Stress-induced changes in magnetite: insights from a numerical analysis of the Verwey transition

Magnetic susceptibility behaviour around the Verwey transition of magnetite ( ≈ 125 K) is known to be sensitive to stress, composition and o xidation. F rom the isotropic point ( ≈ 130 K) to room temperature, decreasing magnetic susceptibility indicates an increase in magne-tocrystalline anisotropy. In this study, we present a model which numerically analyses low-temperature magnetic susceptibility curves (80–280 K) of an experimentally shocked (up to 30 GPa) and later heated (973 K) magnetite ore. To quantify variations of the transition shape caused by both shock and heating, the model statistically describes local variations in the Verwey transition temperature within bulk magnetite. For the description, Voigt proﬁles are used, which indicate variations between a Gaussian and a Lorentzian character. These changes are generally interpreted as variations in the degree of correlation between observed events, that is between local transition temperatures in the model. Shock pressures exceeding the Hugoniot elastic limit of magnetite ( ≥ 5 GPa) cause an increase in transition width and Ver-wey transition temperature, which is partially recovered by heat treatment. Above the Verwey transition temperature, susceptibility variations related to the magnetocrystalline anisotropy are described with an exponential approach. The room temperature magnetic susceptibility relative to the maximum near the isotropic point is reduced after shock, which is related to grain size reduction. Since signiﬁcant oxidation and cation substitution can be excluded for the studied samples, variations are only attributed to changes in elastic strain associated with shock-induced deformation and annealing due to heat treatment. The shocked magnetite shows a high correlation between local transition temperatures which is reduced by heat treatment. The model allows a quantitative description of low-temperature magnetic susceptibility curves of experimentally shocked and subsequently heat-treated polycrystalline magnetite around the Verwey transition temperature. The curves are accurately reproduced within the experimental uncer tainties. Fur ther applications for analysing magnetite-bearing rocks seem possible if model parameters, such as for oxidation are included into the model.

. Under directional strain, positive slopes of d T V / d p were observed experimentally (Nagasawa et al. 2007 ), and for non-hydrostatic pressure both positive and negative slopes were determined theoretically depending on the stress orientation (Coe et al. 2012 ).
The way static external pressure affects the magnetic behaviour on a microscopic scale differs from the mechanisms in decompressed magnetite.Static pressure affects the electronic structure whereas in a post-shock state changes in the magnetic behaviour are mainly controlled by the formation of defects (Biało et al. 2019 ).Carporzen & Gilder ( 2010 ) found a positive slope of d T V / d p in magnetite decompressed from hydrostatic pressure.
Internal stresses must be considered for the post-shock state as they can be related to dislocations or misfits at phase boundaries in magnetite.The influence of microstress on domain wall controlled coercivity was documented by Moskowitz ( 1993 ) by a 1-D micromagnetic model for grains with high and low defect densities (e.g.recrystallized magnetite).Lindquist et al. ( 2015 ) showed that stress-induced dislocations cause pinning of magnetic domain walls in multidomain magnetite and an increase of microcoercivity in dislocation-rich areas.Upon cooling through T V , internal stress fields impede monoclinic twinning and the direction of stress affects the orientation of monoclinic twins (Williams et al. 1953 ;Calhoun 1954 ;Coe et al. 2012 ;Lindquist et al. 2019 ).Coe et al. ( 2012 ) suggested that a complex internal stress field caused by dislocations and other defects results in broadened Verwey transitions and local variations of T V .
At T V , a change in electron order occurs, which is related to a slight lattice distortion (e.g.Senn et al. 2012 ), and therefore the transition itself causes spontaneous strain (Coe et al. 2012 ).The strain dependence of T V as function of small cation substitution has been investigated by Biało et al. ( 2019 ).These authors observed that pressure cycling to ≈5 GPa has an impact on T V and coerci vity b y pinning the magnetic domains on interstitial cations and structural lattice distortions.Deviations from stoichiometry, either by cation vacancies (Arag ón et al. 1985 ) or by substitution of iron b y v arious metal cations (e.g.Kozłowski et al. 1996b ;Brabers et al. 1998 ;Wieche ć et al. 2005 ) shift T V to lower temperatures.Non-stoichiometry also depresses and broadens the transition (e.g.Honig 1995 ).Jackson & Moskowitz ( 2021 ) observed a bimodal distribution of T V between about 100 and 110 K in a large set of magnetite samples of various natural and synthetic origins, which is explained by variations in non-stoichiometric magnetite (Honig 1995 ).
At about 130 K, which is slightly above T V , magnetite passes through a magnetically isotropic point, where the first magnetocrystalline anisotropy constant K 1 changes sign and the magnetic easy axes reorient within the cubic spinel lattice.Typically, the effect of the isotropic point on the magnetic property changes with temperature is minor compared to the effect of T V , but the commonly observed decrease in magnetic susceptibility χ from the isotropic point to room temperature is expected to be related to an increase in magnetocrystalline anisotropy over that range (Bickford et al. 1957 ;Muxworthy & McClelland 2000 ).According to Arag ón et al. ( 1985 ), a more pronounced curvature in magnetic susceptibility above the isotropic point is related to the polycrystalline nature of the material, and Muxworthy & McClelland ( 2000 ) emphasize that a stronger increase of the magnetocrystalline anisotropy above the isotropic point is related to an increasing contribution of the second cubic anisotropy constant K 2 , which is nearly insignificant above room temperature (Bickford et al. 1957 ;Syono 1965 ).
Because of its sensitivity to changes in stress state and stoichiometry, the Verwey transition morphology can serve as a possible indicator of shock pressure in impact-related rocks.Two Verwey temperatures were identified in basement rocks of the Vredefort impact crater and attributed to a pre-impact magnetite population and a population with impact-related formation showing significantly reduced T V (Carporzen et al. 2006 ).A lower T V attributed to o xidation probab l y related to post-shock alteration w as found in impacted crystalline basement rocks from the Chesapeake Bay impact structure (Mang & Kontny 2013 ) and in volcanic rocks from the El'gygytgyn impact structure (Kontny & Grothaus 2017 ).Reznik et al. ( 2016 ) identified an increase of T V by around 4 K and a broadening of the transition in experimentally shocked magnetite (Fig. 1 a).Kontny et al. ( 2018 ) studied the same samples after a heat treatment up to 973 K and observed a recovery of both T V and the transition width related to annealing of shock-induced defects and dislocations by recrystallization (Fig. 1 b).The results of these authors indicate that the χ -T behaviour of shocked and annealed magnetite is also sensitive to the decrease of χ between the isotropic point and room temperature.
A more detailed description of magnetic susceptibility in the region of the Verwey transition may be useful to better distinguish between these effects.The determination of T V and the transition width often considers d χ/ d T , focusing on the sharp susceptibility increase near T V (e.g.Carporzen & Gilder 2010 ;Reznik et al. 2016 ;Kontny et al. 2018 ).The model introduced in this study uses the experimental temperature range from about 80 to 280 K for the determination of model parameters.To test the approach and to quantify and visualize the effect of shock deformation on the Verwey transition, the model is applied to the χ -T curves measured by Reznik et al. ( 2016 ) and Kontny et al. ( 2018 ) on shocked and subsequently heat-treated magnetite ore (Fig. 1 ).Fur ther more, a χ -T curve of the same ore shocked at 3 GPa using the same experimental setup and heat treatment is e v aluated.

