Edinburgh Research Explorer Multi-scale 3-dimensional characterisation of iron particles in dusty olivine: implications for paleomagnetism of chondritic meteorites

Dusty olivine (olivine containing multiple sub-micrometer inclusions of metallic iron) in chondritic meteorites is considered an ideal carrier of paleomagnetic remanence, capable of maintaining a faithful record of pre-accretionary magnetization acquired during chondrule formation. Here we show how the magnetic architecture of a single dusty olivine grain from the Semarkona LL3.0 ordinary chondrite meteorite can be fully characterized in three dimensions, using a combination of focused ion beam nanotomography (FIB-nT), electron tomography, and finite-element micromagnetic modeling. We present a three-dimensional (3D) volume reconstruction of a dusty olivine grain, obtained by selective milling through a region of interest in a series of sequential 20 nm slices, which are then imaged using scanning electron microscopy. The data provide a quantitative description of the iron particle ensemble, including the distribution of particle sizes, shapes, interparticle spacings and orientations. Iron particles are predominantly oblate ellipsoids with average radii 242 ± 94 × 199 ± 80 × 123 ± 58 nm. Using analytical TEM we observe that the particles nucleate on sub-grain boundaries and are loosely arranged in a series of sheets parallel to (001) of the olivine host. This is in agreement with the orientation data collected using the FIB-nT and highlights how the underlying texture of the dusty olivine is crystallographically constrained by the olivine host. The shortest dimension of the particles is oriented normal to the sheets and their longest dimension is preferentially aligned within the sheets. Individual particle geometries are converted to a finite-element mesh and used to perform micromagnetic simulations. The majority of particles adopt a single vortex state, with “bulk” spins that rotate around a central vortex core. We observed no particles that are in a true single domain state. The results of the micromagnetic simulations challenge some preconceived ideas about the remanence-carrying properties of vortex states. There is often not a simple predictive relationship between the major, intermediate, and minor axes of the particles and the remanence vector imparted in different fields. Although the orientation of the vortex core is determined largely by the ellipsoidal geometry (i.e., parallel to the major axis for prolate ellipsoids and parallel to the minor axis for oblate ellipsoids), the core and remanence vectors can sometimes lie at very large (tens of degrees) angles to the principal axes. The subtle details of the morphology can control the overall remanence state, leading in some cases to a dominant contribution from the bulk spins to the net remanence, with profound implications for predicting the anisotropy of the sample. The particles have very high switching fields (several hundred millitesla), demonstrating their high stability and suitability for paleointensity studies. 100 = 22 x10 - 6 (Radeloff 1964), this occurs only for d > 9 m m such that magnetostriction can safely be neglected for all modeled particles. Also magnetoelastic interaction is neglected because it is assumed that any strain involved in the formation of the particles has been relaxed by plastic deformation such that no noticeable internal stress field is present. Simulations were performed using an Apple iMac with a 3.4 GHz Intel i7 processor and 24 GB of RAM. Each particle was initialized with uniform magnetiza- tion along either the X, Y, and Z axes of the reconstructed volume. Fields varying from 1000 mT to –1000 mT in steps of 10 mT were applied along X, Y, and Z. The converged set of magnetic moments obtained after each field step was subjected to small random rotations (maximum angle 20 ° ) and then used as the basis for the starting condition for the next field step. This step is to ensure that the energy minimization is not trapped in a local energy minima. The average magnetization projected on each the X, Y, and Z axes were calculated at each step to generate the upper branch of the hysteresis loop. Lower branches were not calculated directly using micromagnetics, but are presented for visualization purposes under the as- sumption that these are symmetrically equivalent to the upper branch.


233!
Here we present bright-field TEM and STEM imaging, electron diffraction data as 234! well as dark-field STEM tomography results. Both TEM and bright-field STEM modes 235! allow us to image crystallographic features such as dislocations and sub-grain boundaries 236! (Williams and Carter 2002;Crewe and Nellist 2009). Dark-field STEM tomography of a 237! single particle was performed using a camera length of 130 mm to produce a strong 238! material contrast between the Fe particles and the olivine crystal. The tomographic series 239! was collected at a magnification of 28500X (giving a pixel size of 3.26 nm) using the 240! high angle annular dark field (HAADF) detector. The tilt series consisted of an image 241! taken every 2° for tilts from -76° to +76°. Alignment and reconstruction of the tilt series 242! was achieved using Inspect 3D with the SIRT algorithm (Gilbert 1972). Visualisation 243! was performed using Avizo Fire. 244! 245!

