# Coupling nanoscopic tomography and micromagnetic modelling to assess the
stability of geomagnetic recorders

Ualisson Donardelli Bellon  University of São Paulo, Institute of Astronomy,
Geophysics and Atmospheric Sciences (IAG), Department of Geophysics, São
Paulo, 05360020, Brazil  University of Edinburgh, School of Geosciences,
Edinburgh, EH9 3FE, Scotland  Correspondent authors<EMAIL_ADDRESS><EMAIL_ADDRESS>Wyn Williams  University of Edinburgh, School of
Geosciences, Edinburgh, EH9 3FE, Scotland  Correspondent authors:
<EMAIL_ADDRESS><EMAIL_ADDRESS>Ricardo Ivan Ferreira Trindade
University of São Paulo, Institute of Astronomy, Geophysics and Atmospheric
Sciences (IAG), Department of Geophysics, São Paulo, 05360020, Brazil  Ana
Diaz  Paul Scherrer Institute, 5232 Villigen PSI, Switzerland  Douglas Galante
University of São Paulo, Institute of Geosciences, Department of Sedimentary
and Environmental Geology, São Paulo, 05508080, Brazil

###### Abstract

The recording of planetary magnetic fields is often attributed to uniformly-
magnetised nanoscopic iron oxides, called single-domain (SD). Yet, the main
magnetic constituents of rocks are more complex, non-uniformly magnetised
grains in single or multi-vortex states. We know little about their behaviour
due to limitations in defining their precise shape and internal magnetic
structure. We propose a novel approach combining non-destructive synchrotron-
based ptychographic nanotomography with micromagnetic modelling to explore the
magnetic stability of remanence-bearing minerals. Applied to a microscopic
rock sample, we identified hundreds of nanoscopic grains of
magnetite/maghemite with diverse morphologies. For some grains, shape
irregularities near the transition from SD to the single-vortex state allow
for multiple domain states, some unstable and unable to record the field for
significant periods. Additionally, some other grains exhibit temperature-
dependent occupancy probabilities, potentially hampering experiments to
recover the intensity of past magnetic fields.

Keywords: nano-tomography, micromagnetic modelling, vortex state

## Introduction

Measuring the magnetic properties and identifying the magnetic minerals in
rocks is essential to palaeomagnetism, which studies Earth’s magnetic field
strength and direction throughout the planet’s geological history [1]. While
directional palaeomagnetic data is the only quantitative measurement that
allows geoscientists to study palaeoconfigurations of ancient continental
landmasses [2], databases of the past magnetic field strength
(palaeointensity) [3] provide links to the complex magnetohydrodynamic
processes occurring deep in Earth’s core [4, 5, 6]. Quantifying the magnetic
phases in rocks also provides essential information on globally scaled
geological/climate events [7]. Rocks usually contain a mixture of magnetic
particles in various sizes, including very stable nanoscopic uniformly
magnetised grains (said to be in a single-domain (SD) state) and larger non-
uniformly magnetised grains, either in pseudo-single-domain (PSD) or in multi-
domain (MD) states. The palaeomagnetic theory is mainly based on Néel[8]
ferromagnetic theory for SD grains, which relies on the capacity of these
particles to retain their original remanence for billions of years but
accounts only for a small proportion of magnetic particles contained within a
typical palaeomagnetic sample.

Palaeomagnetic analysis of a samples’ magnetic mineralogy and stability is
usually based on bulk magnetic measurements, often capable of detecting trace
amounts of magnetic minerals. Although the data obtained through these
analyses use statistical procedures to ensure their reliability, critical
magnetic phenomena with geological/geophysical implications can sometimes be
masked in bulk magnetic measurements. The complexity of natural geological
materials and the diversity of composition/geometry of natural remanence
carriers in rocks indicate that the conditions assumed by classical
palaeomagnetic experiments can be far from ideal. A detailed understanding of
the magnetic recording reliability requires a nanoscale interpretation of the
magnetic fidelity of magnetic mineral grains.

Micromagnetic modelling, which explores the magnetic properties of a given
grain (or a population of them) as a function of their size, shape,
composition and temperature [9, 10], is an important tool that can bring a far
greater understanding of the magnetic recording process. Micromagnetic
calculations have revealed, for example, i) that PSD structures (also referred
to as vortex states due to their magnetisation being dominated by single or
multiple vorticity centres), which is probably the most common type of
magnetic domain structure in natural iron oxides [11], often are capable of
retaining their remanence for timescales longer than the age of the Solar
System [12]; and ii) the theoretical discovery of a non-stable magnetic
configuration in grain sizes beyond the SD threshold [13], which can produce
unexpected magnetic properties (e.g, the reduction of coercivities and thermal
stability) and has the potential to greatly affect palaeointensity data [14].

Particle size is a very important controller of domain state and,
consequently, of magnetic stability [15], but geometry is also a critical
factor. Although micromagnetic modelling of irregular particle morphologies
through the discretization of three-dimensional finite element meshes is
possible [10], most of the theoretical knowledge built on micromagnetics of
iron oxides is based on regular euhedral crystalline habits [16, 9, 17, 12,
18, 14, 13]. Although natural processes can trigger regular crystal
growth/alteration of iron oxides (such as biosynthesis of magnetite by
magnetotatic bacteria [19, 20]), tomographic techniques of iron oxides bearing
rocks very often reveal irregularly shaped grains [21, 11, 22]. Consequently,
it is important to understand how these irregular morphologies of particles
might affect their magnetic properties, especially for grains within the
palaeomagnetically significant PSD threshold.

In this study, we introduce the use of Synchrotron-based Ptychographic X-ray
Computed Nano-tomography (PXCT) [23] (Figure 1) to image the magnetic minerals
within a microscopic sample of a Neoproterozoic remagnetised carbonate, and
further model their micromagnetic behaviour through finite element numerical
models. The primary remanence (acquired during pre-to-syn diagenetic
conditions) of carbonate rocks is very often substituted by a secondary
chemical remanence (CRM) [24, 25] when new magnetic minerals grow or are
chemically altered below their respective Curie temperature (Tc) [15]. These
processes result in magnetic grains with a wide range of sizes and shapes,
most of them within the SD/PSD grain size range, making them a target for
micromagnetic studies of individual particles at the micro/nanoscopic scale.
To image them, PXCT was chosen for being a non-destructive coherent
diffractive imaging method that allows a quantitative mapping of the
tridimensional electron density distribution of the specimen at unmatched
nanoscale resolution [26, 27]. Our results provide new insights into the
magnetic properties of natural remanence carriers in rocks and their
thermomagnetic stability.

Figure 1: Experimental setup for the acquisition of PXCT data. X-rays are
focused by a Fresnel Zone Plate (FZP), in combination with a central stop and
an order sorting aperture (OSA), to define the illumination. The radiation
reaches the microscopic pillar sample, mounted at the tip of an OMNY
(tOMography Nano crYo) pin [28]. A scanning electron microscopy (SEM) image of
the sample is shown in the upper corner of the figure. The sample stage allows
a three-axial movement of the sample on top of a rotation to perform raster
scans at different angles with respect to the X-ray beam. Diffraction patterns
at each scan position are captured by a pixelated detector. The components
sketched in this figure are not to scale. Details can be found in the text (or
in the Supplementary File).

## Results

### Nano-tomography

The processed tomograms representing the 3D electron density distribution of
the specimen resulted in artefact-free volumes with a voxel size of $34.84\
\text{nm}$. An example is shown in Supplementary File, Figure S1a, while
details about the PXCT measurements and data processing can be found in the
Methods section and in the Supplementary File. By using the accumulated
histogram distribution of all of the data and the natural breaks in the
distribution (Supplementary File, Figure S1c), we have segmented five
compositional classes (Supplementary File, Figure S1d, Table S1). Our
interpretations of the imaged materials presented here are also supported by
previous chemical/magnetic analyses performed by Bellon et al. [29].

