Magnetic Anisotropy in Single Crystals: A Review

Abstract

Empirical relationships between magnetic fabrics and deformation have long served as a fast and efficient way to interpret rock textures. Understanding the single crystal magnetic properties of all minerals that contribute to the magnetic anisotropy of a rock, allows for more reliable and quantitative texture interpretation. Integrating information of single crystal properties with a determination whether or not mineral and magnetic fabrics are parallel may yield additional information about the texture type. Models based on textures and single crystal anisotropies help assess how the individual minerals in a rock contribute to the rock’s anisotropy, and how the individual anisotropy contributions interfere with each other. For this, accurate and reliable single crystal data need to be available. This review paper discusses magnetic anisotropy in single crystals of the most common rock-forming minerals, silicates and carbonates, in relation to their mineralogy and chemical composition. The most important ferromagnetic minerals and their anisotropy are also discussed. This compilation and summary will hopefully lead to a deeper understanding of the sources of magnetic anisotropy in rocks, and improve the interpretation of magnetic fabrics in future structural and tectonic studies.

1. Introduction

The preferred alignment of minerals in a rock can provide important information in geodynamic, structural and tectonic studies, e.g., to investigate transport or deformation processes. A fast and efficient way of characterizing this alignment is based on magnetic fabrics, e.g., anisotropy of magnetic susceptibility (AMS) or anisotropy of magnetic remanence (AMR), which are commonly used as proxies for rock textures [1,2,3,4,5,6,7]. Compared to direct texture-determining techniques such as U-stage and X-ray texture goniometry or electron backscatter diffraction, magnetic methods have the advantage that they provide a time-efficient integrated measure of the mineral alignment over a large volume and can capture several mineral types and a wide range of grain sizes. Because magnetic data acquisition and processing are fast, the number of samples that can be analyzed increases significantly compared to direct texture-determining techniques, thus allowing to characterize both small-scale and regional variations within a single study.

Traditionally, magnetic fabrics have been interpreted based on empirical relationships. For example, AMS has been shown to reflect the macroscopic foliation and lineation in many rocks, i.e., the maximum susceptibility ($k_1$) indicates lineation, and the minimum susceptibility ($k_3$) is normal to foliation [8,9,10,11,12,13,14,15,16]. However, the magnetic and mineral fabrics are not always parallel. In amphibolites, for example, $k_1$ can be deviated from the macroscopic lineation by up to 90°, even though both magnetic and mineral fabrics are defined by hornblende [17,18,19]. A second empirical relationship that is often used is that the degree of magnetic anisotropy increases linearly with strain [9,13,20,21,22,23,24,25,26,27,28]. The details of this relationship have to be established for each rock type separately, because it also depends on mineralogy [29,30]. For example, AMS-strain relationships exhibit different slopes in different regions of the Alpes Maritimes [21]. This observation is related to the fact that the degree of anisotropy intrinsic to the carrier mineral defines the upper limit of anisotropy that can be observed in a rock in the extreme case of perfect mineral alignment. Hence, observed anisotropy is controlled by both the degree of mineral alignment, as well as intrinsic mineral anisotropy, and mineralogy plays an important role in defining magnetic fabrics.

In the meantime, magnetic fabrics have been linked to crystallographic preferred orientation of rock-forming minerals such as phyllosilicates [9,31,32,33,34,35], or shape preferred orientation and distribution of magnetite [36,37,38,39]. Furthermore, quantitative models have been developed to predict anisotropy based on the mineral alignment [19,40,41,42,43,44]. The accuracy of calculating anisotropy based on crystallographic preferred orientation largely relies on well-defined single crystal properties as input parameters.

Single crystal magnetic anisotropy has been investigated since the 1850s [45], and attracted increased interest once the importance of mineralogy, and hence each mineral’s intrinsic properties, became evident in magnetic fabric studies. Early studies carefully linked magnetic anisotropy to crystallographic directions, often measuring spherical samples (thus avoiding any influence of shape anisotropy) on instruments similar to today’s torquemeters [45,46,47,48,49,50,51,52,53]. Many of these early studies focused on magnetic anisotropy in diamagnetic crystals, correcting for the susceptibility of the air surrounding the sample [47]. The same method was used to determine the anisotropy of organic molecules [54,55,56]. With the advent of low-field AMS methods, large datasets of single crystal properties could be generated [57,58,59,60]. However, this method has the disadvantage that it measures a superposition of both dia/paramagnetic contributions to the anisotropy (crystal itself), and the ferromagnetic anisotropy related to iron oxide exsolutions within the crystal. Therefore, high-field methods are preferred again in the most recent single crystal studies [61,62,63,64]

