upper limits for Ban and K1 > 0 and lower limits for K1 + K2/3 < 0 estimated from Equations (3a) and (3c) using BLFB0(g) – Ban with g=2.1 for synthetic greigite samples and two natural samples (“Taiwan,” “Italy”) containing greigite as dominating magnetic mineral. X-band data (v ∼9 GHz) for SYN519 and all Q-band (v ∼34 GHz) FMR data are from Chang et al. [2012]. The estimated limit values exceed the true values in magnitude because of line width broadening and have an uncertainty range due to the unknown intrinsic g-factor of greigite, which most likely is 2.09 (see also Table 1). The respective values for g=2.04 are in parentheses. The values obtained from model fits for sample SYN702 are given in the second row for comparison (in bold).

SYN7029.072.10 (2.04)103 (113)+12.5 (13.6)−18.7 (−20.3)
model fits2.09 ± 0.0190 ± 6+10.5 ± 1−15 ± 1
SYN70634.02.10 (2.04)109 (143)+13.1 (+17.2)−19.7 (−25.8)
SYN5199.402.10 (2.04)178 (187)+21.4 (+22.5)−32.1 (−33.8)
34.02.10 (2.04)134 (168)+16.2 (+20.3)−24.2 (−30.4)
SYN50434.02.10 (2.04)140 (174)+16.8 (+20.9)−25.3 (−31.4)
“Taiwan”34.02.10 (2.04)120 (154)+14.4 (+18.5)−21.6 (−28.0)
“Italy”34.02.10 (2.04)124 (158)+14.9 (+19.1)−22.5 (−28.6)
Mean (Q-band data)34.02.10 (2.04)125 ± 12+15.1 ± 1.5−22.7 ± 2.2
  (159 ± 12)(+19.2 ± 1.5)(−28.8 ± 2.2)

4.4 Intrinsic Magnetic Hardness

The reduced anisotropy constant Q, defined by inline image, with inline image, where Kd is the stray-field energy constant, is a measure of the intrinsic magnetic hardness of a given material [Hubert and Schäfer, 1998, p. 397] and is inversely proportional to the intrinsic magnetic susceptibility of single domains (cf. section 4.8.2). For inline image, we have Q = 0.33; for a inline image, using inline image with inline image, we obtain inline image. For comparison, inline image for magnetite at room temperature with inline image and inline image [Bickford, 1950]. The significantly higher magnetic hardness of greigite compared to magnetite implies that magnetocrystalline anisotropy has a much greater influence on the magnetic properties of greigite particles compared to shape anisotropy, as will be shown below.

4.5 Shape Versus Magnetocrystalline Anisotropy

The self energy density for a particle with uniaxial shape anisotropy Ku is given by

display math(15)

[e.g., Tonge and Wohlfarth, 1958] where the h, k, and l are the (normalized, i.e., h2 + k2 +l2 = 1) Miller indices of the direction of the elongation and the αi represent the direction of the magnetization, where both are defined with respect to the cubic <100> system, say the [001] direction (see also equation (2)b)). For an effective demagnetization factor inline image where Nlong and Nshort are the demagnetization factors along the long and short axis, respectively, the uniaxial anisotropy constant is given by

display math(16)

A particle with elongation along [100] will have uniaxial characteristics (i.e., two stable minima, which implies a remanence ratio of 0.5 for the randomly oriented assemblage) once Ku exceeds inline image [Tonge and Wohlfarth, 1958], i.e., once inline image, which for Q = 0.33 translates into an ellipsoid with aspect ratio greater than 3. However, for the case with elongation along [111], uniaxial behavior sets in already at inline image (for K1 > 0) and inline image (for K1 < 0) [Geshev et al., 1998; Newell, 2006b], which corresponds to an axial ratio of 2.1 (for K1 > 0) and 1.33 (K1 < 0), respectively, or for bars with square cross section to an axial ratio of 2.5 (for K1 > 0) and 1.4 (K1 < 0), respectively [see Sato and Ishii, 1989] for expressions of ΔN in this case). By comparison, for magnetite at room temperature, the critical aspect ratio leading to uniaxiality in [100] or [111] elongated particles is 1.22 or 1.04, respectively.

