Iron oxide inclusions in pyroxene crystals from the Modipe Gabbro, Botswana, have been studied to determine their recording fidelity. Hysteresis parameters, first-order reversal curves, isothermal remanent magnetization acquisition curves, and thermomagnetic data all indicate that the iron oxide occurs as stoichiometric magnetite in the form of single-domain and/or small pseudo single-domain particles. Using distributions for the grain shape and nearest-neighbor spacings determined from optical micrographs as input, a first-order reversal curve (FORC) diagram was simulated using a numerical micromagnetic model. The simulated FORC diagram was found to closely match the measured FORC distributions. Analysis of the interaction fields in the model found that the standard deviation of the interaction field distribution was 2.7 mT compared to a bulk coercivity of ~20 mT, suggesting that the majority of particles are unlikely to be affected by magnetostatic interactions. Thellier analysis of the samples induced with an initial laboratory thermoremanence yielded near-perfect behavior. It is suggested that the Modipe Gabbro dated at 2784.0 ± 1.5 (2σ) Ma is potentially an ideal recorder of the geomagnetic field.
 Ever since Néel's seminal work on thermoremanent magnetization [Néel, 1949, 1955], the natural occurrence of assemblages of non-interacting single-domain iron oxide particles has been the holy grail of rock magnetism. But the recognition of such assemblages in real rock samples has turned out to be rather difficult. One early candidate was provided by the Modipe Gabbro, a Neoarchaean intrusion in southern Africa, in which it was discovered that some of the silicate grains—the basic building blocks of all igneous rocks—contain arrays of ultrafine iron oxide inclusions resulting from natural diagenetic evolution during cooling from the melt [Evans and McElhinny, 1966; Evans et al., 1968]. Other examples have come to light from time to time [Palmer and Carmichael, 1973; McClay, 1974; Davis, 1981], but it is only in recent years that their significance has been fully appreciated [Feinberg et al., 2005; Letts et al., 2009; Usui et al., 2009]. This is partly due to the development of the so-called single-crystal technique [Tarduno, 2009], wherein the enhanced sensitivity of modern cryogenic magnetometers allows individual silicate crystals to be used to determine the strength of the Earth's magnetic field in the geological past. In Tarduno's graphic phrase, such “natural magnetic time capsules” are ideal recorders because their iron oxide inclusions are protected from chemical alteration during their geological lifetime and during the necessary laboratory heating steps. But this renewed interest was also spurred on by significant progress in both theory and practice: improved computing power has driven enormous advances in numerical micromagnetics so that fine particle magnetism can now be properly modeled from a theoretical point of view, and the introduction of first-order reversal curve (FORC) diagrams has revolutionized the interpretation of experimental hysteresis data.
 The persistent scarcity of reliable paleomagnetic results from the deep Precambrian means that the Modipe Gabbro retains considerable importance, particularly now that it has been accurately dated at 2784.0 ± 1.5 (2σ) Ma [Feinberg et al., 2010]. Its high-coercivity remanence was exploited in an early paleointensity study [McElhinny and Evans, 1968], and it provides one of the oldest well-dated paleomagnetic poles available. Thus, it has ramifications for the evolution of both the core and the crust: it helps constrain the early geodynamo, and it provides one of only a handful of results that permit the Earth's first supercontinent—Vaalbara—to be reassembled [de Kock et al., 2009; Reddy and Evans, 2009]. In view of this continuing interest, we felt that a new investigation using modern rock magnetic techniques would be worthwhile. Here, we report the findings of such a study.
 The starting material for the present study consisted of mineral separates obtained from the most magnetically stable samples discovered during a routine paleomagnetic survey of the Modipe Gabbro (site BO in Figure 1 of Evans and McElhinny ). Four pyroxene powders—with their magnetite inclusions—had been obtained by a sequence of increasing current settings on a magnetic separator in order to progressively approach a “clean” aliquot. As a final step, the last sample in the sequence was handpicked under a binocular microscope [Evans et al., 1968]. In the early work, convenient laboratory samples were made by mixing each powder with a suitable amount of KBr and pressing them into pellets in a 25 mm-diameter cylindrical mold (samples A1, A2, A3, and A4HP, respectively). The first step in the present study was to recover the pyroxene grains by dissolving the KBr in an ultrasonic bath. From each recovered powder, 200–300 mg fractions were removed and packed into small glass holders for the FORC and hysteresis measurements (these samples are labeled B1, B2, B3, and B4HP). In the case of sample B4HP, the entire 204.7 mg aliquot was used. Thus, to obtain more “clean” grains for the other experiments planned, the grains recovered from the penultimate sample in the original sequence (A3) were subjected to renewed magnetic separation using a Frantz Isodynamic separator, again followed by handpicking. This new aliquot was pressed into a 1 cm-diameter KBr pellet (sample C). A typical photomicrograph of the grains is given in Figure 1.
