The Philosophy of Rock Magnetism What we do, how we do it, and why...

Richard J. Harrison and Ioan Lascu Department of Earth Sciences, Cambridge The philosophy of rock magnetism

• Introduction to rock magnetism and paleomagnetism • First-order reversal curve (FORC) diagrams for rock magnetic applications • Case study: dusty olivine – measuring magnetic signals from the solar nebula Magnetic Minerals

Oxygen (O) and iron (Fe) are the 1st and 4th most abundant elements in the Earth’s crust. These elements readily combine to form magnetic iron oxides such as magnetite (Fe3O4) and hematite (Fe2O3). Magnetic minerals occur everywhere in the natural environment (rocks, sediments, soils, meteorites). These minerals turn rocks into magnetic hard drives: they retain a memory of the direction and intensity of the geomagnetic field that was present when the rock formed. The magnetic memory of rocks

The direction and intensity of the Rocks retain a memory of field varies systematically with the field that was present at latitude its formation

Memory is caused by presence of magnetic minerals The magnetic memory of rocks How good is this memory?

Southern hemisphere of Mars displays magnetic anomalies 1-2 orders of magnitude stronger than those observed on Earth. The magnetic field on Mars switched off around 4 billion years ago, yet the crust still has not forgotten... What do we need to know to interpret the remanence with confidence?

Region of Interest Which magnetic minerals? Positions? Separations? Volumes? Shapes? Crystallographic orientations? Chemical Compositions? Internal microstructures/defects? Domain States? Coercivities? Reversal mechanisms? Magnetic domain state versus particle size Arguably the most important concept in rock magnetism is the transition from superparamagnetic (SP) to single-domain (SD) to pseudo-single-domain (PSD) to multi-domain (MD) behaviour as function of particle size.

Non-interacting uniaxial single-domain (SD) particles: the holy grail of rock magnetism!

Single domain d < 100 nm

Pseudo single domain 100 nm < d < 2 μm

Multi domain d >> 2 μm 266

9. Experimental estimates of the critical size for single- 06 domain behaviour 05

A comparison of the observed hysteretic properties 4 with the values predicted by the Stoner-Wohlfarth 0.4 ~ x model shows that the critical size increases with in- - 03 creasing x. The magnetite samples show no single- . domain properties, henceIdealit hasvaluesto be concluded for single-domainthat 02 grains the single-domain size is well below 1 pm. The results from the wet-ground1.0 samples indicate that the single-Ms Similar calculations0 I lead to the•

domain size is closer toM0.1rs pm. Forx = 0.2, the criti- classic Stoner-WohlfarthI resultsI 0.5 cal size is above 0.1 pm but below 1 pm. Nc and i- varia- for the coercivityI0~ and2 coercivity3 4 5 6 7 8 9 IO~ 2 3 4 5 tions indicate that the critical size is just below 1 pm of remanence: ~ M/Ms 0 Hc whilefR/Js indicates a lowervalue. The composition .

x = 0.4 shows single-domainHcr behaviourin the two smallest Hc =0.48HFig.K5. -‘~/~ -0.5 versus5 size fractionsplacing the transition at about 1 pm. Forx -1.0 Hcr =0.524HK = 0.6, the transition-1.5 occurs-1.0 at-0.5about0 2pm.0.5 Soffel1.0 (1971)1.5 cate that this transition size is about 10—20 pm for looked at the domain patterns on hthe surface of titano- magnetite, depending on the property considered. This magnetite grains of composition x = 0.55 and con- compares favourably with Stacey’s (1962) value of 18 cluded that the single domainMcriticalrs size could be Has cr pm for magnetite. The transition size increases with high as a few microns. =0.5 =1increasing.1 x to a value of about 30—40 pm for x = 0.6. Ms Hc The transition from single-domain to truly multi- At these sizes r is nearer to 3.5—4 than 3 or 5. domain behaviouris not abrupt; there is a transition Anothermethod of estimating the transition sizes region. Multi-domainValues of Mgrainsrs/Ms !are 0.5characterized (values up toby: 0.87 are obtainedis a graphical when theapproach anisotropyusing two or more magnetic has cubic rather than uniaxial symmetry) and 1 properties.< Hcr/Hc < 2Fig. are4 highlyshows JRS/Js against r where it can diagnostic of non-interacting single-domainbe seen grains.that the samples follow a definite trend from 0.05 and r 5.0 truly multi-domain behaviour at the bottom right to single-domain behaviour at the top left. Many of the although Parry (1965) suggests that multi-domain be- samples fall in the transition region between SD and haviour begins for r 3.0. Using these values for truly MD behaviour. multi-domain grains, the minimum size for multi- Fuller (1974), while investigating the magnetic be- Bulk hysteresis measurementsdomain behaviour can be estimated. The results mdi- haviour of lunar samples, suggested that a plot ofJRs/ The Day PlotJsagainst x/Js should show the same trends. Fig. 5 shows this graph for the titanomagnetites. The same 1.0 Ms Small The ‘Day’ plot is a plot of Mrs/Ms versus Hcr/ Hc, and is a trendcommonis diagnosticevident buttool innote rockthat the graph also shows a magnetism. The plot reveals a systematic Mrs 0.6 SD variationcompositional in hysteresis parametersdependence. as a This reduces its use as an 0.5 Intermediate indicatorfunction of grainof the size.domain state if the composition is un- 0.5 ~ known, but when used in conjunction with Fig. 4, it The smallest grains (d << 1 "m) plot within 04 •I the expectedmay limitsbe possible for SD behaviour.to determine the approximate compo- M/Ms 0 sition of the sample. Hc 0.3 PSD The largest grains (d >> 20 "m) have very Js .• low Mrs/Ms and very high Hcr/Hc. These 02 samples are10. multiDiscussion domain (MD), and are Hcr : -0.5 characterised by the presence of mobile •. S magnetic domain walls. 0.1 1. Large ______~. MD Intermediate Thegrain single-domainsizes show a smoothcritical sizes found experimen- 001 4 5 transition betweentally are classicalgenerally SD andhigher classicalthan the calculated sizes. -1.0 MD behaviour. This region is termed -1.5 -1.0 -0.5 0 0.5 1.0 1.5 ‘~ pseudo-singleAssuming domaina mean (PSD).elongation of 1.4, the calculated h Day,Fig. R., 4.M. J~/~JFuller, andversus5 V. A. SchmidtHRc/HC. (1977), Hysteresis critical sizes for x = 0, 0.2, 0.4 and 0.6 are 0.08—0.15, properties of titanomagnetites: Grain size and PSD and MD behaviour is the subject of the composition dependence, Phys. Earth Planet. Inter. next lecture. 13, 260 – 267. Plotting Mr/Ms vs Hcr/Hc gives some indication of where a rock lies within the SD-PSD-MD spectrum.

Only non-interacting SD particles have a theory that is of practical use for paleomagnetism. The problem is that the vast majority of rocks plot in the PSD range.

This can be interpreted in a number of different ways. Rocks are a complex mixture of interacting and non-interacting particles with a range of particles sizes and separations.

Hysteresis loops on their own are of limited use - we need much more detailed information... First-order reversal curves (FORCs)

20 -6 weakly interacting uniaxial SD grains 40x10 10 0.12

20 0 0.10

0.08 -10 0 0.06

Hu (T) -20 0.04 Moment -20 -30 0.02

0.00 -40 -40 -3 -50x10 -0.10 -0.05 0.00 0.05 0.10 -3 0 20 40 60 80x10 Field (T) Hc (T)

FORC diagrams are now a standard tool for rock magnetic studies:

– Domain state fingerprinting – Coercivity distributions – Interactions – Magnetic mixtures – Quantitative modelling FORC measurements FORC measurements FORC surface

The FORC surface is a plot of magnetisation M as a function of Ha and Hb.

The FORC distribution is defined as the mixed second derivative of M with respect to Ha and Hb.