Because of their abundance, and their importance in defining fabrics, the magnetic anisotropy of phyllosilicates has been studied extensively. Numerous studies report that the minimum susceptibility of phyllosilicates is normal to their basal plane (001) [57,61,65,66,67,68,69,70]. These studies also show that the degree of anisotropy depends on mineralogy, i.e., biotite has a higher anisotropy than chlorite and muscovite. Further, it was demonstrated that the anisotropy of biotite increases more strongly with decreasing temperature than for other phyllosilicates [69,70]. Other minerals, e.g., olivine, amphiboles and pyroxenes, have been investigated, but yielded inconsistent results with regard to principal anisotropy directions, degree and shape of anisotropy [51,58,59,60,62,63,71,72,73,74,75,76,77,78]. Lagroix and Borradaile [60] attributed these inconsistencies to the presence of oriented ferromagnetic inclusions within the crystals. Even single crystals may contain iron oxides, which have a preferred orientation because they exsolved parallel to distinct crystallographic directions or planes [79,80,81]. These exsolutions can contribute to, or completely dominate the crystal’s low-field anisotropy [62,63,73]. To overcome this problem, techniques have been developed to separate the ferromagnetic contribution to anisotropy (inclusions) from the para- and diamagnetic components carried by the crystal itself, or to enhance the paramagnetic contribution [82,83,84,85,86,87]. Based on these techniques, it has been possible to characterize the isolated magnetocrystalline anisotropy of many common rock-forming minerals, which then relates to the crystal structure, chemical composition, and the oxidation state of iron [62,63,64,70,73,88].

In contrast to dia/paramagnetic minerals that possess solely magnetocrystalline anisotropy, ferromagnetic minerals (sensu lato) may possess shape anisotropy in addition [89]. In fact, shape and distribution anisotropy dominate in magnetite [37,38,90]. For pyrrhotite, both shape and magnetocrystalline anisotropy have been described, and anisotropy is field-dependent [91,92]. Hematite possesses magnetocrystalline anisotropy that is also field-dependent [93,94,95]. Note that although a small amount of ferromagnetic minerals may dominate the mean susceptibility of a crystal or rock, it does not necessarily dominate their anisotropy [7,30,32,96,97,98,99].

A solid understanding of the magnetic anisotropy of single crystals is an essential prerequisite for interpreting magnetic fabrics quantitatively. In particular when magnetic fabrics are used to infer mineral alignment, it is important to know how the magnetic anisotropy in a crystal is related to the crystal lattice and the composition of the crystal. This review paper discusses single crystal magnetic anisotropy for rock-forming minerals in the silicate and carbonate group, and also summarizes data on accessory minerals that are important in magnetic studies, i.e., iron oxides and sulfides.

2. Theory

2.1. Diamagnetic, Paramagnetic, and Ferromagnetic Anisotropy

Even what we call single crystals may consist of several anisotropy carriers, e.g., the silicate host and oxide exsolutions. A careful characterization of the magnetic anisotropy tensor will therefore separate out the diamagnetic, paramagnetic and ferromagnetic contributions to AMS. These individual contributions as well as different sub-populations of ferromagnetic grains can be separated based on a variety of methods, e.g., temperature-dependence, high-field methods, anisotropy of full or partial remanence, or by comparing in-phase and out-of-phase susceptibility [85,100,101].

Diamagnetic susceptibility is a property of all materials, and related to the orbital moments of the electrons. Diamagnetic anisotropy arises when the electron orbits cover different areas in different directions. For example, in pure calcite, the most negative susceptibility is parallel to [001], because the electron clouds of the CO₃²⁻ groups are aligned in the (001) plane [64]. Diamagnetic susceptibility and anisotropy are independent of field and temperature.

Paramagnetic susceptibility is associated with partially filled electron orbitals, and is particularly strong when many orbitals are only partially filled. In minerals, it is mainly Fe (and Mn, …) that causes paramagnetic susceptibility. Therefore, paramagnetic anisotropy is related to the site occupancy, distribution, concentration, and oxidation state of Fe. For example, the Fe in distorted octahedral sites in phyllosilicates and amphiboles causes a minimum susceptibility normal to the octahedral planes or bands [62,65,67,70]. Paramagnetic susceptibility and anisotropy are independent of field, but susceptibility increases with decreasing temperature according to the Curie-Weiss law.

