By continuing to browse this site you agree to us using cookies as described in About Cookies
Notice: Wiley Online Library will be unavailable on Saturday 7th Oct from 03.00 EDT / 08:00 BST / 12:30 IST / 15.00 SGT to 08.00 EDT / 13.00 BST / 17:30 IST / 20.00 SGT and Sunday 8th Oct from 03.00 EDT / 08:00 BST / 12:30 IST / 15.00 SGT to 06.00 EDT / 11.00 BST / 15:30 IST / 18.00 SGT for essential maintenance. Apologies for the inconvenience.
We present a highly sensitive and accurate method for quantitative detection and characterization of noninteracting or weakly interacting uniaxial single domain particles (UNISD) in rocks and sediments. The method is based on high-resolution measurements of first-order reversal curves (FORCs). UNISD particles have a unique FORC signature that can be used to isolate their contribution among other magnetic components. This signature has a narrow ridge along the Hc axis of the FORC diagram, called the central ridge, which is proportional to the switching field distribution of the particles. Therefore, the central ridge is directly comparable with other magnetic measurements, such as remanent magnetization curves, with the advantage of being fully selective to SD particles, rather than other magnetic components. This selectivity is unmatched by other magnetic unmixing methods, and offers useful applications ranging from characterization of SD particles for paleointensity studies to detecting magnetofossils and ultrafine authigenically precipitated minerals in sediments.
Single domain (SD) magnetic particles have been the subject of intense research in material sciences and in paleomagnetism and environmental magnetism for more than 70 years because of their stability as remanent magnetization carriers. Strictly quantitative theories on recording the Earth's magnetic field in rocks exist only for noninteracting or weakly interacting SD particles [Shcherbakov and Sycheva, 1996]. Therefore, rocks and single crystals containing such particles as the predominant magnetic phase are ideal for paleointensity studies [Tarduno et al., 2006]. Ultrafine magnetic particles with grain size distributions typically covering the superparamagnetic (SP) and SD range also form authigenically in sediments and soils where they can be an important source of a stable natural remanent magnetization (NRM), as well as being a paleoenvironmental tracer. Since discovery of biomineralized magnetite in Chiton teeth [Lowenstam, 1962], in magnetotactic bacteria [Bellini, 1963; Blakemore, 1975; Frankel et al., 1979] and by dissimilatory iron reducing bacteria [Lovley et al., 1987], biogenic magnetic minerals have been found in sediments and sedimentary rocks globally. Magnetotactic bacteria are of particular interest because they synthesize chains of SD crystals (magnetosomes) with extremely well-controlled sizes and shapes. Magnetosomes can be preserved over geologically significant times, if they survive diagenesis, in which case they are called magnetofossils [Kirschvink and Chang, 1984]. Magnetofossils have been found in a variety of marine and freshwater sediments and sedimentary rocks [e.g., Petersen et al., 1986; Chang et al., 1987; Snowball, 1994], where they can contribute >60% to the saturation remanent magnetization [Egli, 2004a, 2004b]. Magnetofossils are also of interest in paleoenvironmental studies because they provide information about past geochemical conditions that favored growth of magnetotactic bacteria and/or that controlled the preservation or dissolution of magnetosomes [e.g., Hesse and Stolz, 1999].
When magnetosomes are preserved over geological times they become exceptionally stable NRM carriers provided that the chain structure is prevented from collapsing [Kirschvink, 1982; McNeill and Kirschvink, 1993; Shcherbakov et al., 1997; Kobayashi et al., 2006]. The importance and efficiency of magnetofossils as NRM carriers, along with the physical process they record, are largely unknown. A depositional remanent magnetization model suggests that aggregation of ultrafine magnetic particles during deposition does not lead to formation of chain structures [Shcherbakov and Sycheva, 2008]. This implies that isolated chains of magnetite particles in sediments are exclusively of bacterial origin. On the other hand, nonisolated magnetic chain structures occur in silicate-hosted (titano)magnetite inclusions, where the host crystal provides a template for their growth [Feinberg et al., 2006], and in heated Fe-rich carbonates [Golden et al., 2001]. Such structures are, however, magnetically distinct from biologically produced chains.
An exact FORC model for Stoner-Wohlfarth (SW) particles [Stoner and Wohlfarth, 1948] has been elaborated by Newell , followed by theoretical analysis of weak magnetostatic interactions in diluted dispersions of such particles [Egli, 2006a]. These works provide the basis for understanding the FORC signatures of well dispersed SD particles. One of these signatures, consisting of a narrow ridge concentrated on the Hc axis of a FORC diagram, is expected to be clearly identifiable even in complex mixtures with other magnetic components. This signature has sometimes been recognized in natural samples [e.g., Roberts et al., 2000; Yamazaki, 2008; Abrajevitch and Kodama, 2009] and cultured magnetotactic bacteria [Li et al., 2009], without further quantitative interpretation, but more often than not it has been overlooked, due to insufficient measurement resolution or incorrect data processing.
We present a precise FORC method for quantifying noninteracting or weakly interacting uniaxial SD particles, hereafter collectively referred to as UNISD particles. Although FORC measurements are not particularly rapid, they are completely automated by the software that controls the Micromag® vibrating sample magnetometer (VSM) and the alternating gradient magnetometer (AGM). This method can therefore be used to calibrate rapid magnetic characterization methods for analysis of larger numbers of samples. Importantly, we believe our approach provides a uniquely diagnostic method for characterizing UNISD particles, which opens up new opportunities for detecting fossil magnetosome chains in sediments.
