Ferromagnetic resonance (FMR) spectroscopy has become an increasingly useful tool for studying the magnetic properties of natural samples. Magnetite (Fe3O4) is the only magnetic mineral that has been well characterized using FMR. This limits the wider use of FMR in rock magnetism and paleomagnetism. In this study, we applied FMR analysis to a range of magnetic minerals, including greigite (Fe3S4), monoclinic pyrrhotite (Fe7S8), magnetically non-interacting titanomagnetite (Fe3-xTixO4), and synthetic magnetite chains to constrain interpretation of FMR analysis of natural samples and to explore applications of FMR spectroscopy. We measured the FMR signatures of a wide range of well-characterized samples at the X- and Q-bands. FMR spectra were also simulated numerically to compare with experimental results. The effects of magnetic anisotropy, mineralogy, domain state, and magnetostatic interactions on the FMR spectra are discussed for all studied minerals. Our experimental and theoretical analyses of magnetically non-interacting tuff samples and magnetically interacting chains enable quantitative assessment of contributions of magnetostatic interactions and magnetic anisotropy to the FMR spectra. Our results also indicate that intact magnetosomes are a unique system with distinct FMR signatures. While FMR analysis is useful for characterizing magnetic properties of natural samples, care is needed when making interpretations because of overlaps in a range of FMR signatures of different magnetic minerals with different magnetic properties. Our analyses will help to constrain such interpretations in rock magnetic studies.
 In this study, we performed FMR analyses on a range of magnetic minerals, including magnetic iron sulfides (greigite (Fe3S4) and monoclinic pyrrhotite (Fe7S8)), non-interacting titanomagnetite (Fe3-xTixO4) and synthetic magnetite chains, to constrain interpretation of FMR analysis of natural samples and to explore applications of FMR spectroscopy. Iron sulfide minerals are widespread in nature and are considered to be the most important minerals in ore deposits. The thiospinel greigite and monoclinic pyrrhotite are two important magnetic iron sulfide phases [e.g., Pearce et al., 2006]. Greigite commonly forms in anoxic diagenetic sedimentary environments [see Roberts et al., 2011b, and references therein], and as a biomineralization product [e.g., Konhauser, 1998; Bazylinski and Frankel, 2004; Suzuki et al., 2006]. Pyrrhotite is common as an authigenic mineral in sediments [e.g., Weaver et al., 2002], as a detrital mineral in sediments [e.g., Horng and Roberts, 2006], and in igneous and metamorphic rocks [e.g., Rochette, 1987; Horng and Roberts, 2006]. Magnetic iron sulfide minerals are also commonly present in extraterrestrial materials in the solar system [e.g., Rochette et al., 2001]. Iron sulfides are important carriers of remanent magnetizations and therefore significantly contribute to paleomagnetic and paleoenvironmental records in many geological settings. In this study, we analyzed a wide range of well-characterized greigite and pyrrhotite samples. We also analyzed a set of standard samples from the Tiva Canyon (TC) ash flow tuff that contains non-interacting single-domain (SD) titanomagnetite (Fe3-xTixO4) grains. The TC Tuff has long been of interest in rock magnetism because of its narrow magnetic grain size distribution and well dispersed magnetic particles that lack magnetostatic interactions [e.g., Schlinger et al., 1991; Rosenbaum, 1993; Worm and Jackson, 1999; Roberts et al., 2000; Till et al., 2011]. These tuff samples appear therefore ideal for studying the FMR signal of an interaction-free system other than bacterial cells. Finally, we analyzed synthetic magnetite chains to compare measured FMR signatures with those from intact magnetite-producing magnetotactic bacteria and magnetofossils.
 The pure synthetic and natural greigite samples analyzed here (Figures 1a and 1b) have been subjected to detailed magnetic characterization previously [Chang et al., 2007, 2008, 2009a, 2009b; Roberts et al., 2011b]. The pure synthetic greigite samples (labeled “SYN-XXX”) were prepared by hydrothermally reacting ferric chloride (FeCl3·6H2O) with thiourea (CH4N2S) and formic acid (HCOOH) at 170°C for eight hours [Tang et al., 2007; Chang et al., 2008]. After synthesis, the greigite samples were sealed in small glass sample vials and were stored in a desiccator to prevent oxidation. These synthetic greigite samples contain nearly equi-dimensional crystalline particles, of mostly cubo-octahedral morphology, in the 10 μm size range. These synthetic greigite samples are dominated by pseudo-single-domain (PSD)/multidomain (MD) magnetic properties [Chang et al., 2007, 2008]. The natural greigite samples are iron sulfide nodules from the Valle Ricca section near Rome [van Dongen et al., 2007] and from the Lower Gutingkeng Formation in southwestern Taiwan [Jiang et al., 2001] (and are labeled “Italy” and “Taiwan,” respectively). Greigite is the only magnetic phase in these natural samples, which have SD magnetic properties. Scanning electron microscope (SEM) observations indicate that these samples contain equi-dimensional greigite grains (mostly cubo-octahedral) with lengths of several hundred nanometers [Roberts et al., 2011b]. The grain size distribution of these samples has not been determined because the greigite crystals are too small to be accurately resolved with SEM instruments.