4.6 Superparamagnetic Limit

For pure cubic magnetocrystalline anisotropy, the free energy barrier (per unit volume) separating easy <111> directions is obtained from equation (14) as

display math(17a)

The edge length of a cubic particle with volume V = a3 for which the energy barrier is equal to inline image, where kB is the Boltzmann constant, is obtained as inline image for inline image. The inline image values in Table 1 ranging from 1.2 to 1.5 kJ/m3, correspond to inline image values of 44…41 nm. The energy barrier separating easy <100> directions is given by

display math(17b)

which yields inline image for inline image. From inline image (Table 1), we obtain inline image. Using the relaxation rate equations derived by Newell [2006a, (equations (24) and (25) therein] and neglecting the role of K2 on the shape of the energy surface about the saddle point, we calculated the edge length for which the relaxation rate amounts to 1 Hz as inline image and inline image, respectively, in good agreement with the critical edge lengths obtained from the simple inline image criterion. An increase of these edge lengths by 10% leads to relaxation times larger than 1000 s. As discussed by Diaz Ricci and Kirschvink [1992], Uda [1965] failed to impart a stable remanence on his synthetic greigite samples with particle sizes ranging from 30 to 50 nm. This observation tallies with the SD/SP limits obtained here from our experimental K1 values.

4.7 Partial Superparamagnetism Due to Competing Anisotropies

Combining the results from section 'Shape Versus Magnetocrystalline Anisotropy' and 'Superparamagnetic Limit', we are now addressing the problem of magnetic stability against thermal fluctuations in single domain particles where the uniaxial anisotropy is oriented along a hard magnetocrystalline axis, which implies competing anisotropies. We first consider the case where the hard <100> axis (negative K1) is the axis of positive uniaxial anisotropy, a situation that may be relevant to greigite magnetosomes that are elongated along a <100> direction [Heywood et al., 1990, 1991, cf. Discussion 5.2.1]. The effect of shape anisotropy here is to push the orientations for minimum energy closer to the axis of elongation (compare Figure 4a and Figure 4b), thereby breaking up the eight minimum directions into two groups of four. Within a group, the minima are closer together and separated by lower saddle points compared to the purely cubic case. The energy barrier for transitions from one group to the other (i.e., from one hemisphere to the other) increases simultaneously. For the example depicted in Figure 4b, where inline image, which with inline image translates into an axial ratio L/w of 1.53 (prolate ellipsoid of revolution) or 1.68 (bar with quadratic cross section), the energy barrier between adjacent energy minima within a group is reduced from inline image (isometric case and K2 = 0) to 0.037 inline image, while the energy barrier for switching between the groups rises to inline image. In other words, elongation along a hard <001> axis has a stabilizing effect on the polarity of the remanent magnetization in the crystal and a slightly destabilizing effect on its colatitude (z component), but promotes randomization of the azimuthal orientation. Such a situation leads to more than one SP critical size, as emphasized by Newell [2006a], who developed the concept of partial superparamagnetism. In our example, the SP size related to azimuthal randomization would increase from ca. 41 nm (section 'Shape Versus Magnetocrystalline Anisotropy') to nearly 50 nm, while the SP size related to polarity flipping would decrease to 26 nm (short edge length). Thus, in the partial SP range between 26 nm and 50 nm, the azimuth would be random, but the average of the z component would be about cos(42°)=0.75, where 42° is the angle between the minimum orientations and the [001] direction.

Figure 4.

Energy surfaces for competing cubic magnetocrystalline inline image and uniaxial anisotropy due to elongation along a magnetic hard [001] axis. (a) Pure cubic magnetocrystalline anisotropy inline image and (b) mixed anisotropy inline image, where the axis of uniaxial anisotropy is oriented along [001]. Because of mirror symmetry about the (001) equatorial plane, only the Northern hemisphere is shown, in stereographic projection, centered on [001]. Highlighted contours: energy barrier for switching in case (a) (solid white line in Figures 4a and 4b); energy barrier for partial superparamagnetism in case (b) (dashed white line in Figure 4b); energy barrier for polarity switching in case (b)) (black solid line in Figure 4b). The cubic <111> directions, here [111], [−111], [1–11], and [−1−11], are indicated by white dots. Note that compared to Figure 4a, the minima in Figure 4b are shifted from the former <111> positions (oriented at an angle of 54.7° relative to [001]) closer to the [001] direction (now making an angle of 42° relative to [001]), with adjacent minima now separated from each other by a smaller energy barrier inline image compared to a) inline image.

