The loss in remanence at the Verwey transition (TV) was modeled for elongate stable single domain magnetite for two experiments: 1) thermal cycling of room temperature saturation isothermal remanent magnetization (RTSIRM), 300 → 10 → 300 K, and 2) warming of zero-field cooled and field-cooled remanences from 10 K to 300 K. The RTSIRM simulations used magnetocrystalline anisotropy constants for stoichiometric magnetite and aspect ratios (AR) from 1 to ∞, for assemblages of inorganic particles and 10-magnetosome chains. The results match the experimentally observed behavior of reversibility. The second set of simulations was conducted with low-temperature magnetocrystalline anisotropy constants for varying degrees of non-stoichiometry, and AR = 5. Minor non-stoichiometry lowers the drop in remanence at TV and increases the “delta ratio” (δfc/δzfc) to values as high as ∼6. New experiments demonstrate that maghematization (non-stoichiometry) can partly explain the low-temperature magnetic behavior observed in magnetotactic magnetite to date.
 Magnetite has a phase transition, the Verwey transition, at ∼120 K (TV) where its crystal structure changes from cubic to a lower symmetry structure, likely monoclinic or triclinic [Verwey and Haayman, 1941; Medrano et al., 1999]. The magnetocrystalline anisotropy of the low-temperature phase is much stronger and has a different form than the cubic phase. On cooling through TV in a zero field, a random cube edge becomes the c-axis of the new crystal; the a and b-axes are formed from two face diagonals of the high temperature phase which are orthogonal to one another and to the cube edge which was transformed into the c-axis.
 Rock and paleomagnetic applications that exploit the low-temperature properties of magnetite include: 1) identification and quantification of trace amounts of magnetite [e.g., Nagata et al., 1964; Muxworthy and McClelland, 2000] and 2) low-temperature demagnetization (LTD) of remanence for application in paleomagnetic and paleointensity studies [e.g., Dunlop and Özdemir, 1997]. Details of the Verwey transition's effect can yield information concerning the exact nature of the magnetite present: e.g., domain state [e.g., Carter-Stiglitz et al., 2002], stoichiometry, i.e., maghemitization [Özdemir et al., 1993; Smirnov and Tarduno, 2002; Kosterov, 2002], and genesis—biogenic versus inorganic—[Moskowitz et al., 1993]. Moskowitz et al.  suggested that measurement of low-temperature remanent magnetization (LTRM) on warming (typically 20–300 K) from two initial states—zero-field cooled (ZFC), and field cooled (FC, typically μ0H = 2.5 T)—provides a diagnostic signature of the presence of chains of stable single domain (SSD) magnetite (magnetosomes) produced by magnetotactic bacteria. Specifically the so-called “delta-ratio” δfc/δzfc, where
appears to exceed 2.0 only for magnetite produced by magnetotactic bacteria. Modeling by Carter-Stiglitz et al.  suggested that the elevated delta ratios were due to the interaction caused by the chain configuration. Unfortunately, the magnetosome simulations were in error; corrected results show ratios only slightly larger than those possible with acicular inorganic material. Thus the cause of delta ratios greater than two is not understood.
 The second application, LTD, removes unstable multi-domain (MD) moments on low-temperature cycling, leaving reversible SSD-like moments [Dunlop and Argyle, 1991]. They suggested that domain reorganization completely demagnetizes the MD grains, while the SSD grains are unaffected because their magnetic anisotropy is dominated by temperature invariant shape anisotropy. The second supposition is incorrect as the low-temperature magnetocrystalline anisotropy is a factor of three larger than the maximum shape anisotropy [Carter-Stiglitz et al., 2002]. Why then are SSD remanences acquired at T > TV unaffected on cycling through TV?
 It has long been known that maghemitization modifies the magnetic effects of the Verwey transition [Özdemir et al., 1993]: the temperature of the transition decreases and its effects diminish with oxidation. Kakol and Honig  demonstrated a strong dependence of the monoclinic magnetocrystalline anisotropy constants on stoichiometry, e.g., the constant for the a axis drops by 75% for Δ = 0.01 (where Δ is the cation deficiency in Fe3(1−Δ)O4). Note that this non-stoichiometry parameter is commonly referred to as δ, but here we use Δ to avoid confusion with the remanence-drop parameter.