2 ⇥ M(Ha, Hb) (Ha, Hb) = ⇥Ha⇥Hb

What does the FORC distribution tell us? After M smoothing

dM/dHa

Preisach Interpretation

Ha

b Hu a dM/dHb b a a b

Hb

b Preisach space a a b

a d2M/ b dHaHb a b

Hc

FORC space Original Ha-Hb axes Transformation of axes

‘Coercivity’ axis

Hc = (Hb - Ha)/2

‘Interaction field’ axis

Hu = (Ha + Hb)/2 (b) (a) (a) (b) +M +M +MS S +MS S Rotated Hc-Hu axes

H H -H H H -H -Hc +H -H c +Hc c +Hc c c +Hc

-M -MS -M S -MS S

(c) (d) (c) +M (d) +M +MS S +MS S

H Hu u H H H Ha Hb -H -Hc +H a Hb c +Hc c

-M -M -MS S -MS S FORC fingerprints Non-interacting uniaxial single domains 3.3 Study of magnetotacte.g.ic bact ersedimentia containing magnetotactic bacteria 2.2 High-resolution electron microscopy 3.3.3 Electron tomography Information about the relative orientations of the magnetosomes in a single chain can High angle annular dark field (HAADF) be obtained from either zone-axis selected area electron diffraction (SAED) patterns electron tomography allows the three- or high-resolution (HR) images of dimensional morphology of a sample to individual crystals. Figure 3 shows a 20 bright-field image of a double chain of be deduced from a series of two- magnetite crystals. By inspection, and with reference to previous literature on dimensional images taken at dierent Egli 2012 magnetite magnetosomes [3], apart from tilt angles, typically from plus 70 to 10 the crystals at the ends of the chains, [111] Geochemistry is nearly parallel to the chain axis in all of minus 70 (17). Geophysics 3 newell: forc 10.1029/2004GC000877 the magnetosomes, as indicated by the line Central ridge Electron tomography has its origins 300 of white arrows. Geosystems 0 Figures 4 and 5 show zone-axis HR in the biological sciences where conven- G images and SAED patterns acquired from tional bright-field micrographs were used crystals 4 and 7, respectively. One of the sample [5,6]. The approach relies on being able to reverse the direction of magnetisation in the tilt axes of the double-tilt specimen holder to reconstrsample ucextacthrtlye.e Fdimensionalor chains ofmor cry-stals this condition is likely to be met, and can be checked by 200 (y) was approximately parallel to the chain pholoregies.peatingF otrhedenser same emateriaxperimelntsp seeci-veral times. The approach also relies on diffraction contrast in the -10 +ve x10 axis, while the other (x) was perpendicular crystals being identical in each pair of images. Artefacts arising when this is not the case can usually to it. The small tilts about x that were mens,bes identifieduch as the visually.magnetosomes in bac- [47] The FORC function is obtained by integrating -6