Micromagnetic Modelling 246!
A selection of Fe metal particles, representing the range of sizes and shapes 247! within the ensemble, were chosen to perform detailed micromagnetic simulations. Each 248! particle was cropped from the segmented FIB-nT stack and converted to a tetrahedral 249! ! 12! finite-element mesh in a multi-step process. An example of the initial geometry of a 250! particle defined by the FIB-nT is shown in Fig. 3a. Each rectangular block represents a 251! single 9.8 x13.2 x 20 nm voxel. This representation of the particle was used to generate a 252! bounding polyhedron that best approximates the actual particle surface, where each point 253! of the resulting mesh must solve a Poisson boundary condition (Fig 3b). We further 254! refine the surface mesh by passing it through a surface smoothing routine (Fig. 3c). We 255! coarsen the smoothed surface using a Delaunay triangulation routine to produce a surface 256! mesh at the desired resolution of 5 nm (Fig. 3d). The final triangular surface mesh was 257! imported into the software package CuBit (KitWare), where it was turned into a 258! tetrahedral volume mesh. We used an initial surface mesh with average node spacing 5 259! nm, which was then used to generate tetrahedral nodes on average every 5 nm throughout 260! the volume. Although this resulted in a mesh size slightly bigger than the 3.4 nm 261! exchange length for iron, it enabled the number of elements in the model to be kept 262! below approximately 300,000 and provides acceptable resolution for modelling simple 263! vortex micromagnetic structures.

264!
Micromagnetic modeling was performed using MERRILL (Micromagnetic Earth 265! Related Rapid Interpreted Language Laboratory), a micromagnetics package optimized 266! for the rock magnetic community developed by K. Fabian and W. Williams (Williams 267! and Fabian 2016). MERRILL uses a Finite Element Method/Boundary Element Method 268! (FEM-BEM) to solve for the magnetic scalar potential inside the particle and thereby 269! calculate the demagnetizing energy of the system. The use of FEM-BEM avoids the need 270! to discretize the non-magnetic volume outside the particle. Simulations were performed 271! by minimizing the total micromagnetic energy. This consists of summing the exchange, 272! ! 13! cubic anisotropy, magnetostatic and demagnetizing energies. Energy minimization was 273! performed using a conjugate gradient method, specially adapted to micromagnetic 274! problems. MERRILL has been successfully tested against µMAG Standard Problem #3 275! (http://www.ctcms.nist.gov/~rdm/mumag.org.html).

276!
Material parameters used were appropriate for pure iron at room temperature: 277! saturation magnetization M s = 1715 kA/m, exchange constant A = 2 x 10 -11 J/m, and 278! cubic anisotropy with K 1 = 48 kJ/m 3 (Muxworthy and Williams 2015). We arbitrarily set 279! the cubic <100> axes parallel to the X, Y and Z axes of the volume reconstruction. We 280! will show in Section 3.3 that this does not greatly influence our analysis, as the magnetic 281! behavior of the particles studied is dominated by shape rather than magnetocrystalline 282! anisotropy.

283!
Using the method of Hubert (1967), it can be estimated that magnetostrictive 284! effects become important for iron only when the magnetostrictive energy density 9/2 (c 11 -285! c 12 ) λ 100 2 is of similar size as the energy density ~2(A K 1 ) 1/2 /d generated by a 180 o -domain 286! wall, where d is the dimension of the particle (Hubert 1967;Fabian et al. 1996;Hubert 287! and Schäfer 1998). For iron with elastic constants c 11 =241 GPa, c 12 =146 GPa (Lee 1955), 288! and λ 100 =22 x10 -6 (Radeloff 1964), this occurs only for d >9 µm such that 289! magnetostriction can safely be neglected for all modeled particles. Also magnetoelastic 290! interaction is neglected because it is assumed that any strain involved in the formation of 291! the particles has been relaxed by plastic deformation such that no noticeable internal 292! stress field is present.