The situation for the opposite case, where the hard <111> axis (positive K1) is the axis of positive uniaxial anisotropy, is shown in Figures 5a and 5b, where again inline image, which with Q = 0.33 translates into an axial ratio of 1.335 (ellipsoid) or 1.423. It can be seen that the uniaxial anisotropy breaks up the six easy directions into two clusters of three. Compared to the purely cubic case, the energy barriers between the three minima within a group decrease (from 1/4 K1 to about 1/7 K1 in this example), while those between the two groups increase (from 1/4 K1 to about 3/8 K1), which leads to partial superparamagnetism in the grain size range from 26 to 37 nm (short edge length), within which azimuthal randomization occurs. Increasing the axial ratio from 4/3 to 3/2 inline image leads to partial superparamagnetism in the range 23 nm to 42 nm. Figure 6 shows that for competing anisotropy, the lower boundary of partial superparamagnetism is close to the SP boundary for the ideal situation where one of the magnetocrystalline easy axes is coaxial with the axis of elongation (Figure 6a, for K1 < 0, and Figure 6b for K1 < 0).

Figure 5.

Energy surfaces for competing cubic magnetocrystalline inline image and uniaxial anisotropy due to elongation along a magnetic hard [111] axis. (a) Pure cubic magnetocrystalline anisotropy inline image and (b) mixed anisotropy inline image, where the axis of uniaxial anisotropy is oriented along [111], which here defines the z axis inline image. The solid white contour in Figures 5a and 5b is the energy barrier K1/4 for case (a). In Figure 5b, the dashed white contour is the saddle point energy 0.15 K1 for transitions between adjacent easy directions, the black dashed contour is the saddle point energy 0.39 K1 for polarity switching between the hemispheres.

Figure 6.

Partial SP grain size range (shaded area) for greigite particles with competing magnetocrystalline and uniaxial shape anisotropy compared to the ideal SP limit (thick solid line), both defined in terms of energy barriers amounting to 25 kT. (a) inline image, partial SP range defined by elongation along hard [100], ideal SP defined by elongation along easy [111]; (b) inline image, partial SP range defined by elongation along hard [111], ideal SP defined by elongation along easy [001].

4.8 Derived Magnetic Properties for SD Greigite

In the following calculations, we use our FMR-derived anisotropy values (Table 1) for sample SYN702 to compute magnetic properties for SD greigite. The effective magnetocrystalline anisotropy constants, however, may differ from these values if hysteresis measurements are carried out under constant stress so that the crystal lattice is allowed to deform under the action of the magnetostriction forces [Mason, 1954]. If magnetostriction constants of greigite are large, then the FMR-derived anisotropy values will underestimate the effective ones.

4.8.1 Magnetic Hysteresis Parameters

For an assemblage of noninteracting, randomly distributed SD particles with cubic anisotropy in the absence of thermal fluctuations, Joffe and Heuberger [1974] gave the following expressions for the ensemble averages of coercive force and remanence ratio:

display math(18a)
display math(18b)

Hysteresis loops (Figure 7a) simulated for two {K1, K2} models with inline image (a) and inline image (b), have remanence ratios in agreement with equations (18a) and (18b), but slightly larger values for Bc due to the additional K2 term,

display math(19a)
display math(19b)
Figure 7.

(a) Hysteresis loops and (b) switching field distributions (SFDs) simulated for randomly oriented cubic particles for the two (K1, K2) solutions: (i) inline image (orange), and (ii) inline image (blue dashed), each with inline image. Thermal fluctuations were not included, but would reduce both the coercive force and saturation remanence in Figure 7a and would shift the SFD (Figure 7b) toward lower values of Bc, where the low Bc part of the SFD would be affected more than the high Bc tail. Solid black line in Figure 7b represents the SFD obtained from FORC data measured at room temperature on sample SYN06 [Chang et al., 2009b].