100 required to achieve zone axis orientations teria, bright-field imaging is unsuitable in crystals 3-7 from a zone axis orientation (a) the function Hu (T) for identical(b)-20 particles (9) over N: in crystal 2 suggest that their [111] as the2.4.con Off-axistrast of electronthe imag holographyes is not mono of magnetite- chains directions are approximately parallel to Off-axis electron holograms of bacterial magnetite chains were recorded in magnetic-field-free tonically dependent on the thickness; ef- one another (and to the chain axis), the conditions, both at room temperature and with the sample cooled usin+g Ma double tilt liquid nitrogen +M 0 largest difference being 4.5º between fectscforlomd hodilderar. c tAion thoermForceosneluplefring indiecsated that the cold holder nominally cooleSd the sample to 116 K, 1 H H S dN crystals 3 and 7. In contrast, much larger 1-30 a b can wcahuseich prois inble thms.e vicHAADFinity of thimage Veingrwey Transition (119 K), so we can be confident of beHing beHlow the N 26 tilt angles were required about y to achieve m a; b; S m~ ; r ; S 2 : zone axis orientations. The measured is thereforeisotropiusedc poinint oaf smcaanninggnetitetr (ansmis-130 K). However it was not possible to acquire diffractionð patterns of the Þ ¼ Ms 0 NMs NMs ð Þ N ð Þ -100 crystal orientations (relative to crystal 2) crystals to assess the true temperature of the crystals under the conditions used for electron Z  -ve  are plotted on a stereogram in figure 6, sion helectronolographmicy. rToscophe pree(STEsent eM)xpewrimhereent can therefore be regarded as a preliminary study of the effect of -40 which highlights this difference. Assuming the intemperaturetensity is onprop magneticortional microstructureto thick- in biogenic magnetite crystals. that sample preparation has not altered the Figure 8 shows a representative magnetic induction map of two double chains[ o48f m]agnTheetite crysintegrandtaHls is only nonzero for N1 HN relative orientations of the crystals, the ness squared, the contrast is strong and -3 determined from holograms acquired at room- Htemperature with the chains magnetised parallel and -H   chain is therefore analogous to beads on a there are little or no diraction eects. N+H, where N =-50x10H c/M and N = +Hin(1/2, 2 H / string that are allowed to rotate freely. antiparallel to their length. The contours are highly cconstrained to be parallel to eac2h octher within the 1 a s 2 M c a Biological control over the orientations of Reconstrcrystals ucantdion to ofof lthelow 3thDe mocharphol-in axis, although they deviate when breaks in thMe cha)in (seeoccur. EAppendixach j Aj ). The0.00new coef0.02ficients0.04ajre j 0.06 0.08 0.10 0.12 the crystals appears to be stricter in setting ogy cfromhain 2Dis semicroen togra bephshave isredonelativelus-y independently, however there is some transfer sof magnetic flux [111] parallel to the chain axis than in between adjacent chains, as well as between the two pairs of chains in the figure. constraining their orientation about this ing specialised software, the resulting Hc (T) direction. model containing information not only -M N2 From a magnetic perspective, the Figure 3. Bright-field image of a double S Ha -MdN alignment of the crystals ensures that their on thechashain opf e,mabutgnetiatelso magonnettohesomvearias, acqtionuired aFigurt e 3.6: Reconstruction of the 3D Figure 8. MagnetiAc Ha; S a r N; S S 27a magnetocrystalline easy axes are closely 400 kV using a JEOL 4000EX TEM. The NM N in density, and even chemical composi- morpohology of the double magneto- phase contours ð Þ ¼ N1 s ð Þ ð Þ parallel to the chain axis at room orientations of the crystals marked 1-7 are measured using Z   temperature. However, this relationship tion ofrefetherred sttoructure in the tifextene anrdgy in filteredsubsequensomet chain shown in figure 3.3, from no longer holds below the Verwey electron holography TEMfiisgurused.es. The white arrows are approximatelelecy tron tomography. Tomography by transition, as discussed below. parallel to [111] in each crystal. from two pairs of R K K Chong. bacterial magnetite N2 chains at 293 K, Ha Hb dN B Ha; Hb; S b ; r N; S : 27b (c) after mðagnetising Þ ¼ (d)NM NM ð Þ N 2 ð Þ the sample parallel ZN1  s s +M and antiparallel to +M 17 Sthe direction of the S w[4hi9te] arTrowh. eThetheory in this article applies only to colours, which were dparticlesetermined from twhe ith uniaxial anisotropy, but many geo- H local gradient of the u plogicallyhase image, showinteresting materials such as magnetite the direction of the H H Hmhaveagnetic ainducubicction magnetocrystalline-H anisotropy+H . Fortu- a abccording to the c c cnatelyolour whee,l theshown cubic anisotropy can be neglected if the below. The contour sparticlespacing is 0.a25re elongated. For such particles it is best Newell 2005 radians. -MSto start with a pdf for the particle-MaspectS ratio q and use it to derive the pdf for the demagnetizing factor (Appendix A). The integral is evaluated using adaptive Gauss/Lobatto quadrature [Gander and 6 Figure 8. The components of the FORC function for Gautschi, 2000] with a relative accuracy of 10À an isotropic sample with a lognormal distribution of or better. aspect ratios (q = 2 and s = 0.25). (a) A. The function for identical particles with aspect ratio q is shown as a [50] Many of the properties of a system of identical dashed line. (b) B. The same color scale is used as in particles are still true of a system with a shape Figure 1b. distribution. For example, in a set of identical particles, B(H , H ) = B(H , H ). In (27) this a À b À a b implies that B(Ha, Hb; S) = B(Ha, Hb; S). Thus À À positive region near the Hc axis, the delta function the antisymmetry of B about Hu = Hc is À on the Hc axis, and the equal distances of these preserved. Also, the positive and negative peaks peaks from the origin. Also, the FORC function is have the same relative weights as in section 3.3. identically zero for Hu > 0. These predictions are [51] The arguments are plotted in Figure 8 for a robust because they do not depend on the distribu- lognormal distribution of aspect ratios with mean tions of particle orientations or shapes. The other q = 2, corresponding to a demagnetizing factor kind of prediction does depend on these distribu- tions. The distribution of particle orientations has N0(q) = 0.24. Equation (27b) predicts that the shape distribution will spread the FORC function little effect on the shapes of the positive and equally in all directions, and this is consistent with negative peaks, but it has a strong effect on the Figure 8. One result of this spreading is that the relative size of these peaks. The distribution of slanted ridges are replaced by humps that are particle shapes affects the shapes of the peaks roughly symmetric about the peaks. equally, with a realistic distribution tending to smear them out and remove the ridge between the peaks. 4. Discussion [53] In this section I discuss two robust predictions [52] This model makes two kinds of predictions and their significance. The first, a negative peak about FORC functions of uniaxial SD particles. near the Hu axis, is one of the surprises to come out Some predictions follow directly from the proper- of plots of experimental FORC functions. I clarify ties of the single-particle hysteresis loops. These its physical significance in section 4.1. The second include the negative region near the Hu axis, the is that the function is identically zero for Hu > 0. 9 of 14 FORC fingerprints Weakly interacting uniaxial single domains 3.3 Study of magnetoe.g.tactic concentratedbacteria sample of magnetotactic bacteria 2.2 High-resolution electron microscopy 3.3.3 Electron tomography Information about the relative orientations of the magnetosomes in a single chain can High angle annular dark field (HAADF) be obtained from either zone-axis selected area electron diffraction (SAED) patterns electron tomography allows the three- or high-resolution (HR) images of dimensional morphology of a sample to individual crystals. Figure 3 shows a 20 bright-field image of a double chain of be deduced from a series of two- magnetite crystals. By inspection, and with reference to previous literature on dimensional images taken at dierent magnetite magnetosomes [3], apart from tilt angles, typically from plus 70 to 10 the crystals at the ends of the chains, [111] 0.12 is nearly parallel to the chain axis in all of minus 70 (17). the magnetosomes, as indicated by the line Electron tomography has its origins of white arrows. 0.10 Figures 4 and 5 show zone-axis HR in the biological sciences where conven- 0 images and SAED patterns acquired from tional bright-field micrographs were used crystals 4 and 7, respectively. One of the sample [5,6]. The approach relies on being able to reverse the direction of magnetisation in the tilt axes of the double-tilt specimen holder to reconstrsample ucextacthrtlye.e Fdimensionalor chains ofmor cry-stals this condition is likely to be met, and can be checked by 0.08 (y) was approximately parallel to the chain pholoregies.peatingF otrhedenser same emateriaxperimelntsp seeci-veral times. The approach also relies on diffraction contrast in the -10 axis, while the other (x) was perpendicular crystals being identical in each pair of images. Artefacts arising when this is not the case can usually 0.06 to it. The small tilts about x that were mens,bes identifieduch as the visually.magnetosomes in bac- required to achieve zone axis orientations teria, bright-field imaging is unsuitable in crystals 3-7 from a zone axis orientation Hu (T) -20 in crystal 2 suggest that their [111] as the2.4.con Off-axistrast of electronthe imag holographyes is not mono of magnetite- chains 0.04 directions are approximately parallel to Off-axis electron holograms of bacterial magnetite chains were recorded in magnetic-field-free tonically dependent on the thickness; ef- one another (and to the chain axis), the conditions, both at room temperature and with the sample cooled using a double tilt liquid nitrogen largest difference being 4.5º between fectscforlomd hodilderar. c tAion thoermForceosneluplefring indiecsated that the cold holder nominally cooled the sample to 116 K, 0.02 crystals 3 and 7. In contrast, much larger -30 tilt angles were required about y to achieve can wcahuseich prois inble thms.e vicHAADFinity of thimage Veingrwey Transition (119 K), so we can be confident of being below the zone axis orientations. The measured is thereforeisotropiusedc poinint oaf smcaanninggnetitetr (ansmis-130 K). However it was not possible to acquire diffraction patterns of the 0.00 crystal orientations (relative to crystal 2) crystals to assess the true temperature of the crystals under the conditions used for electron are plotted on a stereogram in figure 6, sion helectronolographmicy. rToscophe pree(STEsent eM)xpewrimhereent can therefore be regarded as a preliminary study of the effect of -40 which highlights this difference. Assuming the intemperaturetensity is onprop magneticortional microstructureto thick- in biogenic magnetite crystals. that sample preparation has not altered the Figure 8 shows a representative magnetic induction map of two double chains of magnetite crystals relative orientations of the crystals, the ness squared, the contrast is strong and -3 determined from holograms acquired at room temperature with the chains magnetised parallel and chain is therefore analogous to beads on a there are little or no diraction eects. -50x10 string that are allowed to rotate freely. antiparallel to their length. The contours are highly constrained to be parallel to each other within the Biological control over the orientations of Reconstrcrystals ucantdion to ofof lthelow 3thDe mocharphol-in axis, although they deviate when breaks in the chain occur. Each 0 20 40 60 -3 the crystals appears to be stricter in setting ogy cfromhain 2Dis semicroen togra bephshave isredonelativelus-y independently, however there is some transfer of magnetic flux 80x10 [111] parallel to the chain axis than in between adjacent chains, as well as between the two pairs of chains in the figure. constraining their orientation about this ing specialised software, the resulting Hc (T) direction. model containing information not only From a magnetic perspective, the Figure 3. Bright-field image of a double alignment of the crystals ensures that their on thechashain opf e,mabutgnetiatelso magonnettohesomvearias, acqtionuired aFigurt e 3.6: Reconstruction of the 3D Figure 8. Magnetic magnetocrystalline easy axes are closely -6 in densit400 yk,Va ndusinevg aen JEcOhemicalL 4000EXco mpTEMo.si- Thmore pohology of the double magneto- phase contours parallel to the chain axis at room orientations of the crystals marked 1-7 are measured using 40x10 temperature. However, this relationship tion ofrefetherred sttoructure in the tifextene anrdgy in filteredsubsequensomet chain shown in figure 3.3, from no longer holds below the Verwey electron holography TEMfiisgurused.es. The white arrows are approximatelelecy tron tomography. Tomography by transition, as discussed below. parallel to [111] in each crystal. from two pairs of R K K Chong. bacterial magnetite chains at 293 K, 20 after magnetising the sample parallel and antiparallel to 17 the direction of the white arrow. The 0 colours, which were determined from the local gradient of the Moment phase image, show the direction of the -20 magnetic induction according to the colour wheel shown below. The contour -40 spacing is 0.25 radians. -0.10-0.05 0.00 0.05 0.10 Field (T) FORC fingerprints

True PSD particles with intermediate grain size between SD and MD

Wright Industries Fire obsidian (1st ever magnetite 3006 (particle FORC measured on our size: 1.06 +/- 0.71 new Lakeshore AGM!) microns)

60 0.05 1.0 40 150

0.8 20

100 0.00 0 x10

0.6

-6

x10 50 Hu (T)

-20

-3

Hu (T) 0.4 -40 0 -0.05

0.2 -60

-3 -80x10 0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 -0.10 Hc (T)

0.00 0.02 0.04 0.06 0.08 0.10 Hc (T) FORC fingerprints Mixture SD + PSD: Central ridge superimposed on PSD background

0.02

0.00 14

12 -0.02 10

-0.04 8 x10

6 -3

Hu (T) -0.06 4

-0.08 2

0

-0.10 -2

-0.12

0.00 0.02 0.04 0.06 0.08 0.10 Hc (T) FORC fingerprints True MD behaviour: titanomagnetite

R.T. 50 K 100 100

Weak Pinning Strong Pinning

-6 200x10 50 50 -6 200x10

100 100

0 0 Moment Moment 0 0 Hu (T) Hu (T)

-100 -100

-200 -200 -0.10 0.00 0.10 -50 -50 -0.10 0.00 0.10

Field (T) Field (T)

-3 -3 -100x10 -100x10 -3 -3 0 10 2030x10 0 10 2030x10 Hc (T) Hc (T) FORC fingerprints Highly-correlated interacting system

‘Wishbone’ structure with vertical offset peak FORC fingerprints Highly-correlated interacting system

‘Wishbone’ structure with vertical offset peak FORC fingerprints Highly-correlated interacting system

‘Wishbone’ structure with vertical offset peak FORC fingerprints

Random distribution of 1000 uniaxial single domain magnetite particles (70x70x70 nm) with log-normal distribution of switching fields and randomly oriented easy axes inside a gradually shrinking box.