Model f orm ulation f or an idealized small magnetite volume
The presented model approach is tested on χ -T curves of a wellcharacterized magnetite ore exhibiting stress-induced features in magnetite without cation substitution or oxidation (Reznik et al. 2016 ;Kontny et al. 2018 ).For consideration of other effects like cation substitution or oxidation, adjustments to the model will be necessary.Contributions to the bulk susceptibility by minerals other than magnetite (quartz, amphibole, chlorite, biotite and pyrite) can be neglected, either because of their low susceptibility compared to magnetite (e.g.Hunt et al. 1995 ) or because of their low content in the studied samples.In the examined temperature range, background susceptibility shows negligible thermal variations compared to those due to magnetite.For a better comparability of the curve shape between the initial, shocked and reheated samples, the χ -T curves are normalized to values between zero and one.
The model assumes that a small magnetite v olume, w hich can be considered homogeneous with regard to factors that influence the transition temperature, has a certain transition temperature T Vd at which it transforms instantaneously.This assumption corresponds to a first order phase transition.A change to a second order transition is related to non-stoichiometry (e.g.Arag ón et al. 1985 ) and is observed at static pressures above 6 GPa (Rozenberg et al. 1996 ).The susceptibility was measured in the decompressed post-shock state and therefore it is not likely that the magnetite underwent a second order phase transition during the susceptibility measurement.Parameters related to such a model volume fraction are denoted by subscript d to distinguish them from bulk magnetite properties.For the magnetite analysed in this study, shock-related deformation and dislocations can be considered as the main factors affecting the size of the volume.The model volume idealizes the microscopic processes involved in the Verwey transition and cannot be directly related to, for example a magnetic domain, although it is assumed to be of a similar scale.Below T Vd , the susceptibility χ d of the volume fraction is assumed to be constant and equal to zero due to normalization: (1) Especially the magnetite ore shocked at 10 GPa and above shows a significant susceptibility increase already at the lower experimental limit ( ≈80 K).Thermal susceptibility variations below the Verwey transition are attributed to changes of easy magnetic axes orientation in twins of the monoclinic phase, which can be induced by an external magnetic field (e.g.Kosterov 2003 ;Kosterov & Fabian 2008 ).The frequency of orientation changes increases with temperature due to the higher thermal energy.For the analysed χ -T curves, a clear numerical distinction between these low-temperature processes and the Verwey transition is not possible.At temperatures above of the Verwey transition, the magnetic susceptibility of magnetite exhibits a maximum related to the isotropic point at about 130 K that is especially in single crystals, often exhibited as a distinct peak.The analysed magnetite does not show a distinct peak close to 130 K but a rather broad susceptibility maximum.Therefore, the numerical approximation of these χ -T curves does not require a specific consideration of the isotropic point.
Especially for magnetite shocked at 5 GPa and above, the susceptibility at room temperature relative to the isotropic point is significantly lower and partially recovers in form of a relative increase of susceptibility after heating.To describe these changes, χ d at temperatures above T Vd is assumed to be the sum of a temperatureindependent susceptibility contribution χ 0d and a temperaturedependent part χ 1d ( T ) : (2) Parameters affecting the constant part of χ are denoted with subscript 0 and parameters of the temperature-dependent part with 1.In the model, the decrease of the magnetic susceptibility above T Vd is not regarded as directly related to the Verwey transition.The The susceptibility decrease between the isotropic point and room temperature is related to the increase of the magnetocrystalline anisotropy in multidomain magnetite grains (Muxworthy 1999 ).The model uses an exponential approach for a basic numerical description of the observed decrease above the transition: (4) The dimensional number a is required for an exponent without dimension of a temperature.Temperatures in this study are given in Kelvin ( a = 1 K).The rate of the exponential susceptibility decrease is described by the parameter r 1d .An example of a magnetite volume with T Vd = 125 K is shown in Fig. 2 .Downloaded from https://academic.oup.com/gji/article/238/2/794/7688477 by KIT Library user on 17 June 2024 Stress-induced changes in magnetite 797