293!
Simulations were performed using an Apple iMac with a 3.4 GHz Intel i7 294! processor and 24 GB of RAM. Each particle was initialised with uniform magnetization 295! ! 14! along either the X, Y and Z axes of the reconstructed volume. Fields varying from 1000 296! mT to -1000 mT in steps of 10 mT were applied along X, Y and Z. The converged set of 297! magnetic moments obtained after each field step was subjected to small random rotations 298! (maximum angle 20°) and then used as the basis for the starting condition for the next 299! field step. This step is to insure that the energy minimisation is not trapped in a local 300! energy minima. The average magnetization projected on each of the X, Y, and Z axes 301! were calculated at each step in order to generate the upper branch of the hysteresis loop.

302!
Lower branches were not calculated directly using micromagnetics, but are presented for 303! visualisation purposes under the assumption that these are symmetrically equivalent to 304! the upper branch.

FIB-nT 307!
The reconstructed dusty olivine volume is shown in Fig 4a-c. A qualitative 308! analysis (see movie in supplemental information) reveals that i) the particles are loosely 309! arranged in planar sheets, ii) the particles tend to be flattened in the direction 310! perpendicular to the sheets, and iii) there is a preferred orientation of particle elongation 311! along a direction within the sheets. We will present the crystallographic analysis in 312! section 3.2 when we present the TEM and STEM results. Particles are widely distributed 313! in terms of their size and aspect ratio. Figure 5 summarises this distribution by plotting a 314! histogram of the best-fit ellipsoid diameters for the major (Fig. 5a), intermediate (Fig. 5b) 315! and minor (Fig. 5c) (Flinn 1962). This 'Flinn' plot also scales the size of each data point to the size of the 320! major radius. The line y = x separates prolate particles (above the line) from oblate 321! particles (below the line). There are significant populations of small, flattened particles 322! that plot close to the horizontal (uniaxial oblate) and elongated particles that plot close to 323! the vertical (uniaxial prolate) axes. As seen qualitatively in the Flinn plot and 324! quantitatively in the histogram in Fig. 6b, 95% of the particles are classified as oblate in 325! aspect ratio with a mean Flinn ratio of 0.76. This gives the majority of particles a tri-axial 326! symmetry. This non-spherical and non-uniaxial aspect ratio has profound implications for 327! the magnetic anisotropy of each particle, which we will explore in more detail using FEM 328! models (Section 4.3). Of the reconstructed particles, only 11 possess a prolate aspect 329! ratio. The three smallest of these are close to the limitations of our reconstruction 330! resolution. For these particles, the radii are between 2 to 6 voxels long in any one 331! direction, which means that the uncertainty in the ellipsoid-fitting algorithm can be on the 332! order of the size of the particle. We found that extracting these individual particles from 333! the larger tomographic volume and rerunning the BoneJ analysis lead to small changes in 334!  Fig. 7 show the principal axes of the anisotropy of susceptibility of 344! anhysteretic remanent magnetization (ARM) for the whole dusty olivine chondrule, as 345! determined by scanning SQUID microscopy (Fu et al. 2014

371!
The lower right hand image in Fig. 8b is a 3D visualisation of the small Fe 372! particle obtained using STEM tomography (Movie of particle in Supplemental 373! Materials). The voxel size is 3.3 nm, at least a factor of 3 smaller than the voxel size 374! obtained via FIB-nT. The particle dimensions are 232 nm by 205 nm by 232 nm (X, Y, Z 375! with respect to the image plane), giving it a slightly oblate profile. These dimensions 376! mean that the particle is one of the smaller ones in the population. It is representative of 377! the largest complete particle that could be imaged using STEM tomography (larger 378! particles were truncated by the surfaces of the TEM foil). Particles of this size (and 379! smaller) are observable using the FIB-nT approach (albeit at considerably lower spatial 380! resolution), so there is some overlap between the size ranges accessible by the two 381! techniques. 2D particle analysis of this lamella does not reveal any particles smaller than 382! those observed in the FIB-nT volume. 383! Figure 8a shows sub-grain boundaries in the olivine running parallel and 384! perpendicular to [001]. As the diffraction pattern for the olivine in Fig 8b is  intersecting the 100 nm thick lamellae surface with an angle of around 9° as noted above.