0 10 -3 20 30 x10 5 Volcanic glass 0 -5

-3 x10 -10 -15 ∞ -20 FORC fingerprints

Random distribution of 1000 uniaxial single domain magnetite particles (70x70x70 nm) with log-normal distribution of switching fields and randomly oriented easy axes inside a gradually shrinking box.

0 5 10 15-3 20 25 30 x10 Volcanic glass 4 2 0 -2 -4 -6

-3 -8 x10 -10 -12 -14 -16

-18 20000 nm -20 FORC fingerprints

Random distribution of 1000 uniaxial single domain magnetite particles (70x70x70 nm) with log-normal distribution of switching fields and randomly oriented easy axes inside a gradually shrinking box.

0 5 10 15-3 20 25 30 x10 Volcanic glass 4 2 0 -2 -4 -6

-3 -8 x10 -10 -12 -14 -16

-18 10000 nm -20 FORC fingerprints

Random distribution of 1000 uniaxial single domain magnetite particles (70x70x70 nm) with log-normal distribution of switching fields and randomly oriented easy axes inside a gradually shrinking box.

0 5 10 15-3 20 25 30 x10 Volcanic glass 4 2 0 -2 -4 -6

-3 -8 x10 -10 -12 -14 -16

-18 6000 nm -20 FORC fingerprints

Random distribution of 1000 uniaxial single domain magnetite particles (70x70x70 nm) with log-normal distribution of switching fields and randomly oriented easy axes inside a gradually shrinking box.

0 5 10 15-3 20 25 30 x10 Volcanic glass 4 2 0 -2 -4 -6

-3 -8 x10 -10 -12 -14 -16

-18 4000 nm -20 FORC fingerprints

Random distribution of 1000 uniaxial single domain magnetite particles (70x70x70 nm) with log-normal distribution of switching fields and randomly oriented easy axes inside a gradually shrinking box.

0 2 4 6 8 10 12 14 -3 16 18 20 22 24 26 28 30 x10 Volcanic glass 4 2 0 -2 -4 -6

-3 -8 x10 -10 -12 -14 -16

-18 3000 nm -20 FORC fingerprints

Random distribution of 1000 uniaxial single domain magnetite particles (70x70x70 nm) with log-normal distribution of switching fields and randomly oriented easy axes inside a gradually shrinking box.

0 5 10 15-3 20 25 30 x10 Volcanic glass 4 2 0 -2 -4 -6

-3 -8 x10 -10 -12 -14 -16

-18 2400 nm -20 Case study: dusty olivine – measuring magnetic signals from the solar nebula

Richard J. Harrison1 Sophie Lappe1 Sara Russell2 Josh Feinberg3 Geoff Bromiley4 Nathan Church1 Alice Bastos da Silva Fanta5 Rafal Dunin-Borkowski5

1. Earth Sciences, Cambridge 2. Natural History Museum, London 3. Institute for Rock Magnetism, Minnesota 4. Earth Sciences, Edinburgh 5. Centre for Electron Nanoscopy, Copenhagen REPORTS leointensity as that of the NRM) exhibits the and indicates that the magnetization in the un- HC magnetization is highly unlikely to be the same high AF-related noise (fig. S3B). If we baked interior is preterrestrial (Fig. 2). This in- product of nebular lightning (21), which cannot assume this is the characteristic magnetization, terior magnetization consisted of a HC primary produce the observed low NRM/IRM values. On then we obtain paleointensities of ~3 to 8 mT magnetization component (extending from 15.8 to the other hand, the paleointensities are within the (ARM method) and ~2 mT(IRMmethod)(fig. >290 mT) trending to the origin (Fig. 1C), usually range expected for the disk dynamo excited by S3, C and D, and table S3). D’Orbigny’s and overprinted by a weak LC component that is magnetorotational instabilities (22), T Tauri flares A-881371’sgreatage,excellentpreservationstate, probably a terrestrial VRM (see SOM). AF at ~0.2 AUs, magnetic fields generated by large and nearly instantaneous (10 to 50°C hour−1) demagnetization of five mutually oriented sub- impacts (2), strong crustal ferromagnetic anom- primary igneous cooling rates (19)indicatethat samples of a second chip of Angra dos Reis taken alies, and a core dynamo. However, our data their HC magnetization is a truly ancient thermo- from the interior (≥6mmfromthenearestfusion indicate that the paleofields on the APB lasted for remanence (20). crust) of the main mass in Museu Nacional, Brazil at least 10 My after CAIs, beyond the likely The coarse-grained younger angrite Angra (MNB) revealed an intense LC overprint from lifetime of a circumstellar disk dynamo (23). The dos Reis (Pb/Pb age of 4557.7 ± 0.1 Ma) (3)also previous sample handling and a unidirectional HC absence of shock textures in angrites means that has a preterrestrial magnetization acquired in a component trending to the origin interpreted as a it is highly unlikely that the HC magnetization, similarly substantial magnetic field. Our AF analy- primary thermoremanence (fig. S1). Two addi- which is blocked to coercivities >290 mT in ses of mutually oriented subsamples from a chip tional subsamples on the opposite side of the chip Angra dos Reis, can be a shock remanence of Angra dos Reis from the American Museum were nearly fully overprinted by the high intensity created in an impact-generated field (24). Addi- of Natural History (AMNH) traversing the fusion component and did not yield primary remanence. tionally, the slow cooling rates of the coarse- crust to the interior revealed a unidirectional Paleointensity experiments on the HC component grained angrites like Angra dos Reis (25, 26) magnetization in the interior. Subsamples from for seven subsamples from both the AMNH and mean that they would have acquired their thermo- within 2.7 mm of the fusion crust have directions MNB samples gave mean values and SDs of 17 ± remanence over a period of thousands to millions divergent from (coincidentally nearly antipodal 11 mT(ARMmethod)and19±9mT(IRM of years, far longer than the expected lifetime of to) the interior and Earth-strength paleointensi- method) (table S3). any impact-generated fields [which last just ~1 ties. This is consistent with the outer few milli- Three angrites record magnetic fields on the day, even for basin-scale impacts on a Moon- meters having been baked by atmospheric passage order of 10 mTontheAPBextendingfromat sized body (27)] or T Tauri flares [lifetimes of least 4564.4 ± 0.1 (Pb/Pb age of the oldest several hours (28)]. Such slow cooling rates also on May 13, 2009 angrite, D’Orbigny) to ≥4557.7 ± 0.1 Ma (Pb/Pb make it highly unlikely that these angrites could age of the youngest studied angrite, Angra dos have been magnetized by the spatially complex Reis) (Fig. 3). Our fusion-crust baked contact test fields expected from magnetorotational instabil- on Angra dos Reis and unidirectional magneti- ities while situated on the translating, spinning zation trending to the origin observed in the in- APB. Crustal field sources couldMotivation potentially ac- teriors of Angra dos Reis and D’Orbigny (with count for angrite magnetization, but such strong Angra dos Reis having an especially low-noise crustal fields would probably demand a core dy- signal) are collectively strongly indicativeChondrules of pri- namo provide for their formation. evidence of the mary thermoremanence. The implied paleointen- This reasoning implies that the source of processes that operated within the www.sciencemag.org sities are ~20% of Earth’sfieldtodayandfar these fields was internal, most likely a convecting larger than that of the galactic field,protoplanetary solar wind, metallic corenebula, and dynamo. yet Angritesthe exact contain geo- Mercury’spresentsurfacefield,andtheexpectedenvironmentchemical and evidence mechanism for the formation of of an Fe-Ni time-averaged fields of the T Tauri Sun outside core with a mass ~8 to 60% of the APB (7, 29)by 0.2 astronomical units (AUs) (Fig.chondrule 3). Angrite 4MyafterCAIs(formation 4is, 30 still), possibly debated. coincident with