The numerical values for the prefactors (0.23 and 0.34, respectively) agree with those of Geshev et al. [1998, 2001, Figure 6 therein]. The two models (a and b) yield coercive forces with similar values (cf. equation (19)). The switching field distribution (Figure 7b) for the case K1 > 0 is more concentrated than the one for K1 < 0. Depending on the activation volume, thermal fluctuations may reduce the coercivity values considerably.

4.8.2 Magnetic Susceptibility κ and SIRM/κ

Generalizing the calculation of Stacey and Banerjee [1974, p. 72], we obtain the initial magnetic susceptibility for randomly oriented SD particles dominated by cubic anisotropy as

display math(20)

The apparent magnetic susceptibility for a sample consisting of such SD particles depends on the demagnetization factor N of the sample and the volume concentration f of particles,

display math(21)

Thus, the SIRM/κ ratio (SIRM=saturation isothermal remanent magnetization) can be written as

display math(22)

For N = 1/3, we obtain inline image inline image or inline image inline image. Given that these Q values here are from the lower end of the uncertainty range for our FMR estimates (see Table 2), the SIRM/κ ratios above may be minimum values for a given f. For 30% larger Q values, the SIRM/κ ratio increases by about 33 kA/m. SIRM/κ ratios reported for sediments dominated by greigite often exceed 50 kA/m [e.g., Snowball and Thompson, 1990; Sagnotti and Winkler, 1999], with maximum values up to 95–105 kA/m [Florindo and Sagnotti, 1995; Snowball, 1997], which are in good agreement with our SIRM/κ estimates, given that natural samples may well have paramagnetic contributions, which reduce the SIRM/κ ratio. For pure synthetic greigite powders, Roberts et al. [2011] measured inline image, which for these pseudosingle-domain samples ( inline image, inline image) yields SIRM/κ = 34 kA/m, significantly lower than for SD greigite. Importantly, isometric magnetite SD particles inline image would have inline image Therefore, high SIRM/κ ratios primarily reflect SD particles with high saturation magnetization and predominant magnetocrystalline anisotropy.

5 Discussion

Measured FMR spectra for greigite powder samples can be fitted with two different anisotropy models (Table 1), which yield similar anisotropy fields. One is a more conventional model with inline image, which applies to most types of cubic ferrites, such as magnetite above the isotropic point (135 K) [Bickford, 1950; Kakol and Honig, 1989], although there are some important exceptions with inline image, e.g., Fe2.4Ti0.6O4 [Sahu and Moskowitz, 1995] or CoFe2O4 [Krupicka and Novak, 1982, Table 37 therein). The other model is more exotic with inline image, which would be trivially the case near an isotropic point (if K2 remains finite when K1 is changing sign), but otherwise is rarely to be seen. For the sulphospinel MnCr2S4 (normal spinel structure), Tsurkan et al. [2002] obtained K2 > K1 > 0 between 40 K and 7 K, but the origin of this anomalous anisotropy has not been fully elucidated yet. van Stapele et al. [1971] determined inline image and K2K1 at 4 K for daubréelite (cubic FeCr2S4), another ferrimagnetic sulphospinel, but observed Moessbauer spectra (quadrupole splitting due to uniaxial electric field gradient, which is along the [100] direction) which differ so essentially from the ones recorded for greigite [cf. Chang et al., 2008] that we cannot consider FeCr2S4 (normal spinel, i.e. Fe2+ on tetrahedral sites) an analogue system for greigite (inverse spinel, Fe3+ on tetrahedral sites). Without additional information from independent observations, the sign of K1 for greigite at room temperature remains ambiguous. It may be possible to constrain the sign of K1 from powder samples with a multiband FMR spectrometer over a range of temperatures. Magnetite, for which K2 is small compared to K1, closely follows the Akulov tenth-power law in the temperature range from room temperature to 500°C, i.e., inline image [Newell et al., 1990]. Greigite appears to have no low-temperature transition [Moskowitz et al., 1993; Roberts, 1995; Chang et al., 2008, 2009a, 2009b; Lyubutin et al., 2013] and provided that the general Akulov-Zener power law (see Appendix section A1c) holds for greigite, the conventional anisotropy model inline image implies an increase in the magnitude of K1 with decreasing temperature, while the exotic model inline image leads to a strong increase in K2, but decrease in K1.