2. Model Description
 The model used in this study is the same as in Carter-Stiglitz et al.  including corrections [Carter-Stiglitz et al., 2003]. The simulations begin from a saturated state; each grain's moment iteratively rotates in the direction of the steepest energy gradient until an energy minimum is reached. Energy minima are recalculated at each temperature step as anisotropy constants vary discontinuously at TV and continuously elsewhere. For more details see Carter-Stiglitz et al. . This study uses a simplified formulation for the low-temperature magnetocrystalline anisotropy where the K〈111〉 is neglected [Kakol and Honig, 1989]. Changes in Ms with temperature are included, both in terms of the shape anisotropy constants, and the magnitude of the individual grain moments.
 Two experiments were simulated for this study: 1) ZFC and FC LTRM warming, described above, and 2) room temperature saturation isothermal remanence magnetization (RTSIRM) measured during the temperature cycle 300 → 10 → 300 K. Anisotropy constants from Kakol and Honig  were used to explore the effects of non-stoichiometry. The data were fit with polynomial functions to allow intermediate Δ values to be used. Data for the temperature dependence of the constants for all Δ do not exist for two reasons: 1) easy axis switching for higher values of Δ interferes with the measurement of magnetization curves, and 2) single-phase single-crystal samples with Δ > 0.04 could not be made because hematite is the stable phase for Δ > 0.04 at the temperatures used in the fabrication (>1273 K). Thus the polynomial fits for Δ = 0.002 data were used to calculate the temperature dependence of the anisotropy constants for all values of Δ, i.e., K(Δ, T) = K(Δ, 4.2) × K(0.002, T)/K(0.002, 4.2). Δ = 0.002 was chosen as it is an intermediate value of Δ, and the temperature variation was well defined by the data. Note that this approach causes the suppression of TV with increasing Δ to be absent from the modeling results. Variations in spontaneous magnetization of magnetite with temperature were modeled by fitting the data shown in Dunlop and Özdemir  with a polynomial.
3.1. Modeling Results
Figure 1a shows RTSIRM simulations for 105 randomly oriented particles of magnetite (Δ = 0.000) with varying aspect ratios (AR). The remanence lost at TV decreases with increasing AR. The maximum lost at TV is 50% for the fictitious case of an infinitely small shape anisotropy constant which dominates above TV (exactly matching the analytical result for this special case). The remanences are reversible on cycling through TV. Figure 1b shows the same experiment simulated for 103 randomly-oriented 10-magnetosome chains comprised of particles with Δ = 0 and AR = 1.5, showing the same reversibility as the inorganic simulations. The magnitude of the remanence drop is approximately the same as that for the inorganic material with AR = 10.
Figure 2a shows ZFC simulations for 105 randomly oriented inorganic particles with AR = 5, and varying Δ. The initial LTSIRM at 10 K is always half of the saturation magnetization, as expected for randomly-oriented uniaxial SSD assemblages. ZFC simulations show a decrease in δ with increasing Δ, with ∼1% < δ < 30%. In the FC case the initial remanence ratios exceed 0.5, due to the non-random orientation of easy axes after FC (Figure 2b), but the same decrease in δ is seen with increasing Δ. However, δ decreases with increasing Δ more rapidly in the ZFC case than in the FC case.
3.2. Experimental Results
Figure 1b shows three RTSIRM curves for the same sample of freeze-dried magnetotactic bacteria (strain MV1). One curve was measured for a fresh sample obtained in 1999, the second five months later, and the third roughly three years later, in 2002. The first curve shows a drop in remanence at TV that is, within error, the same magnitude as that seen in the stoichiometric magnetosome simulation. As the sample aged the drop in remanence at TV became smaller. The curve measured 3 years later shows a drop of only ∼2%, down from 12%. The same reversibility is, however, seen in all of the measurements.