389!
These observations about the arrangement of the sub-grain boundaries are in line with the 390! previous study by Kirby and Wegner (1978), which demonstrated that the dislocation 391! arrays in olivine concentrate along the {100} lattice planes. The dislocations defining the 392! sub-grain boundaries are more clearly visible in the bright-field TEM image (Fig. 8c), 393! which was taken from the region outlined in blue. The sub-grain boundary seen on the 394! right of this image is parallel to (001) of the olivine and lies parallel to a prominent (100) 395! facet of the Fe crystal. This sub-grain boundary is also parallel to the plane containing the 396!

major and intermediate axes of the ensemble (inset upper left). Previous studies of natural 397!
and synthetic dusty olivines (Leroux et al. 2003;Lappe et al. 2013) have suggested that 398! the Fe nanoparticles arrange along dislocation arrays associated with the sub-grain 399! boundaries. By measuring the crystallographic information of the lamella from the same 400! region, we are able to demonstrate that the particle ensemble does indeed arrange in 401! sheets related to crystallographic planes of the olivine host crystal. 402! 403!

Remanence states and magnetic moments.
The results of micromagnetic 406! simulations for 9 selected particles (8 particles extracted from the FIB-nT stack plus the 407! STEM tomography particle shown in the red inset of Fig. 8) are summarized in Table 1.

408!
A range of oblate (Flinn ratio less than 1) and prolate (Flinn ratio greater than 1) 409! ! 19! ellipsoids are represented, with volumes ranging from the smallest in the ensemble 410! (Particle 48) to those closer to the average (Particle 165). Remanence states were 411! obtained after applying a saturating field of 1 T along the X, Y and Z directions of the 412! reconstructed volume, and then stepping the field down to zero in steps of 10 mT. The 413! squareness M rs /M s is the magnitude of the total remanence vector normalized to the 414! saturation moment of the particle. To aid comparison and to give a better sense of the 415! magnitude of the magnetic moment of each particle, we define 'relative M r ' as the total 416! remanent moment of the particle divided by the total remanent moment of a uniformly 417! magnetized 25 nm diameter sphere. This size corresponds to the upper threshold for 418! single-domain (SD) Fe (Muxworthy and Williams 2015).

419!
No particles were small enough to adopt an SD state. Instead all particles adopt 420! either pseudo-single domain (PSD) or emerging multi-domain (MD) states, consisting of 421! either a single vortex or multiple vortex/wall-like structures (Fig. 9). To highlight the 422! orientation and nature of vortex cores, the magnitude of the vorticity of the moment 423! vector field (|∇×!|) is plotted as an isosurface (green in Fig. 9). The choice of isosurface 424! magnitude is somewhat arbitrary (too large and only the ends of the core are highlighted; 425! too small and the surface extends too far from the core region). We chose the largest 426! value that would produce a continuous trace of the core from one surface termination to 427! another. The remanent states of the particles studied can be divided into four general 428! categories: I) single vortex with core oriented close to the minor axis of the best-fitting 429! ellipsoid (Fig. 9a); II) single vortex with core oriented within the plane defined by the 430! minor and major axes of the best-fitting ellipsoid (Fig. 9b); III) single vortex with core 431! ! 20! oriented parallel to the major axis of the best-fitting ellipsoid (Fig. 9c); IV) multiple 432! vortex/wall-like structures containing two or more cores (Fig. 9d).

433!
Type I behavior was typically observed in oblate particles with lower Flinn ratios 434! (flattened ellipsoids). Type I particles display straight cores located at the center of the 435! largest face. The relatively strong demagnetizing field of the vortex core in this case can 436! be shielded by antiparallel spin tilting at the outer rim of the oblate particle. Type II 437! behavior was typically observed in oblate particles with higher Flinn ratios (triaxial 438! ellipsoids). Type II particles display curved cores that adopt a sigmoidal trajectory 439! through the center of the particle. The ends of the core lie normal to their surface 440! terminations, as requested by the micromagnetic boundary conditions (e.g., Hubert and 441! Schäfer 1998). Type III behavior was observed in the three prolate particles. Type III 442! particles display cores that track the major ellipsoid axis in the central section of the 443! particle. The ends of cores again lie normal to their surface terminations, causing 444! deviations and distortions of their trajectory. Type IV behavior was observed in the two 445! large prolate particles and already represents a diamond-shape domain pattern with 446! preference for 90° walls, as previously observed for iron thin films and whiskers (e.g., 447! Hubert and Schäfer, 1998). The remanent state of these particles was more 448! sensitive to the direction of the saturating field than oblate particles with similar volume.