The classes are named according to their increasing electron density (Figure
2). Class 1 yields the lowest electron density of $\mu=0.31\pm 0.11\
e^{-}\cdot\text{\r{A}}^{-3}$, also showing restricted volumetric occurrence.
This material appears in small clusters within the sample (Figure 2a, green
colour), also showing a spatial correlation to localised porosities. We
attribute this class to organic matter, specifically to bitumen. The
geological formation (from which our sample comes) is naturally enriched in
hydrocarbons [30], as biodegradation of the organic matter is also considered
partially responsible for the growth of authigenic ferromagnetic minerals.
Natural bitumen density might vary depending on its formation and
preservation, but the density of natural bitumens [31] has been reported to be
$1.02-1.03\ \text{g/cm}^{3}$, which reflects an electron density (Equation 2)
of $0.30-0.31\ e^{-}\cdot\text{\r{A}}^{-3}$.

Class 2 $(\mu=0.65\pm 0.08\ e^{-}\cdot\text{\r{A}}^{-3})$ material occupies a
large volume of the sample (Figure 2a, yellowish colour), and it is
interpreted as a phyllosilicate. Smectite-illite has been previously
identified as a major mineralogical constituent of these samples, see Ref.
[29], as their conversion process (smectite to illite) is believed to release
iron that sequentially reacts with the available oxygen to form iron oxides
[32, 33]. The density of such phyllosilicates will depend on their purity.
Smectite’s (end member smectite, _EMS_ , $<2.3\ \text{g/cm}^{3}$)
transformation to illite might be partial (illite-rich mixed-layer clays,
_IML_ , $2.4-2.7\ \text{g/cm}^{3}$), or complete (end-member illite, _EMI_ ,
$>2.7\ \text{g/cm}^{3}$) [34]. The spread distribution of Class 2 matches the
expected density of a partial smectite-illitisation, as _EMS_ and _IML_
electron densities are between $0.69$ and $0.72\ e^{-}\cdot\text{\r{A}}^{-3}$
(respectively).

Classes 3 $(\mu=0.80\pm 0.01\ e^{-}\cdot\text{\r{A}}^{-3})$ and 4
$(\mu=0.88\pm 0.03\ e^{-}\cdot\text{\r{A}}^{-3})$ comprise the majority of the
rock matrix and are interpreted, essentially, as calcium carbonates (Figure
2b). Class 3 matches the expected electron density of calcite
$(\text{CaCO}_{3},0.81\ e^{-}\cdot\text{\r{A}}^{-3})$, while Class 4 matches
that of dolomite $(\text{CaMg(CO}_{3}\text{)}_{2},0.85\
e^{-}\cdot\text{\r{A}}^{-3})$.

Figure 2: Reconstruction of the PXCT 3D electron density distribution after
segmentation, highlighting different materials in (a) and (b). Most of the
iron oxide grains have a major elongation axis between $80-600\ \text{nm}$ (c)
and an axial ratio ($\text{P}_{1}/\text{P}_{3}$) $<2$. In (d), the shape
parameter ($\epsilon$, revisit Equation 4) is quite scattered, showing a well-
distributed population of triaxial ($\epsilon\approx 0$), prolate
($\epsilon<<0$) and oblate ($\epsilon>>0$) grains.

Class 5 gathers the highest mean electron densities of our sample
$(\mu=1.08\pm 0.09\ e^{-}\cdot\text{\r{A}}^{-3})$, almost 20% larger than
Class 4. The volumetric distribution of the material represented by this class
is also quite distinct from the other ones, as these appear as individual
grains spread in the matrix (Figure 2b). They also represent a wider electron
density distribution, with values that reach as far as $1.24\
e^{-}\cdot\text{\r{A}}^{-3}$. We interpret this material as the iron oxide
particles in our sample. The electron density upper limit of Class 5 was also
observed for partially oxidised magnetite/maghemite in previous PXCT studies
[35], but is slightly lower than expected from their theoretical densities.
Several aspects could implicate such discrepancies, for example: (i) chemical
impurities, crystalline imperfections and oxidation; (ii) density is size-
dependent in nanomaterials, being lower/greater than a correspondent
macrometric material depending on the lattice parameter [36]. The most likely
reason though is related to the partial volume effect, _i.e_ , for a given
small volume the measured electron density has a large contribution from the
voxels at the interface with its surrounding material, which can affect the
final average density value. In our case, we have particles of a few hundred
nanometers sampled with a 3D resolution, therefore a large proportion of the
selected volume for density quantification may have a contribution from the
density of the surrounding material. If the surrounding material has a lower
density, the density will be underestimated. Partial volume effects have been
considered before, e.g., in the segmentation of volumes of nanoporous material
obtained by PXCT [37]. Prior applications of synchrotron-based techniques,
specifically X-ray Fluorescence (XRF) and X-ray Absorption Near Edge
Spectroscopy (XANES), have been utilised to identify magnetite/maghemite in a
sister sample currently undergoing nano-tomographic analysis [29]. This
additional evidence further supports our interpretation, as we follow the
micromagnetic modelling of their properties.

Following the processing, segmentation, and mesh generation (refer to
_Methods_ for clarity), we have identified 118 distinct grains of iron oxides
with varied shapes and sizes. Due to their high irregularity, we will discuss
their sizes based on the ratios of the grains’ main axes
($\text{P}_{1}>\text{P}_{2}$ > $\text{P}_{3}$). More than 95% of the major
axis have an elongation of $80-600\ \text{nm}$ (see Figure 2c), and 68% is
smaller than $250\ \text{nm}$. The axial ratio $(\text{P}_{1}/\text{P}_{3})$
evidences that although these grains are irregular, the size of major and
minor axes of most of them do not show big discrepancies, which is reflected
in the overall rounded morphologies (Figure 2c). The smallest particles of our
sample are within the spatial resolution of the tomography and might carry
uncertainties in the imaged morphology. Regardless, these morphologies still
yield scientific validity, as any irregular shapes are valid for our
micromagnetic models. We can more effectively access the morphologies by
evaluating all three axes through the shape parameter ($\epsilon$ as per
Equation 4). The great majority of them are triaxial to oblate-like particles
($\epsilon\geq 0$), but there is still a considerable amount of prolate-like
grains ($\epsilon<0$, Figure 2d). The largest particle identified in our
sample, quite isolated in size, is a very oblate ($\epsilon>0$) grain, which
also corresponds to the biggest axial ratio in Figure 2c at a value just over
7.0. The presence of larger grains poses a computational challenge for
micromagnetic modelling. Generating finite element meshes for their morphology
while adhering to the exchange length ($\text{I}_{\text{e}}$) [38] as the
maximum element size leads to excessive processing times. Consequently, we
have focused exclusively on the finest grains ($\leq 1000\ \text{nm}$), which
are more likely to exhibit the vortex state. As a result, from the initial set
of 118 grains, we have excluded five particles in the subsequent micromagnetic
modelling steps, working with a total of 113 grains. A spreadsheet with the
morphological characteristics and summarised magnetic properties (which are
further discussed) of all of these grains is also provided in a public
repository [39].

The magnetic grains we image in this work result from growing mechanisms that
occurred below the magnetite’s Curie temperature in a sedimentary rock, i.e.,
they originally bear a chemical remanence (CRM). In terms of micromagnetic
modelling, the origin of these grains is not relevant to our calculations, as
the major parameters influencing our results are the dimensions, morphology,
and composition of these grains [10]. In our case, grains were discretised as
finite element meshes of a material with a homogeneous chemical composition.
Comparing morphologies is not a direct task, but the range of morphologies
depicted in Figure 2d is similar to that of inclusions of magnetic minerals
hosted in igneous rocks [22]. Therefore, we consider the implications of our
further discussions do apply to any kind of rocks bearing irregularly shaped
nanomagnetic grains.