Fig. 3. Summary of paleo-

Fig. 2. Fusion-crust baked contact test on Angra intensity estimates for an- Downloaded from dos Reis parent sample AMNH. HC magnetization grites. Each point is derived directions of mutually oriented subsamples rang- from the HC magnetization ing from the fusion-crusted exterior to the interior in a single subsample. Cir- are shown and plotted on an equal-area projec- cles, D’Orbigny; triangles, tion. Closed and open symbols represent projec- A-881371; squares, Angra tions of vector directions onto the lower and upper dos Reis (with approximate hemisphere, respectively. Ellipses give estimated magnetization age labeled orientation uncertainty (either MAD of least- next to each meteorite). Solid squares fit or estimated sample positioning uncer- symbols, IRM method; open Chondrules may acquire a pre-accretionary tainty, whichever is larger). Distance from fusion symbols, ARM method using thermoremanent magnetisation (TRM) crust in millimeters is listed next to each sample. 50-mTbiasfield.Meanpaleo- Only sample AMC5 contains fusion crust. The seven intensities from IRM and ARM during their formation. The intensity of the remaining samples are from the interior, with methods are given by thick magnetising field could yield important clues AMC8 and AMC10 apparently baked by atmo- black and gray lines. For com- spheric passage. Fisher mean direction (gray star) parison, the surface fields of to the mechanism and location of chondrule and associated 95% uncertainty confidence esti- Earth and Mercury, the solar forming events. However, to make any firm mate (a95 =10.7°)areshownforinteriorsubsam- wind field at Earth’sorbit(1AU ples. The shallow depth of divergent magnetization from the sun), the galactic field, conclusion the uncertainties must be directions (<3 mm) and the fact that measured and the inferred paleofields of the typical T TauriWeiss sun et and al. short-lived 2008 Science flares at 0.2322 AUs are also shown. A samples have NRM/IRM < 1% throughout their full magnetorotational instability (MRI) protoplanetary disk dynamo, impact plasma-generated fields, and core reduced... coercivity range indicate that the exterior has been dynamos can produce paleointensities over a wide range of values,713-716 including values consistent with angrites. thermally remagnetized by atmospheric passage An estimate of the uncertainty range for an individual angrite paleointensity datum (primarily because of rather than isothermally remagnetized by a magnet. uncertainty in the ratio of ARM and IRM to thermoremanence) is shown at right. www.sciencemag.org SCIENCE VOL 322 31 OCTOBER 2008 715 Dusty olivine: a potential carrier of pre-accretionary remanence

•Olivine grains containing submicron particles of ~pure metallic Fe

•Found in unequilibrated and carbonaceous chondrites

•Formed from reduction of Type I (Mg- rich) chondrules

•May appear as relict grains that survived the chondrule forming event without melting

•Alignment and clustering controlled by crystallography of host olivine First-order reversal curves (FORCs)

60 0.1 a b 1 40 0.0 3.0 )

2 20 2.5

2.0 x10 3

0 (T) -0.1 1.5 -3

u -3

2.0x10

H 1.0 1.5 -20 M /M = 0.036 2 Moment(Am rs s 1.0 0.5 -0.2 0.5 0.0 -40 H /H = 7.45 cr c Centralridge profile 0.0 0.0 0.2 0.4 0.6 0.8 -6 Hc (T) -60x10 -0.3 -1.0 -0.5 0.0 0.5 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Field (T) Hc (T) 1 = Non-interacting SD particles with coercivity distribution extending to 600 mT

2 = Positive satellite peak at average values Hc ~ 56 mT Hu ~ 117 mT

3 = Negative satellite peak at average values Hc ~ 180±30 mT

Relative proportions of signals 1 vs 2/3 vary from sample to sample !!14 1999; Dumas et al. 2007 a, b). The FORC coordinates of the positive peak are related to the

nucleation (HN) and annihilation (HA) fields associated with the transition from SD to SV states and J. Appl. Phys., Vol. 85, No. 9, 1 May 1999 C. Pike and A. Fernandez 6669

from SV to SD states, respectively (Pike and Fernandez 1999): !!14 1999; Dumas et al. 2007 a, b). The FORC coordinates of the positive peak are related to the

Single! vortex (SV) nucleation{Hc, Hu} = {(andHA -annihilation HN)/2, -(HA + HN)/2}! (3) FIG. 3. IRM and DCD remanence curves for field aligned with easy particle nucleation (HN)axis. andThe annihilation annihilationfield value is probably a(betterHAestimate) fieldsthan that associated with the transition from SD to SV states and FIG. 1. A scanning electron micrograph of a portion of the Co dot array. given by the major hysteresis loop. The dots, which are produced by interference lithograph and a lift-off pro- cess, are nominally 450 nm3250 nm in size and 30 nm thick. Signal 2 is associated with the nucleation (SD to from SV to SD states,For the applied respectivelyfield aligned with SV)the hard(Pike and~short annihilation! axis,and Fernandez (SV to SD) 1999): of vortex ~DCD! remanence curves with the applied field aligned with the hysteresis loop has negligible coercivity or remanence the particle easy axis ~Fig. 3!. ~The IRM remanence curve @Fig. 2~b!#. This is because when a saturatingmagnetisationfield is applied states in particles that are just begins in a demagnetized state and the DCD curve begins in along the hard axis and removed, the particles relax into a 5 a negatively saturatedRearrangingstate.! From these curves we Eqn.estimate 3,single-vortex HN andstate, as shownHAby canMFM investigations.above be calculated theFrom critical size from for SD the behaviour. FORC coordinates of the positive peak: the vortex annihilation field at HA592.6 mT. The nucleation the major hysteresis loop we estimate the median values: field cannot be estimated from the IRM curve because all the HN546 mT and HA5169 mT. However, as described above, particles begin in a vortex state, so no nucleation takes place. these estimates have considerable uncertainty. The IRM and The DCD curve begins in a negative saturation state, so DCD remanence curves are of no use when the applied field nucleation does take place, but!it is difficult to discern where, is aligned with the hard axis, since{Hthec, remanentHu} state= {(is HA - HN)/2, -(HA + HN)/2}! (3) in the neighborhood of H50, this curve has a maximum always a vortex state. slope; hence it is difficult to estimate HN . ! III. FORC DIAGRAMS HN = -Hc - Hu! (4) A FORC diagram is calculated from a class of hysteresis curves known as first order reversal curves. As shown in Fig. 4, the measurement of a FORC begins with the saturation of Rearranging Eqn.the sample 3, byHaNlarge andpositive HappliedA canfield. The befield calculatedis then from the FORC coordinates of the positive peak: ! ramped down to a reversal field Ha . The FORC consists of aHA = Hc - Hu! (5) measurement of the magnetization as the applied field is in- creased from Ha back up to saturation. The magnetization at the applied field Hb on the FORC with reversal point Ha is denoted by M(Ha ,Hb), where Hb>Ha . A FORC distribu- Pike and Fernandez 1999 SV nucleation and annihilationtion is defined inas Cothe mixed nanodotssecond derivative For dusty olivine: 6674 J. Appl. Phys., Vol. 85, No. 9, 1 May 1999 C. Pike and A. Fernandez ! HN = -Hc - Hu! (4) Values of Hc = 56 ± 14 mT and Hu = -114 ± 41 mT were calculated from the average positive peak HN ~ 58 mT ! HA = Hc - Hu! (5) position of ~ 30 individual samples, yielding HN = 58 ± 55 mT and HA = 170 ± 55 mT. The negative HA ~ 170 mT