Figure 3 shows ZFC and FC remanences on warming for the same sample measured in 1999 and 2002. Similar to the RTSIRM data, a decrease in δ with aging is seen in the ZFC and FC data. Moreover, the delta ratio increases strongly with aging, from 1.55 to 4.80.
 RTSIRM simulations corroborate the idea that SSD moments are completely reversible on cycling through TV. Consider the simple case where shape anisotropy is ignored below TV, and only shape anisotropy is included above TV (AR = 1, Figure 1a). In the RTSIRM state the magnetic moment of each particle is parallel to the particle's long axis, and the moments are evenly distributed over the hemisphere whose pole is parallel to the applied field. On cooling through TV each moment rotates from the long axis of the particle to the c-axis of the low-temperature phase, as far away as 90° but no more. For any point on the RTSIRM hemisphere, say the direction defined by θ1 and ϕ1, the magnetizations parallel to that point fan out onto a second hemisphere whose pole is the direction (θ1, ϕ1), thus reducing the total remanence by 50%. On warming through TV the moments which began in the direction (θ1, ϕ1) rotate back to (θ1, ϕ1) and a reversible curve results. For more complicated simulations where shape anisotropy is included, on cooling through TV the moments do not rotate as far away from the room-temperature easy axis, as the average angle between the low-temperature and high temperature easy axes decreases with increasing shape anisotropy. Therefore the remanence drop will always be less than 50%, and the reversibility is independent of AR, as well as non-stoichiometry which has the same net effect as increasing AR (see below). Moreover, this basic relationship should not be affected by the magnitude of the remanence, e.g., a weak field thermoremanent magnetization, since the only aspect changed would be the starting distribution of magnetic moments.
 Though the magnetotactic samples show complete reversibility as predicted by the modeling, some inorganic samples of SSD magnetite show more complicated behavior [e.g., Özdemir et al., 2002]. In particular, complications could arise if the particles are magnetically interacting.
 The simulations demonstrate two ways in which remanence behavior across TV is sensitive to variation in the low-temperature magnetocrystalline anisotropy constants: 1) a decreasing drop in remanence at TV (on both warming and cooling) with increasing Δ and 2) an increase in the delta ratio with increasing Δ. This pattern is the same as that seen with increasing AR [Carter-Stiglitz et al., 2002]. The cause is the same as well. Consider the ratio of anisotropy which is present at all T, i.e., shape and interaction, to the anisotropy that is only present below TV, i.e., the low-temperature magnetocrystalline anisotropy (Ks/KLT). As Ks/KLT increases, either by increasing the magnitude of the shape anisotropy or decreasing the magnitude of the low-temperature magnetocrystalline anisotropy, the average angular distance between the low-temperature easy axis and the high-temperature easy axis decreases [Carter-Stiglitz et al., 2002]. Thus δzfc and δfc decrease as Ks/KLT increases. Moreover, δfc decreases more slowly with increasing Ks/KLT than δzfc, as there is an easy-axis bias in the FC case for all Ks/KLT, so as Ks/KLT increases so does the delta ratio.
Figure 4 shows the delta ratio plotted against Ks/KLT for three simulation sets: the inorganic simulations from Carter-Stiglitz et al. , the ZFC and FC simulations from this study fully accounting for changes in spontaneous magnetization (as in Figure 2), and the same simulations using the temperature dependence of Ms to calculate the shape anisotropy but otherwise assuming a constant spontaneous magnetization. Error bars were calculated assuming counting error, relative standard deviation equal to (n)−1/2, where n is the number of particles (chains) in the calculation. Maximum shape anisotropy and low-temperature magnetocrystalline anisotropy constants appropriate for stoichiometric magnetite only yield delta ratios as high as ∼1.6. However, a decrease in the magnetocrystalline anisotropy constants, e.g., Δ ≠ 0, yields higher delta ratios. Consider the data unadjusted for spontaneous magnetization. Although the counting error becomes quite large, delta ratios as high as 12.5 are indicated (Δ = 0.0109). The delta ratios appear to saturate as the easy axis bias for the FC case begins to lose its potency. The delta ratios calculated after the MS(T)-adjustment only reach maximum values of ∼6, because as the drop in remanence at TV decreases, the drop in remanence due to a decrease in spontaneous magnetization becomes more important; since this drop in remanence is the same for both FC and ZFC cases it suppresses the delta ratio towards 1. This limits the delta ratio to values less than ∼6, in good agreement with magnetotactic bacteria data which show delta ratios up to ∼5 [Moskowitz et al., 1993].