 Sample “EOR2” is a natural hydrothermal pyrrhotite sample collected from mine dumps at Ortano on the east coast of Elba [Dekkers, 1988]. This pyrrhotite sample was obtained by crushing the pyrrhotite-bearing rocks and magnetically concentrating the pyrrhotite. The pyrrhotite was then sieved. This sample contains equidimensional pyrrhotite grains in the 100–150 μm size range [Dekkers, 1988]. Samples “9–47” and “20–131” are pyrrhotite-bearing metamorphic rocks from Taiwan (C.-S. Horng, Metamorphic pyrrhotite as a tracer for denudation of orogenic belts, manuscript in preparation, 2012). These samples represent the parent material from which detrital pyrrhotite is supplied to marginal marine basins in Taiwan [Horng and Roberts, 2006; Horng and Huh, 2011]. SEM observations indicate that these samples contain hexagonal-shaped and irregular pyrrhotite crystals in the micrometer size range. Sample “syn_ph” is a synthetic pyrrhotite sample formed by heating the synthetic greigite sample in argon [Chang et al., 2008]. X-ray diffraction, thermomagnetic and low-temperature measurements indicate that this sample contains single-phase monoclinic pyrrhotite.
 Synthetic magnetite chain samples were prepared following the method of Liu and Chen . Reagents used include 0.5 g of ferrocene (Fe(C5H5)2) (98%), 0.75 g of polyvinylpyrrolidone (PVP), and 12.0 g of dry ice (99.9%). Hydrothermal reaction occurred in a 20 ml steel autoclave at 450°C for 800 min. After reaction, the solution was cooled to room temperature naturally and the remaining CO2 was vented. The synthetic product was then washed alternately with toluene and ethanol several times, and dried at 60°C in air for several hours. The final product is composed of linear chains of SD magnetite (octahedra) that range from 40 to 120 nm, covered by a thin amorphous carbon coating (Figure 1d).
 In a typical EPR experiment, a sample is subjected to a DC magnetic field and microwave radiation and is placed in a resonating cavity that can absorb photons generated by a microwave radiation source. The microwave energy can be absorbed by the sample due to the Zeeman effect that splits the energy of the unpaired electron within an atom or molecule in a magnetic field. In a magnetic field, the spin of the unpaired electron can align either along or in the opposite direction to the magnetic field. A resonant absorption condition occurs when the incident photon energy is equal to the energy separation between the two electronic energy levels. The resonance condition is given by:
where h = 6.626 × 10−34 Js is Planck's constant, ν is the microwave frequency, B is the intensity of the magnetic field and μB = 9.274 × 10−24 J/T is the Bohr magneton. In FMR, absorption of microwave energy is due to exchange-coupled spin assemblages precessing coherently around the local effective field vectorBeff when the Larmor precession frequency associated with Beff is equal to the microwave frequency ν. This process produces intense and broad signals [Griscom, 1980; Kittel, 1996]. The resonance condition in FMR is also given by equation (1), where B is replaced by Beff. The effective magnetic field Beffis the vector sum of the applied field and internal fields due to sample geometry, magnetocrystalline anisotropy, stress-induced anisotropy, and magnetic interactions (dipolar and exchange interactions). The FMR signature therefore contains much useful information about the magnetic properties of samples. A detailed description of the FMR theory is given byKittel . A detector measures the change in FMR absorption as a function of a sweeping magnetic field produced by an electromagnet (Figure 2a). FMR spectra are usually measured and displayed as the first derivative of absorption (Figure 2b). This enhances the signal-to-noise ratio [e.g.,Pawse et al., 1998] and magnifies the fine structure of the FMR absorption signal. The FMR spectra can be used to quantify some simple magnetic systems, e.g., a non-interacting SD assemblage. Other systems, e.g., assemblages with magnetostatic interactions, so far can only be understood in a general way.
 We use the following parameters to describe the FMR spectra (Figure 2). Beffis the zero-crossing field in the derivative spectra, which is also the maximum absorption field in the integrated spectra. The effectiveg value (geff) is given by geff = hν/μBBeff. Blow, Bhigh, ΔBFWHM and A are defined in a typical absorption spectrum (integration of the first derivative spectrum) (Figure 2a) [e.g., Weiss et al., 2004; Kopp et al., 2006a, b]. Blow and Bhighare the magnetic fields where the absorption is half the maximum value at the low- and high-field ends, respectively, of the absorption peak. ΔBlow and ΔBhighare the low- and high-field linewidths, respectively. ΔBFWHM is the sum of ΔBlow and ΔBhigh and is the width of the absorption peak at half of its maximum value. The asymmetry ratio A is defined as A = ΔBhigh/ΔBlow. We also define a set of FMR parameters in a typical first derivative spectrum (Figure 2b) that are similar to those used by Griscom  and Fischer et al. . Bp1 and Bp2are the magnetic fields at the maximum and minimum peaks, respectively. The peak-to-peak linewidth ΔBpp is the distance between Bp1 and Bp2, and A′ = ΔBp1/ΔBp2 = (Beff − Bp1)/(Bp1 − Beff).
 The FMR spectra were measured using an X-band Bruker EMX micro spectrometer and a Q-band Bruker E500 FMR spectrometer at the School of Chemistry, the University of Manchester, and at the Research School of Earth Sciences, Australian National University. For each measurement, small samples (from several mg to a few tens of mg) were loaded into FMR glass tubes. For X-band measurements, we set the microwave frequency at ∼9.4 GHz and the microwave power at ∼0.632 mW. All spectra were integrated over several magnetic field sweeps from 0 to 700 mT. At the Q-band, the microwave frequency was set at ∼34 GHz and the microwave power at ∼0.12 mW. Field scans were made from 10 to 1410 mT and from 600 to 2000 mT for Q-band measurements. There is not much useful information below 600 mT, therefore we only present data from the 600–2000 mT scan at Q-band.