peak corresponds toFIG. 4.theDefinition lowerof a first order halfreversal curve. ofThe measurementa ‘butterfly’of a structure that is diagnostic of SV states (Pike and FIG. 14. FORC diagram for two uncorrelated annihilation fields having FORC begins by saturating the sample with a large positive applied field. identical distributions. Calculated using Eq. ~15!. rN and rA have means 0.2 FIG. 15. FORC diagramThecalculatedappliedusingfieldEq. is~17then! with kdecreased50.09, wheretor a reversal field Ha . The FORC is and 2.0, respectively, and standard derivations of 0.42. N FIG. 2. Major hysteresis loop for fieldValuesaligned with ~a! easyofand andH~brA!1chardhave =means 56comprised0.2 and 2.0,±respectively,of 14a measurementand mTstandard derivationsof andthe magnetizationof Hu as=the -114applied field ±is 41 mT were calculated from the average positive peak 0.42, and where the distribution of HA2 is given by rA1(HA21k). particle axis. The values for nucleation and annihilation fields are only increased from Ha back up to saturation. The magnetization at Hb on a rough estimates. FORC with reversal field Ha is denoted by M(Ha ,Hb), where Hb>Ha . D. Two annihilation fields;FernandezHA2 distributed at smaller 1999). The butterfly is centred on FORC coordinates {Hc, Hu} = {HA, 0} and is caused values tively. On a FORC diagram, after changing variables to H Downloaded 24 Nov 2005 to 131.111.41.167. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jspc In Sec. V C, it was assumed that the two annihilation and Hu , this lower peak will have coordinates $Hc ,Hu% position of5 ~ 230 individual2 1 samples, yielding H = 58 ± 55 mT and H = 170 ± 55 mT. The negative field values, HA1 and HA2 , had the same statistical distribu- $(^HA1& ^HN&)/2, (^HA1& ^HN&)/2%. Conversely, we N A tion. But in actuality, a distinction between these two distri- can calculate ^HN& and ^HA1& from the coordinates of the butions is expected for the following reason: When a vortex lower peak: nucleates, it will pick that site on the particle surface where by a small difference^HN&52Hc2Hu ; ^inHA1&5 Hthec2Hu annihilation~18a! field for a vortex nucleating on opposite sides of a particle. irregularities cause the greatest reduction in the nucleation energy barrier. It is reasonable to assumepeakthat the samecorrespondsir- The first term in Eq. to~17! generatesthea peaklowerin the upper half of a ‘butterfly’ structure that is diagnostic of SV states (Pike and regularities which lower the energy barrier to vortex nucle- quadrant. This peak will be located at Ha5^HN& and Hb ation will also lower the energy barrier to vortex annihila- 5^HA1&2k. Since rA1(HA21k) is the distribution of the tion. Therefore, the location on the particle where vortex annihilation field HA2 , we can write ^HA2&1k5^HA1&, so nucleation takes place will also tend to be a favorable loca- Hb5^HA2&. On a FORC diagram, after changing variables tion for vortex annihilation.TheSince Haverageis defined as the an- position of the negative peak was determined to be H = 183 ± 30 mT, which agrees A2 to Hc and Hu , this upper peak will have coordinates A nihilation field when a vortex exits on the same side it enters, Fernandez$(^ H1999).A2&2^HN&)/2, (^H AThe2&1^HN&)/ 2%butterfly. Conversely, we can is centred on FORC coordinates {Hc, Hu} = {HA, 0} and is caused then it follows that HA2 will tend to be smaller than HA1 . calculate ^HN& and ^HA2& from the coordinates of the upper To model this effect, we shift the distribution of HA2 to peak: smaller values. We will write the distribution of HA1 as ^HN&5Hu2Hc ; ^HA2&5Hu1Hc . ~18b! rA1(HA1) and the distribution of HA2 as rA1(HA21k) where k is positive. Then Eq.well~14! becomes with the Tovalueillustrate the features ofof thisHmodel,A wedeterminedchanged variables from the position of the positive peak. This agreement by a smalland differencecalculated a FORC diagram ofinr(H c the,Hu) for kannihilation50.09 field for a vortex nucleating on opposite sides of a particle. r~Ha ,Hb!5 dHN dHA1 dHA2 using the distribution functions of Sec. V. C and letting E E E2`,` rA15rA . As seen in Fig. 15, the negative region of the 3rN~HN!rA1~HA1!rA1~HA21k! butterfly structure has been shifted leftward and downward relative to the positive region. 3r~Ha ,Hconfirmsb ,HN ,HA1 ,HA2!. that~16! both peaks are related to the nucleation and annihilation of vortices. This integration is evaluated in AppendixTheA. We averageget VI. COMPARISON positionOF THEORY ANDofEXPERIMENT the negative peak was determined to be HA = 183 ± 30 mT, which agrees

r~Ha ,Hb! On the basis of the above-described models, we can 5r ~H !r ~H 1k! identify the peaks in our FORC diagrams as manifestations N a A1 b of the nucleation and annihilation of single-vortex states. The 2rA1~2Ha!rA1~Hb1k!1rA1~Hb!d~Ha1Hb! presence of the butterfly structures in Figs. 8 and 12 is con- sistent with the model of Sec. V C and is an indication that 1r ~2H !r ~2H !. well with~17! the value of HA determined from the position of the positive peak. This agreement N b A1 a there are two distinct annihilation processes. The region of The last term in Eq. ~17! generates a peak in the lower positive values contained in this butterfly structure is shifted quadrant of a FORC diagram. This peak will be located at to the left and downward relative to the region of negative Ha52^HA1& and Hb52^HN&, where ^HN& and ^HA1& de- values. This is most evident when the applied field is aligned note the medians of the4.2.3.distributions confirmsof H NElectronand HA1 , respec- thatwith the holographyparticle botheasy axis peaks~Fig. 8!. This shift areis consistent related to the nucleation and annihilation of vortices. Downloaded 24 Nov 2005 to 131.111.41.167. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

! Both SD and SV states were imaged directly via electron holography (Fig. 8). SD states were

typically4.2.3. observed Electron inholography elongated particles and were easily recognised by their distinctive external ! Both SD and SV states were imaged directly via electron holography (Fig. 8). SD states were dipolar stray fields and closely spaced interior flux lines parallel to the particle length (Particle 1, typically observed in elongated particles and were easily recognised by their distinctive external Fig. 8). SV states were observed in both “in-plane” and “out-of-plane” orientations, according to dipolar stray fields and closely spaced interior flux lines parallel to the particle length (Particle 1, whether the vortex core lies in or out of the specimen plane (Particles 2 and 3, respectively, Fig. 8). Fig. 8). SV states were observed in both “in-plane” and “out-of-plane” orientations, according to

whether the vortex core lies in or out of the specimen plane (Particles 2 and 3, respectively, Fig. 8). SEM/TEM SEM/TEM SEM/TEM SEM/TEM

SEM/TEM

95 nm

45 nm 3D imaging using FIB slice-and-view

Schaffer & Wagner (2008) Microchim Acta. Alice Bastos da Silva Fanta, Sophie Lappe, Richard Harrison 3D imaging using FIB slice-and-view

Image resolution: 1024 x884 Pixel size: 1,92 x 10-8 µm = 19.2 nm Slice thickness: 20nm Number of slices: 61 3D imaging using FIB slice-and-view

Image resolution: 1024 x884 Pixel size: 1,92 x 10-8 µm = 19.2 nm Slice thickness: 20nm Number of slices: 61 Electron holography: magnetic imaging at the nm scale

 Electron holography: magnetic imaging at the nm scale

Single Domain (SD)

Double Vortex (DV)

Single Vortex (SV)

 Electron holography: magnetic imaging at the nm scale

 Electron holography: magnetic imaging at the nm scale

 Predicting Domain States

4 Domain states, volumes and 3

coercivities can be 2 determined. Together these VORTEX ONLY are the necessary 100 parameters to apply Néel’s 9 8 SD theory to model the 7 6 acquisition of thermal 5 remanence. 4 SD ONLY 3

2 Particlelength (nm) Single domain Vortex (core out-of-plane) Vortex (core in-plane) 10 9 Calculated SD/vortex boundary 8 (Butler & Banerjee 1975) 7 0.0 0.2 0.4 0.6 0.8 1.0  Axial ratio Chapter 5 - FORC Method for Palaeointensity Determination ! and Dunlop (2002), suggest that the interaction field is directly related to the magnetisation, i.e. hs(T) ∝ MS(T). The PreisachUsing distribution, FORC p(hc, hs), andinformation the thermal critical tocurves predict react to changes in temperature in opposite wayspaleomagnetic (Fig. 5.3). While the critical behaviour curves contract with decreasing temperature, the Preisach distribution expands from the origin and grows in the hc and ± hs Using Néel theory, plus an empirical relationship between coercivity and volume, Muxworthy and directions. Heslop (2011) proposed a method to use the information in a FORC diagram to predict thermal B04102 MUXWORTHY AND HESLOP:remanence PREISACH acquisition PALEOINTENSITY PROTOCOL—THEORY B04102

References Morris, M.A., and S.J. Desch (2010) Thermal histories of chondrules in solar nebula shocks. Astrophys. J., 722, 1474-1494, doi:10.1088/0004-637X/722/2/1474.