Model Verwey transition temperature distribution in bulk magnetite
The model uses statistical distributions of T Vd over the magnetite volume to describe the effect of, for example local stress states on the local transition temperature (Fig. 3 a).The model uses symmetric distributions to reflect an overall balanced stress state in the sample during the χ -T measurements.A Voigt profile is the convolution of a Gaussian and a Lorentzian curve and these curve shapes represent boundary conditions.A Gaussian distribution results from uncorrelated events whereas a Lorentzian distribution indicates a correlation between events (e.g.Succi & Coveney 2019 ).In terms of a physical interpretation, a Gaussian distribution indicates that for all volume fractions, T Vd is independent from T Vd of other volume fractions whereas a Lorentzian distribution results from a correlation of T Vd between volume fractions.
In various spectroscopic methods, Gaussian and Lorentzian distributions are used to characterize variations in line shape that may be related to dislocations and point defects (e.g.Stoneham 1969 ;Major et al. 2020 ).Voigt distributions of lattice strain are used to describe diffraction patterns (e.g.Csikor & Groma 2004 ).Shapes of X-ray diffraction (XRD) line profiles are related to the distribution of internal lattice strain.The convolution integral of a Voigt profile can be expressed through the real part of the Faddee v a function w( z) .The Faddee v a function is a scaled complex complementary error function: (5) For computation of the Faddee v a function, the Python module scipy.special.wofz is used.The source code of the package uses a combination of different algorithms (see Johnson 2012 ).To calculate the bulk susceptibility from the idealized χ -T curves for specific local transition temperatures, they are scaled by a dimensionless value v T d ( T ) , corresponding to the value of the distribution of T Vd .Based on the Faddee v a function, the expression for v T d ( T ) that is used in the Verwey transition model is where z is: The centre T V of the distribution is the model temperature at which the largest volume fraction experiences the susceptibility increase.In this study, T V is determined by the fitting process.The parameter T V is close to T V , determined from the maximum of d χ/ d T .In the following, the model parameter is referred to as T V , while T V is used for other e v aluation approaches and the Verwey temperature in general.
The shape of the Voigt profile is defined by the ratio of the scale parameters σ V and γ V of the convolved Gaussian and Lorentzian distributions, which are determined by fitting.For γ V / σ V 1 the profile approximates a Lorentzian distribution, whereas for γ V / σ V 1 it approximates a Gaussian distribution.For σ V = 0 and γ V = 0 the Voigt profile is a Lorentzian and Gaussian distribution, respecti vel y (e.g.Armstrong 1967 ).A Gaussian distribution leads to a more pronounced peak at the transition, whereas a Lorentzian distribution leads to a significant increase in susceptibility at low temperatures and a smoother transition.
The bulk χ -T curve is the sum of all contributions to χ , since the bulk susceptibility is the sum of the magnetic moments of all volume fractions of magnetite.The summation leads to a broadening and a smoothing of the model curve, similar to the experimental curve shape (Fig. 3 b).In the temperature range where a significant model volume passes through the Verwey transition, the resulting bulk curve is affected by both the susceptibility increase and the decrease above T Vd .This range can exceed the sharp increase in susceptibility (see Figs 3 a and b).At higher temperatures, the model curve is not significantl y af fected b y the Verwey transition, but is controlled b y the susceptibility decrease ( c 1d , r 1d ).