 FMR spectra were simulated using the model described by Charilaou et al. . Elongated crystals were modeled as prolate ellipsoids of revolution, with aspect ratio q = c/a, where c and a denote long and short major axes, respectively. The demagnetization factors along these axes are Nc < Na, and ΔN = Na − Nc. For cubic minerals and for particles elongated along a 〈100〉 axis (hard axis), the cubic magnetocrystalline anisotropy energy density is given by:
where (ϑ, φ) denotes the polar and azimuthal angle of the magnetization with respect to the long particle axis. For particles elongated along the 〈110〉 (intermediate axis) and 〈111〉 (easy axis) axes, the respective expressions are given by:
For each of the three crystallographic scenarios, we computed FMR powder spectra for several axial ratios q, following the approach of Charilaou et al. , who dealt with crystals elongated along the 〈111〉 axis. By taking advantage of symmetry relationships, we can restrict the applied field range to ϑH ∈ [0, 90°], φH ∈ [0, 45°] for the 〈100〉 case, to ϑH ∈ [0, 90°], φH ∈ [0, 90°] for the 〈110〉 case, and to ϑH ∈ [0, 90°], φH ∈ [0, 60°] for the 〈111〉 case. For numerical computation, the (ϑH, φH) interval is approximated by a discrete grid with equidistant spacing in φH (5° mesh size) and equidistant in the cos ϑH (0.02 mesh size). For each point on the (ϑH, φH) grid, the resonance condition is evaluated and the resulting resonance field (a delta peak) is convolved with a Lorentzian of 20 mT half-width field at half maximum.
 We use the following magnetic parameters for our FMR simulations. For magnetite, we use a saturation magnetization (Ms) of 470 kA/m and a first-order cubic magnetocrystalline anisotropy constant (K1) of −12 kJ/m3 [Kakol and Honig, 1989]. For TM10, we use Ms of 435 kA/m and K1 of −23 kJ/m3 determined with the torque method [Syono and Ishikawa, 1963]. Strictly speaking, K1values obtained from torque magnetometer measurements represent stress-freeK1′ values and need to be corrected for magnetostriction to obtain the intrinsic strain-freeK1 [Ye et al., 1994]. However, we do not know whether the titanomagnetite crystals in the TC Tuffs are strain-free. Regardless, the difference betweenK1′ and K1is small for low-x titanomagnetite. From equation (9) in Ye et al.  and the magnetostriction data for x = 0.1 [Syono and Ishikawa, 1963], we find that K1′ − K1 = ∼3 kJ/m3, so that K1 = −20 kJ/m3, in accordance with the K1 data obtained by Kakol et al.  and interpolated for TM10.
 While the magnetocrystalline anisotropy energy of ideal monoclinic pyrrhotite is characterized by a purely sixfold symmetry in the basal plane [e.g., Martín-Hernández et al., 2008], our pyrrhotite samples are not ideal and therefore are best described by a uniaxial (twofold symmetry) anisotropy in the basal plane. We therefore modeled monoclinic pyrrhotite with uniaxial magnetocrystalline anisotropy in the basal plane. In this case, the expression for the magnetocrystalline anisotropy energy is given by Bin and Pauthenet :
where ϑ′ is the angle of the magnetization with respect to the hard  direction and φ′ is the azimuthal angle of the magnetization projected onto the easy (001) plane, where φ′H = 0 defines the easy axis in the plane. The anisotropy model with purely sixfold symmetry in the basal plane will be dealt with in more detail in a separate paper. The room temperature magnetocrystalline anisotropy coefficients for monoclinic pyrrhotite are K4 = 32.2 × 105 J/m3, K3 = 1.18 × 105 1J/m3, K1 = 0.35 × 105 J/m3 [Bin and Pauthenet, 1963; Martín-Hernández et al., 2008]. These parameters were used as they appear to reflect typically real samples. We simulate equant particles because the shape anisotropy (Ms = 80 kA/m) is of minor importance compared to magnetocrystalline anisotropy for pyrrhotite.
4. Results and Discussion
4.1.1. FMR Spectra at X-Band
 Room temperature X-band FMR spectra for selected greigite samples are presented inFigure 3 and FMR parameters are listed in Table 1. Pure synthetic greigite samples give rise to a single absorption line with maximum and minimum absorption peaks at ∼130 and ∼330 mT, respectively, with ΔBFWHM of around 200 mT (Figure 3a). The peak intensity of low-field absorption is consistently larger than that of the high-field peak. These synthetic samples have highgeff values (2.9–3.1), which is consistent with the expectation that MD assemblages typically have large geff values well above the free electron value of 2 [e.g., Weiss et al., 2004]. The spectra are asymmetric with A values larger than 1 (Table 1). The large geff, ΔB and Avalues can be explained by enhanced low-field absorption due to absorption in magnetic domains within MD particles in directions different to that of the applied field. The small nonzero absorption near zero field (Figure 3a) may also be a reflection of a MD effect. Compared to the pure synthetic greigite samples, the sedimentary greigite samples have different FMR spectra (Figure 3b): geff values are much reduced (2.02), linewidth is lower (∼180 mT) and A is less than 1 (0.7) (Table 1). The low-field peak intensity is also smaller than that of the high-field peak. The small absorption peak at ∼155 mT (Figure 3b) corresponds to a g value of 4.3, which is typical of paramagnetic high-spin Fe3+ in a low symmetry environment, such as in feldspars [Hofmeister and Rossman, 1984], and here is related to paramagnetic Fe3+ within the sediments [e.g., Kopp et al., 2006a].