449!
In the case of Particle 233 (the largest and most elongated prolate particle), states III, IVa 450! and IVb were adopted for saturating fields applied along X, Y and Z, respectively (Table  451! 1). IVa contains two curved cores corresponding to Bloch walls that split the particle into 452! magnetic domains; IVb is an efficient flux closure structure containing ten magnetic 453! domains (Fig. 9d).
The morphology of the vortex core evolves with increasing volume of particle 455! (Fig. 10). Small particles contain well-defined cylindrical cores (Fig. 10a). Larger 456! particles develop cores with a 'winged' structure, with wings protruding along the 457! directions of emerging domain walls (Fig. 10b). In larger prolate particles, the core is 458! poorly defined, becoming flattened and developing off-shoots (Fig. 10c) Fig. 11c. The cross section reveals the presence of 4 wall-470! like structures associated with the rotation of spins around a central vortex core. Due to 471! its alignment with a perpendicular easy axis, the vortex core appears as the cylindrical 472! hole in the center of the anisotropy surface in Fig. 11c (Fig. 12a). The remanence 491! states in each pair are antiparallel to each other, but lie at some angle to the other pair.

492!
Remanence directions for two Type I particles are shown in Figs. 12b and 12c. The angle 493! between remanence pairs is 32° for Particle 364 (Fig. 12b) and is 80° for Particle 155 494! (Fig. 12c). In Particle 364 (a small oblate particle), the remanence lies close to the core 495! direction, i.e. close to the minor axis of the best-fitting ellipsoid (Fig. 12b). Note also that 496! the remanence states obtained for this particle after applying a saturating fields along X 497! and Z are identical (only the X state is plotted in Fig. 12b). In Particle 155 (the most 498! extreme oblate particle), the remanence obtained after magnetizing along Y lies much 499! closer to the major axis of the best fitting ellipsoid than the minor axis (Fig. 12c). A Type 500! ! 23! II particle is shown in Fig. 12d. The angle between remanence pairs is small (16°) and the 501! remanence lies at an intermediate angle between the minor and major axes of the best 502! fitting ellipsoid, parallel to the average core orientation. A Type III particle is shown in 503! Fig. 12e. The angle between remanence pairs is 56°, with the remanence lying close to 504! the minor axis for fields applied along X and close to the major axis for fields applied 505! along Z. In Type IV particles (not shown), the remanence generally lies closest to the 506! major axis of the best-fitting ellipsoid. Those particles that show a change of domain type 507! with field direction (Table 1) display a correspondingly wider range of possible 508! remanence directions.

509!
It is possible to access each of the four states by applying a suitably oriented 510! saturating field. For example, applying saturating fields to Particle 155 along X and Y 511! switches the sense of vortex rotation while retaining the direction of core magnetization 512! (Fig. 12c). In small fields, the four states are separated by energy barriers that could, in 513! principle, be overcome by thermal fluctuations. However, the energy barriers associated 514! with switching the sense of vortex rotation while retaining the core direction (or 515! switching the core direction while retaining the sense of vortex rotation) are likely to be 516! very high compared to the barriers associated with switching both together. This is 517! because the former process will require considerable internal disruption to the 518! micromagnetic state and a correspondingly high exchange energy penalty, while the latter 519! can be achieved simply by 180° rotation of the micromagnetic state against the shape 520! anisotropy of the particle. Calculating these energy barriers is the next computational 521! challenge, and will ultimately enable the acquisition of remanence during cooling of 522! these particles to be modelled. 523! ! 24! 524! Fig. 13 for selected particles and 525! applied field directions. The magnetic response of all particles is dominated by reversible 526! magnetization processes (e.g., the rotations of bulk spins towards the field). The 527! reversible component of magnetic susceptibility is highest (lowest) for fields applied 528! parallel to the major (minor) axis of the best-fitting ellipsoid. Irreversible magnetization 529! processes (e.g., nucleation of vortices, irreversible switching of vortex core position, core 530! orientation or core magnetization, changing sense of bulk spin rotation, denucleation of 531! vortices) produce small steps in magnetization superimposed on the large reversible 532! component. This leads to loops characterized by very low values of coercivity (H c ) and 533! squareness (M rs /M s ). Highest coercivities (30-40 mT; Table 1) are observed in the small 534! Type 1 particles when fields are applied along X (e.g., antiparallel to the core direction, 535! close to the minor axis of the best-fitting ellipsoid; Fig. 13a). Typical coercivities are of 536! the order of a few mT or less (Fig. 13d). Negative values of coercivity listed in Table 1 537! highlight an unusual behavior (e.g., Fig. 13c), whereby the upper branch of the hysteresis 538! curve reaches the M = 0 axis at a positive applied field. This behavior leads to a self-539! reversal of saturation isothermal remanent magnetization (SR-SIRM), in which a 540! component of saturation remanent magnetization is antiparallel to the saturating field 541! direction.