Effect of model parameters
To illustrate the effect of c 1d , σ V and r 1d on a bulk χ -T curve based on a Gaussian distribution of T Vd , individual parameters were varied while the other parameters were set to fixed values (see Fig. 4 ).Variations in T V cause a temperature shift of the χ -T curve without affecting the shape of the curve.The effect of variations in the relative susceptibility decrease c 1d is shown in Fig. 4 (a).Without a temperature independent component ( c 1 d = 1 ), susceptibility at high temperatures approaches zero.For c 1d = 0 , the susceptibility does not decrease above the transition.An increase in σ V or γ V results in a broadening of the transition (Fig. 4 b).When the rate of the exponential decrease r 1d is large, the susceptibility decreases rapidly above the Verwey transition (Fig. 4 c).For the e v aluation, c 1d and r 1d were constrained to a common variable value for all volume fractions.Curve scaling is used to compensate for varying peak heights (Fig. 4 ) The fitting was done using Python 3.9.13.

Experimental
Metamorphosed banded quartz-magnetite ore from the Sydvaranger mine (Norw ay) w as used for the shock experiments (Reznik et al. 2016 ;Kontny et al. 2018 ).The ore contains about 80 weight per cent magnetite, 18 weight per cent quartz and 2 weight per cent accessory minerals including amphibole, chlorite, biotite and pyrite in bright quartz-rich and dark magnetite-rich bands (Reznik et al. 2016 ).The magnetite bands consist of polycrystalline and elongated grains.Scanning electron microscopy-electron backscatter diffraction (SEM-EBSD) shows recover y (subg rains) and recrystallization as well as mineral inclusions due to the metamorphic history (Mamtani et al. 2023 ), but no oxidation.X-ray absorption near edge structure (XANES) analysis of iron oxidation states shows only a slight change at high shock pressure.After shock, part of the samples were heat-treated to 973 K to exceed T C in an argon atmosphere to prevent oxidation (Kontny et al. 2018 ).The pure magnetite composition is confirmed by a Curie temperature of T C = 861 ± 2 K for all samples.
The experimental χ -T curves are shown in Fig. 1 (a).After shock ≥5 GPa, T V is increased by about 5 K (Table 1 ) and the susceptibility at temperatures above the transition is significantly lower (Kontny et al. 2018 ).Reheating causes a recovery of the shockinduced changes towards the initial state.Similarly, hysteresis measurements showed a decrease in saturation magnetization M s after shock and an increase after reheating (Kontny et al. 2018 ).The hysteresis measurements indicate that the initial sample is in the range of multidomain (MD) behaviour.After shock, the authors identified pseudo-single domain (PSD) behaviour with a tendency to MD behaviour after reheating.Changes in domain state are more pronounced at higher shock pressures.Reznik et al. ( 2016 ) identified grain fragmentation, lattice defects and micro-shear bands indicating inhomo geneousl y distributed brittle and plastic deformation.Table 1.Parameters from the e v aluation of the measured magnetite χ -T curves before heat treatment assuming a Voigt profile for the distribution of T Vd . (1)Data from Kontny et al. ( 2018 ). (2)Standard deviation σ between the model susceptibility and the normalized χ -T curve.

Sample
T The authors also reported lattice distortion induced by the shock wave associated with elastic strain.Reznik et al. ( 2016 ) reported an inhomogeneous distribution of deformation features in all shocked samples with pressure-dependent variations in the identified features of brittle and plastic deformation.
The shock reverberation experiments were performed at the Ernst-Mach-Institute in Freiburg (Germany).Detailed information about the experimental setup and the method are given in M üller & Hornemann ( 1969 ), Fritz et al. ( 2011 ) and Reznik et al. ( 2016 ).The samples were shocked at maximum pressures of 3, 5, 10, 20 and 30 GPa.Up to 10 GPa, an air gun was used to accelerate a flyer plate for momentum transfer on the sample.For higher pressures, high e xplosiv es were used to accelerate the flyer plate.The Hugoniot elastic limit (HEL) of magnetite was exceeded for at least 5 GPa shock pressure, resulting in both brittle and plastic deformation (Reznik et al. 2016 ).After shock at 3 GPa, no significant changes in domain state from the initial magnetite were observed, indicating that the HEL was not yet reached (Mendes & Kontny 2024 ).
The low-temperature susceptibility measurements were performed with an AGICO KLY-4S Kappabridge (ef fecti ve field intensity: 300 A m −1 ; frequency: 875 Hz) equipped with a CS-L low-temperature cryostat (83-300 K).The accuracy of the recorded temperature is within ±1 K (according to thermometer supplier JUMO) and the displayed resolution is 0.1 K.The sample is placed in a test tube with the cryostat cooled with liquid nitrogen.After purging the nitrogen with argon, χ is measured in ambient atmosphere while warming to room temperature.Measurement with the AGICO KLY-4S instrument is described by Pokorn ý et al. ( 2004 ).

