Single particle multipole expansions from Micromagnetic Tomography
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Single particle multipole expansions fromMicromagnetic Tomography
David Cort´es-Ortu˜no , Karl Fabian , and Lennart V. De Groot Paleomagnetic laboratory Fort Hoofddijk, Department of Earth Sciences, Utrecht University,Budapestlaan 17, 3584 CD Utrecht, The Netherlands. Norwegian University of Science and Technology (NTNU), S. P. Andersens veg 15a, 7031 Trondheim,Norway
Corresponding author: David Cort´es-Ortu˜no, [email protected] –1– a r X i v : . [ phy s i c s . g e o - ph ] J a n anuscript submitted to G-cubed
Abstract
Micromagnetic tomography aims at reconstructing large numbers of individual magne-tizations of magnetic particles from combining high-resolution magnetic scanning tech-niques with micro X-ray computed tomography (microCT). Previous work demonstratedthat dipole moments can be robustly inferred, and mathematical analysis showed thatthe potential field of each particle is uniquely determined. Here, we describe a mathe-matical procedure to recover higher orders of the magnetic potential of the individualmagnetic particles in terms of their spherical harmonic expansions (SHE). We test thisapproach on data from scanning superconducting quantum interference device microscopyand microCT of a reference sample. For particles with high signal-to-noise ratio of themagnetic scan we demonstrate that SHE up to order n = 3 can be robustly recovered.This additional level of detail restricts the possible internal magnetization structures ofthe particles and provides valuable rock magnetic information with respect to their sta-bility and reliability as paleomagnetic remanence carriers. Micromagnetic tomographytherefore enables a new approach for detailed rock magnetic studies on large ensemblesof individual particles. Key Points: • Micromagnetic Tomography uniquely recovers higher-order multipole terms forseveral individual grains in a sample. • Higher order multipole moments are an expression of the internal domain struc-ture of magnetic grains. • Ultimately, this enables to select individual grains for rock- and paleomagnetic stud-ies based on domain configuration.
Initially, the development of micro- to nanoscale scanning magnetometers for rock-and paleomagnetism aimed at recovering statistical information about the average mag-netization of a sample by measuring the surface magnetic signal of the remanence car-rying mineral grains (Egli & Heller, 2000; Weiss et al., 2007). One approach is to recoverthe total dipole moment of a larger sample volume by upward continuation of the mag- –2–anuscript submitted to
G-cubed netic measurements on the surface above this volume to suppress higher-order terms (Fuet al., 2020). Another approach is the spatial domain unidirectional inversion of the mag-netization by means of least-squares fitting, which relies on constraints to the magne-tization vector (Weiss et al., 2007; Myre et al., 2019). A recently developed method ofde Groot et al. (2018) improves upon the total moment measurement by aiming to re-cover the dipole moments of all individual magnetic particles. Although this appears tobe unnecessarily complicated, this approach potentially solves almost all problems thathaunt paleomagnetism from its early beginnings until today. The most prominent amongstthese problems is mineral alteration, by which the original carriers of the paleomagneticinformation change in chemistry or shape and either loose their primary magnetization,or acquire a new magnetization in a different field. Another critical problem is the oc-currence of multiple minerals as remanence carriers. If these minerals acquire their mag-netization by different processes and with different efficiencies, the mixed natural rema-nent magnetization may not reflect the paleofield in a straightforward way. The thirdcommon problem is a large variation in grain size or domain state of the remanence car-rying magnetic mineral by which less reliable multidomain carriers may overprint andinvalidate the paleomagnetic signal represented by the more reliable small grain-size frac-tions of pseudo-single domain or single-domain carriers. By individually determining themagnetic moments of all magnetized grains in a sample, it