### Micromagnetics

#### Local Energy Minimum (LEM)

In this section, our objectives are (i) to assess the stability of remanence
by investigating domain states; and (ii) we seek to explore the conditions
under which different domain states are favoured at varying temperatures.
Furthermore, we identify the temperature intervals during which these domain
states exhibit dominance.

Identifying the magnetic structures of each grain involves running several
models for each grain’s geometry to search for local energy minimum (LEM) in
the magnetic domain states (See _Methods_) [10]. The LEM with the lowest
energy will usually represent the most stable domain state for each grain.
Model LEM solutions were obtained using MERRILL [10] (an open-source 3D
micromagnetics software package), where the magnetic domain states structures
are represented by a unit vector at each node of the model mesh geometry. The
net magnetizations of grains are reported here as values normalised by their
saturated magnetic state. Both the remanence (Mr) values and the topology of
the magnetic domain structures observed for magnetite and maghemite were very
similar (Supplementary File, Figure S2a), as were the corresponding values of
their magnetic free energies (Supplementary File, Figure S2b). This is to be
expected since the minerals share very similar magnetic material properties
(Table 1). Thus although most of the results we describe here are for
magnetite, we anticipate very similar behaviour for maghemite.

We observe a great variety of domain states in the studied grains (Figure 3),
which we categorise following the same approach of other studies [12, 11, 18,
14]. As expected, SD states dominate the smaller particles, which in our case
are also the most rounded morphologies (Figure 3a,b). With increasing grain
sizes and as morphologies become more irregular, non-uniformly magnetised
domain states arise as: single-vortex (SV) structures (Figure 3c,d), with a
vortex core aligned along a specific axis; and multi-vortex (MV) structures,
which only appear for the largest grain sizes considered in our study. As
grain size increases it becomes possible to nucleate multiple different domain
states. These different domain structures can sometimes simply be SV states,
where different LEM vortex core alignments may occur within an irregular grain
morphology (Figure 3c). Larger grains ($\text{P}_{1}>200\ \text{nm}$) might
result in a variety of domain states with differently twisted vortex cores, or
high energy LEMS’s with multiple vortex cores in the same grain (Figure 3e).

Figure 3: Examples of different types of domain states (for magnetite) and
their behaviours are presented here. On the left side, we have compiled the
magnetisation (green diamonds) and the corresponding magnetic energy (black
dots) for each of the 50 LEM models of a specific grain. These values are
normalised by the respective maximum values (of either magnetisation/energy)
among the fifty models. The stereographic projections within the plots display
the direction of the magnetisation for all models by converting the
magnetisation components in the X ($\text{M}_{\text{x}}$), Y
($\text{M}_{\text{y}}$), and Z ($\text{M}_{\text{z}}$) directions to
declination $(\text{DEC}=\arctan(M_{y}/M_{x}))$ and inclination
$(\text{INC}=\arctan(M_{z}/\sqrt{M_{x}^{2}+M_{y}^{2}}))$. The inclination is
coloured according to its polarity (red for positive and blue for negative),
and the triangle indicates the direction of the LEM state with the lowest
energy (Min). On the right, the minimum free energy model of the corresponding
grain is depicted, where the arrows indicate the direction of the
magnetisation, and the vortex structures inside are the isosurface hued by the
helicity. SD grains show (a,b) practically no difference between the models,
whereas SV can exhibit a variety of possible domain states (c,d) with distinct
magnetisation (and associated energies), especially when the morphology of the
grain is highly irregular (e,f).

The normalised magnetisation of magnetic particles has important implications
for the identification of a given domain stature through bulk magnetic
properties [15], particularly for non-uniformly magnetised grains, which
likely dominate the assembly of stable magnetic carriers in rocks [11]. When
we examine the range of a grain’s LEM magnetisation that can be nucleated in
each grain, our results indicate that most grains have $\text{Mr}<0.5$, where
SV structures are dominant. None of the lowest energy states observed in the
LEM models of grains with multi-vortex structures, whether composed of
magnetite or maghemite, show a Mr exceeding $0.5$. Remanences greater than
such threshold for MV particles have only been observed for higher energy
states, which are less stable, while $\text{Mr}>0.5$ (as in Supplementary
File, Figure S2a) are essentially displayed by either SD or SV-like
structures.

The temporal stability of each grain was determined using the Nudged–Elastic-
Band (NEB) method [12]. This allows us to determine the minimum energy path
(MEP) between any two given domain states. For each grain, we took the lowest
energy state (from a set of 50 random LEM states) and calculated the minimum
energy path to its anti-parallel state. In the case where a significant new
minimum was observed along the MEP, we subdivided the path at the new
intermediate minimum and searched the grain for a new MEP. In this way, we
obtained the energy barriers only for the lowest energy LEM state. In this
study, we have chosen not to investigate the stability of higher energy (and
thus less abundant) states we encountered. The energy barriers from the MEP
were converted into magnetisation relaxation times $(\tau)$ using the Néel
Arrhenius (Equation 9).

In Figure 4 we show a spectrum of energy barriers, where the vast majority of
grains yield relaxation times that far surpass the age of the Solar System.
The "eternal" stability includes SD, SV and MV states, the latter having the
highest energy barriers. However, four grains have small $\tau$ values. Grain
32 ($\epsilon=0.78,\ \text{P}_{1}=122\ \text{nm},\
\text{P}_{1}/\text{P}_{3}=1.27$), for example, yields a single vortex
configuration (Figure 4), but our simulation indicates that thermal energy
would overcome magnetic energy in $\tau\approx 3\ \text{hours}$. The other
three grains are also elongated but of grain sizes close to our tomography
pixel size (e.g., Grain 113, $\epsilon=-0.35,\ \text{P}_{1}=49\ \text{nm},\
\text{P}_{1}/\text{P}_{3}=1.21$). Their relaxation times are mostly
instantaneous (Figure 4), which points out that not only size but morphology,
is critical in determining domain stability for such small grain sizes.

Figure 4: Compilation of the computed magnetic energy barriers for each of the
lowest energy LEM states across a minimum energy path to its anti-parallel
state, using the Nudged–Elastic-Band _(NEB)_ method [12]. This energy barrier
is then applied to the Néel–Arrhenius [8] equation to calculate the relaxation
time (at 20 ${}^{\circ}C$) for the transition $\tau$ from one domain state to
another. The magenta curve represents the density of the state’s distribution.
The vertical dashed lines correspond to $\tau$ for $100$ seconds, $1$ million
years, and $4.5$ billion years. Most of the states exhibit stabilities that
far surpass the age of the Solar System (as seen in the MV states for Grains
19 and 115). However, a minority of them, such as Grain 32, are unstable
magnetic recorders, representing the magnetically unstable zone (MUZ) between
the transition of the SD to the ESV state. The colours shown for the arrows
within the grains (as well as the vortex structures) have the same meaning as
the examples shown in Figure 3 (see colour bar for helicity). Calculations
were performed assuming magnetite as the mineral phase.