Ferromagnetic susceptibility and anisotropy are caused when the magnetic moments of strongly magnetic ions (e.g., Fe) interact with each other. They are observed in e.g., iron oxides and iron sulfides. Ferromagnetic susceptibility and its anisotropy are independent of temperature in a first-order approximation. They can show field-dependence in low fields, and saturate in high fields.

2.2. Description of Magnetic Fabrics

Magnetic fabrics in minerals and rocks are generally represented by symmetric second-order tensors. Tensors can be determined for the diamagnetic, paramagnetic and ferromagnetic components of the anisotropy, or the superposition of all contributions. The eigenvalues, $k_1\ge k_2\ge k_3$, and eigenvectors of each tensor, referred to as principal susceptibilities and principal susceptibility directions, define the magnitude ellipsoid of the respective contribution [102]. Note that the susceptibility of some minerals can contain higher-order components; for example, a threefold symmetry has been described in the basal planes of some hematite and pyrrhotite crystals [92,93,103]. These higher-order components are not reflected by second-order tensors, but they can be analyzed, e.g., by torque magnetometry.

Susceptibility can be measured as full or deviatoric tensor. For full tensors, the average of the eigenvalues defines the mean susceptibility, $\frac{(k_1+k_2+k_3)}{3}=k_{mean}$. Therefore, they are often presented in their normalized form, with $(k_1+k_2+k_3)=3.$ Deviatoric tensors describe the deviation from mean susceptibility, so that $k_i$ of the deviatoric tensor equals $k_i-k_{mean}$ of the corresponding full tensor, and for the deviatoric tensor $k_1+k_2+k_3=0$. Many parameters have been defined to describe the degree and shape of anisotropy [3,104]. Only few of them are universally applicable to both full and deviatoric tensors. A selection of these will be used throughout this review paper: anisotropy degree will be described by $k^{\prime }=\sqrt{\left({\left(k_1-k_{mean}\right)}^2+{\left(k_2-k_{mean}\right)}^2+{\left(k_3-k_{mean}\right)}^2\right)/3}$ [105], and anisotropy shape by $U=\left(2\ast k_2-k_1-k_3\right)/\left(k_1-k_3\right)$.

2.3. Symmetry Constraints

Neumann’s principle [106,107] states that any physical property of a crystal must include all the symmetry elements of the crystal itself. In other words, tensorial properties must have equal or higher symmetry than the crystal lattice. This poses important constraints on the allowed eigenvector orientations for second-order tensors, and may require that several eigenvalues be equal, especially for high-symmetry minerals. For example, for cubic minerals $k_1=k_2=k_3$ and the susceptibility is isotropic. Hexagonal, tetragonal and trigonal minerals have $k_1=k_2$ or $k_2=k_3$, and the unique axis ($k_3$ or $k_1$) must be aligned with the crystallographic [001] axis. The eigenvalues can differ for orthorhombic crystals ($k_1\ge k_2\ge k_3$), but each eigenvector has to be aligned with a crystallographic axis. Monoclinic crystals must have one eigenvector parallel to the [010] axis. There are no symmetry constraints on triclinic crystals.

Whether or not Neumann’s principle is fulfilled can give a first estimate on the data quality when measuring single crystal magnetic anisotropy. Similarly, Neumann’s principle can help evaluate whether previously reported anisotropy tensors are accurate. When symmetry constraints are violated, this may indicate that a superposition of several components of anisotropy was measured, the data is affected by noise, or the crystal may have been misoriented.

3. Measurement Methods

Because susceptibility is represented by a symmetric second-order tensor, at least 6 independent measurements are needed to describe it. More measurements will allow for assessing data quality, and defining confidence intervals for both the eigenvalues, and the directions of the eigenvectors. Common measurements schemes determine (1) directional susceptibilities for a predefined set of orientations, or (2) susceptibility differences in three mutually perpendicular planes. These methods in general yield full and deviatoric tensors, respectively, but the latter can be integrated with at least one directional susceptibility measurement to calculate the full tensor. Measuring deviatoric tensors is more accurate than measuring full tensors [102,108].