becomes possible to calcu-late statistical averages over specifically chosen subsets of optimal remanence carriers.For example one could remove all particles above a certain grain-size threshold from thestatistical ensemble, or disregard all particles with certain unwanted physical or chem-ical properties, such as all particles below a certain density threshold. The fundamen-tal idea that enables to recover individual particle moments independent from all othermoments, is to combine micro X-ray computed tomography (microCT) and scanning mag-netometry for rock magnetic measurements. This technique, called Micromagnetic To-mography, has been demonstrated by de Groot et al. (2018). In theory, by adding mi-croCT information, the individual magnetic potentials of topologically separated par-ticles can be recovered from surface measurements of the normal field component on anenclosing sphere, and the corresponding inverse problem even is well-posed in the senseof Hadamard (Fabian & de Groot, 2018). Here, this general result is exploited and ex-perimentally tested by inverting not only for the magnetic dipole moments of the indi-vidual particles, but also for higher spherical harmonics, or multipole moments. One ad- –3–anuscript submitted to
G-cubed vantage of this approach is that it provides additional information about the internal mag-netization structure of the particles, even though the well-known non-uniqueness of po-tential field inversion problems (Zhdanov, 2015) makes it impossible to invert for the mag-netization structure itself. For example, multipole inversion may indicate that the par-ticle magnetization is carried by a multidomain structure rather than by a single-domainstructure if the magnetic scanning measurement reveals that the quadrupole and octupolecoefficients are much larger than expected for a homogenously magnetized particle. Astudy by Fu et al. (2020) has recently accounted for the deviation from dipolar behav-ior (i.e. contribution of higher order moments) in magnetic field maps via the combina-tion of high resolution Quantum Diamond Microscopy and upward continuation of thefield data. Here, multipole moments are fully recovered from the inversions which is fa-cilitated by knowledge of the grain positions. Multipole inversion thus may enable to in-trinsically select statistical ensembles of magnetization carriers, based on their internalmagnetization structure.
Tomography and scanning data in this study were acquired from the synthetic sam-ple described in (de Groot et al., 2018). It contains natural magnetite particles preparedand described by Hartstra (1982) with diameters between 5 and 35 µ m. The particleswere embedded in epoxy at approximately 2,800 grains per cubic millimeter. Sizes, shapes,and positions of the magnetite grains are recovered from the three-dimensional densitydistribution within the sample, acquired by microCT (Sakellariou et al., 2004). As de-scribed in de Groot et al. (2018), a 1.5 mm × T = 4 K has been measured above thesample using scanning superconducting quantum interference device microscopy (SSM)(Kirtley & Wikswo, 1999). SSM and microCT data are available from the PANGAEAdata repository at https://doi.org/10.1594/PANGAEA.886724. The inverse modeling of the magnetic flux density is based on a forward model ofthe sources, where each particle center is the center of a spherical harmonic expansion –4–anuscript submitted to
G-cubed up to a maximal order n , where n = 1 denotes a pure dipole potential, because mag-netic potentials contain no monopole contribution. In the test sample the individual par-ticles are spherically isolated in the sense that each particle is contained in a sphere, suchthat no two of these spheres intersect. This is an essential requirement to ensure thatthe spherical harmonic potentials are in principle uniquely defined (Fabian & de Groot,2018). In the following it is assumed that all sources are spherically isolated. If the in-dividual particles do not fulfill this condition, they have to be grouped, such that the re-sulting groups are spherically isolated, and the recovered potentials then represent thepotentials of these groups.For the forward calculation, the potentials of all sources are added and the verti-cal derivative of the combined potential in the