From unblocking curves, Vasilenko et al. [2010] estimated K1 as 31 … 198 kJ/m3 at T ∼ 60 K for greigite nanoparticles with Ms = 14 Am2/kg (compared to 59 Am2/kg reported by Chang et al. [2008]). However, surface anisotropy due to “frozen” surface spins may significantly exceed bulk anisotropy in nanoparticles so that the actual intrinsic bulk anisotropy may easily be overestimated by one or two orders of magnitude [Martinez et al., 1998; Komorida et al., 2009]. Therefore, with the surface contribution to the total anisotropy not being corrected for, the K1 (60 K) estimates of Vasilenko et al. [2010] are not helpful for constraining the sign of the FMR-based anisotropy model for greigite.

5.1 Independent Constraints on the Crystallographic Easy Axis Orientation

Yamaguchi and Wada [1970] inferred easy <100> directions from electron diffraction powder patterns of colloidal greigite nanocrystals, which were presumably aligned with their easy axis along an applied magnetic field of 2 mT (but surprisingly not so in higher fields, which would produce higher torques and therefore more effective alignment). However elegant the method may otherwise be, the authors presented no compelling data necessary to support their claims on the texture and failed to furnish proof of principle in the form of calibration measurements for control materials with known easy axis orientation. The basic idea is that a preferred crystallographic orientation (here a fiber texture along the magnetic field axis) should produce a characteristic azimuthal intensity variation along each diffraction ring. For a <100> texture, the azimuthal variations for the rings due to the strongly diffracting planes (311), (400), and (440) are different from those for a <100> texture, as seen in field aligned magnetosome chains) [Körnig et al., 2014]. However, the problem with the diffractograms shown in Yamaguchi and Wada [1970] is their incompleteness, which precludes a statistically robust texture reconstruction. Our analysis of the diffractograms (see supporting information, Figure SI), suggests a different interpretation than the one put forward by Yamaguchi and Wada [1970]. In conclusion, the purported <100> easy axis orientation is not supported by the data.

A physically compelling argument for easy <100> axes would be the observation of 90° domain walls in a {100} plane. Reconstructed magnetization directions in Hoffmann [1992, Figures 9a and 9b] indicate mostly 180° orientations, which do not favor either sign of K1. Again, lack of a large single crystal that can be cut in a certain plane hampers direct domain observations.

5.1.1 Argument for <100> Easy Axes From <100> Elongated Magnetosomes

Another argument that has been put forward to support a <100> easy axis orientation for greigite [Roberts, 1995; Moskowitz, 1995] comes from transmission electron microscopic (TEM) studies on magnetotactic bacteria (MTB), reporting greigite magnetosomes with particle elongation along a <100> axis [Heywood et al., 1990; 1991; Pósfai et al., 1998b], which in a few cases appeared to be oriented with their long axis along the chain axis [Bazylinski et al., 1995]. The <100> based argument is one from analogy, because some magnetite-producing MTB grow crystals with elongation along an easy <111> axis, which also defines the chain axis [Matsuda et al., 1983; Mann et al., 1984; Buseck et al., 2001; Lins et al., 2005; Abracado et al., 2010]. For <111> elongated magnetite crystals, an aspect ratio of 1.04 is sufficient to reduce the number of easy axes from four to one (section 'Shape Versus Magnetocrystalline Anisotropy'), with the remaining easy axis being the axis of elongation. Although there are other types of MTB with magnetite magnetosomes of arrowhead shape and elongation along <100>, <112>, or <114> directions [Taylor et al., 2001; Hanzlik et al., 2002; Kasama et al., 2006; Pósfai et al., 2006; Lins et al., 2007; Li et al., 2010], magnetite is magnetically soft enough for shape anisotropy to override magnetocrystalline anisotropy in these elongated magnetosomes [Körnig et al., 2014]. In contrast, the observed aspect ratio of <100> elongated greigite magnetosomes (between 1.0 and 2.2) [Heywood et al., 1990; 1991; Posfai et al. 1998b] is well below the critical value of 3 (cf. section 'Shape Versus Magnetocrystalline Anisotropy') and therefore results in a shape anisotropy that is too small to override the magnetocrystalline anisotropy. Thus, for K1 > 0 and elongation along a <100> easy axis, the effect of shape anisotropy is not to reduce the number of local minima in the magnetic energy landscape, but to select the easy axis that coincides with the elongation magnetization, which then represents the most stable state that can be reached. For Q = 0.33, the edge length for superparamagnetism could be reduced from 32 nm (pure cubic case, cf. section 'Shape Versus Magnetocrystalline Anisotropy') down to 22 nm for an aspect ratio of 1.5.