A P P L I C AT I O N O F T H E M O D E L T O χ -T D ATA O F S H O C K E D A N D R E H E AT E D M A G N E T I T E O R E
The full width at half maximum ( F W H M Voigt ) is determined to quantify the transition width.The FWHM of a Gaussian distribution is F W H M Gauss = 2 σ V √ 2 ln 2 and the FWHM of a Lorentzian distribution is F W H M Lorentz = 2 γ V .The FWHM of the Voigt profile is approximated using an approach of Whiting ( 1968 ), modified b y Oli vero & Longbothum ( 1977 ): to characterize the Voigt profile.The parameter varies between 0 for pure Gaussian and 1 for pure Lorentzian.
The parameters obtained, assuming a Voigt profile, are shown before heat treatment (Table 1 ) and after heat treatment (Table 2 ).Results for a Gaussian and Lorentzian distribution are shown in the supplements.For comparison, T V , determined by the maximum of d χ/ d T given in Kontny et al. ( 2018 ), is shown.For the magnetite ore shocked at 10 GPa, glmi x = 0 .72 was determined for the Voigt profile (Fig. 5 ), corresponding to a predominantly Lorentzian distribution.The differences resulting from using a Voigt profile or a Lorentzian distribution of T Vd are insignificant at 10 GPa and both accurately describe the experimental observations.In contrast, the best Gaussian fitting does not adequately describe the measured curve, especially the rather flat susceptibility increase at low temperatures The quality of the fits was assessed by the standard deviation σ between the model susceptibility and the normalized χ -T curve over the whole measured temperature range ( σ Gauss , 10 GPa = 8 .3 × 10 −3 , σ Lorentz , 10 GPa = 3 .4 × 10 −4 , σ Voigt , 10 GPa = 3 .3 × 10 −4 ).

Effect of shock and heat treatment
The measured χ -T curves of the initial, 3, 5 and 10 GPa magnetite ore before and after heat treatment are shown in Figs 6 (a)-(d) along with the best approximation for a Voigt profile.The χ -T curves for 20 and 30 GPa are similar to the 10 GPa curve (see Fig. 1 ).All measured χ -T curves were well approximated with a Voigt profile over the entire temperature range.The corresponding distributions of T Vd over the magnetite volume are shown before heat treatment (Fig. 6 e) and after heat treatment (Fig. 6 f).The initial and 3 GPa magnetite ore show a narrow, almost pure Lorentzian distribution, which changes to a narrow Gaussian distribution after heat treatment (see Tables 1 and 2 ).The change towards a Lorentzian character is reflected by a smaller susceptibility increase below T V (Figs 6 a and b).
The T V of the initial (120 K) and 3 GPa magnetite ore (122 K) is within the typical range of T V for stoichiometric magnetite with low internal stress.At 5 GPa and above, shock causes a large increase in transition width, which varies little with shock pressure (Fig. 6 e).For these samples, T V is approximately 125 K.After heating, T V is between 121 and 123 K for all samples, with a slight increase at pressures below the HEL and a slight decrease above.These differences are within the uncertainty of the Kappabridge measurements.

Shock pr essur e dependence of the Verwe y transition
The changes in T V , which is the centre of the distribution of T Vd , the transition width ( FWHM ) and c 1d of the shocked and subsequently heat-treated magnetite are compared with the results of Reznik et al. ( 2016 ) and Kontny et al. ( 2018 ) in Figs 7 (a)-(f).The pressure dependence of T V (Fig. 7 a) and T V (Fig. 7 b), determined from the maximum of d χ/ d T (Kontny et al. 2018 ), are similar.Ho wever , the slight decrease in T V at 20 and 30 GPa was not observed with the current model, which may be due to uncertainties in the different e v aluation procedures.
Figs 7 (c) and (d) compare the model transition width of the shocked samples with the results of Reznik et al. ( 2016 ) from fitting a Gaussian distribution to dχ/ dT around T V .Assuming a Gaussian distribution of T Vd , the shock pressure dependence of the FWHM is very similar to the model results, showing a maximum at 10 GPa and a slight decrease at higher shock pressures.A Gaussian distribution cannot describe the flat susceptibility increase from 80 K upwards (see Fig. 5 ) and therefore the increase does not significantly affect the FWHM compared to the determination based on d χ/ d T .Assuming a Voigt profile, the FWHM of the shocked magnetite ore is significantly lower and more independent of shock pressure.Since the Voigt profile allows an accurate fitting at low temperatures, it follows that the inclusion of low-temperature susceptibility does not cause an increase in the FWHM , but a change from a Gaussian to a more Lorentzian character.At 3 GPa, the transition width is not significantly affected by shock and heat treatment.After heat treatment, the transition width in shocked magnetite decreases but remains above that of the initial magnetite ore.
Table 2. Parameters from the e v aluation of the measured magnetite χ -T curves after heat treatment assuming a Voigt profile for the distribution of T Vd . (1): Data from Kontny et al. ( 2018 ). (2): Standard deviation σ between the model susceptibility and the normalized χ -T curve.Similar to T V P = χ max / χ 10 • C (Kontny et al. 2018 ), c 1d depends on the ratio of the susceptibility at ambient temperature to the maximum near the isotropic point.After shock and after heat treatment, c 1d (Fig. 7 e) and T V P (Fig. 7 f) show a similar dependence on shock pressure.The larger relative decrease in susceptibility after shock is reflected by an increase in c 1d for pressures above the HEL.The increase is largest at 10 GPa.Heat treatment causes a strong recovery of c 1d in magnetite ore, shocked above the HEL, to the value of the 3 GPa magnetite ore.A decrease of c 1d was also observed in the original magnetite.