2. Magnetic Methods for Magnetofossil Identification
The simplest nondestructive methods for magnetofossil identification are based on two bulk magnetic parameters: the ratio between anhysteretic remanent magnetization (ARM) and isothermal remanent magnetization (IRM) or low field susceptibility χ, and the crossover Raf between remanent acquisition and alternating field (AF) demagnetization curves [Cisowski, 1981; Moskowitz et al., 1993; Snowball et al., 2002]. ARM ratios are sensitive domain-state indicators, with highest values for SD particles (typically χARM/IRM > 1 mm/A and χARM/χ > 10), while crossover values close to 0.5 are characteristic of noninteracting SD particles and magnetosome chains. The disadvantage of these methods is that bulk magnetic parameters do not allow unique interpretation with nonnegligible additional contributions from other magnetic components.
A specific test for the presence of magnetite magnetofossils is based on the distinctive Verwey transition features of linear chains of SD magnetite particles. The ratio δFC/δZFC enables characterization of magnetization losses across the Verwey transition using thermal demagnetization curves of field-cooled (FC) and zero-field-cooled (ZFC) induced remanence, and is systematically higher for magnetosome chains (δFC/δZFC > 2) than for other magnetite particles of any domain state [Moskowitz et al., 1993; Carter-Stiglitz et al., 2002]. Mixture models can be used to convert δFC/δZFC into a magnetosome fraction [Moskowitz et al., 1993; Kim et al., 2005; Housen and Moskowitz, 2006]. Unfortunately, δFC/δZFC is reduced by low-temperature oxidation of magnetite and the chain concentration may be underestimated [Smirnov and Tarduno, 2000; Passier and Dekkers, 2002].
Precise magnetosome shape and volume constraints mean that magnetofossil chains are characterized by a narrow distribution of switching fields, which is not matched by other known natural magnetic components. This property has been exploited by coercivity analysis [Egli, 2003, 2004a] and ferromagnetic resonance (FMR) spectroscopy [Weiss et al., 2004; Kopp et al., 2006]. With coercivity analysis, high-resolution AF demagnetization curves of ARM and IRM are used to calculate the corresponding switching field distributions, which are fitted using appropriate model functions. Each model function is identified with a magnetic component for which ra = χARM/IRM is calculated. In most cases, ra of magnetofossil components is close to theoretical values for SD magnetite [Egli and Lowrie, 2002; Egli, 2004a]. This suggests that magnetofossils or magnetofossil chains must be extremely well dispersed in the sediment matrix, given the sensitivity of ARM to magnetostatic interactions [Egli, 2006b]. Therefore, magnetofossil-related UNISD signatures are expected to be common in sediments. Coercivity analysis gives strictly quantitative results; however, it is time consuming and becomes unstable with increasing numbers of magnetic components. FMR spectra can be analyzed in a manner similar to magnetization curves, by isolating different anisotropy contributions and fitting spectra with model curves. These methods have limitations when interpreting complex magnetic mixtures, which are overcome with FORC measurements. We propose a suitable FORC measurement protocol for samples containing UNISD particles, and provide a theoretical framework for interpreting the corresponding FORC diagrams.
3. FORC Measurement and Data Processing Protocol
3.1. FORC Diagrams
FORCs are partial hysteresis curves that originate from the same branch of the major loop, which is the descending branch in most measurement protocols. To measure a single FORC, a positive saturation field, Hs, is first applied and it is then decreased to a so-called reversal field, Hr. Starting from Hr, the field is increased in steps of size δH and the in-field magnetization M(Hr, H) is measured at each field step H. A family of N ascending FORCs consists of a collection of these partial hysteresis curves starting with different Hr values that are offset by δH from one curve to the next. The FORC function,
[Wilde and Girke, 1959; Mayergoyz, 1986; Pike et al., 1999], is defined on a half plane, called FORC space, that is occupied by the measurement coordinates (Hr, H ≥ Hr). The transformed coordinates Hc = (H − Hr)/2 and Hu = (H + Hr)/2 are commonly used for graphical representation. The (Hc, Hu) coordinate system is motivated by the phenomenological Preisach theory, in which Hc and Hu correspond to the coercivity and the bias field, respectively, of elemental rectangular hysteresis loops called hysterons [Preisach, 1935]. A limitation of interpreting FORC measurements in terms of the Preisach theory is that hysteresis loops of individual particles generally cannot be approximated by rectangular hysterons [Pike and Fernandez, 1999; Newell, 2005; Dumas et al., 2007].
High-precision FORC models have been developed for UNISD particles [Newell, 2005], and for weak dipolar interactions [Egli, 2006a]. The FORC function of UNISD particles can be effectively decomposed into (1) an infinitely sharp ridge ρcr(Hc, Hu = 0) concentrated along the Hc axis, to which we refer as the central ridge, and (2) a continuous function ρur(Hc, Hu < 0) that is antisymmetric about the Hr axis and negative in the lower left-hand domain of FORC space [Newell, 2005]. The central ridge corresponds to irreversible magnetization changes (i.e., moment switching of individual particles), while the continuous part represents differences in reversible magnetization changes that depend on Hr (i.e., rotation of magnetic moments in the applied field). The central ridge can be broadened by thermal relaxation effects [Egli, 2006a]. Other effects of thermal activations, observed in samples containing viscous particles, include a shift of the central peak to lower coercivities and the onset of a positive contribution near the Hu axis [Pike et al., 2001a].