The distinct field- and temperature-dependencies of ferromagnetic, paramagnetic and diamagnetic susceptibilities can be used in order to enhance certain contributions, or separate the individual components of the anisotropy. For example, measuring samples at liquid nitrogen temperature (77 K) instead of room temperature enhances the paramagnetic contribution to the anisotropy. High-field measurements, conducted on a torquemeter or vibrating sample magnetometer, allow for separating the ferromagnetic from the para/diamagnetic anisotropy components. High-field torque measurements in combination with different temperatures allow for full separation of the ferromagnetic from the paramagnetic from the diamagnetic sub-fabrics [82,83,84,85,86,87,109]. Additionally, several methods have been developed to assess the anisotropy of remanence carriers [101,110,111].

4. Single Crystal Properties of Silicate Minerals

The most common rock-forming minerals belong to the silicate group. Depending on their structure, defined by the bonding between different SiO₄⁴⁻ tetrahedra, silicates are categorized as phyllosilicates (sheets), nesosilicates (isolated), inosilicates (single and double chains), cyclosilicates (circles) and tectosilicatets (3D framework). The magnetocrystalline anisotropy of minerals from these silicate groups will be described in this chapter.

4.1. Phyllosilicates: Mica and Chlorite

The main characteristic of phyllosilicates are their tetrahedral sheets, Si₂O₅²⁻. Within these sheets, each SiO₄⁴⁻ is linked to three neighbors with its three basal oxygen atoms. They are stacked with octahedral sheets and interlayer cations, and different stacking sequences define the mica, chlorite, or clay minerals. Most of these minerals are flaky, with a prominent cleavage parallel to the sheets, which makes them good tectonic markers. Magnetic anisotropy has been characterized for micas and chlorite (Figure 1a). The iron atoms are located within the octahedral sheets [112,113]. Small distortions of these octahedral sites lead to an easy-plane anisotropy with the unique axis normal to the plane of the sheets, the so-called basal plane, (001) [65]. This leads to a minimum susceptibility $k_3$ normal to the basal plane in biotite, phlogopite, muscovite and chlorite. If there is any anisotropy within the basal plane, it is negligible. Therefore, principal susceptibility directions are usually reported with respect to the orientation of the sheets, without determining the exact orientation of [100] and [010] [57,61,65,66,67,68,69,70]. The observed magnetic anisotropy has a higher symmetry than the monoclinic crystal lattice, which is in accordance with Neumann’s principle.

The degree of anisotropy is higher in biotite than other sheet silicates. It also shows the strongest temperature dependence in biotite [69]. These observations were explained with varying Fe contents, and the fact that at 77 K, the Fe atoms interact within the basal planes of biotite, but not between them. For the other phyllosilicate minerals, the Fe concentration is too small for interactions to occur [70]. The increase in anisotropy degree at 77 K compared to room temperature, ${p^{\prime }}_{77}=\frac{k{\prime }_{77K}}{k{\prime }_{RT}}$, is 11.1 to 12.2 for biotite, and 6.6 to 8.7 for muscovite, chlorite, and phlogopite [70].

4.2. Nesosilicates: Olivine, Chloritoid

Isolated SiO₄⁴⁻ tetrahedra bonded by interstitial cations define the nesosilicates. Iron is present as interstitial cation [114]. Nesosilicates whose magnetic anisotropy has been characterized, include olivine and chloritoid.

Chloritoid occurs in Al-rich metapelitic rocks. Although chloritoid is a nesosilicate, its structure resembles that of sheet silicates, because the isolated tetrahedra are arranged in layers, with octahedral layers in-between. Similar to phyllosilicates, the minimum susceptibility is normal to the layers, and the anisotropy within this basal plane is negligible (Figure 1b) [115]. Chloritoid possesses monoclinic crystal symmetry, and the anisotropy has higher symmetry, consistent with Neumann’s principle. At this time, no statement can be made about the relationship between iron concentration and anisotropy parameters, because chemical compositions have not been reported.

