4.1. Mechanism of the Remanence Transition
Various mechanisms have been proposed to explain sharp remanence transitions like those in Figure 4 between 50 and 70 K: (1) the presence of an isotropic point, Ti, where the anisotropy constant, K1, changes from negative to positive and the magnetocrystalline easy axis switches from <111> to <100> [Moskowitz et al., 1998; Özdemir and Dunlop, 2003]; (2) the presence of a structural phase transition, such as the Verwey transition in magnetite [Özdemir and Dunlop, 1999]; (3) the presence of a spin glass transition (e.g., as proposed for titanohematite by Burton et al. ); and (4) sudden unpinning of domain walls caused by rapid changes in anisotropy as a function of temperature [Carter-Stiglitz et al., 2006]. The role of the isotropic point in causing the remanence transition depends critically on the sample composition. Estimates of Ti were obtained from the data of Kąkol et al. [1991a] and are indicated on Figure 4. The isotropic point occurs at 130 K in magnetite, falls to below liquid nitrogen temperatures in the compositional range 0.05 < x < 0.35, then rises again to 230 K at x = 0.5. In more titanium-rich compositions, K1 is always positive and there is no low-temperature isotropic point. Hence, for x = 0.2 and 0.3 (Figure 4a) the isotropic point has the potential to play a role in the remanence transition. However, for x = 0.4, 0.46 and 0.5 (Figures 4b–4d) the remanence transition temperature is well below Ti, and remains constant as Ti changes from 110 to 230 K. There is, instead, a good agreement between Ti and the temperature at which the slow remanence decay above 70 K reaches completion. We propose, therefore, that this upper remanence decay temperature is coincident with the isotropic point, giving revised estimates of Ti = 112, 188, and 195 K for samples with x = 0.4, 0.46, and 0.5, respectively. The isotropic point in these samples is responsible for the frequency-independent cusp in the in-phase component of susceptibility (Figure 5c) noted in section 3.3.
Given the evidence for a dramatic transition in domain wall pinning between 50 and 70 K presented in section 3.1, we favor a pinning-related model for the remanence transition [e.g., Carter-Stiglitz et al., 2006]: elevated remanences are obtained when a saturating field is applied below 50 K due to strong domain wall pinning (Figure 3a); this remanence is lost suddenly during the pinning transition between 50 and 70 K as domain walls respond to the demagnetizing field by moving into the sample (Figure 3b). For x = 0.4, 0.46, and 0.5 the remanence that remains above 70 K is due to weaker domain wall pinning by extrinsic defects, such as dislocations. This remanence is lost gradually as domain walls broaden and unpin on approaching the isotropic point.
4.2. FORC Diagrams
An abrupt pinning transition on cooling from 100 to 50 K produces a characteristic crescent-shaped FORC diagram for samples with 0.3 ≤ x ≤ 0.5. The form of the FORC diagram is similar to that predicted by Pike et al. [2001a] using a one-dimensional model of domain wall pinning in a random field [Néel, 1955]. Here we adapt the Pike et al. [2001a] method and demonstrate that the key features of the crescent-shaped FORC diagram can be explained by taking account of how the parameters of the one-dimensional model are modified by the three-dimensional nature of the magnetization process close to saturation.
The sample is considered as a one-dimensional grain with cross-sectional area A and length L. A domain wall with position x (0 ≤ x ≤ L) is able to move through the grain under the influence of an applied field, H (dynamic effects, such as those discussed in section 4.3, are ignored). The magnetization of the grain is M(x) = AMS (L−2x) and the demagnetizing field, HD, is approximated by a uniform field HD(x) = −NM(x), where N is the demagnetizing factor. The total energy of the grain is the sum of pinning, magnetostatic and demagnetizing energies:
where EW(x) denotes the variation in domain wall energy as a function of position due to the presence of pinning potentials. A dimensionless version of equation (2) can be written
where et = ET/μ0A2NMs2, ew = EW/μ0A2NMs2, and h = H/NAMs. The domain wall sits at a local minimum in total energy:
where hp = −1/2 dew/dx is the pinning field. The domain wall is stationary when h = hp + (L−2x), i.e., when the applied field equals the sum of the pinning field and the demagnetizing field. If the pinning field is known, then the position of the wall can be calculated (see Pike et al. [2001a] for details). The partial hysteresis loops that make up FORC diagrams can be modeled starting at positive saturation and calculating the domain wall position reached for a given reversal field, Ha. The progress of the wall is then tracked as the field, Hb, is ramped back to positive saturation.