 First-order reversal-curve diagrams and standard hysteresis parameters (saturation magnetization, Ms; saturation remanence, Mrs; coercive force, Hc; and coercivity of remanence, Hcr) were determined using a Princeton vibrating sample magnetometer (VSM). Laboratory magnetic remanences were studied using an ASC pulse magnetizer, an Agico JR5A spinner magnetometer, and an Orion 3-axis low-field VSM. Synthetic “paleointensity” determinations were made using the double-heating Thellier-Thellier-Coe protocol with pTRM and pTRM-tail checks [Coe, 1967; Walton, 1984: Riisager and Riisager, 2001], using a laboratory field of 100 A/m (125.7 μT).
 Three-dimensional micromagnetic calculations were carried out using a procedure similar to that of Muxworthy and Williams , in which a minimum energy conjugate-gradient algorithm is combined with a dynamic algorithm that solves the Landau-Lifshitz-Gilbert equation. Together, these provide computational efficiency and robust, realistic hysteresis behavior. The demagnetizing energy was calculated using fast-Fourier transforms, which provide the high resolution needed to examine arrays of interacting grains.
3.1 Hysteresis Measurements
 Bulk magnetic properties of the four B samples are summarized in Table 1 and shown as a hysteresis-ratio “Day plot” [Day et al., 1977] in Figure 2. They exhibit a systematic trend, with Mrs/Ms ranging from 0.22 to 0.32 and Hcr/Hc from 1.93 to 1.72, placing them in the area associated with fine-grained pseudo single-domain (PSD) behavior.
Table 1. Hysteresis Parameters of the Samples in This Study
 The FORC results are illustrated in Figures 3a–3d. They are all typical of samples dominated by single-domain (SD) particles, with a well-defined peak and a long tail extending to high coercivities (hc). In all four samples, there is little spread in the vertical axis (hu), suggesting that magnetic interactions are minimal [Muxworthy and Williams, 2005; Egli et al., 2010].
3.2 Micromagnetic Modeling of FORC Data
 We used the grain-size analysis software GIAS [Beggan and Hamilton, 2010] (http://www.geoanalysis.org) to analyze a photomicrograph of the magnetite inclusions in the Modipe Gabbro. For this purpose, we scanned and digitized Figure 1 of Evans et al. . The GIAS software provides distribution data for the standard parameters such as grain size and shape, abundance, and nearest-neighbor distances. The results are conveniently summarized in terms of the corresponding cumulative distributions. For grain size, aspect ratio, and nearest-neighbor distances, we obtain the following values for the lower quartile/median/upper quartile: grain size, 0.4/0.5/1.6 µm; nearest-neighbor distances, 0.8/1.9/4.4 µm; and aspect ratio, 0.3/0.5/0.7. These parameters were used to construct a numerical micromagnetic model. A three-dimensional array of 26 × 26 × 26 nodes was populated with 581 SD magnetite particles using the nearest-neighbor distribution above, each one being given an appropriate uniaxial shape anisotropy oriented at random determined from the aspect ratio distribution.
 The calculated FORC diagram resulting from this procedure (Figure 4) corresponds well with the measured FORC diagram for sample B4HP (Figure 3d). The calculated hc peak and vertical (hu) spread are close to the observed values. However, there is a paucity of low-coercivity points, and the high-coercivity tail is rather restricted. The low-coercivity deficit is almost certainly due to the assumption that all the grains are true SD particles, whereas it is highly likely that some of the larger ones lie in the PSD field. The abbreviated high-coercivity tail probably results from the limitations of the optical microscope. Electron microscopy has shown that the grain-size distribution extends well below the optical limit [Evans and Wayman, 1970], and some of the very small grains present are likely to have high coercivities.
 Within the model, it is possible to directly determine the distribution of interaction fields within the particle assemblage. For such distributions, the mean interaction field is generally close to zero [Muxworthy and Williams, 2005]: it is the width of the distribution that is more important. Throughout the entire FORC simulation, the arithmetic standard deviation of the combined interaction field distribution (4.9 million points) was 2.7 mT, much less than the mean coercivity, which is ~20 mT. The absolute maximum interaction field observed was 16.4 mT. This would suggest that the majority of the grains are effectively non-interacting.