Olivine makes up large parts of the Earth’s mantle. Its magnetic anisotropy has been studied repeatedly; however, conflicting results have been reported [71,72,73,74,78,116,117]: The maximum susceptibility $k_1$ has been attributed to the crystallographic [100] or [001] axes, $k_2$ to [100], [010], or [001], and $k_3$ to [100], [010], or [001]. Note that all of these options are consistent with the symmetry requirements of olivine’s orthorhombic crystal system. Most studies agree that $k_1$ is parallel to [001] [71,73,74,78,116], and Biedermann, Pettke, Reusser and Hirt [73] have shown that the orientation of $k_2$ and $k_3$ at room temperature depends on the Fe content (Figure 1c). The minimum susceptibility aligns with [100] for 3–5 wt.% FeO, and with [010] for 6–10 wt.% FeO. The latter orientation was also observed in weathered olivine with 16 to 18 wt.% FeO. At 77 K, $k_3$ is parallel to [010], independent of iron content, and $k_1$ is parallel to [001] except for the weathered olivine, where it aligns with [100]. The degree of anisotropy increases with Fe content both at room temperature and at 77 K, and the shape changes from highly prolate to less prolate or oblate. At 77 K, the anisotropy degree is 6.0 to 8.4 times stronger than at room temperature (${p^{\prime }}_{77}$).

4.3. Inosilicates: Pyroxenes and Amphiboles

Inosilicates consist of either single chains (pyroxenes, Si₂O₆⁴⁻) or double chains (amphiboles, Si₈O₂₂¹²⁻) of SiO₄⁴⁻ tetrahedra. In pyroxenes, cations can occupy the M1 sites located in the octahedral chains in-between the tetrahedral chains, or the M2 sites which have an irregular six- to eightfold coordination. In amphiboles, the octahedral bands contain three sites, M1, M2 and M3, which are characterized by variable distortions of the octahedra. Polyhedral M4 sites have at least a six-fold coordination, and A sites are coordinated by 12 oxygens. Depending on the size of the cations in their M2 or M4 sites, respectively, pyroxenes and amphiboles are either monoclinic or orthorhombic [118,119].

In clinopyroxenes, Fe²⁺ is mainly located in M1 sites, whereas in orthopyroxenes, it prefers the M2 sites. Fe³⁺ is found in the M1 sites for both clino- and orthopyroxenes. The different site occupancy of Fe²⁺ leads to distinct magnetic anisotropies. Most clinopyroxenes have their $k_2$ aligned with [010], and $k_1$ and $k_3$ are in the [100]-[001]-plane (Figure 1d) [63,75]. Note that some crystals have one principal susceptibility axis close to, but not parallel to [010] [60]. This is not allowed by crystal symmetry constraints, and highlights the importance of separating subfabrics. In diopside and augite, $k_1$ is at a 45° angle to [001]. On the contrary, in aegirine, $k_1$ is parallel to [001], $k_2$ parallel to [010], and $k_3$ parallel to (100). Aegirine contains mainly Na and Fe³⁺ rather than Ca and Fe²⁺, which explains its different magnetic anisotropy. No consistent behavior was observed in spodumene, possibly due to the weak susceptibility and anisotropy of these crystals [63]. The degree of anisotropy generally increases with iron content, and is additionally affected by the Fe²⁺/Fe³⁺ ratio. Namely, aegirine has a lower degree of anisotropy than augite, because Fe²⁺ contributes to anisotropy, but Fe³⁺ is expected to be isotropic based on their electron structure [63]. The values for ${p^{\prime }}_{77}$ cover the range from 2.7 to 20.8, where the extremes on both ends are likely due to unseparated diamagnetic components, or a low signal-to-noise-ratio.

Due to their higher symmetry, orthopyroxenes need to have each of their principal susceptibilities parallel to a crystallographic axis, which again illustrates that subfabrics need to be separated [60]. The maximum susceptibility has been reported parallel to [010] [76], or [001] [60,63], and $k_3$ parallel to [100] [76] or [010] [63] (Figure 1e). This seeming inconsistency may be related to different iron contents. The degree of anisotropy increases with iron content, and $U$ appears to change from prolate to oblate. ${p^{\prime }}_{77}$ varies between 7.3 and 8.6.

In amphiboles, iron can occupy the octahedral M1-M3 sites (clinoamphiboles), or the M4 site (orthoamphiboles). Whether Fe²⁺ prefers the M1, M2, or M3 sites, depends on the exact composition of each amphibole. The smaller Fe³⁺ is preferentially located in M2. These differences again result in distinct magnetic anisotropies for different minerals of the amphibole group. Initial studies on amphibole anisotropy reported maximum susceptibilities either close to [001] or [010], and minimum susceptibilities parallel to [100] or [010] [51,59,60,75]. In the meantime, amphibole magnetic tensors have been determined in relation to mineral group: $k_1$ is parallel to [010], and $k_3$ normal to (100) for actinolite, ferroactinolite, pargasite, and hornblende. In richterite, $k_2$ is parallel to [010], and $k_1$ and $k_3$ lie in the [100]-[001]-plane (Figure 1f). Note that both sets of directions are consistent with monoclinic crystal symmetry. Results for tremolite are inconsistent due to a low signal-to-noise-ratio. Gedrite, the only orthoamphibole that has been characterized, displays $k_1$ parallel to [001], $k_2$ parallel to [010], and $k_3$ parallel to [100] (Figure 1f) [62]. The degree of anisotropy generally increases with Fe content. Additionally, the degree of anisotropy is slightly higher for (ferro)actinolite compared to the other amphiboles, which may be related to different site preference of Fe²⁺, or oxidation state [62]. ${p^{\prime }}_{77}$ covers the range from 5.3 to 13.4.