Magnetostatic interactions convert the central ridge into a function of finite width [Pike et al., 1999]. The case for SD particles homogeneously diluted in a nonmagnetic matrix is well understood and has been modeled analytically [Egli, 2006a]. FORC functions of such particles are roughly symmetric about the Hc axis, and have typical teardrop shaped contours [Pike et al., 1999; Egli, 2006a]. Strong interactions, on the other hand, are difficult to model because interaction fields depend on the magnetization [Muxworthy and Williams, 2005]. A variety of FORC diagrams with oval contours is obtained in this case [Pike et al., 1999]. Dense magnetosome aggregates, such as those originating from chain collapse or magnetic extraction, are characterized by a FORC diagram with oval or teardrop shaped contours that extend well into the Hu ≥ 0 region by up to 60 mT [Chen et al., 2007]. FORC diagrams of sediments containing SD greigite usually have strong interaction features, often accompanied by a downward shift of the central maximum [Roberts et al., 2000, 2006; Rowan and Roberts, 2006; Florindo et al., 2007; Vasiliev et al., 2007]. FORC diagrams of pseudo–single domain (PSD) or multidomain (MD) particles have a large spread along Hu, and are roughly reflection symmetric about the Hc axis [Roberts et al., 2000; Pike et al., 2001b; Muxworthy and Dunlop, 2002]. A narrow ridge along the Hc axis has never been observed for such particles, which supports the conclusion that it is a unique feature of SD particles.
The FORC signature of authigenically precipitated ultrafine magnetite or greigite depends on how well these particles are dispersed in the sediment matrix, and is largely unknown. There is some evidence that pedogenic magnetite is not affected by strong magnetostatic interactions and could contribute to a narrow ridge centered along the Hc axis [Geiss et al., 2008].
Fossil magnetosome chains can be expected to occur in isolated form in the sediment matrix, because the magnetotactic bacteria cell body should have prevented the formation of clusters, providing enough distance to make magnetostatic interaction effects negligible, as observed in cultured magnetotactic bacteria samples [Moskowitz et al., 1993]. Isolated chains of mature magnetosomes are magnetically equivalent to UNISD particles because all crystals in a chain switch at the same critical field [Penninga et al., 1995; Hanzlik et al., 2002]. An exception is represented by magnetotactic bacteria containing magnetosome chain bundles [Hanzlik et al., 2002] or magnetosome clusters [e.g., Faivre and Schüler, 2008, Figure 1c]. Sediments containing abundant magnetofossils are thus expected to display all FORC signatures of UNISD particles. This is confirmed for the first time in Figure 1 on a sample from Lake Ely (Pennsylvania) [Kim et al., 2005].
Suitable measurement parameters must be chosen to correctly resolve the FORC signatures of UNISD particles. The FORC acquisition procedure is automated by the Micromag® software that controls the VSM or the AGM. At the start of the experiment, the user is prompted to input: the saturating field Hs, the Hu range (given by “Hb1” and “Hb2”), the Hc range (given by “Hc1” and “Hc2”), the averaging time, the field increment δH, the number N of FORC curves to be measured, and other parameters that are not discussed here. Best choice of the measurement parameters depends on the sample. Given the low concentration of magnetic minerals in typical sediments, it is important to select the smallest possible measurement range. Typical averaging times are between 0.2 and 1 s. Increasing the averaging time helps to reduce measurement noise, except for the noise deriving from the electromagnets, but it also increases instrumental drift effects. Therefore, averaging multiple FORC runs is more effective than increasing the averaging time in case of particularly weak samples. Care should be taken to avoid drift artifacts, which are particularly pronounced during the first 20 min of instrument operation and are not completely removable by data processing.
The most critical parameters are the Hc and Hu ranges, which determine the FORC space covered by the measurement, and the field increment δH (Figure 2). If c is the largest switching field of interest, and if the FORC diagram is expected to extend by u above the Hc axis, sufficient FORC space coverage with minimum amount of measurements is obtained by choosing Hc1 = 0, Hc2 ≥ c, Hb1 = −c − u, and Hb2 = u. The last two important measurement parameters are the field increment δH and the number N of FORC curves. The Micromag® software automatically calculates either of these two parameters depending on which one the user keeps fixed. The choice of δH dictates the smallest details that can be resolved in FORC space, and is of particular importance for resolving the central ridge. Measurements with resolution δH > 0 unavoidably widen the central ridge. This effect, which also depends on the smoothing factor SF chosen for data processing, must be maintained within acceptable limits, thereby constraining δH to the values listed in Tables 1 and 2.
Table 1. Optimal Choice of FORC Parameters to Measure the Central Ridgea
SF = 3
SF = 4
SF = 5
SF = 6
SF = 7
Maximum switching field sw = 110 mT and FORC resolution ΔH = 2.5 mT.
Hc1, Hc2 (mT)
Hu1, Hu2 (mT)
Estimated measurement time (hours)
Table 2. Optimal Choice of FORC Parameters to Measure the Entire FORC Functiona
SF = 3
SF = 4
SF = 5
SF = 6
SF = 7
Maximum switching field sw = 110 mT and FORC resolution ΔH = 2.5 mT.
Hc1, Hc2 (mT)
Hu1, Hu2 (mT)
Estimated measurement time (hours)
3.3. FORC Data Processing
To address the specific processing requirements of FORC diagrams related to UNISD particles, the FORC processing code written in MATLAB® [Winklhofer and Zimanyi, 2006] was extended to a new version called “UNIFORC.” In UNIFORC, the mixed derivative of equation (1) is calculated either by polynomial fits over squared arrays of (2SF + 1) × (2SF + 1) measurement points, or directly by finite differences. In the latter case, finite differences are filtered by averaging over the same square arrays (Figure 2). The direct method based on finite differences is less prone to artifacts when dealing with noisy data.