Table 1. Room Temperature FMR Parameters of the Samples Measured in This Study
FMR Parameters Defined in the Absorption Spectra
FMR Parameters Defined in the Derivative Spectra
4.1.2. FMR Spectra at Q-Band
 We measured several greigite samples at Q-band (Figures 3c and 3d). All measured synthetic and natural greigite samples have a single absorption line, while the spectra for natural samples are more spread out. The spectrum for a pure synthetic greigite sample indicates a higher maximum peak intensity compared to the minimum peak intensity, while this is opposite for the spectrum from a natural greigite sample. These observations are similar to X-band results. Because of the higher microwave frequency at Q-band, the resonance condition occurs at much higher magnetic fields, i.e., ∼1150 mT (Table 1). geffvalues are almost indistinguishable for different samples at Q-band (2.08–2.12). TheA value for synthetic greigite is higher than 1 (∼1.3), while for sedimentary greigite it is lower than 1 (0.8).
 To interpret the FMR signature of greigite, we first discuss the nature of the studied greigite samples and consider the simple case of a non-interacting, equidimensional SD particle assemblage. SEM observations indicate that sedimentary greigite-bearing rocks often contain nearly equidimensional cubo-octahedral greigite crystals (Figure 1b) [e.g., Jiang et al., 2001; Roberts and Weaver, 2005; Hüsing et al., 2009; Sagnotti et al., 2010; Roberts et al., 2011b]. SEM images of the studied natural greigite confirm the presence of equidimensional greigite crystals (Figure 1b) [van Dongen et al., 2007; Roberts et al., 2011b]. Contributions from shape anisotropy compared to magnetocrystalline anisotropy can therefore be ignored. Dominance of magnetocrystalline anisotropy in sedimentary greigite is also confirmed by hysteresis data from natural greigite samples, which often indicate hysteresis squareness ratios higher than 0.5 [e.g., Roberts, 1995; Sagnotti and Winkler, 1999; Vasiliev et al., 2007; Roberts et al., 2011b]. The measured synthetic greigite samples are also composed of equidimensional cubo-octahedral crystals, although elongated particles (e.g., plates and prisms) were occasionally observed (Figure 1a) [Chang et al., 2008]. The natural greigite samples are dominated by SD properties, while the synthetic samples are dominated by PSD/MD grains [Chang et al., 2007, 2008, 2009b]. These PSD/MD grains will have a significant effect on the X-band spectra, for example, largegeffvalues at X-band are observed. However, because the magnetization of greigite saturates at ∼300 mT [e.g.,Dekkers and Schoonen, 1996; Roberts et al., 2011b], the large scanning field range of 600 to 2000 mT during Q-band FMR measurements is high enough to saturate the magnetization. This is one reason that we carried out Q-band measurements to saturate the PSD/MD grains so that they effectively display SD behavior. For magnetic interactions, SEM observations indicate that sedimentary greigite often forms in close-packed particle clusters (Figure 1b) [e.g., Jiang et al., 2001; Roberts and Weaver, 2005], which produce strong magnetostatic interactions [e.g., Roberts et al., 2000, 2006; Vasiliev et al., 2007]. During sample preparation, we mixed the synthetic greigite powders with eicosane (C20H42) to dilute the magnetic grains. Although this procedure can significantly reduce magnetostatic interactions, many magnetic particles will still clump and produce magnetic interactions [Kopp et al., 2006a].
 Greigite has cubic crystal symmetry. K1 for greigite has been inferred to be positive at room temperature [e.g., Yamaguchi and Wada, 1970; Bazylinski et al., 1995]. However, this inference needs further investigation. Non-interacting SD greigite with magnetocrystalline anisotropy should give rise to FMR spectra withA < 1 and geff < 2.12 if K1 > 0, and with A > 1 and geff > 2.12 if K1 < 0 [e.g., Griscom, 1974; Weiss et al., 2004]. X-band FMR measurements on some sedimentary greigite samples yieldgeff values close to 2 and A ∼0.7 (Table 1). The observed large geff value above 2 is probably caused by magnetostatic interactions. Currently, we cannot quantitatively measure the effects of magnetostatic interactions on FMR spectra for greigite due to the difficulty in obtaining greigite samples without magnetostatic interactions. In principle, magnetostatic interactions can broaden FMR spectra and shift geff to higher values [e.g., Valstyn et al., 1962; Kopp et al., 2006b]. FMR parameters with A < 1 and geff < 2.12 have been suggested to provide strong evidence for magnetite magnetosome chain structures within samples [Weiss et al., 2004]. Our FMR measurements indicate that diagenetic greigite (Figure 1b) can sometimes also have A < 1 and geff < 2.12. Despite the large overlap in A and geff values for diagenetic greigite and magnetite magnetosome chains, which might indicate ambiguity in their discrimination, we observe a large difference in their Beff values (Table 1). Combined with other FMR characteristics for magnetite magnetosome chains (e.g., multiple low-field absorption peaks), FMR analysis should still enable effective discrimination between them. On the other hand, no FMR measurements have been reported for greigite magnetosomes [Vasiliev et al., 2008]. Such direct measurements on greigite magnetosomes are much needed to determine whether they give rise to distinctive FMR spectra.