As stated, when a particle assumes a multiplicity of possible domain states,
the issue regarding which of the LEM states is the more likely to be assumed
can be resolved in the presence of an isolated energy minimum, as observed in
Figure 3c,e. However, some grains may exhibit a more complex behaviour, where
the LEM states yield distinct Mr but similar energies. Boltzmann probability
distribution of states (Equation 8), shows that for a small number of states
at a given temperature, even a small difference of $2\%$ in magnetic energy
(e.g., Grain 25 in Supplementary File, Figure S3) between the lowest energy
LEM state and other states will result in a $\gg 99.99\%$ occupancy rate of
lowest energy state. However, the prevalence of a given state is sometimes
also a function of temperature, with the most favoured domain state changing
on heating or cooling. We discuss below the case of SV/SD grains with LEM
states of similar energies and distinct magnetisations. We intend to
investigate how the magnetic energy of these states evolves with an increase
in thermal energy, up to the Curie temperature of magnetite. Our observations
indicate that the balance between magnetic and thermal energy will also depend
upon both the grains’ size and morphology. The SV behaviour can be classified
into three groups. The first corresponds to the well-behaved type of grain,
where one domain state will prevail up to the Curie temperature (Supplementary
File, Figure S3). An example is Grain 25 ($\epsilon=0.43,\ \text{P}_{1}=270\
\text{nm},\ \text{P}_{1}/\text{P}_{3}=3.03$), where $p_{i}$ indicates that a
flowering state (of higher magnetisation and lower energy) will prevail over
the other competing SV states. The second group represents grains where the
energy of one state dominates at lower temperatures, but the magnetic energy
becomes similar to other LEM states as the temperature increases (Figure
5a,b). For Grain 68 ($\epsilon=0.60,\ \text{P}_{1}=124\ \text{nm},\
\text{P}_{1}/\text{P}_{3}=1.53$), e.g., a single vortex state (Figure 5b’)
dominates until $450\ ^{\circ}\text{C}$ but thereafter $p_{i}$ becomes equally
distributed between the other domain states, indicating that any of these
states with distinct Mr could be assumed. Finally, the third group behaves
poorly. In these grains, the magnetic energies of different domain states are
almost indistinguishable with increasing temperature (Figure 5c,d). For grain
48 ($\epsilon=0.07,\ \text{P}_{1}=91\ \text{nm},\
\text{P}_{1}/\text{P}_{3}=1.17$), up to $\sim 300\ ^{\circ}\text{C}$ there is
a frequent redistribution of domain states (Figure 5d’), and for T above $300\
^{\circ}\text{C}$ $p_{i}$ is equally occupied by the three different states.

Figure 5: Multiplicity of domain states for grains of magnetite. States with
similar energies but distinct magnitudes of magnetisation and their
temperature dependence are explored. In the first example (a), States 2 and 3
exhibit slightly higher energy with a temperature increase (b). The Boltzmann
probability ($p_{i}$, Equation 8) demonstrates that State 1 will prevail over
most of the evaluated temperature range (b’). However, after reaching a
certain temperature threshold (approximately $450\ ^{\circ}\text{C}$), the
probabilities of these domain states to occur become equal. In the example in
(c), this effect is more problematic, as the energies of the states remain
practically the same throughout the entire range of temperatures (d), leading
to swapping of probabilities between the states (d’) up to nearly $300\
^{\circ}\text{C}$, from which the $p_{i}$ becomes equally distributed.

#### Magnetic field-dependent simulations

We have also modelled the hysteresis loops and backfield curves for each of
the studied grains (see Methods), accounting for shape anisotropy effects by
running models in different field directions and averaging their results. In
terms of normalisation, the LEM grain’s magnetisation (Mr) differs from those
obtained from the saturation of remanence (Mrs) (see Figures 6, 7). Both of
these represent the remanence at zero-field. However, Mrs is determined from
the interception of the averaged hysteresis loop, while the Mr is obtained
through an energy minimisation starting from a random initial guess. An
important effect of averaging over distinct field directions is that
saturation (i.e. complete alignment of the magnetisation in the direction of
the applied field) at a given field strength will strongly depend on the field
direction with respect to the magnetic anisotropy axes of the grain. A
particle’s anisotropy is often controlled by its morphology, which then
determines how easily a grain becomes magnetically saturated. Therefore, we
consider the saturation parameter of magnetic hysteresis, Ms, as the
corresponding value of magnetisation for each averaged curve at $250\
\text{mT}$. We apply boxplot statistics to discuss some of the magnetic
properties in terms of all grains. As also highlighted in the previous
subsection, magnetite and maghemite properties are very similar producing
practically indistinguishable results (Figure 6). To avoid redundancies, we
henceforth discuss only the properties of magnetite.

Except for a few outliers, the Ms is practically the same as the theoretical
one for magnetite ($0.99\pm 0.01$). Mrs span from $0.0035-0.70$, whereas only
SD and SV states occupy values greater than $0.5$. Significant discrepancies
emerge when comparing Mr with Mrs. Firstly, there is an important influence of
the field direction in the whole averaged loop, which can generate very
different curves with distinct Mrs for each field direction (Figure 7a,c). For
grains whose minimum energy models yield $\text{Mr}\gtrapprox 0.5$, the
resulting Mrs from the simulated random distribution of grains can be 50%
smaller (Supplementary File, Figure S4a,b). The difference is even greater for
particles whose $\text{Mr}\lessapprox 0.5$, which can result in Mrs up to
$80\%$ smaller than the magnetisation of its lowest LEM (Mr) (Figure 7). This
effect increases with grain size, the larger grains being able to accommodate
a larger number of domain states each with a distinct remanence, but this
effect is consequently averaged during the field-dependent magnetic
experiments or numerical simulations of random distributions of such grains.
Bulk coercivities (Bc) span over a large interval ($0.64-80.59\ \text{mT}$,
Figure 6), most of them within expected [1] values for magnetite ($11.41\pm
11.86\ \text{mT}$). The coercivity of remanence (Bcr) obtained from the
backfield curves is also within the expected range ($46.27\pm 25.32\
\text{mT}$), but several grains present higher values (e.g., Figure 7b), some
of these being greater than $100\ \text{mT}$.

The remanence (Mrs/Ms) and the coercivity (Bcr/Bc) ratios are important
indicators of magnetic domain states [40]. In Figure 8, we further explore
these ratios by making use of the bilogarithmic Day plot (Bcr/Bc vs Mrs/Ms)
[41, 40], as well as the Néel’s diagram (Mrs/Ms vs Bc) [8] and its sister plot
(Mrs/Ms vs Bcr). The Day plot carries markings of conventional divisions [41,
40] for SD ($\text{Mrs/Ms}\geq 0.5,\ \text{Bcr/Bc}\leq 2$), PSD
($0.5>\text{Mrs/Ms}>2x10^{-2},\ 5>\text{Bcr/Bc}>2$) and MD
($\text{Mrs/Ms}>2x10^{-2},\ \text{Bcr/Bc}>5$) states (Figure 8a). These
limits, as well as the interpretation of such parameters, have been questioned
[42, 11] to be non-representative of the many processes that can lead to
variations of such ratios. Regardless, we keep those fields simply as guides
to the eye. On the Day plot, there are two well-defined clusters: for the
first one, SD grains and some SV grains group near $\text{Mrs/Ms}=0.5$; for
the second one, grains follow a linear trend, $\text{y}=(-0.765\cdot
x)-0.682,\ \text{R}^{2}=0.93$. The smallest grains ($100\ \text{nm}\leq$) are
within the first cluster (Figure 8a). The grains aligned along the linear
trend, which includes both SV and MV states, extend from the original PSD down
to the MD region. Besides such an obvious mismatch with the traditional PSD
field, some SV states exhibit extremely high Bcr/Bc ratios (Figure 8a),
especially associated with larger grains. When all of the backfield and
hysteresis are averaged together, our sample produces averaged ratios of
$\text{Mrs/Ms}=0.117$ and $\text{Bcr/Bc}=2.45$, which falls within the
original PSD field of the Day plot (Figure 8a).

On Néel’s diagram (Figure 8b), the same clusters are also present, which is
expected since they share the same ordinate axis. Bc of the first cluster has
a quite spread behaviour, but the other grains show a linear increase of
remanence with the bulk coercivity, $\text{y}=(1.215\cdot x)-2.282,\
\text{R}^{2}=0.91$. This relationship is inversely proportional to the
coercivity of remanence (Figure 8c) for which, although more dispersed, Bcr
increases as the remanence ratio decreases, $\text{y}=(-1.212\cdot x)+0.619,\
\text{R}^{2}=0.56$. For all the properties here discussed, there is no clear
relationship between a magnetic property and the respective elongation
parameter of a grain. However, even for these irregularly shaped grains, there
seems to be a correlation between the magnitude of Bcr and the grain size.
When we put together experimental data [41, 43, 44, 45, 46, 47, 48, 49, 50,
51, 52, 53] and the results of our models, despite a large spreading, the
coercivity of remanence seems to increase and peak around grain sizes between
$600-700\ \text{nm}$, from which it tends to decay again (Figure 8d).