The mixed derivative calculation is particularly critical near the edges of the FORC space (e.g., the Hu axis). Edges are problematic for numerical differentiation methods requiring a rectangular set of data points, which is, by nature, incomplete near H = Hr (Figure 2). Pike  suggested extending each FORC into H < Hr (magnetization-extended FORC) by extrapolating a constant magnetization (Figure 3). The ideal magnetization-extended FORC function contains an infinitely sharp ridge, called the reversible ridge, on the Hu axis, which accounts for reversible magnetization processes that are not recorded in FORC space. The finite resolution of real measurements, however, shifts this ridge into FORC space, where it often overshadows low-coercivity FORC contributions (e.g., Figure 4a). This effect is particularly evident in samples where reversible magnetization processes are dominant, which leads to the widespread practice of clipping the region near Hc = 0 [e.g., Roberts et al., 2006]. This problem is avoided in UNIFORC by assuming that each FORC is point symmetric about H = Hr. A “slope extended” FORC, defined as
does not inflect sharply at H = Hr, and the reversible ridge is removed from FORC space. The information conveyed by this ridge can be plotted separately [Winklhofer et al., 2008].
3.4. FORC Measurements for Two Sediment Samples
FORC diagrams for two natural samples illustrate our approach. They include a sediment from Lake Ely (Pennsylvania), which is known to contain abundant magnetofossils [Kim et al., 2005], and a marine greigite-bearing sediment recovered off the southwestern Taiwan coast (core ORI-758 GH3, 22°16.8N, 119°48.6E, see Horng and Chen  for description of cores from the same site). The three FORC diagrams shown in Figure 4 were obtained from the same Lake Ely sample, and, with the exception of the Hu range, Hc range, and δH, all other measurement parameters were the same. The main feature of the FORC diagrams is a distribution centered on the Hc axis whose width along Hu is different in each case. This difference does not come from the choice of the Hu and Hc ranges or from data processing, which are identical in Figures 4b and 4c. This demonstrates the role of δH in resolving the central ridge. FORC measurements in Figure 4 were processed using the magnetization extended method discussed in section 3.3, in which the reversible ridge encroaches into the low-Hc region of FORC space: this effect is particularly evident with low measurement resolution (Figure 4a). FORC diagrams shown in Figure 5 were obtained from the Taiwan greigite-bearing sample. In this case, unlike the Lake Ely sample, a more than fivefold difference in δH has negligible effect on the shape of the FORC diagram, except for the amplitude of measurement noise.
Why should the FORC function width along Hu be controlled by δH for the Lake Ely sample but not for the Taiwan greigite-bearing sample? This question can be answered by inspecting a plot of the full width at half maximum (FWHM) of the Hu profile taken at the FORC distribution peak, versus ΔH = δH × (SF + 1/2) (Figure 6). FWHM for the Lake Ely sample is proportional to ΔH over the entire range of values and converges to zero as ΔH → 0. This indicates that the true FWHM along Hu is zero or almost zero for UNISD particles. On the other hand, FWHM for the Taiwan greigite-bearing sample converges to 19.5 mT as ΔH → 0, which is the intrinsic width of the FORC function. It is evident from the Lake Ely example that ΔH is the effective resolution of the FORC diagram, which corresponds to the size of the smallest detail that can be resolved with the user-chosen δH and SF. Note that ΔH does not depend on the density of the final grid of (Hc, Hu) coordinates used to represent the FORC function, provided that this is not coarser than the original measurement field increments.
Insufficient measurement resolution not only leads to overlooking the UNISD central ridge, but also to incorrect interpretation. For example, the FORC diagram in Figure 4a could be interpreted as a signature of weakly interacting SD particles. While SF optimization has been discussed [Heslop and Muxworthy, 2005; Harrison and Feinberg, 2008], the importance of δH has not previously been emphasized. The ideal case of ΔH = 0 is not measurable, due to a lower limit for δH imposed by the finite field control precision (≈10 μT ± 0.1% of c for the Princeton Measurement Corporation VSM or AGM). A minimum resolution of 2.5 mT is necessary for correct characterization of the central ridge, as discussed in section 4.6. SF values for sedimentary samples usually range from 3 to 7, therefore δH should not exceed 0.3–0.8 mT (Tables 1 and 2).
After identifying a central ridge in the Lake Ely sample, we performed high-resolution measurements to obtain full coverage of the FORC space (Figure 1). Although Figure 1 is a mosaic of measurements that focus on different domains of the FORC space (for reasons that became obsolete, see the auxiliary material), the same result can be obtained using the measurement parameters listed in Table 2. The sample is magnetically weak; therefore, four identical runs were averaged to improve the signal-to-noise ratio. Individual FORC data sets were preprocessed using UNIFORC and merged into a single diagram. All features of the FORC model calculated by Newell  are clearly distinguishable in Figure 1: a central ridge on the Hc axis and a continuous function over Hu < 0 that is negative in the lower left-hand part of the diagram. Positive values of the FORC function above the Hc axis reflect magnetic contributions in addition to UNISD particles. The teardrop shaped contours are a typical feature of weakly to moderately interacting SD particles [Pike et al., 1999; Egli, 2006a]. Nevertheless, an unquantifiable contribution of larger lithogenic particles cannot be excluded. Contributions from these non-UNISD particles suppress the perfect symmetry of the Hu < 0 region about the Hr axis (Figure 1). This can be seen by following the departure of the white region between positive and negative contributions from the Hu = −Hc diagonal. We model the FORC diagram of Figure 1 as the sum of three distinct contributions: (1) a central ridge ρcr on the Hc axis, produced by irreversible processes associated with UNISD particles; (2) a contribution ρur from reversible processes in the same particles, which is antisymmetric about the Hr axis and zero for Hu > 0; and (3) a contribution ρother from other magnetic components, which is roughly symmetric about the Hc axis. Interpretation of these contributions is discussed below.