4.2. Monoclinic Pyrrhotite
 Most pyrrhotite samples were measured at X-band with one sample at Q-band (Figure 4 and Table 1). At X-band, sample “EOR2” has a dominant single absorption line (Figure 4a) with a geff value of 2.36. Ais close to 1, which indicates nearly symmetric absorption spectra. The absorption line is sharp with a narrow peak-to-peak linewidth (39 mT). The small broad peak at low fields may reflect an MD effect. First-order reversal curve results indicate that sample “EOR2” is dominated by MD particles [Wehland et al., 2005], which may contribute to the observed higher geff value. Complex FMR spectra were observed for samples “20–131” and “9–47” (Figures 4b and 4c). The small peak at ∼160 mT and the multiple sharp lines are characteristic of the absorption of Fe3+ and Mn2+, respectively [e.g., Kopp et al., 2006a]. The sharp Mn2+ lines overprint the FMR signal for pyrrhotite, but can be readily removed by Fast Fourier Transform (FFT) smoothing [Roberts et al., 2011a]. After FFT smoothing, the background FMR signal is clear (Figures 4b and 4c). ΔBpp is ∼57 mT and geff values are close to 2 (Table 1). At Q-band, thegeff value for the synthetic pyrrhotite sample is 2.03 and ΔBpp is ∼134 mT.
 We simulated FMR spectra for equant pyrrhotite powders at different bands (Figure 5). When solving the FMR equation for ideal monoclinic pyrrhotite at X-band frequencies (9.4 GHz), solutions occur only at (φ′H = ±π/2), with low-field absorption at 840 mT atϑ′H = ±π/2 (i.e., in the (110) plane) and high-field absorption at 30.5 T atϑ′H = 0 (i.e., along the  axis), which results in a mean geff value of 0.72 (Beff = 0.92 T) for the two-dimensional powder FMR spectrum obtained by averaging theBeff (cosϑ′H; φ′H = ±π/2) curve, convoluted with a Lorentzian intrinsic line shape. At Q-band frequencies (34 GHz), the directional range that satisfies the resonance conditions becomes larger (φ′H within 10 degrees of ±90°). While the FMR “powder” spectrum has clear gaps because of the narrow directional range within which resonance events can occur (blue curve in Figure 5), the obtained geff = 2.07 (Beff = 1.17 T) is nevertheless in good agreement with our Q-band data for the synthetic sample (geff = 2.03, Beff = 1.2 T). At W-band frequencies (68 GHz), all field orientations produce at least one resonance event. The corresponding powder FMR derivative spectrum (magenta curve inFigure 5) has a strongly asymmetric shape and a geff = 12.90 (Beff = 0.377 T). The g-values are 0.22 for the high-field event (along the hard  axis) and 14.6 for the low-field resonance event along the easy axis (φ′H = 0 in the (110) plane). Such pronounced anisotropy in the g-value is commonly observed in strongly uniaxial antiferromagnets [e.g.,Koonce et al., 1971].
 Published FMR data for monoclinic pyrrhotite are sparse. Mikhlin et al. measured FMR spectra for an air-ground pyrrhotite sample, but no clear FMR signature could be extracted from their spectrum.Fujimura and Torizuka  measured the FMR spectrum of a pyrrhotite single crystal. They observed an extremely large linewidth measured in the easy direction of magnetization, which they attributed to line broadening caused by possible microcrystals within their sample. No resonance was observed when the applied field deviated >10° from the crystallographic c-plane probably because of the large magnetic anisotropy energy along thec-axis in monoclinic pyrrhotite [Fujimura and Torizuka, 1956]. Our FMR modeling indicates that FMR absorption in powdered monoclinic pyrrhotite is small at low frequencies. This is mainly due to the large uniaxial anisotropy in monoclinic pyrrhotite, i.e., the large magnetocrystalline anisotropy (the K4 term) along the c-axis. This large anisotropy only allows resonance to occur at much larger frequencies, i.e., at W-band. At X- and Q-bands, resonance events do not occur in general, but only for some particular directions. This explains why we observed extremely weak FMR signals (note the strong paramagnetic Mn2+ signals in Figures 4b and 4c) despite the fact that the measured pyrrhotite samples are magnetically strong. The X-band spectra of the natural samples probably reflect real structural features (e.g., magnetic domain walls, intergrowths, and twinning etc.) and impurity effects, rather than intrinsic properties of pyrrhotite. Our results demonstrate that pyrrhotite is not a good FMR absorber, at least at the low frequencies (i.e., X-band) that are commonly used for FMR measurements. FMR analysis of more samples and oriented pyrrhotite single crystals and angular dependent FMR spectra are needed to better constrain the FMR signature of pyrrhotite.
4.3. Non-interacting Titanomagnetite and Synthetic Magnetite Chains
4.3.1. FMR Spectra at X-Band
 Similar FMR spectra are observed for all three TC tuff samples at X-band (Figure 6). These spectra are extremely asymmetric and contain three low-field maxima and two high-field minima.geff values are low and range from 2.03 to 2.06 (Table 1), which is close to that of a free electron and for the applied field parallel to an easy axis of magnetization in magnetite [Bickford, 1950]. ΔBFWHM values are between 232 and 243 mT. A is between 0.66 and 0.76. It should be noted that, although SD grains dominate these tuff samples, small portions of PSD/MD and superparamagnetic (SP) grains are also present [e.g., Carter-Stiglitz et al., 2006; Jackson et al., 2006]. SP grains may contribute to the small peak at ∼200 mT and the shoulder near g = 2 (Figure 6). This SP contribution should be small as room temperature frequency dependent susceptibility measurements indicate small changes (only ∼2–3% per decade) [e.g., Carter-Stiglitz et al., 2006; Jackson et al., 2006]. Because of absence of magnetostatic interactions among the magnetic grains, the observed FMR spectra should be simply a linear superposition of different components [e.g., Weiss et al., 2004]. The observed FMR spectra should therefore represent mainly the signature of non-interacting SD titanomagnetite.