Figure 6: A compilation of parameters obtained from the magnetic hysteresis
loops and backfield curves for magnetite (blue) and maghemite (brown) is
presented using boxplot statistics. These parameters correspond to averaged
curves obtained for each grain in twenty different directions (refer to
Methods for details). Bc represents the bulk coercivity, Bcr is the coercivity
of remanence, Ms indicates the magnetisation saturation of the averaged loop
at the maximum field, and Mrs is the remanence of saturation. Black diamonds
signify statistical outliers. As illustrated, magnetite and maghemite exhibit
practically indistinguishable properties.  Figure 7: Examples of magnetic
hysteresis (a, c), backfield curves (b, d), and their parameters (for
magnetite). The grey dots represent the field-dependent data, and the red
curves depict their averages. Notably, for both types of simulated
experiments, different field directions result in distinct curves as a result
of the grain’s shape anisotropy. The particles situated at the upper side of
the backfield curves (b, d) correspond to the LEM states obtained at zero
field. Here, we highlight how the Mr of these states might differ
significantly from the Mrs of magnetic hysteresis (compare the Mrs values
within the boxes in a and c with the respective Mr values within the grains).
Figure 8: a) Remanence (Mrs/Ms) and coercivity (Bcr/Bc) ratios are compared in
the Bilogarithmic Day plot [41, 40]. Two distinct clusters are evident: the
first, located near $\text{Mrs/Ms}\approx 0.5$, and the second, following a
linear trend (green line with shaded extension representing $\alpha 95$). For
a, b and c, the circles are colour-coded based on the shape parameter
($\epsilon$), with blue/red indicating more prolate/oblate grains. The circle
size represents grain size, and the black cross (in a) indicates the average
ratio of all modelled grains. Neél’s diagram [8] (b) and its sister plot (c)
depict linear tendencies with confidence areas. Similar clusters are observed
in both plots, indicating an increase in remanence with coercivity (Bc) and a
decrease with the increase of coercivity of remanence (Bcr). In plot d, Bcr
values obtained in our models (triangles) are compared with their respective
grain sizes, while green squares represent experimental data [41, 43, 44, 45,
46, 47, 48, 49, 50, 51, 52, 53]. A third-order polynomial fit serves as a
guide, revealing a trend where Bcr peaks around $600-700nm$ before decreasing,
consistent with experimental findings.

## Discussion

### On the magnetic properties of natural nanoscopic grains

Although not surprising, an important aspect of our results is that the
magnetic properties of magnetite and maghemite, as demonstrated in Figure 6,
as well as their domain states, are practically indistinguishable at ambient
temperatures. The challenge of differentiating between these two magnetically
soft minerals can be addressed through the use of temperature-dependent
experiments [1], which are proven to be essential for properly interpreting
environmental magnetism data and inferring exogenous processes [54], but often
expose the samples to chemical or domain state change.

Mrs/Ms and Bcr/Bc ratios derived from magnetic hysteresis and backfield curves
can be characteristic for an entire assemblage of grains. It is natural,
therefore, that various processes impact the comparison of these ratios in the
Day plot [41, 40]. Such processes include mixtures of magnetic minerals with
distinct coercivities, and/or mixtures of grains in distinct domain states
[55, 56], as well as surface oxidation, mineral stoichiometry, and the
intrinsic magnetic properties of each grain [42]. Nevertheless, our data
demonstrates that for both magnetite and maghemite (Figure 8a), two well-
defined clusters exist. One is occupied by higher remanences (solely featuring
SD and SV states), and the other displays a linear trend cutting across the
Day plot towards lower remanence ratios and higher coercivity ratios. The Day
plot trend we observe for SV grains closely follows that found in idealised
grains by Williams et al [57], although, for our irregular grains, we do not
observe a well-defined correlation with grain shape. As grain sizes increase,
so does the internal demagnetising field [9, 15], until it grows sufficiently
strong to collapse into a lower energy domain state. Lower energy domain
states often imply a decrease in remanence, which we observe in our analysis
with the increase in Bcr (Figure 8b). Our data also suggests a peak for Bcr
with the increase of grain size around $600-700\ \text{nm}$ (Figure 8d),
aligning considerably with experimental data of grains with distinct
morphologies and sizes [41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53].
Generally, our results also align well with data derived from iron oxide
inclusions [22].

In our simulations, the influence of superparamagnetic (SP) grains is not
resolved. The combination of all grains in hysteresis and backfield curves
suggests that a high quantity of SP particles could potentially decrease
Mrs/Ms and increase Bcr/Bc [40]. Careful interpretation of these diagrams is
necessary. However, with well-established magnetic mineralogy supported by
other analyses such as thermal demagnetisation of synthetic remanences,
thermomagnetic curves, first-order reversal curves (FORCs), and
chemical/imaging analyses, the Day plot, Néel’s diagram, and related plots can
provide useful insights.

Magnetic hysteresis, including FORCs[58, 59, 60], heavily rely on the Mrs to
infer field-induced states and not remanence states. One of our crucial
observations is that unless the grains are SD, the true remanence of a grain
will be potentially underestimated. While this may not be a significant
concern for very fine particles exhibiting SD-like behaviour
($\text{Mrs}\gtrapprox 0.5$), it can become a pronounced phenomenon for larger
grain sizes. Experimentally, magnetic hysteresis, backfield curves, isothermal
remanence curves (_IRM_ s), and FORCs are performed using a fixed field
direction. As illustrated in Figure 7, the applied field’s direction can
induce variable magnetic properties due to the grains’ shape anisotropy. For a
population of irregular grains randomly oriented, changing the field direction
may not significantly decrease this effect because rocks might have millions
of magnetic grains. However, for materials with a well-oriented magnetic
fabric (e.g., inclusions in minerals, and highly deformed rocks), conducting
these measurements in variable field directions (or changing sample
orientation) might enhance the determination of Mrs.

### Temporal and thermal stability of irregularly shaped grains

The capacity of magnetic minerals to retain their remanence over geological
time-scales is crucial for planetary studies, as rock-magnetic recordings
provide information on Earth and Planetary evolution such as tectonics [2],
the timing of inner core nucleation [61], and even the early stages of solar
nebula development [62]. Néel’s theory [8] has long been employed to
demonstrate that SD grains can retain their remanence over billions of years.
Although this domain state is highly stable, it spans only a very limited
range of sizes, and most terrestrial and extraterrestrial magnetic materials
will contain a much broader range of grain sizes.

Nagy et al. [12] have calculated the temporal stability of equidimensional truncated octahedron grains of magnetite ($80-1000\ \text{nm}$) in the SV state and have demonstrated that, except for grains between $80-100\ \text{nm}$, their stability surpasses the age of the Solar System. Our calculations for non-euhedral morphologies, which are more realistic for oxide grains in most natural systems, generally align with this interpretation (Figure 4), demonstrating that most SV and MV grains can also retain eternal stability at environmental temperatures. Nagy et al. [12] also pointed to a magnetically unstable zone (MUZ) for grain sizes between $80-100\ \text{nm}$, for which the relaxation times are near instantaneous at room temperature. In the MUZ for magnetite, the SV domain state is aligned with the hard magnetisation axis (HSV), while large SV grains align their magnetisation with the easy magnetic anisotropy axis (ESV). Wang et al. [13], have shown that within the MUZ ($68-100\ \text{nm}$) the grains’ evolution proceeds as SD $\rightarrow$ HSV $\rightarrow$ ESV. For such, grains within the MUZ were shown to occupy about $20\ \text{nm}$ for euhedral and also for rounded particles [18, 63]. Four of our modelled grains within the MUZ size range proposed by Nagy et al. [12] are unstable at room temperature, e.g. Grain 32 (Figure 4) with an elongation axis of $122\ \text{nm}$. Furthermore, several of the grains (with variable morphology) that we studied have $\text{P}_{1}$ within the proposed range dimensions of the MUZ (e.g., Grain 135: $\epsilon=-0.93,\ \text{P}_{1}=80\ \text{nm},\ \text{P}_{1}/\text{P}_{3}=1.39$ | Grain 48: $\epsilon=0.07,\ \text{P}_{1}=91\ \text{nm},\ \text{P}_{1}/\text{P}_{3}=1.17$), and yet they are highly stable. Therefore, we suggest that the transition of SD to HSV (consequently at the onset of the MUZ) is strongly affected by the morphology of the grain.