4. Analysis of a FORC Diagram Dominated by SD Particles
The general model described in this section for the FORC distribution of UNISD particles enables establishment of a precise link to other magnetic measurements. This is useful for obtaining the switching field distribution and the saturation remanence of UNISD or weakly interacting SD particles, even in samples with mixed magnetic components. Application of this theory for identifying SD minerals, and for discriminating among magnetofossils and authigenically precipitated ultrafine magnetic particles, is thoroughly discussed in a companion paper (R. Egli et al., Magnetic characterization of magnetofossils and ultrafine magnetic particles of diagenetic origin, manuscript in preparation, 2010). Proofs of all equations are provided in the auxiliary material.
4.1. FORC Function for Strictly Noninteracting, Uniaxial SD Particles
The FORC function for UNISD particles calculated by Newell  using the hysteresis model of Stoner and Wohlfarth  can be generalized for uniaxial SD particles requiring more complex models (e.g., including thermal activation effects, or fanning reversal modes for linear particle chains). We consider a sample containing a large number of UNISD particles characterized by a switching field distribution f(Hsw), where Msf(Hsw)dHsw is the contribution of all particles with switching field between Hsw and Hsw + dHsw to the sample's saturation magnetization Ms. Because f(Hsw) > 0 and its integral over all switching fields is equal to 1, we can formally consider the switching field distribution as a probability density function (PDF). The hysteresis loop for all particles with the same switching field, normalized by MsdHsw, is composed of an upper and a lower branch, m+(H; Hsw) and m−(H; Hsw), with discontinuities of amplitude s(Hsw) at H = −Hsw and H = +Hsw, respectively. Because of inversion symmetry, the two branches can be expressed by the same function m = m+, with m−(H) = −m+(−H). Unlike remanent magnetization measurements, where the change in remanence produced by switching is proportional to the particle moment, in-field magnetization jumps depend in detail on the switching mechanism. Magnetization jumps can be negative, acting against the applied field, as for the case of SW particles whose easy axis is almost at right angles to the applied field [Newell, 2005, Figure 4].
The FORC function of the particle assemblage described above is given by ρ = ρcr + ρur, with the central ridge:
where δ is the Dirac delta function. The reversible contribution is given by:
where θ is the Heaviside unit step function and m′ is the derivative of m with respect to H. The contribution to the FORC function of all particles with switching field Hsw is concentrated on a diagonal line connecting the FORC coordinates (Hc = 0, Hu = −Hsw) on the Hu axis with (Hc = Hsw, Hu = 0) on the Hc axis (e.g., a line of points in Figure 2). The continuous part of the FORC diagram along this diagonal is the antisymmetric function ρur(−Hsw, H) of H with −Hsw < H < +Hsw. In the SW model of Newell , ρur diverges at both ends of the diagonal (i.e., H = ±Hsw), which reflects the fact that m′(H; Hsw) becomes infinite as the discontinuity at −Hsw is approached on the upper branch of the loop. This is not the case for real SD particles because switching is assisted by thermal activations and occurs where the slope of m(H; Hsw) is finite.
4.2. Interpretation of the FORC Function in Terms of a Switching Field Distribution
The central ridge in equation (3) is proportional to both the switching field distribution f(Hc) and the magnetization jump amplitude s(Hc). The proportionality to f(Hc) is of interest because it allows interpretation of the central ridge in real FORC measurements (e.g., Figure 1) as the switching field distribution of UNISD particles, while advantageously excluding contributions from non-SD or interacting particles. Coercivity analysis can then be used to identify different magnetic components, as is done with switching field distributions obtained from remanent magnetization curves [e.g., Egli, 2003, 2004a]. The important difference between FORC-based and other coercivity analyses is based on the ability to discriminate UNISD particles.
The approach discussed above is possible only if we can show that s(Hc) in equation (3) can be considered a constant. This is not obvious, because both Hsw and s are functions of the angle ϕ between the easy axis and the applied field direction; however, s is practically independent of Hc for typical switching field distributions in natural samples. A switching field distribution is the product of two intrinsic properties: (1) the distribution of easy axis orientations and (2) the distribution of particle anisotropies, expressed by their microcoercivity HK, which is related to a distribution of particle elongations. Accordingly, the switching field of one particle is written as Hsw = HKhsw(ϕ), where hsw(ϕ) is a function that accounts for the dependence of Hsw on the easy axis orientation. For SW particles, hsw(ϕ) is between 0.5 and 1, with most particles having hsw values close to 0.5 if the easy axes are randomly oriented. Given the limited range of hsw, the switching field distribution of natural particle assemblages is largely controlled by the microcoercivity distribution, which we express by a PDF K(HK) that is defined in the same way as f(Hsw).
Below, we derive a generic expression for the FORC function of UNISD particle assemblages with microcoercivity distribution K(HK) and a distribution p(ϕ) of easy axis orientations. Our only limiting assumption is that the shape of the hysteresis loop of individual particles is independent of HK, as in the SW model. A normalized function μ(h; ϕ) of the applied field h = H/HK describes the upper branch of the “elemental” hysteresis loop of any particle with easy axis orientation ϕ. The function is normalized to yield μ → ±1 for h → ±∞ (Figure 7a). The assumption that μ describes all elemental hysteresis loops will hold reasonably well for particles from the same magnetic component (e.g., ultrafine magnetite or magnetosome chains), which are expected to switch by the same mechanism (e.g., coherent rotation). We now consider a generic additive property G(ϕ) of the normalized hysteresis loop μ, such as saturation remanence μ(0; ϕ) = μr(ϕ), or amplitude S(ϕ) of the magnetization jump at h = −hsw (Figure 7a). We denote with g(Hsw) the total contribution of G(ϕ) corresponding to all particles with the same switching field Hsw. In all cases, g(Hsw) is given by:
where Ω is the range of ϕ. Randomly oriented uniaxial particles are characterized by p(ϕ) = sin ϕ with 0 ≤ ϕ ≤ π/2, where hsw(ϕ) for SW particles is given by Stoner and Wohlfarth . The function G(ϕ) is plotted in Figure 7b for examples of saturation remanence and amplitude of the magnetization jumps, using the SW model.