Sample
For a linear representation of the distribution character, glmix (Major et al. 2020 ) is shown (Fig. 7 g).Except for 5 GPa, where the distribution character is Gaussian both after shock and after heat treatment, the pressure dependence of glmix shows a clear trend.Without shock at pressures below the HEL, the character is almost pure Lorentzian before heat treatment.From 10 GPa onwards, the character becomes increasingly Gaussian.Heat treatment causes a shift towards a more Gaussian character for all samples except 5 GPa.The change is largest in the original magnetite, which shows an almost pure Gaussian character after heat treatment, whereas the magnetite subjected to the highest shock pressures retains a more Lorentzian character.The 3 GPa sample has a slightly more Lorentzian character, probably related to the small transition width, which increases the uncertainty of the determination.
A higher value of the exponential decrease rate r 1d indicates that the initial magnetite ore shows a significantly stronger susceptibility decrease above the isotropic point both before and after heating than the shocked samples (Fig. 7 h).Heat treatment and shock pressure variations have no clear effect on r 1d , although r 1d is slightly lower at 5 GPa, close to the HEL.

Shock and heat-induced changes of low-temperature behaviour and transition shape
Low-temperature χ -T curves of a tectonically deformed magnetite ore, shocked between 3 and 30 GPa and later heat-treated up to 973 K, show strong v ariations, especiall y near the Verwey transition.Reznik et al. ( 2016 ) and Kontny et al. ( 2018 ) suggested that the parameters calculated from these curves contain information on magnetic domain state.For these particular samples, the differences are interpreted in terms of different strain and lattice defects and annealing.
Magnetic susceptibility from about 80 to 280 K is numerically approximated under the assumption of a probability distribution of local transition temperatures T Vd within the bulk magnetite.This approach considers the decrease in magnetic susceptibility above the Verwey transition for the shape approximation, unlike methods that use d χ/ d T to determine T V (e.g.Reznik et al. 2016 ;Kontny et al. 2018 ).The extended fitting range improves the description of the shape and transition width with a minor effect on the determination of T V .By using a distribution of T Vd , thermal variations of the bulk susceptibility near the Verwey transition affect the parameter determination and increase the accuracy.The numerical approximation describes the measured curves well, even above T V , where both the transition and the decrease related to magnetocrystalline anisotropy might affect the susceptibility.At temperatures well above the transition, the model curve is controlled by the exponential decrease corresponding to variations in magnetocrystalline anisotropy.
On a microscopic scale, the processes associated with the Verwey transition are complex.Interactions between monoclinic twin walls and magnetic domain walls that characterize the transition are strongly influenced by, for example, dislocation structures and therefore cause sensitivity to deformation (e.g.Kasama et al. 2010 ;Lindquist et al. 2019 ).The idealization by an instantaneous susceptibility increase in a homogeneous volume does not reflect these processes, but allows a good characterization of the bulk behaviour without increasing the complexity of the model.The samples can be well approximated with a single temperature distribution.A change in processes controlling the susceptibility below T V , namely variations of the magnetization in the monoclinic phase (Kosterov 2003 ) and the transition to the cubic phase, can be suspected but is not visible in the χ -T behaviour.It is suggested that both processes are controlled by shockinduced internal stresses.Kosterov & Fabian ( 2008 ) assumed that defect-related internal stresses cause pinning of monoclinic twin boundaries.
The centre T V of the distribution of T Vd is interpreted as an average value during warming through the transition, corresponding to T V determined for bulk magnetite.The values of T V are close to T V determined by Kontny et al. ( 2018 ).The FWHM of the model distribution is a measure of the transition width.The FWHM corresponds to the spread of T Vd in the bulk magnetite and reflects the distribution of internal stress.The difference between the transition width determined by Reznik et al. ( 2016 ) and Kontny et al. ( 2018 ) and the distribution width of T Vd is small.Especially at shock pressures above the HEL, variations in internal stress state can be attributed to dislocations.The sharp transition in initial magnetite and magnetite shocked below the HEL is consistent with the stoichiometric composition and pre-existing tectonic deformation (Reznik et al. 2016 ;Mamtani et al. 2023 ).The broad transition above the HEL could be interpreted as an increase of microstress in magnetite associated with a higher dislocation density.
The strong shock-induced susceptibility variations together with the high susceptibility of the magnetite ore allow to distinguish between a Gaussian and Lorentzian character of a Voigt profile.Variations in the distribution character are mainly attributed to outliers of T Vd and not to the distribution width (e.g.Succi & Cov ene y 2019 ).The Lorentzian distribution character of the initial magnetite ore indicates a correlation of T Vd caused by internal stress associated with tectonic deformation, which is not significantly altered by shock up to 3 GPa.Slightly above the HEL (5 GPa), the transition width is broadened without the low-temperature susceptibility increase observed after shock at higher pressures, resulting in a Gaussian character, that may indicate a more brittle deformation.The behaviour at 5 GPa may be related to differences in the dominant deformation mechanisms, or to a more heterogeneous dislocation distribution.Above the HEL the internal stress field is expected to be strongly influenced by dislocations.The increasingly Gaussian character after shock at highest pressures indicates a more distribution of internal stresses, presumably related to dislocation annealing by shock-induced heating (e.g.Tong et al. 2015 ).The occurrence of higher temperatures due to higher shock pressures is supported by microstructural observations, such as local amorphization (Reznik et al. 2016 ).Reheating caused a shift towards a Gaussian character after shock both below and above the HEL, indicating further annealing.This is supported by a decrease in coercivity H c (Kontny et al. 2018 ).The smaller shift in magnetite deformed at the highest shock pressures suggests that a larger fraction of dislocations cannot be annealed or that deformation features like deformation twins are harder to anneal.
One might speculate that plastic deformation at shock pressures beyond the HEL causes the formation of subgrains smaller than the original magnetic domains.After heating, dislocations could persist at interfaces between recrystallized subgrains and be related to the lower susceptibility.A change from multidomain to vortex-state and a stable single magnetic domain grain size can be seen as a major reason for the decrease in magnetic susceptibility (Mendes & Kontny 2024 ).The shock pressure and impact heat cause a fragmentation (shock wave) and partial annealing of internal stresses during the shock.
Verwey transition temperatures of magnetite in crystalline rocks are mainly between about 100 and 125 K, rarely exceeding 130 K (Jackson & Moskowitz 2021 ).Even after shock, T V does not exceed 126 K, and only a small proportion of T Vd is about 130 K. Therefore, the results obtained are plausible, although no upper bound on T Vd was defined.Bimodal distributions of T V are observed in many rocks and can be interpreted in several ways, e.g. as significant distinct magnetite populations or caused by a discontinuity in T V beyond a critical point of non-stoichiometry (Jackson & Moskowitz 2021 ).First tests indicate that the presented approach is suitable for unmixing bimodal distributions of transition temperatures.Numerical unmixing of overlapping peaks is also used to separate contributions from different magnetic materials in coercivity spectra (e.g.Heslop 2015 ).Fitting of overlapping peaks is a common issue in the interpretation of M össbauer spectra of Fe-bearing minerals (e.g.Dyar et al. 2006 ).Thermodynamic calculations by Coe et al. ( 2012 ) indicate that d T V / d p can exceed 10 K GPa −1 under directional strain in agreement with observations by Nagasawa et al. ( 2007 ).After shock above 10 GPa, the susceptibility increases already at the lower measurement limit (80 K).Ho wever , these changes may be caused not onl y b y a linear strain relation, but also b y processes in the monoclinic phase or additional effects such as lattice distortion or changes in local stoichiometry.