4.4. Tectosilicates: Quartz and Feldspar

Tectosilicates are made up of infinite three-dimensional frameworks of SiO₄⁴⁻ tetrahedra, where each oxygen is shared by two tetrahedra. The most important tectosilicates in rocks are quartz and feldspar. In its ideal composition, quartz contains only Si and O. In feldspar, Si is partially replaced by Al, and cations are incorporated into the holes of the framework in order to maintain charge balance. Despite their abundance, only few studies discuss the magnetic anisotropy of quartz or feldspar [47,50,51,52,120,121], because they are weakly magnetic, and their contribution to anisotropy is outweighed by small amounts of iron-bearing mafic minerals [121].

The few data available for quartz agree that the most negative susceptibility is aligned with [001], and susceptibility is isotropic in the (001) plane (Figure 1g). This is in accordance with the trigonal symmetry of quartz.

Feldspars can have positive or negative susceptibility, depending on how much their real composition deviates from the ideal formula. Interestingly, even in crystals with positive mean susceptibility, the magnetic anisotropy is dominated by a diamagnetic component with the maximum (or least negative) susceptibility close to [001], and minimum (or most negative) sub-parallel to [010] (Figure 1h). This is related to a preferred plane for electrons along the bases of the SiO₄ or AlO₄ tetrahedra in the (010) plane [121]. Because feldspars can be monoclinic or triclinic, symmetry may require that one principal susceptibility is parallel to [010], or there may be no symmetry constraints. Hence, these results are consistent with Neumann’s principle. No clear relationships were observed between iron content and anisotropy parameters, likely because iron is not always present in the crystal structure, but rather in a variety of inclusions and/or exsolutions.

5. Single Crystal Properties of Carbonate Minerals

Carbonates are the second common group of rock-forming minerals. Similar to feldspars, many carbonate minerals can display a diamagnetic or paramagnetic mean susceptibility, depending on their exact composition. Two categories of anisotropy tensors have been described for carbonates: (1) crystals with low iron content (<~1000 ppm) have their minimum (most negative) susceptibility parallel to the [001] axis, and a highly oblate anisotropy shape; (2) in crystals with higher iron contents, $k_1$ is aligned with [001], and the crystals possess highly prolate anisotropy (Figure 1i) [6,50,53,64,88,120,122,123,124]. Most carbonate minerals (calcite, dolomite, magnesite, siderite, rhodochrosite) have trigonal symmetry, which requires that the unique axis of any second-order tensor property is parallel to [001], and that the property is isotropic within the (001) plane. Some carbonates (aragonite, cerussite, witherite) have orthorhombic symmetry, and azurite is monoclinic. Hence, the principal directions and anisotropy shapes of these minerals are not as strictly constrained. The relationship between degree of anisotropy and Fe content is not straightforward; at low Fe contents, anisotropy decreases with Fe, then increases linearly with Fe once the Fe concentration reaches >1000 ppm [64,88,124]. This relationship is a direct consequence of a composite anisotropy made up of two subfabrics: (1) a diamagnetic component related to preferred planes of electron movement within (001) with its most negative susceptibility parallel [001], and (2) a paramagnetic anisotropy carried by Fe²⁺ cations in the crystal structure, with $k_1$ along [001]. Anisotropy is high when one contribution dominates, and low when they have similar strength so that they cancel each other [64,88].

6. Ferromagnetic Minerals

Ferromagnetic minerals are present as primary or secondary accessory minerals in many rocks, and in some places, they form ore bodies. In rocks, the ferromagnetic minerals occur as individual grains, as unoriented inclusions, and/or as oriented exsolutions within the host silicates. Magnetic fabrics in ores are defined by the shape or crystallographic preferred orientation and distribution of ferromagnetic grains. Of particular interest in fabric studies is when the ferromagnetic accessory grains record different stages of deformation than the bulk rock, or when several sub-populations of ferromagnetic grains record different deformation stages.