 The FMR spectrum of a synthetic magnetite chain sample (Figure 7) does not contain characteristics of magnetosome chain signatures even though abundant linear magnetite chains are present within the samples (Figure 1d) [Liu and Chen, 2008]. FMR parameters for this sample are: A = 0.942, geff = 2.23, and ΔBFWHM = 227 mT. Diluting this sample with eicosane consistently reduced the FMR parameters to A = 0.849, geff = 2.18, and ΔBFWHM = 222 mT (Table 1). This reduction in the FMR parameters after dilution is similar to that found by Kopp et al. [2006b]. However, despite the dilution, the FMR parameters still do not hint at a magnetite chain structure.
4.3.2. Simulation of X-Band FMR Spectra
 The three analyzed TC tuff samples have FMR derivative spectra with a conspicuous high-field double-well feature (at 400 and 460 mT) and a low-field peak at about 220 mT with more or less pronounced side lobes (Figure 6). To our knowledge, such a well-resolved high-field double-well feature has not been observed in published FMR derivative spectra from other geological samples. To study the possible origin of this feature and other FMR signatures, we simulated FMR spectra for Fe3-xTixO4 crystals (x = 0.1, TM10). Schlinger et al.  did not observe any obvious unidirectional orientation of precipitated TM10 microcrystals, therefore we here computed powder spectra by isotropically averaging over all directions in space. It is also not known if the titanomagnetite crystals in the TC tuff are elongated along a preferred crystallographic axis [see Schlinger et al., 1991]. We therefore considered three families of preferred elongation axes: 〈100〉, 〈110〉, and 〈111〉.
 For simulated FMR spectra of magnetically non-interacting TM10 assemblages with different elongations and along different crystallographic directions (〈100〉, 〈110〉 and 〈111〉), the low-field peak consistently shifts with increasing particle elongation to lower field values (Figure 8). The more elongated the particles, the smaller is the external magnetic field required to produce resonance for the specific orientation where the external field is parallel to the long particle axis. The low-field peak for particles elongated along the 〈110〉 and 〈111〉 crystallographic axes splits into double peaks with increasing particle elongation. Two low-field peaks are commonly seen in FMR derivative spectra for intact magnetosome chains [Kopp et al., 2006a, 2006b; Fischer et al., 2008], but their spectra lack the distinct double-well feature on the high-field side of the spectrum. For an axial ratio ofq ∼ 2 (with elongation along the 〈111〉 axis), the double-well positions areB = 410 mT and 470 mT, as is the case in the experimental spectra. Particles elongated along the 〈110〉 axis also produce a double-well structure, albeit at different field values. Despite the good agreement in terms of the double-well feature, there are significant deviations between simulations and experimental data below 400 mT. In particular, the experimentally observed low-field peak at 200 mT is absent in simulations that correctly reproduce the double-well feature. Absence of the 200 mT peak in an otherwise compatible spectrum rules out elongation along the 〈111〉 axes of the TM10 particles in the TC Tuff [Schlinger et al., 1991]. Likewise, a single peak at 200 mT is present in the q ∼ 4.4 spectrum for the 〈100〉 case, however, the high-field spectrum has no suitably located double-well feature. None of the simulated spectra can explain all of the features observed in the experimental data and it is possible that the particles in the TC Tuff are elongated along a crystallographic axis that is different to the three cases simulated here. Use of high-resolution TEM observations would be ideal to resolve this question. Alternatively, in the absence of a single preferred elongation direction, the observed spectra probably represent a more complicated mixture. For example, the experimental spectra have a shallow slope around the effective field (when crossing zero), which is typical of mixture. We also modeled mixtures of grains with different elongation directions and SP grains, which did not give results comparable to the experimental data. This is probably due to the complexity of the studied TC tuff samples, which contain multiple components (different grain elongations and elongation axes). Nevertheless, our modeling results indicate that the lowergeffvalues of the TC tuff are probably due to significant particle elongation and lack of magnetostatic interactions. The high-field double well feature likely develops due to low-field splitting at significant grain elongation as a result of averaging of simulated spectra over all directions in space (Figure 8). Our modeling indicates that FMR spectra for the TC tuff cannot be fully explained by a comparatively simple magnetic distribution. This is probably because there is not a single-preferred crystallographic direction for particle elongation in the TC tuff samples. This indicates that the TC tuff is still a complicated FMR system, even though it is not affected by magnetostatic interactions.
 We also simulated powder FMR derivative spectra at X-band for magnetite chains aligned along the 〈111〉 crystallographic axis (Figure 9) to compare with experimental spectra. Similar to the modeling results of Charilaou et al. , magnetite chains can produce FMR spectra with geff < 2, and A < 1, and also multiple low-field peaks. The simulations indicate that an effective demagnetization factor ΔNas small as 0.2 is sufficient to explain the high-field peak at 390 mT (compareFigures 7 and 9). However, this simulated spectrum has geff < 2, i.e., the effective field is larger than the isotropic resonance field of 336 mT at X-band. Therefore, to explain the experimentally observedgeff > 2 (effective field of 300–310 mT) (Figure 6), a second FMR component centered well below 300 mT has to be added to the synthetic spectrum for non-interacting chains. That broad component does not represent a physical component, but rather a distribution of magnetostatic interactions.