Susceptibility variations between the Verwey transition and room temperature
For PSD to MD magnetite grains, the susceptibility decrease between the Verwey transition and room temperature is inversely related to the thermal variation of the magnetocrystalline anisotropy (Muxworthy 1999 ).Therefore, our model describes the decrease above T Vd by a temperature-dependent susceptibility χ 1d ( T ) .The rate r 1d of the exponential approach describing the susceptibility decrease is high in the tectonically deformed initial magnetite, compared to shocked and reheated magnetite.This indicates a stronger increase of the magnetocrystalline anisotropy near the isotropic point for the tectonically deformed MD magnetite.Shock deformation seems to reduce the development of a magnetocrystalline anisotropy in line with a higher dislocation density and reduced magnetic grain sizes.Ho wever , r 1d is rather in variant to shock pressure variations and reheating, with slightly lower values close to the HEL (5 GPa).
The relative susceptibility loss at room temperature, quantified by c 1d , is sensitive to shock deformation and reheating.After shock above the HEL, high values of c 1d reflect a high relative susceptibility loss.Analogous to the variations of the transition shape ( FWHM , glmix ), the susceptibility loss is largest at 10 GPa and heat treatment causes a partial recov ery.A relativ e decrease in room temperature susceptibility after manual powdering of a natural magnetite single crystal (Fig. 8 ) suggests a relation to grain refinement.Therefore, the increase in c 1d may be related to shock-induced grain fragmentation, consistent with the interpretation of Kontny et al. ( 2018 ).These authors observed an increase in crystallite size after heat treatment, consistent with the decrease of c 1d .The reduction in c 1d in heated initial magnetite suggests a sensitivity to tectonic lattice strain.
The susceptibility decrease is well described by the exponential approach.The susceptibility is almost constant at the upper fitting limit (280 K) and c 1d can therefore be accurately determined.A significant mutual influence in the determination of r 1d and c 1d is therefore unlikely, although both describe the same temperature range.