If K(HK) is a sufficiently wide distribution, g(Hsw) is proportional to the switching field distribution f(Hsw), and equation (5) simplifies to:
is the expected value of G(ϕ), and
is the logarithmically weighted average of hsw over all easy axis orientations. Equation (6) establishes a direct link between the microcoercivity distribution and any additive magnetic parameter g(Hsw). The importance of this link becomes clear if we evaluate equation (6) for SW particles, and three important cases of G(ϕ). The first case is G = 1, which gives the switching field distribution:
with sw ≈ 0.5829. The second case is the saturation remanence G = μr, which gives the derivative of a remanent magnetization curve:
with sw ≈ 0.5463. The third case is the amplitude G = S of the magnetization jumps of the hysteresis loops, which gives the central ridge:
with sw ≈ 0.5348 and ≈ 0.5438. The values of sw are nearly identical in the three cases: therefore, Mr′ and ρcr are both proportional to f(Hsw). Equations (9)–(11) are strictly valid only for UNISD particles: this condition is always fulfilled for the central ridge (equation (11)) because it automatically excludes contributions from other particles, but not for remanence measurements (equation (10)), which respond to all particles that hold a remanent magnetization.
The assumption of a “sufficiently wide” microcoercivity distribution can be tested by comparing equations (9)–(11) with the exact solution of equation (5) for different choices of K(HK). Since f(Hsw) is typically approximated by a logarithmic Gaussian function [Robertson and France, 1994], its width is best expressed by the standard deviation σ of log10Hsw. The maximum difference between the exact and approximated solution is 5% for σ = 0.1, which is the smallest width of a natural component [Egli, 2004a]. Therefore, equations (9)–(11) are valid for any natural UNISD particle assemblage.
A similar approach can be used to calculate the reversible component of the FORC function. Using the same assumptions, namely that the shape of the hysteresis loop of individual particles is independent of HK, and that K(HK) is sufficiently wide, we obtain:
being an odd function of −1 < x < + 1, and μ′ the derivative of μ(h). This means that profiles of ρur along diagonals of the FORC space can be expressed by a function R whose argument is scaled to match the length of the diagonal, while the amplitude is modulated by f(−Hr). An example for randomly oriented SW particles is shown in Figure 8. The shape of R(x) depends on the model used to describe the switching mechanism. It is proportional to the difference between the slope of μ(h; ϕ) evaluated at +xhsw and −xhsw (Figure 7a). The FORC diagram of UNISD particles is completely determined by two functions that reflect the intrinsic properties of the particle assemblage: (1) the switching field distribution f(Hsw), which is a scaled version of the microcoercivity distribution, and (2) a function R that depends on the switching mechanism.
4.3. Irreversible and Reversible Total FORC Contributions
The total contributions of ρcr and ρur to the FORC function convey important information about the switching mechanism of UNISD particles. We define these contributions as integrals of ρcr and ρur over the FORC space, which have the same unit as the measured FORC curves. Integration in (Hr, H) coordinates over any region of the FORC space yields a result that is twice that obtained in (Hc, Hu) coordinates. We choose (Hr, H) as the reference coordinate system, and define Icr as the integral of the central ridge, and Iur as the integral of ρur over the FORC region where it is >0 (i.e., above the Hr axis, see Figure 8b). This choice of Iur is necessary, because the integral of ρur over the entire FORC space is zero for symmetry reasons. The two integrals give the following exact result for any system of UNISD particles with any distribution of microcoercivities and any easy axis orientation:
where is defined by equation (7) with G = S, and Mrs is the saturation remanence. For example, SW particles with easy axes parallel to the applied field (i.e., the rectangular hysterons of the Preisach model), have = S(0) = 2 and Mrs = 1. Accordingly, Icr = Ms and Iur = 0. Uniaxial particles with randomly oriented easy axes are characterized by Mrs = 0.5 and Iur = Ms(1 − )/2, with the ratio
being a pure function of . Equation (15) can be used to compare predicted by hysteresis models (e.g., ≈ 0.5438 for randomly oriented SW particles) with an empirical estimate obtained from FORC measurements.
4.4. Central Ridge Widening by Magnetostatic Interactions
The distance between SD particles in a sample is finite; therefore magnetostatic interactions are expected to convert the central ridge into a function of finite width. If the effective volume concentration (or packing fraction) p is ≪2.7 μ0HK/μs, with HK and μs being the microcoercivity and the spontaneous magnetization of the particles, respectively, we have:
where the interaction field distribution W(.) is the PDF of the interaction field component parallel to the measurement direction [Egli, 2006a]. For magnetite particles with p ≤ 2%, W in equation (16) is approximated by the Lorentzian function:
where 2α ≈ 0.722 pμs is the FWHM of W [Egli, 2006a]. For example, the central ridge of a sample containing 1% homogeneously dispersed SD magnetite particles would have FWHM of ∼4.3 mT, which is slightly below the measurement resolution if the FORC diagram was measured with 1 mT field steps and SF = 5.
