Explorer Magnetic vortex effects on first-order reversal curve (FORC) diagrams for greigite dispersions

First-order reversal curve (FORC) diagrams are used increasingly in geophysics for magnetic domain state identiﬁcation. The domain state of a magnetic particle is highly sensitive to particle size, so FORC diagrams provide a measure of magnetic particles size distributions. However, the FORC signal of particles with nonuniform magnetisations, which are the main carrier of natural remanent magnetisations in many systems, is still poorly understood. In this study, the properties of non-interacting, randomly oriented dispersions of greigite (Fe 3 S 4 ) in the uniform single-domain (SD) to non-uniform single-vortex (SV) size range are investigated via micromagnetic calculations. Signals for SD particles ( < 50 nm) are found to be in excellent agreement with previous SD coherent-rotation studies. A transitional range from ∼ 50 nm to ∼ 80 nm is identiﬁed for which a mixture of SD and SV behaviour produces complex FORC diagrams. Particles > ∼ 80 nm have purely SV behaviour with the remanent state for all particles in the ensemble in the SV state. It is found that for SV ensembles the FORC diagram provides a map of vortex nucleation and annihilation ﬁelds and that the FORC distribution peak should not be interpreted as the coercivity of the sample, but as a vortex annihilation ﬁeld on the path to saturation.

Previous experimental studies on nano-patterned arrays of SV particles (Pike 27 and Fernandez, 1999; Dumas et al., 2007) found that FORC diagrams are signi-28 2 ficatively more complex than for SD signals, with complex off-axis "butterfly" patterns that are related to vortex nucleation/annihilation processes. How-30 ever, it is difficult to relate the behaviour of 2D nano-patterned arrays to the 31 behaviour of natural particle systems found in geological samples. In natural greigite. The relatively high anisotropy of greigite means that the behaviour of this mineral is representative of cubic-anisotropic ferri-and ferro-magnets Such magnetic particle systems will be driven spontaneously toward an equilibrium state with a locally minimal magnetic Gibbs free-energy (Brown, 1963).

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In this study we utilise a modified gradient descent method to find the equilib-  (Bloch, 1932), where C is a 106 function of the spin wave stiffness, they were able to fit the data and obtain an 107 estimate of the spin wave stiffness and therefore the exchange stiffness constant. has been known to produce variable results for magnetite (Chang et al., 2008).

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This places a degree of uncertainty on this measurement for greigite that is hard 113 to quantify in the absence of measurements acquired through means other than 114 low-temperature saturation magnetisation.

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Chemical alteration of greigite at high temperatures has made difficult to 116 measure accurately the Curie temperature; however, there is strong evidence for 117 a Curie temperature T C > 620 K (Roberts et al., 2010). The exchange energy 118 is directly related to T C ; within a mean field approximation (Kouvel, 1956): where K B is Boltzmann's constant, J AB is the exchange integral between A- where µ 0 is the magnetic constant (or vacuum permeability) and as determined by the so-called smoothing factor (SF) and including (2×SF + 1)  Distributions with random orientation of magnetic particles with respect to 161 the applied field were determined by taking 500 field orientations from a sector 162 of the unit sphere ( Fig. 1). We use 500 field orientations as a workable com-163 promise between accuracy and calculation speed. Also, for each particle/field-

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For ensembles with SD particles < 50 nm, hysteresis behaviour is dominated 183 by coherent rotation (Fig. 2). This is seen by comparing FORC diagrams for  Particles with size d ≥ 50 nm switch incoherently (Fig. 3); that is, the 223 FORC diagrams depart from coherent rotation behaviour associated with SD 224 particles as the tight boomerang-shaped FORC diagram pattern exhibited by 225 the SD greigite (Fig. 2) becomes more fragmented (Fig. 4). This change is 226 driven initially by particles with hard axes close to the applied field nucleating FORC diagrams (Fig. 4a,b) are not evident in changes in the saturation rema-248 nence M RS to saturation magnetisation M S ratio up to 74 nm, whereas coercivity 249 decreases sharply above 48 nm (Fig. 5b). The monotonically-decreasing coer- sizes coincide with the anomalous coercivity increase for these sizes (Fig. 5b).

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The increased coercivities can be explained by vortex nucleation, which causes 262 hysteresis loops to become increasingly wasp-waisted (Fig. 6) so that they cross  Particles with hard axes aligned closely with the applied field nucleate hard-282 aligned vortices at high applied field values (Fig. 8); as the field decreases 283 below ∼12 mT these vortices rotate irreversibly to an easy axis alignment. As 284 the field is increased on reversal curves with ∼0 mT ≤ B a ≤ ∼12 mT these 285 vortices switch irreversibly back to a hard alignment at B b ≈ 28 mT to create a 286 local peak at B c ≈ 12 mT, B u ≈ 16 mT (Fig. 7, region 2); this is manifested in 287 the raw hysteresis data by the smoothed discontinuity at B ≈ 28 mT whereas 288 the reversible motion traced by the reversal curves around this region accounts 289 for the tilted, elongated response surrounding the local peak (Fig. 7, region 1).  (Fig. 7, regions 1, 2).

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As the applied field decreases past ∼−52 mT, the vortices of particles with 299 easy axis alignment close to the applied field annihilate (Fig. 8). Reversal  Increasing the applied field to positive values causes the easy-aligned vortices of 308 particles with hard axes close to the applied field to switch to hard alignments at 309 ∼28 mT, creating a negative FORC region (Fig. 7, region 5). The distribution 310 peak at region 6 ( Fig. 7) corresponds to the average annihilation field of the 311 vortices on the reversal paths to positive saturation.

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There is a large spread in the vortex nucleation and annihilation fields (Fig.   313 8). Particles with hard axis alignment close to the applied field nucleate hardaligned vortices for fields as high as ∼200 mT and annihilate on the opposite side 315 of the particle for equally high (absolute) values. However, these nucleation and 316 annihilation events make a negligible contribution to the FORC diagram because 317 the change in magnetisation of a particle nucleating/annihilating a hard-aligned 318 vortex from/to a SD state can be as low as 1%.