Muxworthy, A.R., and D.J. Dunlop (2002) First-order reversal curve (FORC) diagrams for pseudo- single-domain magnetites at high temperature. Earth Planet. Sci. Lett., 203(1), 369-382, doi: 10.1016/S0012-821X(02)00880-4. Figure 3. Schematic showing the propagation of the Preisach distribution into the Preisach plane with Figure 5.3 decreasing temperature (sequence from Figure 3a to Figure 3c) and the positions of the thermal critical CopiedMuxworthy, directlybarriers A.R., from ( yand= Muxworthy D. 45°), Heslop which (2011) and contract HeslopA Preisach with (2011). decreasingmethod for estimating temperature absolute (Figure paleofield 3a then Figure 3b) and shift onto Withintensity decreasing theunderhc theaxis temperature constraint when the of (from fieldusing a isonly to removed c) isothermal the Preisach (Figure measurements: distribution 3b then Figure 1. Theoreticalpropagates 3c). framework. into the Preisach J. plane. The thermal critical barriers (ψ = 45°) contract with decreasing temperature (a to b). They shift onto the hc axis when the field is Geophys. Res. 116, B04102, doi:10.1029/2010JB007843. removed (b to c). and mentally. For the thermal critical curves, the variation of HK(T), is also required; as we are assuming the magnetic Muxworthy, A.R., and W. Williams (2006) Critical3=2 single-domain/multidomain grain sizes in 2=3 2=3 À anisotropy is controlled by the shape anisotropy H (T) HC y HK sin y cos y 10 K / noninteracting andðÞ¼ interacting elongatedþ magnetite particles:ð ImplicationsÞ MS(T)[ forDunlop magnetosomes. and Özdemir J. , 1997]. In Appendix A we  verify this assumption that H (T) M (T) as a first ApplyingGeophys.When equationRes., simulating 111, (10)B12S12, to TRM equation doi:10.1029/2006JB004588. acquisition (9) yields close to the Curie temperature, the mainK part of Sthe Preisach approximation using published experimental/ data. Second, distribution lies within the regions which are superparamagneticfor the Preisach or distribution in the onesp( hwhichc, hs)wedrawon are field Néel’s g y [1953] interpretation of the Preisach model, where h cor- Muxworthy A.R.,hc D.H Heslop,K andha D.h Michalks ðÞ (2009) Thermal fluctuation fields in basalts and the c 1 À ht 11 responds to the coercive force and hs corresponds to the blocked (Fig.2 5.3a).HC y OnÆ cooling,hc it ¼starts to expandð Þ into the memory region (Fig. 5.3b). When the Barbier plot. Earth PlanetsðÞ Space, 61, 111-117. interaction field; therefore h (T) M (T) corresponds in a c / S temperature is reduced to room temperature and the similarexternal manner field tois HswitchedK(T). For off, the the variation critical of hcurvess(T) we The general shapes of the thermal critical curves for y ≠ 0° draw on the experimental findings of Dunlop and West Muxworthy, A.R., D. Heslop, G.A. Paterson, and D. Michalk (2011) A Preisach method for areare centred similar to on those hc = for 0,y resulting= 0° (Figure in the 2b). thermally It should beblocked[1969] region and Muxworthybeing centred and around Dunlop h[2002],c = 0. who both noted,estimating however, absolute that paleofield given the intensity dependency under the on constraint the orienta- of usingfound only isothermalhs(T) M measurements:S(T); that is, the interaction field is directly tionParticles the thermal in critical the superparamagnetic barriers take the form regions of curved are inrelated thermal to the /equilibrium magnetization. and they do not carry a surfaces2. Experimental and only testing. appear J. asGeophys. lines for Res. a single 116, B04103, value of doi:10.1029/2010JB007844.y (for [20] As illustrated in Figure 3, the critical curves and the remanence.example as shown However, in Figure due 2b). to For the the movement general case of there the criticalPreisach curves distribution the thermally respond in theblocked opposite region, sense to chan- are two points of intersection: the first, (hc = 0, hs = ha) and ges in temperature; the critical curves contract (“tightening Muzerolle J., N. Calvet, L. Hartmann, and P. D’Alessio (2003) Unveiling the inner disk structure of whichthe second carriers when thehs = remanence,ha, with a corresponding expands. As value soon of h asc athe particle knot”) aspasses the temperature into this region, decreases, its whereas TRM p(hc, hs) givenT Tauri by stars. Astrophys. J. Lett., 597, L149-L152. expands from the origin, growing outward in both the hc and becomes blocked. The TRM in the equilibrium state± aths blockingdirections. temperature During the simulation (TB) is ofdescribed thermoremanence by HC y acquisition (section 4), at high temperatures approaching TC, (Stancu and Spinu h1998,c 2h t RoshkoðÞ: and Viddal 2004):12 the distribution p(h , h ) is mostly in the field blocking or Nagahara H. (1981) Evidence¼ for HsecondaryK origin of chondrules.ð Þ Nature, 292, 135-136. c s superparamagnetic regions (2, 3, and 4) defined by the crit- ical curves (Figure 3a), moving gradually into the thermally 3.Néel, A L. Thermally (1949) Théorie Activated du traînage Preisach magnétique Model des ferromagnétiques With blocked en grains region fins avec 1 as the temperature decreases (Figure 3b). µ0VMSFinally,(TB )( whenHa theH externals (TB )) field is reduced to zero the critical Thermallyapplications aux Variable terres cuites, Distributions Ann. Geophys., 5, 99-136. TRM = MS tanh curves and memory region are centered along the h axis kB TB c ! [18] To simulate thermoremanence acquisition it✓ is nec- (Figure 3c). ◆! (5.15) essary to incorporate the strong thermal variability of the Néel, L. (1953) Remarques sur la théorie des propriétés magnétiques [des21] substances In the superparamagnetic dures. Appl. regions the magnetic state thermal critical curves (equation (11)) and the Preisach of particles is considered to be in thermal equilibrium, with distributionSci. Res., 4, p13-24,(Ha, Hdoi:10.1007/BF02316465.b) (in Preisach space this is p(hc, hs)) the equilibrium magnetization, Meq, being given by [Néel, into the thermally activated Preisach model described in 1949] !section 2 [Spinu et al., 2001; Borcia et al., 2002a, 2002b]. 117 Néel,[19] L. In (1955) the treatment Some theoretical of the thermal aspects of variability rock magnetism. we make Adv. Phys., 4, 191-243, doi: 0VMS H Meq MS tanh 13 certain necessary assumptions: first, for the thermal critical ¼ kBT ð Þ 10.1080/00018735500101204.  curves, from molecular field theory for crystals greater than ! 0.5 169 a few nanometers in size, MS(T) (1 − T/TC) , where TC is where H is the external field given by equation (3). On the Curie temperature [Aharoni,/ 2000]. This assumption is passing into the thermal blocking region 1 (the remanence required because in our procedure the sample cannot be carrying region), the TRM for identical particles becomes heated, and therefore MS(T) cannot be quantified experi- blocked (frozen) in the equilibrium state at the blocking

5 of 13 Quantitative modelling of thermal remanence using FORC diagrams

First steps: FORCintense Adrian Muxworthy and Richard Harrison

AR Muxworthy and D Heslop (2011) A Preisach method for estimating absolute paleofield intensity under the constraint of using only isothermal measurements: 1. Theoretical framework. Journal of Geophysical Research, 116, B04102, doi: 10.1029/2010JB007843. Quantitative modelling of thermal remanence using FORC diagrams Quantitative modelling of thermal remanence using FORC diagrams Application to dusty olivine

Simulated AF demag at 340 μT (equal to lab field) FORC diagram Electron holography b SD SV

SV SD

Calculated vs observed REM’ acquisition curve compared to Non-linear TRM acquisition REM’ experimental data x103 26 24 22 20 18 16 14 12 10 8 6 4 2 0 0 5 mm 5 2 4 6 Scanning magnetic microscopy - paleomagnetism at at -paleomagnetism microscopy magnetic Scanning Eduardo Lima, Ben Weiss, Sophie Lappe, Nathan Ben Lima, Weiss, Eduardo 8 10 Scanning SQUID microscopy of synthetic dusty olivine dusty synthetic of SQUIDmicroscopy Scanning 12 x10 14 3

16 18 Church, Richard Harrison Richard Church, 20 22 small length scales length small 24 26