 The evolution of the magnetic properties of the sample shown in Figures 1b and 3 is almost certainly due to maghemitization under ambient conditions during the elapsed time period. This is further indicated by a decrease in TV; the derivative of the 1999 FC curve peaks at 116 K, while the derivative of the 2002 curve peaks at 100 K. The original measurement had a delta ratio consistent with the simulations of stoichiometric chains [Carter-Stiglitz et al., 2003]. In fact the drop in RTSIRM and LTSIRM at TV measured in 1999 is the same as that modeled (e.g., Figure 2b). It is not possible to use the simulations presented above as perfect analogues for the magnetotactic data, as the non-stoichiometry in the samples measured by Kakol and Honig  were produced with T > 1000°C and under a controlled fO2, but some critical conclusions can be made nonetheless. Saturation magnetization is not likely to vary much with low levels of maghemitization (Δ < 0.1), and if anything will be lowered. Thus the shape anisotropy and interaction anisotropy can only decrease with increasing maghemitization. Thus the low-temperature magnetocrystalline anisotropy constants likely decreased with maghemitization, driving the loss in remanence at TV down, and the delta ratio up. Using the simulation data to estimate Δ for MV1 (during 2002) indicates values of only ∼1%.
 The modeling presented in this study has provided a clearer understanding of the Verwey transition's effect on the remanence of SSD particles of stoichiometric and non-stoichiometric magnetite. Four conclusions can be drawn from the work presented in this study. 1) The reversibility of elongate SSD remanences after low-temperature cycling can be understood within the context of the discontinuous change in magnetic anisotropy at TV, and is predicted to be independent of stoichiometry. 2) Increasing degrees of non-stoichiometry, by lowering the magnetocrystalline anisotropy constants, decrease δ and increase the delta ratio. 3) The delta ratio is limited to maximum values of around ∼6 due to the decrease in spontaneous magnetization from 80–150 K. 4) Maghemitization is responsible for some of the elevated delta ratios observed in samples of magnetotactic bacteria. This final conclusion leads to two questions concerning the magnetic identification of magnetotactic bacteria: 1) Can maghemitized inorganically produced SSD magnetite have the same elevated delta ratios as those observed in magnetotactic magnetite? 2) Is maghemitization the only mechanism that yields the high delta ratios commonly observed in samples of magnetotactic bacteria? Though the simulations presented above answer the first question with an unequivocal yes, no samples of well characterized inorganic SSD magnetite have shown such elevated delta ratios. The second question is just as ambiguously answered. Samples of magnetotactic magnetite that are likely too fresh to be oxidized show elevated delta ratios suggesting other mechanisms.
 On the utility on the ZFC/FC remanence measurements the following is suggested: 1) ZFC/FC remanence curves are a robust indicator of SSD magnetite, 2) delta ratios greater than 2 indicate small degrees of non-stoichiometry, and 3) the delta ratio is still a good tool to identify the likely presence of magnetotactic bacteria; these are one of the few natural sources of magnetite that produce a particle size distribution (almost) entirely within the SSD size range, and thus one of the few sources that can produce enough SSD material to drive the delta ratio to high values.
 This is IRM contribution 0311. IRM is supported by the Instrumentation and Facilities Program, Earth Science Division, National Science Foundation. B. Carter-Stiglitz and B. Moskowitz were supported in part by NSF grant EAR-0390291. We thank two anonymous reviewers, and editor, Dr. Kristine M. Larson.