Whereas magnetite does possess a weak magnetocrystalline anisotropy, it mainly contributes to anisotropy in rocks through shape and distribution anisotropy [36,37,38,39]. Because of self-demagnetization caused by the strong magnetization in magnetite, sufficiently large grains of magnetite will have their $k_1$ parallel to the grain elongation, and $k_3$ parallel to the shortest direction of the grain (Figure 1j). Note that single domain magnetite particles display inverse fabrics with $k_3$ parallel to the long grain axis, along which they are already magnetized [125]. The susceptibility and anisotropy of pure magnetite are field-independent, but the susceptibility and anisotropy of titanomagnetite display a field-dependence [126,127].

Magnetocrystalline anisotropy has been described in both hematite, hemo-ilmenite, and pyrrhotite (Figure 1k,l) [14,91,92,93,94,103,128,129,130,131]. These minerals display a magnetocrystalline anisotropy with $k_3$ normal to the basal plane, i.e., parallel to (001). Some hematite and pyrrhotite crystals show additional higher-order contributions to anisotropy within the basal plane [103,131]. Furthermore, shape effects have been observed in pyrrhotite [91]. Note that the anisotropy of hematite and pyrrhotite is field-dependent [93,94,95,132].

7. Application of Single Crystal Magnetic Anisotropy to Fabric Interpretation

Even based solely on empirical relationships between principal susceptibility directions or anisotropy degree and strain, magnetic fabrics have been a powerful tool in investigating tectonic and geodynamic processes [1,2,4]. Today, carefully characterized single crystal susceptibility anisotropies, in combination with methods to average tensor properties [43,133], allow for a deeper and quantitative understanding of magnetic fabrics. For example, it is now possible to model individual mineral’s contributions to anisotropy, and to investigate how they interfere with each other [19,34,35,40,42]. These models can help interpret complex fabrics carried by multiple components. Additionally, based on a thorough understanding of single crystal properties, we can now explain why the magnetic fabrics in most rocks are parallel to macroscopic fabrics, and why this is not always true. Some amphibolites, for example, display a maximum susceptibility direction parallel to macroscopic lineation, whereas in other amphibolites it is deviated up to 90°. Previously described as anomalous fabric [17], this observation is now understood as arising from the interplay of the amphibole single crystal anisotropy with texture types, i.e., the grouping of each crystallographic axis with respect to the grouping of other axes [19]. Predicting magnetic anisotropy based on single crystal properties and either simplified model textures or the measured texture of a rock will facilitate the interpretation of magnetic fabrics in future tectonic and structural studies.

8. Conclusions and Outlook

Magnetocrystalline anisotropy is directly linked to the properties of the crystal lattice, e.g., the structure of the electron cloud (for diamagnetic crystals), or the site occupancy, arrangement, and oxidation state of Fe (for paramagnetic and ferromagnetic crystals). Note that some single crystals contain ferromagnetic inclusions, and their contribution to anisotropy needs to be removed in order to describe the susceptibility of the silicate or carbonate alone. Single crystal magnetic anisotropy is constrained by crystal symmetry, because all tensorial properties need to comply with the symmetry of the crystal itself. Therefore, each mineral group has a distinct magnetic anisotropy. In addition, the orientation of principal directions, and the degree and shape of anisotropy generally vary with Fe content. Ferromagnetic minerals may possess shape anisotropy in addition to magnetocrystalline anisotropy, and their anisotropy can contain higher-order components, and be field-dependent.

The AMS is linked to the crystal structure and symmetry, Fe concentration, site occupancy, and oxidation state. While these properties may vary between geological settings, the susceptibility anisotropy per se directly reflects these crystal properties, and therefore does not depend on e.g., locality. Thus, the relationship between crystal orientation and AMS is universal. Magnetic fabrics in rocks, in turn, are directly related to the crystallographic preferred orientation of its constituent minerals. This relationship holds whether or not the magnetic fabric in a rock complies with empirical AMS-strain relationships. Thus, knowing single crystal anisotropies allows establishing a universally applicable, quantitative relationship between AMS and crystallographic preferred orientation, which will lead to more solid and reliable interpretations, and thus greatly benefit future applications of magnetic fabric in studies of e.g., deformation or transport processes.