C O N C L U S I O N S
The presented model allows a quantitative numerical analysis of susceptibility variations from the Verwey transition up to 280 K in magnetite-bearing rocks.In this study, it is applied to a wellcharacterized magnetite ore that exhibits strong variations in lowtemperature susceptibility related to experimental shock and heating.The model includes parameters for the characterization of important features such as T V , the transition width and the relative susceptibility decrease up to room temperature.A distribution of local transition temperatures T Vd in bulk magnetite is assumed.For the particular setting, T Vd is expected to be strongly influenced by internal stress fields.
Voigt profiles, which can have a Gaussian or Lorentzian character, are used for the numerical description of the distribution.The Gauss/Lorentz character shows high sensitivity to shock and heat treatment or to a defect-rich or defect-poor crystal lattice, respecti vel y.A Lorentzian character indicates a local correlation of T Vd with more outliers, suggesting a higher amount of dislocations related to internal stress fields.This behaviour is related to plastic deformation under high shock pressures and a reduction of magnetic domain size.A Gaussian character suggests no correlation between the local transition temperatures T Vd in the bulk magnetite, which can be interpreted as a decrease in internal stresses due to annealing.
The susceptibility decrease above T V , which can be related to the decrease of the magnetocrystalline anisotropy constant K 1 of magnetite in this temperature range (e.g.Muxworthy & McClelland 2000 ), is well described with an exponential temperature dependence for the analysed magnetite ore.The results support the main findings of Reznik et al. ( 2016 ) and Kontny et al. ( 2018 ) of strong shock-induced deformation and annealing effects at high shock temperatures and subsequent heat treatment, and allow a more detailed e v aluation of the effect of internal strain (lattice defects) and internal elastic stress on the magnetic behaviour around the Verwey transition.

Figure 1 .
Figure 1.(a) Normalized χ -T curves of the initial and shocked magnetite ore samples (Reznik et al. 2016 ) and of a sample of the same magnetite ore shocked at 3 GPa.(b) χ -T curves of the same samples after heat treatment to 973 K (Kontny et al. 2018 ).

Figure 2 .
Figure 2. Modelled χ -T curve of a homogeneous magnetite volume fraction with T Vd = 125 K (eq.14).These curves represent an idealized Verwey transition with zero width.Below T Vd , the susceptibility of the volume fraction χ d ( T ) is zero and above T Vd , χ d ( T ) is modelled with a sum of a constant component χ 0 d and of an exponentially decreasing component χ 1 d ( T ) .Model parameters: r 1 d = 0 .035 ; c 1 d = 0 .3 .

Figure 3 .
Figure 3. (a) Distribution of T Vd over the magnetite volume fractions for different ratios of σ V / γ V .(b) Bulk χ -T curves resulting from the distributions shown in (a).The curves represent the sum of the χ contributions of the local transition temperatures.Model parameters: T V = 125 K; r 1 d = 0 .035 ; c 1 d = 0 .3 ; F W H M Voigt = 5 K.

Figure 4 .
Figure 4. Changes in the bulk χ -T curve by varying individual parameters for a Gaussian distribution.Unless indicated in the figures, the following parameters were used: T V = 125 K; r 1 d = 0 .035; σ V = 2 K; γ V = 0 K; c 1 d = 0 .4 .Variations of (a) c 1 d (temperature-dependent fraction of χ at T Vd ); (b) σ V (scale parameter of the distribution of T Vd ) and (c) r 1 d (rate of the exponential susceptibility decrease).

Figure 5 .
Figure 5. -T curve of magnetite ore after shock at 10 GPa (x).Best fits for a Gaussian distribution, a Lorentzian distribution and a Voigt profile for the distribution of T Vd are shown.

Figure 6 .
Figure 6.Measured χ -T curves before (x) and after heat treatment ( + ) and best Voigt fittings before and after heat treatment: (a) initial ore without shock, (b) shock pressure 3 GPa, (c) 5 GPa, (d) 10 GPa.(e) Model distributions of local transition temperatures T Vd before and (f) after heat treatment.The distributions are normalized to the same area, resulting in variations in peak height depending on the FWHM and type of distribution.

Figure 7 .
Figure 7. Pressure dependence of model parameters after shock and subsequent heat treatment of magnetite.The suspected range of the HEL between 3 and 5 GPa is shown.Unless otherwise stated, the fittings are made for a Voigt profile.For comparison, related parameters for the same χ -T curves from Reznik et al. ( 2016 ), Kontny et al. ( 2018 ) are shown.(a) T V ; (b) T V (Kontny et al. 2018 ); (c) FWHM : shock (Gaussian distribution, Voigt profile); shock and heat (Voigt profile); (d) FWHM : shock, Gaussian distribution (Reznik et al. 2016 ) (e) c 1 d ; (f) T V P (Kontny et al. 2018 ); (g) distribution character glmi x ; (h) rate of exponential susceptibility decrease r 1 d .

Figure 8 .
Figure 8. Measured χ -T curves of a natural magnetite single crystal before and after manual crushing.