5.1.2 Counterarguments

The argument for the sign of K1 based on <100> elongated magnetosomes is tempting, but its underlying premise supposes that MTB generally achieve a high level of magnetic optimization. Greigite magnetosomes, however, do not appear to be highly optimized at the levels of crystal perfection and chain architecture. Magnetosome chains made of greigite tend to have a cluttered disposition in contrast to the well aligned, linearly arranged crystals in magnetite magnetosome chains. In the most detailed electron-microscopical study of greigite magnetosome elongation and relative orientation with respect to the chain axis, Kasama et al. [2006] reported highly variable orientations and the absence of a preferred crystallographic direction as the axis of elongation. In this respect, it is interesting to note that Kasama et al. [2006, Table 2] also found greigite magnetosomes with [111] axes parallel to the chain axis. To provide another counterexample, we note that Stanjek and colleagues observed biogenic greigite crystals with elongation along a <111> direction [Stanjek et al., 1994; Stanjek and Schneider, 2000]. It appears that the <100> based argument is biased toward a few early observations then available, which have turned out to be not generalizable.

Even though the argument cannot be fully supported by evidence for it, the conclusion (K1 > 0) may of course still be correct if the opposite (K1 < 0) represents a disadvantage for magnetotaxis. Therefore, we asses the potential advantages or disadvantages that come with magnetosomes elongated along a hard <100> axis (negative K1). From our theoretical treatment of that case in section 'Partial Superparamagnetism Due to Competing Anisotropies', we can conclude that for MTB growing magnetosomes with elongation along a hard axis, the cost would be a somewhat reduced magnetotactic alignment, but the advantage (compared to no or small elongation) would be higher stability against thermal-fluctuation induced polarity flipping. The latter is necessary for bacteria that rely on stable polarity for magnetotaxis, the former is not. Therefore, elongation along a hard <100> axis is no disadvantage even though the optimum solution in this case (K1 < 0) would obviously be <111> elongated magnetosomes (with their elongations oriented along the chain axis). However surprising this result may appear, it makes good sense from an evolutionary standpoint. Evolution is not goal directed (in the sense of absolute optimization), but good at finding solutions for constrained optimization problems in a local neighbourhood of the fitness landscape. The important constraints here are i) biomineralization templates, which initiate and direct growth, ii) shape bags (magnetosome vesicles) that control the final size and shape of the crystals, and iii) magnetic material parameters. If the constraints are K1 < 0 and crystal growth starting on a {100} plane [Heywood et al., 1991], where the precursory phase mackinawite can nucleate [Pósfai et al., 1998a, 1998b], then the obvious local solution to the optimization problem is to elongate the crystals along the axis that is perpendicular to the given {100} truncation face. In summary, it is difficult to reach firm conclusions on the sign of K1 from magnetosome elongations. Whatever the sign of K1, it is likely that MTB found at least a local solution in the optimization space.

5.2 Coercivity of Greigite Magnetosomes

Rod-shaped bacteria (MR) with double-stranded chains of greigite magnetosomes have a mean coercivity of 32 mT [Penninga et al., 1995] when the pulse field is applied (roughly) antiparallel to the chain axis. This is surprisingly low when a coercivity of at least 85 mT (Ban, equation (5), Table 1) could be expected for magnetosomes oriented with an easy axis along the chain axis (cf. section 5.1.1). With elongation along an easy axis, the expected coercivity should be even higher than 85 mT. In view of the detailed TEM measurements reported by Kasama et al. [2006], this discrepancy may be due to lack of coherent relationships between crystal axes and chain axis, and in particular: lack of coherent relationships between crystal axes and elongation in combination with lack of coherent orientation of long axes along the chain axis (see also 5.1.2). Other likely candidates for coercivity reduction may be thermal fluctuations and defects. In particular, defects represented by nonmagnetic inclusions in greigite host crystals due to incomplete conversion of the nonmagnetic precursory FeS to Fe3S4 (see TEM images of Pósfai et al. [1998a, 1998b]) could act as exchange break between two greigite domains in a magnetosome, reducing the effective exchange coupled volume and thereby making it more susceptible to thermoactivated switching. Even without the physical presence of the precursory FeS phase, the remaining lattice defects at the former phase boundary would tend to reduce K1 locally, making it a preferential nucleation site for reversed magnetization.