The form of the pinning field used in this study was the same as that used by Pike et al. [2001a]. A pinning function was generated by dividing the particle into 100,000 discrete points along its length. The pinning function was then generated using
where the index i refers to each of the discrete points in the pinning function (0 ≤ i < 100,000), ξ is a correlation parameter that links the pinning field at each point to the value of the preceding point, R is randomly selected from a normal distribution with mean zero and variance 1, and Ω defines the amplitude of the pinning field. The pinning field for i = 0 was arbitrarily fixed at zero. After generation of a pinning function for the particle, 100 individual FORCs were calculated. To minimize the effect of Barkhausen noise on the simulated FORCs, the pinning field was regenerated 200 times and the resulting FORCs averaged.
The FORC diagram obtained for L = 1, Ω = 0.1 and ξ = 0.01 is shown in Figure 10a, which simply reproduces the original result of Pike et al. [2001a]. The FORC distribution has a well-defined and narrow coercivity distribution that is spread symmetrically along ±Hu due to the demagnetizing field. Two key features of the experimental FORC diagrams (Figures 1 and 2) are not reproduced by the Pike et al. [2001a] model, however. The model distribution has a “cigar” shape rather than a crescent shape, and the intensity of the FORC distribution is evenly spread throughout the cigar, rather than being maximum at Hu = 0 and tailing off toward the tips. The curvature of the FORC distribution can be obtained by modifying the nature of the pinning function. To obtain larger values of Hc at the tips of the distribution (i.e., when ∣Hu∣ is large), it is necessary to increase the amplitude of the pinning field as the sample approaches saturation (i.e., when the domain wall approaches the edges of the particle). An example pinning function and the resulting FORC diagram obtained after multiplying equation (5) by an amplification factor of [1 + 3(x − 0.5)2] is shown in Figure 10b. The cigar is now transformed to a “banana”, but still lacks the appropriate variation in intensity from the center to the tips. This intensity variation can be obtained by relaxing the assumption that the demagnetizing factor N is constant. The effect of increasing the demagnetizing field slightly as the sample approaches saturation, by adding an additional contribution of −5(x − 0.5)3 to the original demagnetizing field of (1 − 2x) (keeping the amplitude of the pinning field constant), is shown in Figure 10b. The distribution of intensity within the cigar now corresponds more closely to the experimental results. The effects of simultaneously increasing both the amplitude of the pinning and demagnetizing fields as the sample approaches saturation is shown in Figure 10d. The simulated and experimental FORC diagrams are compared in more detail in Figure 11 and agree well.
Figure 10. Simulated FORC diagrams based on the Néel  one-dimensional domain wall pinning model. (a) Model corresponding to that of Pike et al. [2001a] with constant amplitude of pinning and demagnetizing fields. (b) Effect of increasing amplitude of pinning field near sample saturation. (c) Effect of increasing demagnetizing field near saturation. (d) Effect of increasing the amplitude of both pinning and demagnetizing fields near saturation.
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Figure 11. Comparison of (a) a measured FORC diagram for a single crystal with composition x = 0.35 at 50 K and (b) a modeled diagram using modifications of the method described by Pike et al. [2001a]. The scaling used in Figure 11b is arbitrary, but the shape and profile of the FORC distribution is similar to that of the experimental diagram.