Figure 3 further highlights that a multiplicity of domain states may occur in
one or more temperature intervals and that the likelihood of a grain being
able to nucleate multiple domain states increases with grain size. This is a
particular problem for the determination of palaeointensities which are often
performed using a stepwise cycle of heating, where the original remanence (a
thermal remanence, TRM) in any temperature interval is replaced by laboratory-
induced partial thermal remanence (pTRM) in a known field. [64, 65]. An
important assumption in these experiments is that the magnetic recorders in
the sample obey the so-called Law of Reciprocity, which states that a magnetic
grain demagnetises (unblocks) and re-magnetises (blocks) at the same
temperature i.e. ($T_{ub}=T_{b}$) [66, 64, 67], which is only valid for the
original SD theory of Néel. When multiple domain states are possible in the
same grain, thermal cycling may produce a pTRM with grains now occupying very
different domain states with different thermal stabilities, such that
$T_{ub}\neq T_{b}$. This issue has been discussed by Nagy et al. [14]. In
their models, using a euhedral magnetite grain ($100\ \text{nm}$, with $30\%$
of elongation along [$100$]), they not only point to a multiplicity of domain
states but also to a preference for a given state as temperature changes,
especially as a function of the cooling ratio. A similar effect has been
identified by Tauxe [67] where "fragile" magnetic remanences behaved
differently during stepwise demagnetisation and remagnetisation after samples
"aged" by sitting in various storage fields over many months. This time-
induced effect is equivalent to the pure thermal effect described by Nagy et
al. [14], which would be caused by the multiplicity of possible domain states.

We now suggest this situation might affect a much broader range of grain
sizes, and be highly dependent on grain morphology. Any failure of blocking
and unblocking reciprocity, as well as the often large change in a grain’s
magnetisation, will make it impossible to obtain reliable paleomagnetic
intensities. In exploring the multiplicity of domain states that co-exist with
similar magnetic free energies for grains of irregular shape we observed this
same problem. Grains whose behaviour is dominated by one or two states over a
large range of temperatures are the ones that will likely provide good
paleointensity results up to a certain temperature, where the difference in
energy is significant. Above such temperature threshold, any of the possible
domain states could be assumed as the probability is equally distributed
(Figure 5b’). The exact occupancy of the different domain states will be
dependent on the thermomagnetic history of the sample. Repetitive cycling
through temperatures, each yielding a variety of domain states, is expected to
gradually shift the overall distribution of domain states within the sample,
deviating from the initial distribution that carried the original (geological)
recording of the geomagnetic field. Thus, thermal cycling and differential
heat/cool rates are likely to further exacerbate the problem [14]. It is
possible that grains exhibiting the two dominant state behaviours (Figure 5b’)
could in some instances be an alternative explanation to high temperature
palaeomagnetic experimental failure that is often attributed to chemical
alteration [68]. Grains with more complex behaviour (Figure 5d’),
characterised by a constant re-balancing of probabilities, are expected to
yield unreliable thermally dependent paleomagnetic and paleointensity data,
similar to the findings presented by Nagy et al. [14].

Magnetic grains whose domain states are either truly SD or ESV but far from
such transition zones (mostly grains $\gtrapprox 200\ \text{nm}$) are likely
to be the most stable magnetic recorders, and also the ones for which
palaeomagnetic information is less affected by thermal experiments. That is
probably because the mechanism for magnetic state transitioning of ESV grains
is the structure coherent rotation [12], which (as well as for SD grains)
requires a uniform rotation of all of the magnetic moments associated with the
grain. Furthermore, because domain state transitions seem to imply several
complications in the magnetic properties of grains, there should be an equal
dedication to studying grains within the ESV $\rightarrow$ MV transition.

## Methods

### Sample preparation

The sample used in our research is a Neoproterozoic cap carbonate rock, part
of the Guia Formation, which spreads over the Amazon craton and the Paraguay
belt (central-West Brazil). These rocks have been targets of palaeomagnetic
studies and suggested to carry pervasive evidence of authigenic growth of
secondary magnetic grains as a result of low-temperature (_< Tc_) chemical
processes [30, 69]. Recently, Bellon et al. [29] performed a mineralogical,
microscopic and magnetic analysis on remagnetized carbonate rocks from South
America. We have chosen a thin-section sample from their data set that has
shown remarkable amounts of micrometric/nanometric iron oxides to build
continuous work. Preparation of microscopic pillars was acquired by firstly
coating the thin sections with carbon and sequentially cutting a $\approx
18\times 40\mu\text{m}$ pillar using a FIBSEM Helios 600i TFS FIB-SEM
(Scientific Center for Optical and Electron Microscopy - ScopeM, EHT Zürich).
Because the pillar sample represents only an infinitesimal part of the
macroscopic thin section, the pillar was prepared in a region that had been
chemically mapped for iron oxides through both SEM-EDS and synchrotron-based
X-ray Fluorescence/X-ray Absorption Near Edge Spectroscopy (XRF/XANES), [29].
Lastly, the pillar was mounted on a PXCT OMNY-pin [28] using a
nanomanipulator.

### Synchrotron-based Ptychographic X-ray Computed Nano-tomography (PXCT)

PXCT was performed at the _cSAXS_ (coherent small-angle X-ray scattering)
beamline of the Swiss Light Source at the Paul Scherrer Institute (PSI,
Villigen, Switzerland). For the acquisition, we used the flOMNI setup [70,
26], a photon energy of 6.2 keV, and we placed the sample downstream of the
focal spot defined by a Fresnel zone plate (FZP). This energy was chosen due
to the major mineral constituent being calcite $(\text{CaCO}_{3})$, which
results in nearly $50\%$ of transmission in the middle of the sample for a
pillar of 20 $\mu$m. Scans were performed in a Fermat spiral pattern[71] with
a field of view of $37\times 10\,\mu\text{m}$, and coherent diffraction
patterns were collected with an Eiger $1.5\text{M}$ detector [72] positioned
$5.229\text{m}$ downstream of the sample. We acquired $650$ projections at
different incidence angles of the beam from the sample, ranging from $0$ to
$180^{\circ}$. During the data collection, we estimate that a dose of
$5.6\times 10^{8}\ \text{Gy}$ was delivered to the sample. Reconstruction of
ptychographic projections were performed with the PtychoShelves package
developed by the Coherent X-ray Scattering group at PSI [73], using a
combination of the difference map algorithm [74], followed by the maximum
likelihood algorithm [75]. This resulted in images with a pixel size of $34.8\
\text{nm}$. Phase images were treated with the removal of offset and linear
terms and were further aligned in the vertical and horizontal directions with
a subpixel precision method [76, 77]. Tomographic reconstructions were finally
performed by applying a filtered back projection [76, 77]. Using Fourier shell
correlation _(FSC)_ [78], we estimated that the average 3D half-pitch
resolution of the whole 3D dataset is $89\text{nm}$. This resolution reflects
the specimen’s overall average. The limited quantity of high-density magnetite
particles could be imaged with better resolution due to their contrasting
density with the surrounding material. More details about the PXCT
experimental setup, data collection and reconstruction can be found in the
Supplementary File.