4.5. Central Ridge Widening by Data Processing
Measurements and data processing with a combined resolution ΔH convert ρcr into a smoothed function cr characterized by a larger FWHM. For example, the numerical FORC calculation procedure originally proposed by Pike et al. , which is based on piecewise polynomial fits over square regions containing an array of regularly spaced measurement coordinates (Hr, H), is formally equivalent to convolution with a kernel function [Heslop and Muxworthy, 2005]. The kernel function works as a derivative operator and as a low-pass filter that reduces measurement errors. The latter operation is conveniently modeled by the convolution of ρcr with a weighting window Q(x, y):
where Q(x, y) depends on the algorithm used for FORC calculation. The simplest example is a moving average window, given by Q(x, y) = 1/(2ΔH)2 for ∣x∣, ∣y∣ ≤ ΔH, and otherwise Q = 0. Other choices of Q(x, y) are possible; for example, Q(r) = (1 − r3)−3 is used by the FORCinel algorithm [Harrison and Feinberg, 2008], where r is proportional to the Euclidean distance between coordinates. If σi is the intrinsic FWHM of ρcr, and σQ is the FWHM of the weighting window, the processed central ridge has a width given by:
It can be shown by numerical simulation that σQ ≈ 0.665 δH(2SF + 1) for the polynomial fitting procedure of Pike et al.  (see auxiliary material and Figure 6). Equation (19) predicts a direct proportionality between FWHM and σQ if the central ridge intrinsic width is ≪σQ, as is the case for the Lake Ely sample, while no appreciable smoothing (FWHM ≈ σi) occurs when the intrinsic width is ≫σQ, as observed for the greigite sample. Intermediate cases are plotted in Figure 6 for comparison. A best fit of the central ridge FWHM versus ΔH for the Lake Ely sample is obtained with equation (19) when σi < 0.4 mT, which we take as the upper limit for the intrinsic width (Figure 6). Assuming magnetite composition and σi < 0.4 mT, equation (17) predicts an upper limit of only 9 × 10−4 for the volume concentration of UNISD particles. This concentration corresponds to an average distance between particles that is >10 times larger than their size.
4.6. Calculating the Switching Field Distribution From the Central Ridge
The amplitude and width of the central ridge depends on extrinsic parameters such as the measurement resolution, data processing, and particle concentration. More useful characterization of the central ridge is obtained by integrating cr over Hu. The resulting central ridge function
depends only on intrinsic properties of the particles. The central ridge function contains all physical information related to ρcr in simplest form, and is proportional to the switching field distribution. It is the only known parameter that is absolutely not responsive to non-UNISD magnetic components.
The FORC diagram for a mixture of UNISD and non-UNISD particles is the sum of the central ridge cr and the reversible contribution ρur of UNISD particles, and a continuous function ρother representing PSD, MD, or interacting SD particles. These contributions are clearly distinguishable in Figure 1. Separation of cr from other contributions is required to calculate μcr from equation (20). Measurement resolution therefore becomes critical because the wider the cr, the more difficult it is to distinguish it from a continuous background.
Rigorous separation of cr is based on evaluation of the FORC diagram around the Hc axis. Figure 9a is a vertical profile of the FORC function at an arbitrary value of Hc. This profile corresponds to idealized measurements with infinite resolution, where ρcr(Hu) is a Dirac delta function. The SW model predicts ρur to diverge at Hu = 0 [Newell, 2005] because of the infinite slope of individual particle hysteresis loops just before switching (Figure 7a). In reality, ρur is regularized by thermal activation effects over the entire SD grain size range, so we can safely assume it to be a finite function. FORC data processing, magnetostatic interactions, and thermal activations convert the ideal profile of Figure 9a into that of Figure 9b. FORC data processing has a negligible effect on the continuous background, because of its wide profile. The central ridge is characterized by a finite FWHM given by equation (19), and is significantly >0 only if ∣Hu∣ ≤ η FWHM, where η ≈ 1 depends on the exact shape of the profile. Outside of this region, the FORC diagram is unaffected by the central ridge. Correct separation of the central ridge is possible if the range of Hu values covered by cr is small enough to allow linear extrapolation of ρur + ρother toward Hu = 0 without introducing significant errors. This range should not exceed a small fraction of the typical width of ρur + ρother along Hu, and defines the required resolution of the FORC diagram. Measurement points located just outside ∣Hu∣ ≤ η FWHM can be used to extrapolate the “ridge-free” FORC function = ρur + ρother near the Hc axis. The central ridge is then obtained by subtracting from the FORC diagram. In the Lake Ely example, FWHM ≈ 2.4 mT, and points with ∣Hu∣ > 5 mT (i.e., η ≈ 1) were used to extrapolate under the central ridge (Figure 10).
The reconstructed ridge-free FORC function = ρur + ρother for the Lake Ely sample is shown in Figure 11. The central ridge, cr = ρ − , calculated by subtracting (Figure 11) from the original FORC diagram (Figure 1), is shown in Figure 12a. Finally, the ridge function was obtained by numerical integration of cr over Hu using equation (20) (Figure 12b). This result is independent of measurement resolution, provided FWHM is sufficiently small for correct isolation of ρcr. Our procedure for isolating the central ridge function is incorporated into the UNIFORC processing code.