-5 -4 -3 -2 -1 0 1 2 3 4 5

z(ir ) ) T T (micro (micro bz bz x103 26 24 22 20 18 16 14 12 10 8 6 4 2 0 0 5 mm 5 2 4 6 Scanning magnetic microscopy - paleomagnetism at at -paleomagnetism microscopy magnetic Scanning Eduardo Lima, Ben Weiss, Sophie Lappe, Nathan Ben Lima, Weiss, Eduardo 8 10 Scanning SQUID microscopy of synthetic dusty olivine dusty synthetic of SQUIDmicroscopy Scanning 12 x10 14 3

16 18 Church, Richard Harrison Richard Church, 20 22 small length scales length small 24 26 -60x10 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

-3

z(ir ) ) T T (micro (micro bz bz x103 26 24 22 20 18 16 14 12 10 8 6 4 2 0 0 5 mm 5 2 4 6 Scanning magnetic microscopy - paleomagnetism at at -paleomagnetism microscopy magnetic Scanning Eduardo Lima, Ben Weiss, Sophie Lappe, Nathan Ben Lima, Weiss, Eduardo 8 10 Scanning SQUID microscopy of synthetic dusty olivine dusty synthetic of SQUIDmicroscopy Scanning 12 x10 14 3

16 18 Church, Richard Harrison Richard Church, 20 22 small length scales length small 24 26

8000 7950 7900 7850 7800 7750 7700 7650 7600 7550 7500 7450 7400 7350 7300 13.20 13.30 100 13.40 x10 13.50 3

μ m 13.60 13.70 13.80 -60x10 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

-3

z(ir ) ) T T (micro (micro bz bz x103 26 24 22 20 18 16 14 12 10 8 6 4 2 0 0 5 mm 5 2 4 6 Scanning magnetic microscopy - paleomagnetism at at -paleomagnetism microscopy magnetic Scanning Eduardo Lima, Ben Weiss, Sophie Lappe, Nathan Ben Lima, Weiss, Eduardo 8 10 Scanning SQUID microscopy of synthetic dusty olivine dusty synthetic of SQUIDmicroscopy Scanning 12 x10 14 3

16 18 Church, Richard Harrison Richard Church, 20 22 small length scales length small 24 26

8000 7950 7900 7850 7800 7750 7700 7650 7600 7550 7500 7450 7400 7350 7300 13.20 13.30 100 13.40 x10 13.50 3

μ m 13.60 13.70 13.80 -60x10 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

-3

z(ir ) ) T T (micro (micro bz bz Scanning magnetic microscopy - paleomagnetism at small length scales

Bx and By can be calculated from a measurement of Bz (Lima and Weiss 2009 JGR 114, B06102).

Bx By Bz 60 60 60 40

40 40

bx (micro (micro bx T)

by (micro (micro by T) bz (micro (micro bz T) 20 20 20 0 0 0

-20 -20 -20 -40 -40 -40 -3 -60x10 -3 -3 -60x10 -60x10

Fitting dipole equation to Bx, B y, and Bz simultaneously provides robust Mx, M y, and Mz (and x, y, z positions of particle). Scanning magnetic microscopy - paleomagnetism at small length scales Observed

0.3 Btotal 0.2 47 9 0.1 (micro bz T) 8 10 63 dipoles 0.0 4844 28 -0.1 39 45 fitted 7 -0.2 51 42 -0.3 41 40 16 46 43 53 11 27 15 26 29 6 30 5455 0 31 13 12 34 57 14 3332 35 56 61 1918 Fit 25 58 59 24 60 17 5 20 62 52 21 4 36 22 23 50 37 38 1 3 2 49 Demagnetising TRM and ARM

Note that ARM is carried predominantly by SD particles, whereas the original TRM is carried by both SD and SV. Demagnetisation spectra agree only when the SD component is isolated above 170 mT Chapter 6 - Scanning Superconducting Quantum Interference Device (SQUID) Microscopy!

a b -12 ] ]

particle 0 2 250x10 particle 5

2 -9 3.0x10

2.5 200

TRM TRM 2.0 150 ARM ARM 1.5 100 1.0 50 0.5 Total Magnetic Moment [Am

Total Magnetic Moment [Am 0.0 0 0 50 100 150 200 250 0 50 100 150 200 250 AF field [mT] AF field [mT] c d ] ] 2 particle 25

2 particle 10 -12 -12 80x10 10x10

8 60 TRM TRM ARM ARM 6 40 4

20 2 Total Magnetic Moment [Am

0 Total Magnetic Moment [Am 0 0 50 100 150 200 250 0 50 100 150 200 250 AF field [mT] AF field [mT] Figure 6.9 Direct comparison of the AF demagnetisation curves of the TRM and ARM for particle 0 (a), particle 5 (b), particle 25 (c) and particle 10 (d). Note that the two demagnetisation curves for all samples are very similar in shape from an AF demagnetisation field of about 160 mT onwards.

Orthogonal plots of the ARM demagnetisation for the four particles used as examples are presented in Figure 6.10. Orthogonal plots for all particles can be found in Appendix B. The first orthogonal plot for the samples (Fig. 6.10a, c, e and g) shows the ARM demagnetisation without having the residual TRM removed. It is observed that in this case the demagnetisation does not actually trend towards the origin. Removing the residual TRM (Fig. 6.10b, d, f, and h) results in the ARM demagnetisation trending towards the origin. For all samples a clear magnetisation along the z-axis is observed, which is the direction in which the field during ARM acquisition was applied. Only particle 10 seems to be magnetised not directly in the z-direction. However, one has to bear in mind that this sample is one of the smallest ones observed. Furthermore, considering the scaling, the offset from the direct z-direction is not larger than for the other three samples. Slight variations in the magnetisation direction might be due to the experimental setup for the ARM acquisition. It might also be a sign of a certain degree of magnetic anisotropy in the sample. However, none of the previous experiments showed indications of magnetic anisotropy for these samples.

146 Calibrating non-heating paleointensity measurements

Remanence behaviour changes dramatically as a function of the SD/SV ratio (identified by FORC diagrams). Best results are when SD component is isolated. Chapter 6 - Scanning Superconducting Quantum Interference Device (SQUID) Microscopy!

8 average fit to all samples cube 18 average fit to all samples ± 1σ used for SQUID average fit to samples made from natural olivine average fit to samples made from natural olivine ± 1σ SQUID slope1 (100-160 mT) 6 SQUID slope2 (180-290 mT)

natural olivine precursor synthetic Ni-free olivine precursor synthetic Ni-olivine precursor 4

cube4 excluded from fit

Slope Slope (TRM lost / ARM lost) 2

0 0 400 800 1200 1600 Applied field [µT] Figure 6.12 Plot of dTRM/dARM vs applied laboratory field for all samples (from chapter 4). Diamonds, open circles and filled circles represent samples made from natural olivine, synthetic Ni-free olivine and synthetic Ni-bearing olivine precursor, respectively. The red solid curve shows the average fit to the data of all samples using Equation 4.1 (chapter 4). Blue dotted curves represent the average fit to all data ± 1σ. Cube4 was excluded from the fit as it showed an anomalous amount of magnetite. The dark green solid curve shows the average fit to the data of the samples made from natural olivine only using Equation 4.1. Purple dotted lines represent the average fit to the data of samples made from natural olivine only ± 1σ. Orange crosses show the values obtained for slope1 for all particles from the SQUID microscope measurements. Note that the majority plots within the average fit ± 1σ range for the fit curve to all data. Light green crosses represent the values obtained for slope2 for all particles from the SQUID microscope measurements. The majority of the values for slope2 plot below the lower margin of the average fit ± 1σ range for the fit to the samples made from natural olivine only.

Figure 6.12 shows the plot of the slope of TRM lost versus ARM lost for all samples introduced in chapter 4. The data obtained from the SQUID microscopy measurements are added. To determine the slope for the TRM lost versus ARM lost curve for all individual samples a straight line was fitted to an AF demagnetising field range from 100 to 150 mT (cf. chapter 4). In the case of the individual cube samples no demagnetisation data beyond 150 mT could be obtained. The TRM lost versus ARM lost plot for the bulk samples showed an approximately constant slope from 100

149 The philosophy of rock magnetism Microscopy

Rock magnetism Tomography

NanoPaleoMag

Scanning Magnetic Microscopy Holography

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Computer simulations