3.3 Magnetic Mineralogy
 The acquisition of isothermal remanent magnetization by sample C, followed by its continuous thermal demagnetization, are illustrated in Figure 5. The former shows that the coercivity spectrum is dominated by a magnetically soft ferrite. The latter yields a Curie point of 578 ± 1°C—compelling evidence that the mineral involved is essentially pure magnetite. Using standard Néel theory, an estimate of the grain volume distribution can be derived from the unblocking temperature spectrum. As a working model, we assume that all the grains are stoichiometric magnetite and are single domain with an aspect ratio equal to the mean value given by the image analysis software, i.e., 0.5. The resulting nominal grain-size distribution is shown in Figure 6. Hence, the thermal evidence suggests that the bulk of the inclusions have dimensions between 50 and 100 nm, in good agreement with published data on sized magnetite powders (see Figure 2).
3.4 Synthetic Paleointensity Experiments
 Three successive synthetic paleointensity experiments were carried out using sample C, a typical result being illustrated in Figure 7. The initial thermoremanent magnetizations (TRMs) have sharp unblocking spectra, which means that the temperature interval over which the paleointensity could be determined on the Arai plot was rather narrow. All the Arai plots are very linear, with no significant pTRM or pTRM-tail check discrepancies (Table 2). The arithmetic mean of the three values is 127.6 ± 2.1 μT, very close to the laboratory field of 125.7 μT.
Table 2. Three Paleointensity Estimates Determined for Sample C, Induced with a TRM Intensity of 125.7 μT (100 A/m)a
a The parameters used to describe the Arai plots are described in more detail in Leonhardt et al.  and references therein.
b ΔT is the temperature range used to make the paleointensity estimate.
c Number of points used in the calculation.
d The fraction of natural remanent magnetization used.
e The gap factor.
f Quality factor.
g Maximum difference produced by a pTRM check, normalized by the TRM (δ(CK)).
h Maximum difference produced by a pTRM-tail check, normalized by the NRM.
 The most significant finding of this work is the striking success of the micromagnetic code in generating a FORC diagram that corresponds well with the experimentally measured version (cf. Figures 3d and 4). To our knowledge, this is the first time that real optical microscope imagery has been used to constrain micromagnetic calculations. The observed agreement establishes a robust synergy between theory and experiment. The next stage is to address the deficits at low and high coercivities. The “missing” magnetically soft particles could theoretically be dealt with by modifying the model to include PSD grains, i.e., particles that have internal magnetic structure; however, to calculate a FORC diagram for a distribution of statistically significant particles (>500) with internal non-uniform magnetic structures is beyond current available computational capabilities. At high coercivities, it will probably be necessary to resort to transmission electron microscopy to extend the grain-size distribution below the optical limit.
 Given the virtually ideal mineralogy of the Modipe Gabbro, an attempt to determine the strength of the geomagnetic field in the deep Precambrian is a high priority. In fact, an early effort to do so was reported by McElhinny and Evans , but it was carried out by a non-Thellier method on the grounds that the blocking temperature spectrum was too restricted. However, the synthetic paleointensity experiments (Figure 7) suggest that modern techniques are much more likely to be successful. Of course, any magnetic grains that do not occur as pyroxene-protected inclusions may complicate Thellier experiments by exhibiting multidomain behavior and/or by undergoing mineralogical alteration during laboratory heating. The severity of these perennial hazards can only be judged from experiments on whole-rock samples. To this end, we plan to carry out a new sampling campaign in the near future.
 Mineral separates (Figure 1) from the Modipe Gabbro confirm that iron oxide inclusions inside pyroxene crystals are an important source of geologically stable thermoremanence. Hysteresis, isothermal remanent magnetization and thermal demagnetization measurements (Figures 2, 5, and 6) indicate that these inclusions are predominantly SD (with some PSD) grains consisting of pure magnetite. FORC diagrams (Figure 3) confirm the importance of SD grains. A spatial model of an array of SD inclusions—derived from microscope observations—was used to obtain a theoretical FORC diagram (Figure 4) using three-dimensional micromagnetic code. The output agrees well with the experimentally determined FORC diagram and thus advances the convergence of rock magnetic theory and experiment.
 We thank Andrew Roberts, Xiang Zhao, and Liao Chang for guidance and assistance with the hysteresis and FORC measurements and David Krása, Larry Heaman, and Judy Schultz for help with the preparation of the pyroxene grains. Figure 7 was prepared with L. Tauxe's PmagPy-2.132 software package. Financial support from the Natural Sciences and Engineering Research Council of Canada and the Royal Society is gratefully acknowledged.