 We suggest that the FMR spectra for the TC tuff samples probably represent the complex sum of multiple components. Our modeling results indicate that it is not possible to fit the measured spectra with a single magnetic component. In addition, slight sample heterogeneity due to particle size distribution may also gives rise to high α values (Table 1) and broadening of the low-field side of the spectra (Figure 6). TM10 has a cubic structure with negative K1. An equidimensional SD assemblage of such particles with no magnetostatic interactions should therefore have A > 1 and geff > 2.12, as is the case for magnetite [e.g., Weiss et al., 2004; Kopp et al., 2006b]. Our FMR modeling indicates a geff value of 2.15 for equidimensional grains (Figure 8). TEM observations indicate that the measured TC tuff samples contain significantly elongated titanomagnetite grains (Figure 1c) [e.g., Rosenbaum, 1993; Till et al., 2011]. Such acicular morphology produces strong shape anisotropy. TEM observations also indicate good dispersion among titanomagnetite particles (Figure 1c) [e.g., Rosenbaum, 1993; Till et al., 2011], which results in nearly no magnetostatic interactions. Such a non-interacting SD assemblage with strong shape anisotropy can shift absorption to lower fields and produceA < 1 and geff < 2.12, and may also produce multiple absorption maxima, as observed in magnetotactic bacteria [e.g., Weiss et al., 2004; Kopp et al., 2006a, 2006b]. These FMR signatures are consistent with our experimental data and simulations. In contrast, basalts containing interacting titanomagnetite grains have high geff and ΔB values and A > 1 [e.g., Weiss et al., 2004]. Our FMR data indicate that the lower FMR parameters (geff, ΔB and A) for these non-interacting tuff samples compared to those in basalts are probably due to a combination of enhanced shape anisotropy and absence of three-dimensional magnetostatic interactions. While three-dimensional magnetostatic interactions (as in the case of particle clumps) broaden the FMR spectra and shiftgeff and A to higher values [e.g., Kopp et al., 2006b], absence of three-dimensional magnetostatic interactions can preserve FMR signatures withA < 1, geff < 2.12 and multiple peaks. In contrast to bacterial cells, the synthetic carbon coating surrounding the magnetite chains is too thin to keep the chains from magnetically interacting with each other. This magnetic particle system of closely spaced chains should be seen as clumps, which therefore has both strong shape anisotropy and three-dimensional magnetostatic interactions. Our measurements of this sample demonstrate that three-dimensional magnetostatic interactions can apparently mask the FMR signature of magnetosome chains, which supports previous observations byWeiss et al.  and Kopp et al. [2006a].
 We conclude that the observed multiple absorption maxima with A < 1 and geff < 2.12 are due to a combination of strong shape anisotropy and lack of three-dimensional magnetostatic interactions. Enhanced shape anisotropy can be attributed to particle elongation effects (in the case of TC tuff samples) or to one-dimensional magnetostatic interactions due to chain structures (in the case of the intact magnetosomes). Absence of three-dimensional interactions is a result of good dispersion of magnetic particles within the TC tuff samples or to separation of magnetosome chains by bacterial cells or other materials. The overlap of some FMR signatures (i.e.,A < 1 and geff < 2.12) between magnetotactic bacteria [e.g., Weiss et al., 2004; Kopp et al., 2006a, 2006b, 2007, 2009; Kind et al., 2011; Gehring et al., 2011a; Roberts et al., 2011a] and other types of samples analyzed here suggests that care is needed when using FMR spectroscopy to detect magnetosome chains within samples. Despite the large overlap in A and geff values that may cause interpretational ambiguity, there is a large difference in Beff values for magnetosome chains compared to other samples (Table 1). For example, the TC tuff samples have much larger Beff values compared to those of intact magnetite magnetosomes [e.g., Weiss et al., 2004; Kopp et al., 2006a, 2006b, 2007, 2009; Fischer et al., 2008; Roberts et al., 2011a]. These distinct FMR signatures should therefore enable discrimination between the different types of magnetic particle assemblage. Our analysis also demonstrates that intact magnetosomes produced by magnetotactic bacteria are a unique model system with distinct FMR signatures [Weiss et al., 2004; Kopp et al., 2006a, 2006b; Charilaou et al., 2011].
5. Summary and Conclusions
 The shape of FMR spectra and the values of FMR parameters (such as geff, ΔBFWHM and A) are sensitive to magnetic mineralogy, magnetic anisotropy (magnetocrystalline, magnetoelastic and shape anisotropy due to crystal morphology and/or to the spatial arrangement of particles), magnetic mineral grain size (SP/MD effects), and magnetostatic interactions. We plot our measured FMR parameters in plots of geff versus A (Figure 10a) and ΔBFWHM versus A (Figure 10b) following Weiss et al.  and Kopp et al. [2006a, 2006b]. In a plot of geff versus A (Figure 10a), some diagenetic greigite samples and the TC tuff samples fall within the region for magnetite magnetosome chains (i.e., A < 1 and geff < 2.12). On the other hand, data for the measured samples are more scattered in plots of ΔBFWHM versus A in which values of the parameter α are also shown (Figure 10b). α is an empirical parameter that combines the FMR parameter A and ΔBFWHM. It is defined as α = 0.17A + 9.8 × 10−4 ΔBFWHM [Kopp et al., 2006a, 2006b], and is an empirical proxy for σ, the Gaussian broadening factor [Kopp et al., 2006b]. Low values of α and σ imply more homogeneous size, shape and arrangement of particles, which Kopp and Kirschvink  interpreted as a fingerprint of natural selection on the biologically controlled mineralization of magnetosomes. Our results support the hypothesis of Kopp et al. [2006a, 2006b] that the FMR parameters of geff < 2.12, A < 1 and α < ∼0.3 strongly suggest the presence of magnetofossils in sediment samples.