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Physical justification for the increase in pinning and demagnetizing fields as the sample approaches saturation can be found in the low-temperature domain images (Figure 9). As argued by Kosterov  for the case of low-temperature magnetite, the microcoercivity due to domain wall displacement varies as sec(θ), where θ is the angle between the applied field and the magnetization within a domain. In Figure 9, discrete regions in low-temperature titanomagnetite have uniaxial easy axes oriented 90° to each other. If the field is applied parallel to the easy axis of one region, the domains within this region will move more easily than those in the perpendicular region. Close to saturation, the sample will be dominated by the perpendicular regions, and hence the average effective pinning field will increase. A similar argument can be used to explain the increase in demagnetizing field. It is well known that the demagnetizing field of a soft ferromagnet increases close to saturation because of the elimination of closure domains [Dunlop, 1983]. Although closure domains are unlikely to form in low-temperature titanomagnetite, due to the high uniaxial anisotropy, the sample can reduce its demagnetizing field by forming surface regions with easy axis parallel to the surface. As before, domains within these regions require a higher critical field before they move, and are more likely to survive until the sample approaches saturation. As these surface domains are removed, the demagnetizing factor will increase, just as it does for a soft ferromagnet.
4.3. AC Susceptibility
In larger bias fields, the in-phase susceptibility for x = 0.4 develops a peak immediately above, and a trough immediately below, the relaxation temperature (Figure 8). Here we demonstrate that these features can be explained by a resonant component to the domain wall relaxation.
The dynamic response of a domain wall to an applied field can be compared to that of a damped harmonic oscillator [Rado et al., 1950]:
where x is displacement, t is time, m is the mass of the domain wall, β is a viscous damping coefficient, α is a spring constant, Ms is the saturation magnetization, and H is the applied field. The first term in equation (6) describes the acceleration of the wall in response to the applied field. The mass of the domain wall is given by [Döring, 1948; Rado et al., 1950]
where γ is the gyromagnetic factor and δ = π is the domain wall width (A is the exchange stiffness constant and K is the anisotropy constant). The second term in equation (6) describes the viscous drag force experienced by a moving domain wall. In conducting ferromagnets this drag force can arise from eddy currents created as the wall moves. In ferrites, however, the drag force is dominated by diffusion-induced MAE (disaccommodation) [Castro and Rivas, 1993; Hubert and Schäfer, 1998]. Rado et al.  defined β as
where η is a viscosity coefficient. The third term in equation (6) describes the restoring force acting on a domain wall that is displaced from the origin. This restoring force may arise from the self-demagnetizing field of the sample or from the presence of a harmonic pinning potential. The solution to equation (6) for an AC field with frequency ω is given by [Rado et al., 1950]
for the complex susceptibility or
for the in-phase and out-of-phase components, where χ0 is the static susceptibility χ0 = Ms2/αd (d is the average domain size), and ω0 and ωc are characteristic resonance and relaxation frequencies of the system, respectively:
Equations (10) and (11) are plotted as two-dimensional functions of ln(ω0) (horizontal axis) and ln(ωc) (vertical axis) in Figure 12 for a fixed applied frequency ω = 1. A horizontal cross section through Figure 12 yields a purely resonant response, while a vertical cross section yields a purely relaxational response (see insets). The measured susceptibility depends on the exact path taken through ln(ω0)−ln(ωc) space. In our experiments, ω is kept constant and we measure the in-phase and out-of-phase response of the system as function of temperature. T is not an explicit variable in equations (10) and (11), therefore we must consider the temperature effect on ω0 and ωc separately. Numerous MAE studies [e.g., Walz et al., 1997], as well as our own AC susceptibility measurements, indicate that domain wall dynamics are controlled by thermally activated electron hopping in the vicinity of the low-temperature relaxation. The viscous term in equation (6) is, therefore, dominated by the MAE (i.e., local enhancement of anisotropy and pinning caused by thermally activated rearrangement of Fe2+ and Fe3+ ions within the wall) and the viscosity parameter (η in equation (8)) is directly proportional to the relaxation time for electron hopping (equation (1) [Castro and Rivas, 1993]). This relation causes ωc to vary as ln ωc = ln ωc0 −Ea/kBT. Assuming that ω0 is constant and that ω < ω0 (i.e., we are performing a subresonant experiment), the response of the system is sampled along a vertical path from positive ln(ωc) (high temperatures) to negative ln(ωc) (low temperatures). The result is a conventional Debye relaxation, as shown in the insets to Figure 12, which provides an explanation for the behavior observed for x = 0.2 and x = 0.4 in low bias fields.