scanning x − y -plane determines the to-tal vertical field component B z ( x, y, h ), where h is the scanning height of the sensor loop.The flux through the sensor is obtained by integrating this field component over the sen-sor area in the x − y -plane. By this procedure the sensor signal in the forward modelis represented by a linear combination of the spherical harmonic expansion coefficientsof all particles. If the number of measurements is larger than the number of these ex-pansion coefficients, the design matrix M of the system becomes over-determined for theexpansion coefficients, and a least-square fit of these coefficients can be obtained via thepseudoinverse of M .A single magnetic particle inside a volume V corresponds to a distribution of vol-ume charges λ ( r ) ( λ = 0 outside V ), where r is the position vector within the source.Its magnetic scalar potential at a location R outside the smallest sphere that contains V reads Φ( R ) = γ B (cid:90) V λ ( r ) | R − r | d r, (1)where γ B = µ M (4 π ) − and M is the particle magnetization. When the observationpoint R is far away from the magnetic source, i.e. R (cid:29) r , with R and r the magnitudesof the position vectors, the expansion of | R − r | − in terms of R − leads to the Carte-sian multipole expansion of Φ as –5–anuscript submitted to G-cubed γ − B Φ( R ) =( − (cid:20)(cid:90) V d r λ ( r ) r i (cid:21) ∂∂R i R (cid:124) (cid:123)(cid:122) (cid:125) dipole + ( − (cid:20)(cid:90) V d r λ ( r ) r i r j (cid:21) ∂ ∂R i ∂R j R (cid:124) (cid:123)(cid:122) (cid:125) quadrupole (2)+ ( − (cid:20)(cid:90) V d r λ ( r ) r i r j r k (cid:21) ∂ ∂R i ∂R j ∂R k R (cid:124) (cid:123)(cid:122) (cid:125) octupole + O (cid:0) r (cid:1) where R i and r i for i = 1 , , R and r , respectively,and repeated indexes follow Einstein’s summation convention. Again, the expansion startswith the dipole term because there are no magnetic monopoles. In Cartesian coordinates,all n th-order derivatives of R − have the form p ( R , R , R ) R − n − , (3)where p is a homogeneous n -th-order harmonic polynomial. The vector space spannedby these polynomials has a basis of 2 n +1 linearly independent elements as a subspaceof the space of all n th-order homogeneous polynomials. By defining a scalar product oftwo polynomials p, q as the average over the unit sphere by (cid:104) p, q (cid:105) = 14 π (cid:90) r =1 p ( r ) q ( r ) dS, (4)it turns out that the n -th-order derivative polynomials are not orthogonal. To correctthis problem, both the terms with derivatives and the terms with integrals are expressedas tensors. The derivative tensor is traceless for n > /R is harmonic out-side the sphere. The integral tensor defines the corresponding multipole coefficients ofthe charge distribution.The product of these two tensors can be transformed to the basis of real spheri-cal harmonics, which transforms the polynomials into a completely orthogonal set of ba-sis functions. The resulting multipole expansions can be written as γ − B Φ( R ) = (cid:88) n =1 2 n +1 (cid:88) α =1 Θ t ( n ) α Q α ( n ) ( R ) + O (cid:0) r (cid:1) , (5)where Θ t ( n ) α are the components of the traceless magnetic multipole tensor of rank n and Q α ( n ) ( R ) are the spherical harmonic polynomials that decay as R − ( n +1) . The here ap- –6–anuscript submitted to G-cubed plied mathematical formalism (Burnham & English, 2019; Applequist, 2002; Stone, 2013)is detailed in Appendix A and Section S8 of the Supplementary Material. The correspond-ing n -th-order multipole terms of the magnetic field B ( R ) = −∇ Φ( R ) decay propor-tional to R − ( n +2) . The first (dipole), second (quadrupole) and third (octupole) order con-tributions B ( n ) k , for n = 1 , , n , an approximation of the field can be constructed,where the field created by each particle at each measurement point is represented by its n ( n +2) independent multipole coefficients. Based on these coefficients, a L × ( n ( n + 2) K )forward matrix is constructed that models the magnetic flux from the K magnetic sourcesto multipole order n at each of the L measurement points i as B (scan) z | i = K (cid:88) j =1 B (particle) z | i,j = K (cid:88) j =1 (cid:104) P α (1) z | i,j · Θ t (1) α | j + P α (2) z | i,j · Θ t (2) α | j + P α (3) z | i,j · Θ t (3) α | j (cid:105) , (6)with