 Various effects (magnetostatic interactions, domain state and crystal morphology) on FMR parameters are illustrated in Figure 10. In summary, enhanced positive anisotropy (originating either from grain elongation or chain structure) increases ΔBFWHM and decreases geff and A.Three-dimensional magnetostatic interactions produce highergeff and ΔBFWHMvalues. This is because three-dimensional magnetostatic interactions affect the effective field and also cause Gaussian line broadening due to the heterogeneity of local magnetic environments produced by interactions [Kopp et al., 2006b]. Compared to SD grains with the same composition, SP grains with relaxation times smaller than the Larmor precession period have geff values close to their intrinsic value and A close to 1, which can be explained by thermally induced magnetization fluctuations averaging out the magnetic anisotropy field over a precession period [Sharma and Baiker, 1981]. The lack of anisotropy in SP grains also reduces ΔBFWHM. The linewidth of SP materials broadens with decreasing temperature because of the diminishing influence of thermal fluctuations. MD grains tend to increase geff, A and ΔBFWHM values due to absorption of microwave energy by domain walls at relatively low field strengths where domain walls still exist. Sample heterogeneity normally increases geff and ΔBFWHM, but seems not to affect A [Kopp et al., 2006a]. It should be noted that the effects of both three-dimensional magnetostatic interactions and domain structure on the FMR line shape are not well understood yet. More work is needed to better explain these effects.
 FMR spectra for greigite are broad with linewidths between ∼180 and 240 mT at X-band at room temperature. Pure synthetic PSD/MD greigite samples have highgeff values (2.9–3.1) and asymmetry ratio A > 1 (1.2–1.4). MD grains tend to decrease the effective absorption field (i.e., increase geff, see paragraph above). Sedimentary (diagenetic) greigite produces lower geff values (2.0–2.2), lower linewidths, and A values less than 1 (∼0.7–0.9), which sometimes fall within the range of some parameters that have been used as indicators of the presence of magnetite magnetofossils (A < 1 and geff < 2.12). geffvalues are almost indistinguishable for all of the measured greigite samples at Q-band (2.08–2.12). Compared to greigite, pyrrhotite has strong uniaxial anisotropy, which requires very high frequency (e.g., W-band) to acquire a full FMR absorption spectrum. This is consistent with the weak measured FMR signals at X- and Q-bands, which probably reflect effects of magnetic domain walls, defects, and impurities, rather than intrinsic properties of pyrrhotite. FMR fingerprints at X-band may be diagnostic of pyrrhotite and greigite.
 The TC tuff samples with non-interacting titanomagnetite grains give rise to multiple local absorption maxima and minima with lowergeff (2.03–2.06), ΔBFWHM (232–243 mT) and Avalues (0.66–0.78) than magnetically interacting samples. These FMR features can be predicted by modeling elongated TM10 grains. Measurements of synthetic magnetite chains do not give rise to FMR signatures that are characteristic of magnetosome chains although magnetite chains are present. Three-dimensional magnetostatic interactions among the magnetite chains can apparently mask the chain signal. Our FMR data demonstrate thatgeff, ΔBFWHM and A decrease with decreasing magnetostatic interactions. The multiple absorption maxima in the FMR spectra with A < 1 and geff < 2.12 result from a combination of enhanced shape anisotropy effects and a lack of magnetic interactions. Our results demonstrate that samples with asymmetric FMR spectra with extended absorption at low fields need not necessarily contain magnetosome chains. Despite the large overlap in a range of FMR signatures between intact magnetite magnetosome chains and the samples analyzed here, magnetite magnetosome chains still have some distinct FMR characteristics (i.e., ΔBFWHM and α values). Intact magnetosome chains therefore remain a rather unique FMR model system [Weiss et al., 2004; Kopp et al., 2006a, 2006b; Charilaou et al., 2011]. Our results support previous studies that concluded that FMR analysis provides a powerful tool for identifying magnetite magnetosome chains within samples [e.g., Weiss et al., 2004; Kopp et al., 2006a, 2007, 2009; Fischer et al., 2008; Faivre et al., 2010; Kind et al., 2011; Roberts et al., 2011a; Gehring et al., 2011a].
 While FMR analysis is useful for characterizing magnetic minerals in natural samples, care is needed when interpreting results because of the overlapping ranges of FMR parameters for different magnetic minerals with different properties. It is therefore not straightforward to interpret FMR signatures because of these overlapping effects. Ambiguity may also arise due to the complexity of natural magnetic samples. Nevertheless, the advantages of FMR analysis, such as its rapidity of measurement (it typically takes only several minutes to measure a sample) and inexpensiveness, make it possible to screen large numbers of samples to provide valuable magnetic information about samples.
 All FMR spectra were measured at the EPR National Service Centre at the University of Manchester, which is funded by the UK Engineering and Physical Sciences Research Council (EPSRC), and at the Research School of Earth Sciences, Australian National University. We thank Joanna Wolowska and Rainer Grün for help with measurements. We thank Mike Jackson at the Institute for Rock Magnetism for providing the TC Tuff samples. We are grateful to Bob Kopp, an anonymous reviewer and the Associate Editor for comments that significantly improved this paper. This work was partially funded through U.K. Natural Environment Research Council (NERC) grant NE/G003319/1 to APR.