Figure 12. (a) In-phase and (b) out-of-phase susceptibility as a function of lnω0 and lnωc for an applied frequency ω = 1 Hz, derived from equations (10) and (11), respectively. Insets represent the purely resonant response obtained by measuring a horizontal cross section and purely relaxational response obtained from a vertical cross section. Insets are not to the same scale.
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The behavior observed for x = 0.4 in high bias fields (Figure 8) can be explained by taking into consideration the T dependence of magnetocrystalline anisotropy. As K1 increases with decreasing temperature, the width of domain walls will decrease and their Döring mass will increase (equation (7)). This increase in mass decreases ω0 (equation (12)) and moves the system closer to the resonance peak. A nonlinear path through ln(ω0)−ln(ωc) space (Figure 13) gives a qualitative explanation for the data acquired at 1 Hz with a 150 mT bias field. For comparison with the data, values of ln(ωc) were converted to temperature values using the activation energy and attempt times measured in Figure 7. The path taken assumes that the rate of increase in domain wall mass accelerates significantly near the relaxation, causing the system to pass through the tip of the resonance peak. The modeled susceptibility has a sharper transition than the data because we assume a single characteristic relaxation time rather than a broad distribution. Nevertheless, the position and magnitudes of the minor peaks and troughs agree well with the measured values, and the area under the out-of-phase curve equals the area under the data.
Figure 13. (a) Nonlinear path that approaches both the resonance and relaxation regions in lnω0−lnωc space, yielding susceptibility curves that qualitatively explain the distinctive behavior highlighted in Figure 8. (b) Comparison of the calculated in-phase and out-of-phase susceptibility-temperature curves obtained from the path shown in Figure 13a with measurements of a sample with composition x = 0.4, measured with a 1 Hz AC field and a 150 mT bias field.
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This modeling suggests that the reduced rate of electron hopping precipitates a rapid increase in anisotropy with decreasing temperature below 77 K, which moves the system from pure relaxation toward a more resonant behavior. The observed behavior, however, depends on the proximity of the system to the resonant peak above 77 K. If ω << ω0, then simple relaxational behavior is expected. This is the case for weakly pinned (broad) walls with low mass and high ω0. If ω0 decreases significantly (or if we increase ω significantly) then resonant behavior becomes more likely. Enhancement of resonant behavior at higher measurement frequencies was noted in Figure 5. The enhancement of resonant behavior in higher bias fields might also be explained if there is a broad distribution of ω0 values in the system: the bias field eliminates weakly pinned walls with low mass and high ω0, leaving the strongly pinned walls with high mass and low ω0. Hence, increasing the bias field sweeps away the weakly pinned domain walls that otherwise dominate the susceptibility signal, effectively moving the path taken during analysis closer to the resonance peak.
Interpretation of Figure 8 in terms of a resonance effect might also provide an alternative explanation for the appearance of a “negative” Debye peak at ∼65 K in the MAE studies of Walz et al. . In MAE studies it is normal to identify positive Debye signals with a relaxation-induced, time-dependent decrease of the initial susceptibility due to development of enhanced anisotropy within domain walls. The negative signal identified by Walz et al. , at a temperature corresponding to the low-temperature resonance seen here, is highly unusual, as it implies a relaxation-induced, time-dependent increase of the initial susceptibility due to the development of reduced anisotropy within the wall. This was explained by Walz et al.  in terms of an electron-supported mechanism of local stress release due to rearrangement of differently sized Fe2+ and Fe3+ ions in the vicinity of Ti4+ ions. Alternatively, the resonance effect can be used to explain the increase in susceptibility, while retaining the usual interpretation of enhanced anisotropy at the domain wall: enhancement of anisotropy causes the system to move closer to the resonance, which enhances the susceptibility ahead of the main relaxation.