P α ( n ) z | i,j = − ∂Q α ( n ) i,j /∂z as the derivative of the spherical harmonic polynomials forparticle j at measurement point i and Θ t ( n ) α | j the rank- n multipole tensor components ofparticle j . The polynomials for the z -component of the magnetic field B k are again or-thogonal. Because only B z is used in the inversion, this orthogonality implies that theSHE bases of each particle are uncorrelated, which should lead to numerically favorableproperties.The best-fit multipole coefficients of the sources are then computed via the pseu-doinverse of this rectangular design matrix, which corresponds to the linear least-squarefit of the measurement data, as long as L > n ( n + 2) K . To analyze the effectiveness of the multipole expansion technique, solutions for theSSM measurements reported in de Groot et al. (2018) are obtained by solving the in-version problem in the three areas depicted in Fig. 1b-d. To validate the calculations ap-plied to real samples, test problems for the inversion of dipole signals as a function ofgrain position, grain depth and field scan noise, have been developed and are describedin Section S1 of the Supplementary Material. –7–anuscript submitted to
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In de Groot et al. (2018) magnetic grains are modelled as aggregations of cubes witha constant magnetization and locations specified by microCT image analysis. This al-lowed to uniquely solve for homogeneous magnetizations in each of these spherically sep-arated grains. Unique source assignment of the potential-field signal in this kind of sys-tem is a well-posed inverse problem (Fabian & de Groot, 2018). The uniqueness is notrestricted to the dipole moments, but extends to the potential field of each grain, andthus to all spherical harmonic expansion coefficients. The essential constraint of the unique-ness theorem is that the complement of the source regions must be simply connected (Fabian& de Groot, 2018). This excludes the possibility to reconstruct signals from source re-gions inside other source regions, and makes it unfeasible to obtain detailed complex mag-netization structures like multi-domain structures. Yet, the multipole expansion of thepotential may suffice to distinguish between a finite number of physically possible localenergy minimum structures that can be modelled based on grain shape and mineralogy.The coefficients of the multipole expansion can be found for particles for which the small-est enclosing sphere is completely below the scan surface to ensure the validity of the far-field potential description (3) at the measurement points. These coefficients describe pointsources representing the magnetic particles. A natural choice for the origins of the mul-tipole expansions are the geometric centers of the particles outlined by their microCTdensity anomaly with respect to the surrounding matrix. Alternatively, the expansioncenter may be chosen as the center of the smallest sphere that contains the particle, butthe choice of the expansion center does not influence the reconstructed dipole coefficients.The three-dimensional image in Fig. 1a visualizes the relation between the scan-ning surface and the position of the magnetite grains for Area 1. The plots below, Fig. 1b-d, depict the original measurement data for three areas from the SSM scan together withthe particle boundaries as determined by microCT. The magnetic states in the magneticsources correspond to the initial state after preparation of the sample, CT measurementand cooling in the SQUID magnetometer, and are considered as random magnetizationstates without prior information in the inversion process. The row of images in Fig. 1e-g shows the modeled B z signals for the inversions of Area 1 to stepwise increasing multipole-expansion orders from n = 1 (dipole, left) to n = 3 (octupole, right). The bottom row,Fig. 1h-j, depicts the maps of the corresponding residuals.A first test for the consistency of the inversions is to compare the dipole momentsrecovered by inversions at different multipole orders. From the inverted dipole moments –8–anuscript submitted to G-cubed Ə Ɛ Ə Ə Ƒ Ə Ə ƒ Ə Ə l Ə Ɛ Ə Ə Ƒ Ə Ə l 0 u ; - Ɛ Ə Ɛ Ə Ə Ƒ Ə Ə l Ə Ɣ Ə Ɛ Ə Ə Ɛ Ɣ Ə 1 u ; - Ƒ Ə Ɛ Ə Ə Ƒ Ə Ə l Ə Ɣ Ə Ɛ Ə Ə Ɛ Ɣ Ə Ƒ Ə Ə 7 u ; - ƒ " " v 1 - m Ŋ &