### Segmentation and processing of tomographic data

Distinguishing from traditional X-ray microscopy techniques, X-ray
ptychography provides images of the complex-valued transmissitivity of the
specimen, where the real part is related to the absorption and the imaginary
part with the phase shift of the X-rays. In combination with tomography, PXCT
provides the 3D distribution of the full complex index of refraction
$n(\vec{r})$ [23]:

$n(\vec{r})=1-\delta(\vec{r})-i\cdot\beta(\vec{r})$ (1)

where $\delta(\vec{r})$ and $\beta(\vec{r})$ are obtained from the tomographic
reconstruction using the phase and the absorption images, respectively. The
phase shift of X-rays is related to refraction, which describes the change in
the direction of the X-rays, it is strongly associated with the material
density, composition and internal structure, while absorption of X-rays is
related to the local structure and composition environment of an adsorbing
atom, apart from its density. In our paper, we rely on the $\delta(\vec{r})$
reconstructed tomography to recover the morphology and segment different
mineral constituents of the pillar due to its higher resolution compared to
the $\beta(\vec{r})$ tomogram. Nevertheless, we compare that data with
$\beta(\vec{r})$ reconstruction to refine our interpretation. The quantitative
$\delta(\vec{r})$ values obtained at each voxel are converted to electron
density $n_{e}(\vec{r})$ as [79]:

$n_{e}(\vec{r})=\frac{\delta(\vec{r})k^{2}}{r_{0}2\pi}$ (2)

where $k=(2\pi/\lambda)$, $\lambda$ is the X-ray wavelength and $r_{0}$ is the
Thomson scattering length. In order to compare with our data, we can use the
theoretical relation between the mass density of a given material
$\rho(\vec{r})$ and its $n_{e}(\vec{r})$ as [80]:

$\rho(\vec{r})=\frac{n_{e}(\vec{r})\cdot A}{N_{A}\cdot\phi}$ (3)

where _A_ is the molar mass, $N_{A}$ is Avogadro’s number and $\phi$ is the
total number of electrons in a molecule of the material.

After converting the $\delta$ values to electron density (Equation 2), we
calculated the histogram distribution of all voxels in the tomogram to segment
the distinct populations composing the sample (Supplementary File, Figure
S1c). Before segmenting the data, we first deal with salt-and-pepper noise
(maximum/minimum intensity voxels randomly distributed throughout the image
[81]) by applying a three-dimensional Gaussian filter to the data (assuming an
isotropic $\sigma=1$, centred at each voxel). Data segmentation was acquired
by processing the natural breaks in the histogram distribution and using them
as thresholds after stripping away the air response. Based on the $n_{e}$
distribution, five classes have been defined (Supplementary File, Figure S1d).
The $n_{e}$ slices were reclassified into binary files, assigning _True_ for
the iron oxides and _False_ for all of the other materials. Binary files were
then imported to the software _Dragonfly_ to generate stereolithographic data
(STL file) of each iron oxide grain. Even aware that filtering would be able
to deal with salt-and-pepper noise, we have removed particles represented by a
single voxel, to give more robustness to our models.

To systematically acquire information on the morphology of the iron oxides we
used a Principal Component Analysis (PCA) of their spatial distribution.
Firstly, we export the STL files as a Cartesian coordinate cloud of points (x,
y, and z) describing the external morphology of the particle. Sequentially, we
use a Python script to calculate the convex hull [82] of each set of
coordinates of a given particle. A convex hull is the smallest possible convex
shape that encloses a given set of points, forming a polygon that connects the
outermost points by ensuring that all of its internal angles are less than
180∘ [83]. We then performed a PCA [84], to get the eigenvectors (main
directions) and eigenvalues (their magnitudes) of a covariance matrix of
standardised variables. Sequentially, we have projected the original convex
hull vertices onto the space defined by the principal components and
calculated the size of the hull along the main directions. In this case, the
maximum variance component $\text{P}_{1}$ represents the maximum elongation
axis, $\text{P}_{2}$ is the intermediary one and $\text{P}_{3}$ is the minimum
variance component (short axis). To more broadly evaluate the morphology of
the particles in our sample, we use the shape parameter (here called
$\epsilon$) of Nikolaisen et al. [22] by using our main components achieved
through the PCA analysis:

$\epsilon=\log\left(\frac{\text{P}_{1}-\text{P}_{2}}{\text{P}_{2}-\text{P}_{3}}\right)$
(4)

for which triaxial morphologies (all of the main axes have different
dimensions) will have $\epsilon\approx 0$, the more positive $(\epsilon>>0)$
more oblate is the grain, and the more negative $(\epsilon<<0)$ more prolate
is the grain.

### Micromagnetic modelling

#### Finite element meshes and magnetic properties

In micromagnetic models, physical processes and their respective energies are
evaluated mathematically through a continuous function, by assuming that at a
given spatial volume the magnetisation is an average of discrete physical
sources of magnetism, and that such volume is representative of the individual
atomic behaviour of the averaged atoms [85, 86, 10]. To solve the local energy
minimum (LEM) for each of the iron oxides identified in our tomography, we
rely on the open-sourced software package MERRILL [10]. This software uses
linear tetrahedral finite elements to describe the geometry of a particle and
solve the LEM stable states minimising the free energy of a magnetic system
$E=\vec{H}_{\text{eff}}\cdot\vec{M}$, for which the total effective field
$(\vec{H}_{\text{eff}})$ is [10]:

$\vec{H}_{\text{eff}}(\vec{M)}=\vec{H}_{\text{exg}}(\vec{M)}+\vec{H}_{\text{ans}}(\vec{M)}+\vec{H}_{\text{Zmn}}(\vec{M)}+\vec{H}_{\text{dmg}}(\vec{M)}$
(5)

where $\vec{M}$ is the magnetisation vector, $\vec{H}_{\text{exg}}$ the
exchange field, $\vec{H}_{\text{ans}}$ the anisotropy field,
$\vec{H}_{\text{Zmn}}$ the Zeeman field, and $\vec{H}_{\text{dmg}}$ is the
demagnetising field. We have used the software _Cubit_ to build the
tetrahedral meshes of each particle (previously exported as a STL file). To
ensure that our finite element meshes were fine enough to resolve the spatial
variation of the magnetisation within the geometry of each grain we determined
the maximum element size as dependent on their exchange length
$\text{I}_{\text{e}}$ [38]. $\text{I}_{\text{e}}$ of magnetite
$(\text{Fe}_{3}\text{O}_{4})$ is $\approx 9\ \text{nm}$ at environmental
conditions of temperature and pressure, and slightly larger than that near its
_Tc_ [10]. The exchange length of maghemite
$(\gamma\text{Fe}_{2}\text{O}_{3})$ is slightly larger than magnetite’s, so it
is not an issue to assume $9\ \text{nm}$ as a maximum element size for both
mineralogies. Magnetic parameters of these minerals are set as in Table 1.