4.4. Mechanism of the Pinning Transition
Our observations provide general constraints on the mechanism of the pinning transition for the studied range of compositions and temperatures where the crescent-shaped FORC is observed. The pinning transition is only observed at 50 K for samples with x > 0.2, which according to O'Reilly and Banerjee  corresponds to the composition where Fe2+ ions first enter the tetrahedral sites of the spinel lattice. The presence of tetrahedral Fe2+ is important due to its role in generating a tetragonal Jahn-Teller distortion [Goodenough, 1964]. The appearance of enhanced pinning only for x > 0.2 hints that magnetoelastic contributions to the magnetic anisotropy, originating from the presence of Fe2+ Jahn-Teller ions on the tetrahedral sublattice, are an essential component of the pinning transition. Sudden loss of pinning on heating indicates, however, that the presence of tetrahedral Fe2+ alone is insufficient to cause the effect, and that localization of Fe2+ ions on the octahedral sublattice due to cessation of electron hopping is also necessary for enhanced pinning. This observation is consistent with the analysis of Kataoka , who explained the origin of giant magnetostriction in ulvöspinel (Fe2TiO4) in terms of an interaction between the Jahn-Teller distortion of Fe2+ on tetrahedral sites and the spin-orbit coupling (i.e., magnetocrystalline anisotropy) of octahedral Fe2+. We suggest that a similar mechanism operates here and that enhanced pinning below the low-temperature relaxation is predominantly magnetoelastic in origin.
Pinning of domain walls in minerals is normally considered to be due to the presence of extrinsic defects, such as dislocations, grain boundaries, impurities, etc. However, close association of the pinning transition and the MAE-driven domain wall relaxation suggests that pinning at low temperatures may be an intrinsic effect, directly related to enhancement of anisotropy within the walls. In this scenario, Fe2+ and Fe3+ ions within the wall rearrange in such as way as to enhance the tetragonal magnetoelastic distortion associated with the interaction between tetrahedral and octahedral Fe2+ [Kataoka, 1974]. Below the relaxation temperature, when electrons are no longer able to diffuse on the time scale of the experiment, this magnetoelastic MAE is no longer responsible for a viscous drag force on the domain walls (i.e., the β term in equation (6)), but instead causes a static, high-amplitude intrinsic pinning field (i.e., the hp term in equation (4)). Above the relaxation temperature, the MAE once again plays the role of a viscous drag force and pinning is controlled by extrinsic defects until the isotropic point is reached.
Evidence in support of the existence of an intrinsic pinning regime at low temperatures may be obtained from the large difference between ZFC and FC remanence below the pinning transition, which implies that application of a magnetic field during cooling can modify the nature of the pinning function (this would not be the case if pinning were entirely caused by extrinsic defects). Modification of the pinning function may be caused by the elimination of regions with differently oriented magnetocrystalline anisotropy axes [e.g., Carter-Stiglitz et al., 2006], or simply through elimination of domain walls themselves (thereby eliminating wall-related pinning sites) when cooling in a strong field. Another prediction of the intrinsic pinning regime is that domain walls will be rigidly pinned along their entire length, whereas in the extrinsic pinning regime walls are only pinned at discrete points, permitting sections of wall in between pinning sites to bow in response to an applied field. These two regimes might be distinguishable using hysteresis loops in the Barkhausen regime. Intrinsic pinning would not permit any change in magnetization between Barkhausen jumps, leading to a series of horizontal steps in the magnetization curve, as observed (Figure 3). Bowing would lead to a more pronounced gradient in the magnetization curve between Barkhausen jumps.