Multicellular magnetic prokaryotes (MMP) from California have even lower coercivity (20 mT) than cells of MR, which was ascribed to nonmagnetic crystals (of pyrite, or more likely, of cubic FeS, the nonmagnetic precursor to greigite) [Pósfai et al., 1998a, 1998b] within the MMP magnetosome chain that act as break gaps to the otherwise stabilizing intrachain magnetostatic interaction, thereby allowing partial reversal to occur in the chain magnetization [Penninga et al., 1995]. This intrachain reversal implies nonsquare remanence curves, which have been observed for MMP as well as for MR cells. This mechanism is therefore unlikely to explain the observed difference in coercivities obtained for MR and MMP. Given the multicellular nature of the MMP, in which each cell has a double-stranded magnetosome chain, which however may not be coaxial with the resulting remanence axis of the multicellular organism [Winklhofer et al., 2007], it is also possible that the nonsquare remanence curves represent switching-field distributions of chains that make different angles to the external magnetic field.

In a pulse-field study on a MMP variety from Brazil, again with greigite magnetosomes, coercivities of remanence were determined as 20–35 mT [Winklhofer et al., 2007], which bridges the coercivity gap between the Californian MMP and MR reported by Penninga et al. [1995]. Holocene sapropels from the Baltic Sea, which contain greigite magnetofossils, have remanence coercivities of 20 mT [Reinholdsson et al., 2013], which are consistent with values from Mediterranean sapropels that contain iron sulfides [Roberts et al., 1999] that are now thought to represent greigite magnetofossils [Reinholdsson et al., 2013].

5.3 Coercivity of Synthetic Single-Domain Greigite

In Figure 7b, we compare the switching-field distribution (SFD) related to the hysteresis curve of Figure 7b with the SFD extracted from the FORC diagram for a fine grained synthetic greigite sample SYN06 shown in Chang et al. [2009b, Figure 1h]. Since thermal activation effects have a greater influence on the low-coercivity part of the SFD than on the high-coercivity tail, we interpret the observation that the peak of the FORC distribution is centered at Bc values of 5–20 mT in terms of a thermal activation induced shift from the expected Bc range of 20–40 mT. At low temperature (10 K), where thermoactivation effects on coercivity are largely removed, coercivity values of inline image and inline image were measured [Chang et al., 2009b, Figure 1e]. With the Ban value determined here and an assumed temperature dependence according to (equation (A7) in Appendix, for K1 < 0), i.e., inline image, we can account for the observed coercivities in synthetic fine-grained greigite from Chang et al. [2009b].