The FORC diagram of the Lake Ely sample contains a central ridge whose intrinsic FWHM is <0.4 mT (Figure 6). We interpret this ridge as the signature of individual SD particles or linear chains of such particles that are extremely well dispersed in the sediment matrix. If these particles are magnetofossils, they must occur in the form of isolated, intact chains. This excludes contributions from magnetotactic bacteria that contain two or more adjacent chains [Penninga et al., 1995; Hanzlik et al., 2002], or magnetosome clusters [Faivre and Schüler, 2008], because of the strong magnetostatic interactions expected in such cases. Collapsed chains [Kobayashi et al., 2006] would also not contribute to the central ridge for the same reason. The existence of a central ridge testifies to the integrity of at least some of the magnetofossil chains in the Lake Ely sample. Chain integrity is presumably a prerequisite for preservation of a paleomagnetic signal carried by magnetofossils, and will be compromised by early diagenetic magnetite dissolution [e.g., Karlin, 1990; McNeill, 1990] and possibly by sediment matrix recrystallization [McNeill and Kirschvink, 1993]. The capability of intact chains to survive these processes is poorly known: experiments indicate that ultrasonic disruption of cell membranes does not lead to chain collapse [Kobayashi et al., 2006], and a rock magnetic study on sediments from the Bahamas banks suggests that intact chains must be present in the majority of the sediments [McNeill and Kirschvink, 1993]. Magnetic components compatible with nondisrupted magnetofossil chains have also been reported in two pelagic limestones [Egli, 2004a]. These interpretations were mainly based on ARM data, while FORC measurements would provide a much more powerful indication for chain integrity.
Three coercivity components can be identified in the central ridge, as suggested by the “bumps” on both sides of the central maximum (Figure 12b). Coercivity analysis of the central ridge will be discussed in detail in another paper (Egli et al., manuscript in preparation, 2010). The low-coercivity component is characterized by a mean switching field of <20 mT, which is small compared to typical values obtained from measurements of whole magnetotactic bacteria cells [Moskowitz et al., 1993; Kobayashi et al., 2006] and magnetofossil components [Egli, 2004a]. It is compatible with magnetotactic bacteria grown in the laboratory under unnatural conditions [Li et al., 2009; Carvallo et al., 2009], which are unlikely in nature. Ultrafine magnetite precipitated in aqueous solution, either inorganically [Maher, 1988], or by dissimilatory iron reducing bacteria [Lovley et al., 1987], is also compatible with a median switching field of 18–20 mT. The occurrence of such a component in the central ridge would imply that the particles are well dispersed in the sediment matrix, in contrast to laboratory products of dissimilatory iron reduction [Moskowitz et al., 1993], and synthetic clay-magnetite assemblages proposed as analogs to natural aggregates in lake and marine depositional environments [Galindo-Gonzalez et al., 2009].
The saturation remanence of UNISD particles can be estimated from Icr using equation (14) and a hysteresis model for SD particles, which allows one to calculate . If we choose SW particles with randomly oriented easy axes (Mrs = 0.5 Ms and ≈ 0.5438), we obtain Mrs = 0.447 μAm2, which is only 12% less than Mrs of the bulk sample (Table 3). However, if the UNISD component is dominated by magnetofossil chains, the SW model is not expected to be appropriate and should be replaced by alternative switching models, such as the chain of spheres fanning [Jacobs and Bean, 1955]. The integral of the central ridge-free FORC function (Figure 11) is much larger than the 12% non-UNISD contribution to Mrs obtained with the SW model, which suggests that Mrs of UNISD particles must be smaller. This implies that > 0.6 for magnetofossil components. In a companion paper (Egli et al., manuscript in preparation, 2010) we will estimate on the basis of equation (15) and compare it with models for linear magnetosome chains.
Table 3. Summary of Measured and Inferred Magnetic Contributions to the Remanent Magnetization and to the FORC Function
FORC Integral (μAm2)
Calculated from Icr using equation (14) and the SW model ( = 0.5438).
By definition of randomly oriented UNISD particles.
Iother = 4(Hc, Hu)dHcdHu, where the factor 4 includes a factor 2 for the variable transformation (Hc, Hu) → (Hr, H) and another factor 2 for the extrapolation of ρother below the Hc axis, based on the assumption that ρother is an even function of Hu.
We have developed a FORC measurement and data processing protocol for characterizing noninteracting or weakly interacting uniaxial SD particles. Our method is insensitive to admixtures of other magnetic components, and its resolution is limited only by the sensitivity of the instrument used for FORC measurements. The method was tested on a sediment sample from Lake Ely, which is known to contain abundant magnetofossils, where we identified for the first time all FORC signatures predicted for noninteracting SD particles. The FORC diagram of this sample is dominated by a narrow ridge on the Hc axis. This ridge indicates that magnetofossils are extremely well dispersed in the sediment matrix, and that they occur in form of isolated, intact chains. Collapsed or clustered chains, if present, would contribute to other parts of the FORC diagram. Authigenically precipitated ultrafine magnetite or greigite [Rowan et al., 2009] possibly contributes to the low-coercivity component of the central ridge, if well dispersed in the sediment matrix.
We have also demonstrated that the central ridge function, obtained by integration of the central ridge over Hu is proportional to the switching field distribution of the magnetic particles. This allows precise comparison with results obtained from other magnetic characterization techniques. Our Lake Ely example demonstrates the highly discriminative power of FORC measurements, with possible applications ranging from characterization of magnetofossils for paleomagnetic and environmental studies, to detection of highly dispersed SD magnetization carriers in rocks as ideal NRM carriers for paleointensity studies.
High-resolution FORC measurements were performed at the Institute for Rock Magnetism, University of Minnesota, which is funded by the National Science Foundation, the Keck Foundation, and the University of Minnesota. A. P. Chen is currently supported by a U.S. Fulbright Fellowship and by the German Research Foundation (DFG project Eg294-1/1). M. Winklhofer is supported by the German Research Foundation (DFG project Wi182814-1). We are indebted to Andrew Roberts, Andrew Newell, and two anonymous reviewers for constructive reviews that significantly improved this paper.