Table 1: Magnetic properties of the iron oxides used in our micromagnetic models at $20\ ^{\circ}\text{C}$. $A_{ex}$ is the exchange constant, $K_{1}$ is the anisotropy constant, and $M_{s}$ is the magnetisation saturation.  Mineral | Anisotropy | $A_{ex}$ | $K_{1}$ | $M_{s}$ | Ref
---|---|---|---|---|---
Magnetite $(Fe_{3}O_{4})$ | Cubic | $1.33\cdot 10^{-11}$ | $-1.24\cdot 10^{4}$ | $4.8\cdot 10^{5}$ | Heider and Williams [87]
Maghemite $(\gamma Fe_{2}O_{3})$ | Cubic | $1\cdot 10^{-11}$ | $-4.6\cdot 10^{3}$ | $3.8\cdot 10^{5}$ | Dunlop and Özdemir [15]

#### Numerical calculations

For any micromagnetic experiment performed through MERRILL, we can achieve a
visualisation of the magnetic energy within the particles by performing simple
mathematical operations to better illustrate the finite element solutions. We
have achieved that through the open-sourced software _Paraview_ [88]. Firstly,
we calculate the curl of the magnetisation
$\vec{M}=M_{x}\hat{i}+M_{y}\hat{j}+M_{z}\hat{k}$, which is basically its
vorticity $(\vec{\omega})$, as:

$\vec{\omega}=\nabla\times\vec{M}$ (6)

Sequentially, we calculate the projection of $\vec{\omega}$ onto $\vec{M}$,
which is the normalised helicity $(\vec{H})$ shown in our figures, as:

$\vec{H}=\frac{\vec{M}\cdot\vec{\omega}}{\lVert\vec{\omega}\rVert}$ (7)

When starting from random initial orientations, LEM calculations might reach
slightly different products resulting from local minima. We calculate fifty
(50) different LEM models for every grain at $20\ ^{\circ}\text{C}$ and then
harvest the states with the lowest minimised energies to reach the most stable
magnetic domains (here called $r$). Upon categorising the states by their
energies and magnetisation intensity/direction, we can determine the
probability ($p_{i}$) associated with a particular state. This probability is
expressed as a function of the state’s energy and the system temperature, in
the presence of other states, utilising a normalised Boltzmann probability
distribution:

$p_{i}=\frac{\exp\left(-(\varphi_{i}-\lambda_{j})\right)}{\sum_{j}^{r}\exp\left(-(\varphi_{j}-\lambda_{j})\right)}$
(8)

Here, $\varphi_{i}=\frac{E_{Mi}}{k_{B}\cdot T}$, where $E_{Mi}$ represents the
magnetic energy (in J) of the state under consideration; $k_{B}$ is the
Boltzmann constant $(1.3806\times 10^{-23}\ \mathrm{m^{2}\cdot kg\cdot
s^{-2}\cdot K^{-1}})$; and the temperature of the system, denoted as $T$ (in
K). Additionally, $\lambda_{j}$ is defined as $\min(\varphi_{j})$. To
ascertain the probability $(p_{i})$ under varying temperatures, we conducted
LEM minimisations for selected examples. In this process, each of the
$r$-states served as the initial solutions. The temperatures ranged from
$40-579\ ^{\circ}\text{C}$, with intervals of $20\ ^{\circ}\text{C}$ up to
$500\ ^{\circ}\text{C}$, followed by intervals of $10\ ^{\circ}\text{C}$ up to
$570\ ^{\circ}\text{C}$, and finally, $1\ ^{\circ}\text{C}$ increments up to
$579\ ^{\circ}\text{C}$.

To assess the temporal stability of magnetic particles at environmental
temperatures, we employed the Nudged–Elastic-Band (NEB) method [12].
Initially, we sampled the minimum energy state of a given particle, and
inverted the direction of magnetisation, resulting in two opposite (mirrored)
domain states. The NEB method was then utilised to determine the magnetic
energy (called energy barrier) required for the transition from one domain
state to the other. For that, we simply gather the energy difference between
the global maximum and minimum of the pathway. Subsequently, we have used this
energy to calculate the relaxation time $(\tau)$ for each grain through the
Néel–Arrhenius [8] equation as follows:

$\tau=C\cdot\exp\left(\frac{Em}{k_{B}\cdot T}\right)$ (9)

where $C=10^{-9}\ \text{s}$ is a frequency factor, and $T$ is fixed at
$293.15\ \text{K}$.

Magnetic hysteresis were evaluated from $-250\ \text{mT}$ to $250\ \text{mT}$
in $2\ \text{mT}$ steps, while backfield curves were evaluated from $0$ to
$-250\ \text{mT}$ using the same field steps, as this range covers the
expected coercivity of such magnetic minerals. In our magnetic hysteresis
simulations, we assess the magnetisation of a material under the influence of
an applied magnetic field. The process initiates with saturation magnetisation
(Ms) in a specific field direction, achieved at the maximum applied field.
Subsequently, the field is gradually reduced to zero, resulting in the
remanence of saturation (Mrs) at zero field. Following this, a magnetic field
in the opposite direction (negative) is induced, where the field eventually
causes the magnetisation to be zero (which is the coercivity, Bc), ultimately
reaching the saturation point again. Backfield curves follow a similar process
with a slight modification. The sample is first saturated, starting at zero
field (Mrs). The field is then increased in steps, and before moving to a new
step, it is decreased to zero. The remaining remanence is recorded, and as the
remanence approaches zero, the coercivity of remanence (Bcr) is determined. To
put it simply, magnetic hysteresis involves measuring magnetisation under the
influence of a magnetic field, while backfield curves are influenced by a
magnetic field, but the field is "turned off" during the actual "measurement"
of the magnetisation. In our simulations of magnetic field-dependent
experiments, the grain’s shape can create anisotropic properties due to
easy/hard magnetisation axes. To account for this, we simulate different
magnetic field orientations by sampling a field-dependent magnetic property
$F_{i}$ from random 3D coordinates (drawn from a Fibonacci sequence) on a unit
sphere as:

$\iint_{s}{f(x,y,z)}\ ds\ \approx\ \sum_{i=1}^{20}\frac{F_{i}}{20}$ (10)

Here (Equation 10), $f(x,y,z)$ evaluates the magnetic property $F_{i}$, and
the integral over the surface $s$ is approximated by the sum over all samples.
Because increasing the number of samples improves the approximation, we have
run field-dependent experiments in twenty ($\text{n}=20$) random directions
for every grain.

## Acknowledgements

### Funding

This study was funded by grants $16/06114-6$, $21/00861-2$, $22/14100-6$ of
the São Paulo Research Foundation (FAPESP). The opinions, hypotheses, and
conclusions or recommendations expressed in this material are the
responsibility of the authors and do not necessarily reflect the views of
FAPESP. We extend our gratitude to Dra. Cilene de Medeiro and Dra. Ingrid
Barcelos for their valuable assistance during the collection of SEM-EDS data
(essential for selecting the area for FIB cutting) at the Microscopic Sample
Laboratory, Brazilian Synchrotron Light Laboratory (LNLS, Proposal
$20220316$). The authors gratefully acknowledge ScopeM for their support &
assistance in this work, especially to Anne Greet Bittermann, for the
preparation of the FIB pillar. We acknowledge the Paul Scherrer Institut,
Villigen, Switzerland for the provision of synchrotron radiation beamtime at
beamline cSAXS of the SLS (Proposal $20221762$). U.D.B. expresses appreciation
to the School of Geosciences at the University of Edinburgh for providing the
computational resources essential for running the micromagnetic models
presented in this paper. W.W. would like to acknowledge support from the
Natural Environment Research Council (NERC) through grants $NE/V001388/1$,
$NE/W006707/1$, and $NE/S011978/1$. D.G. acknowledges CNPq (Conselho Nacional
de Desenvolvimento Científico e Tecnológico) through grant $310817/2020-0$.

### Author contributions

U.D.B. performed sample preparation, data collection, processing, and analysis
of tomography data, and conducted and interpreted the numerical models. W.W.
contributed to running numerical models and supervising the project. R.I.F.T.
conceived and supervised the overall project. A.D. was involved in collecting
and processing tomography data. D.G. provided project supervision. All authors
have actively contributed to the preparation and review of this manuscript.

### Competing interests

The authors declare no competing interests.

### Data, Materials, and Software Availability

Besides the Supplementary File, all of the data related to this research is
preserved at (https://doi.org/10.5281/zenodo.10837481) and can be publicly
accessed from the same repository. The data is subdivided into folders,
including the meshes used in our simulations (.pat files) and the MERRILL
scripts, together with any other relevant file. We are also providing the
averaged hysteresis and backfield curves of each grain. The results reported
here were achieved through MERRILL [10], an open-sourced code for
micromagnetic modelling of tetrahedral finite element meshes. Installation,
guide tutorials and courses are available at https://blogs.ed.ac.uk/rockmag/.

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