542!
Despite the low M rs /M s values, the large volume of the particles means that their 543! total moments are at least equivalent to that of a 25 nm diameter SD particle, and in many 544! cases significantly greater (relative M r values vary from ~1 up to ~44; Table 1). Despite 545! the low H c values, the remanence states are also highly stable with respect to applied 546! ! 25! fields. We define the stability of a remanence sate in terms of the 'minimum irreversible 547! field' (Table 1). To calculate this field, the spin state obtained at each negative field of 548! the upper hysteresis branch (from -10 to -1000 mT) was chosen as an initial 549! configuration, and the micromagnetic energy was minimized under zero field. The 550! minimum irreversible field is the smallest negative field required to produce a change in 551! the remanent state of the particle. Minimum irreversible fields are typically several 552! observed in synthetic dusty olivine samples using FORC diagrams (Lappe et al. 2011(Lappe et al. , 569! ! 26! 2013 and to the high stability of natural remanent magnetization with respect to 570! alternating-field demagnetization observed by Fu et al. (2014). In comparison, FORC 571! diagrams of MD materials typically show irreversible magnetization restricted to fields 572! less than a few mT (Church et al. 2011;Lindquist et al. 2015) Unsurprisingly, the lowest 573! irreversible fields observed here are for Type IV particles, which contain more MD-like 574! structures. Even here, though, irreversible fields of 40-100 mT are typical. 575! 576!

Rock magnetism of realistic ensembles 578!
Conventional characterisation of the remanence carriers in rocks typically relies 579! on either optical or SEM imaging of polished surfaces, or TEM imaging of thin foils. 580! Although such 2D methods are an essential part of the qualitative characterisation 581! process, our numerical simulations emphasise just how important 3D knowledge of 582! particle geometry is for quantitative modelling. Shape and crystallographic orientation of 583! an individual particle controls the orientation of its vortex core, or equivalently the 584! position and type of its domain walls. Near the surface the bulk spins adapt to the 585! faceting of the particle. In symmetric particles, we might expect that the bulk spins cancel 586! each other out and that the total remanence would be dominated by uncompensated spins 587! within the core. A striking example of where this assumption breaks down is shown in 588! Fig. 14 (Particle 155, a Type I particle with a pronounced oblate geometry). In this case, 589! the core is parallel to the minor axis of the best-fitting ellipsoid, but the net remanence 590! lies at a large angle to this, and rotates by 80° as the sense of bulk spin rotation changes 591! ! 27! (Fig. 12c). The explanation for this behaviour is the combination of the short length of 592! the core, which reduces its contribution to the net moment, and the uneven length of 593! opposing surface facets, which creates a significant contribution from uncompensated 594! bulk spins. In Fig. 14, this unbalancing is highlighted by plotting the anisotropy energy of 595! the domain walls. The width of the walls and the sizes of the four resulting quadrants can 596! be seen to be of unequal sizes. In the configuration shown, there are more spins pointing 597! along +Z due to the larger facet on right than there are compensating spins along -Z due 598! to the small facet on the left. A similar situation is known for vortex states in sufficiently 599! large uniaxial particles, where several metastable magnetization states may exist that are 600! related to edge moments that can be aligned parallel or anti-parallel to the global 601! demagnetizing field (Fig. 7 in Rave et al. 1998). The importance of bulk spins in 602! controlling the remanence of vortex states in particles with realistic morphologies is not 603! generally appreciated.