5.4 Coercivity of Authigenic Single-Domain Greigite

The K1 values from the upper end of the uncertainty range in Table 2 ( inline image and inline image), result in Bc values of 50 mT, which is still too low to explain the high room temperature Bc values of 60 mT observed for natural SD greigite concretions (“nodules”) (e.g., samples “Italy” and “Taiwan” in Roberts et al. [2011, Figure 7]; see also Sagnotti and Winkler [1999, Table 2] with Bc values ranging from 41 to 67 mT). Even though these nodules are densely packed and therefore do not strictly meet the noninteraction criterion for equation (18) or (19) to hold, they have high remanence ratios inline image, which indicates that cubic magnetocrystalline anisotropy dominates over shape/interaction anisotropy) [Roberts, 1995]. From coercivity values of 62 mT for authigenic greigite nodules, Roberts et al. [2011] estimated the minimum value of inline image as +21 kJ/m3 inline image. For comparison, we would require the g factor to be at most 2.0 to obtain at least +21 kJ/m3 as an upper bound estimate for K1 from the Q-band FMR spectra recorded by Chang et al. [2012] for the natural samples (see Table 2). As mentioned in section 'Shape Versus Magnetocrystalline Anisotropy', it is possible that the effective magnetocrystalline anisotropy constants relevant for hysteresis properties are larger in magnitude than the FMR-derived ones because of a magnetostrictive contribution [Ye et al., 1994]. Cation impurities in natural greigite, such as nickel, could have an effect on the magnetocrystalline anisotropy. Likewise, unrecognized pyrrhotite lamellae in natural greigite could enhance the coercivity, because pyrrhotite is magnetically much harder than greigite and therefore pin the magnetization due to exchange coupling between hard and soft phase. Analytical techniques beyond energy-dispersive X-ray (EDX) analysis in the scanning electron microscope and bulk X-ray diffraction, such as high-resolution TEM with EDX, are needed to better determine local chemical and structural information in sedimentary greigite crystals and thus to obtain more insight into the possible origins of their poorly understood magnetic properties, which also include the large gyromagnetic effect [Snowball, 1997; Hu et al., 1998] as well as first-order reversal curve diagrams shifted toward negative bias fields [Roberts et al., 2006]. From a theoretical point of view, a sample with an internal bias field is prone to acquire a remanence during alternating-field (AF) demagnetization, because the effect of an internal bias is largely equivalent to that of an externally applied static field in anhysteretic remanence acquisition, and would therefore explain the gyromagnetic effect. In synthetic greigite and samples containing greigite magnetofossils, no such bias fields nor gyromagnetic effects have been reported. We therefore suggest that sediments that have a stable remanence carried by greigite but do not acquire a remanence during AF demagnetization, are likely to contain greigite magnetosomes.

6 Conclusions

From fitting FMR spectra of greigite powder samples, we have determined magnetocrystalline anisotropy parameters, anisotropy field and energy barriers. We assume that each sample represents a sufficiently uniform system in terms of intrinsic material parameters and that a certain spread in these as well as all inhomogeneity effects due to varying particles shapes and dipolar interactions can be accounted for by isotropic symmetric Gaussian line width broadening. The g factor that yields the best fitting anisotropy models amounts to 2.09±0.01, in agreement with our estimate based on independent data from neutron scattering and measurements of Ms. We obtain the anisotropy field as inline image for our new X-band data and higher estimates inline image for other greigite samples measured previously at Q-band frequencies. These values greatly exceed the value of 30 mT for magnetite at room temperature and imply that greigite particles with axis ratios up to at least 4/3 for elongation along a <111> axis (and higher aspect ratios for other elongation axes) should be dominated by magnetocrystalline anisotropy. This can also explain the high SIRM/κ ratios observed for sediments containing greigite. The derived coercive force inline image agrees reasonably well with that measured for fine-grained synthetic greigite samples, but cannot explain the large coercivities measured for authigenic greigite nodules, which should be further investigated to identify the source of the high coercivity and bias mechanism. Values for the first-order magnetocrystalline anisotropy constant K1, which are consistent with FMR spectra and g = 2.09, are inline image or inline image. Curiously, any of the K1 > 0 anisotropy model implies a relatively large value of K2, i.e., inline image. K1 > 0 was inferred by Yamaguchi and Wada [1970] on the basis of rather incomplete diffraction data with a method that was not calibrated on reference materials with known easy axis. We have argued that <100> elongated greigite magnetosomes should not be taken as experimental support for K1 > 0 and we consider a <111> easy axis orientation more likely. The energy barriers determined here yield maximum superparamagnetic grain sizes of 32 nm (K1 > 0) or 44 nm (K1 < 0), which both are in accord with magnetosome data [see Muxworthy et al., 2013]. Our new values for K1 and K2 are not ideal but are as good as can be made with available powder samples. Accurate determination of these parameters will eventually require work on oriented single crystals.

Acknowledgments

We are indebted to Klaus Köhler from the Technical University of Munich for kindly providing us with measurement time on the ESR spectrometer and to Carmen Haeßner for assistance. We are grateful to Andrew J. Newell, Andreas U. Gehring, and the Associate Editor, Bruce M. Moskowitz for their careful reviews and constructive comments. We thank Andrew P. Roberts for critically reading an earlier version of the manuscript. MW acknowledges funding from the DFG (grant Wi 1828/4-2).