604!
The sheet-like arrangement of particles within the reconstructed volume (Fig. 4) is 605! expected to generate significant remanence anisotropy. Such anisotropy is well 606! documented in single-crystal paleomagnetism studies (Feinberg et al. 2004(Feinberg et al. , 2005Fu et 607! al. 2014) and must be corrected before the measured remanence directions can be 608! interpreted quantitatively. Fu et al. (2014) measured the anisotropy of ARM susceptibility 609! for the same grain studied here (Fig. 7), finding normalised values of 0.45, 0.56 and 1 for 610! the minimum, intermediate and maximum susceptibilities, respectively. The maximum 611! ARM susceptibility is observed for fields applied along Z, which corresponds to the 612! average orientation of major ellipsoid axes. This observation is most consistent with 613! remanence carried by prolate Type III and IV particles and oblate Type I particles such as 614! ! 28! those shown in Fig. 14, and suggests that such particles are more prevalent in regions of 615! the grain outside the reconstructed volume. The fact that the minimum susceptibility axis 616! is not aligned with the average orientation of minor axes is explained by the presence of 617! Type I and II particles, which contribute to the remanence when fields are applied normal 618! to the sheets. A full model of the entire ensemble, taking into account the magnetostatic 619! interaction fields between particles and the distribution of shape anisotropy, would be 620! necessary to fully describe the anisotropic response of the system (Hargraves et al. 1991) 621! as well as accounting for the presence of much larger particles that may occur outside the 622! analysed volume. Such calculations could serve to improve dramatically the 623! interpretation of single-crystal paleomagnetic studies, and minimise the number of repeat 624! measurements needed to reach a statistical significance equivalent to bulk paleomagnetic 625! studies. 626! 627!

Implications for chondrite paleomagnetism 628!
This study demonstrates that particles in the lower half of the size distribution 629! adopt a single vortex state (Type I and II), with the larger particles adopting Type III and 630! IV (i.e., emerging MD) behaviour. Lappe et al. (2011) similarly identified a dominance of 631! single vortex states, but noted also a significant number of SD particles. It appears that 632! the size distribution of particles in synthetic dusty olivine is shifted to smaller values than 633! the natural sample studied here. Nevertheless, given the abundance of single vortex states 634! in both cases, calibrations of non-heating paleointensity methods using synthetic dusty 635! olivine (Lappe et al. 2013) remain largely valid. Similarly, given the high field stabilities 636! observed here for single vortex states (Table 1), the conclusion that dusty olivine is 637! ! 29! capable of recording and maintaining a faithful record of pre-accretionary remanence 638! (Lappe et al. 2011) also remains valid. Indeed, the combination of high switching fields 639! and large volumes of single vortex states should translate to carriers with very high 640! thermal stability (i.e. high blocking temperatures). High thermal stability of vortex states 641! at temperatures up to the Curie temperature has recently been demonstrated in magnetite 642! using electron holography (Almeida et al. 2014(Almeida et al. , 2016. In principle, micromagnetic 643! simulations can be used to calculate energy barriers between alternate remanence states, 644! and thereby model the acquisition of thermoremanent magnetization. Such calculations 645! are beyond the scope of the present study, but are an obvious next step in the 646! nanopaleomagnetic approach. 647! 648!

649!
FIB-nT reveals not only the true size and shape distribution of individual 650! particles, but also the required mesoscale information at the ensemble level. The spatial 651! resolution is good enough to detect particles that span the SD to PSD range (the size 652! range of most importance in rock magnetism) and the volume of sample accessible by 653! FIB-nT approaches the volumes that can now be detected paleomagnetically using 654! SQUID microscopy (i.e. tens of microns). A long-term goal of rock magnetism is to 655! understand the collective behaviour of particle ensembles based on fundamental physical 656! principles. Some recent progress in this area has been made (Harrison and Lascu 2014)  the vorticity of the moment vector field. These vary particle to particle and are selected to 867! present the minimum continuous surface. (a) Type I: single vortex with core oriented 868! parallel to the minor axis of the best-fitting ellipsoid for particle 155. (b) Type II: single 869! vortex with core oriented within the plane defined by the minor and major axes of the 870! best-fitting ellipsoid for particle 165. (c) Type III: single vortex with core oriented 871! parallel to the major axis of the best-fitting ellipsoid for particle 75. (d) Type IV: multiple 872! core/wall-like structures for particle 233 with the magnetic field applied along the Y and 873! Z-axis respectively. two Type I particles (b, c); (d) a Type II particle; and (e) a type III particle. Note also that 898! the remanence states obtained for this particle after applying a saturating fields along X 899! and Z are identical (only the X state is plotted in Fig. 12b). 900! 901! Figure 13 Calculated components of magnetization parallel to X (blue), Y (green) and Z 902! (red) as a function of applied field for the upper (solid) and lower (dashed) branches of 903! the major hysteresis loop. (a) Type I particle (364) with field applied along X. (b) Type I 904! particle (48) with field applied along Y. (c) Type I particle (48)