# Anisotropy of Out-of-Phase Magnetic Susceptibility and Its Potential for Rock Fabric Studies: A Review

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Physical Principles of Out-Of-Phase Magnetic Susceptibility

_{o}cos(ωt)

_{o}is amplitude, ω is angular frequency, and t is time, and the magnetic response is measured, most conveniently represented by magnetization. In diamagnetic, paramagnetic, and many ferromagnetic minerals, the magnetization also varies sinusoidally; because it is in phase with the applied field and the magnetization-to-field ratio, then it maintains a constant value that represents the susceptibility. In some ferromagnetic and/or electrically conductive minerals, however, the magnetization is not in phase with the applied field, but it lags behind the field (Figure 1a) and may therefore formally resolve into in-phase and out-of-phase components (Figure 1b). The ipMS and opMS are defined correspondingly:

_{o}k′cos(ωt) + H

_{o}k″sin(ωt)

_{o}is amplitude of the magnetizing field, ω is angular frequency, t is time, k′ and k″ are ipMS and opMS components, respectively, related as tan δ = k″/k′, with δ being referred to as the phase angle or simply phase.

#### 2.2. Effect of Mineral Fractions with Exclusive In-Phase Response on the Whole Rock Phase Angle

_{w}= c

_{d}k

_{d}+ c

_{p}k

_{p}+ c

_{f}k

_{f}= K

_{d}+ K

_{p}+ K

_{f}

_{w}is the whole rock susceptibility, k

_{d}, k

_{p}, k

_{f}are susceptibilities of dia-, para-, and ferromagnetic fractions, respectively, and c

_{d}, c

_{p}, c

_{f}are the respective percentages; K

_{d}, K

_{p}, K

_{f}are called the respective contribution susceptibilities. Analogously, we can introduce another susceptibility resolution:

_{w}= K

_{ip}+ K

_{mix}

_{ip}is the contribution susceptibility of the mineral fraction that possesses only the in-phase response (its opMS is zero) and K

_{mix}is the contribution susceptibility of the mineral fraction whose opMS is non-zero. Re-writing this equation in terms of in-phase and out-of-phase contribution susceptibilities yields:

_{w}= K

_{ip}+ K′

_{mix}

_{w}= K″

_{mix}

_{w}= k″

_{w}/k′

_{w}= tan δ

_{mix}/(1 + r)

_{mix}= K″

_{mix}/K′

_{mix}and r = K

_{ip}/K′

_{mix}. It is obvious that while the whole rock in-phase contribution susceptibility is controlled by all mineral fractions present in the rock, the out-of-phase contribution susceptibility is solely controlled by the fraction whose opMS is non-zero. However, the phase angle is affected by both the fractions present. Figure 2 shows the tan δ

_{w}vs. tan δ

_{mix}plot for several r ratios, indicating that the increasing amount of the fraction with only in-phase response results in decreasing the whole rock phase angle.

#### 2.3. Accuracy of opAMS Determination

## 3. Physical Mechanisms That Produce Out-Of-Phase Susceptibility

- (1)
- viscous relaxation,
- (2)
- electrical eddy currents (induced by AC field in conductive materials),
- (3)
- weak field hysteresis (non-linear and irreversible dependence of M on H).

#### 3.1. Viscous Relaxation

_{sd}= 2M

_{s}/3H

_{k}, β = K

_{a}V/k

_{B}T and ω = 2πf; M

_{s}is saturation magnetization, H

_{k}is microscopic coercivity related to macroscopic coercivity (H

_{c}) as H

_{k}= 2.09H

_{c}[16], K

_{a}is the aniso- tropy constant, V is the particle volume, k

_{B}is the Boltzmann constant, T is the absolute temperature, τ

_{o}≈ 10

^{–10}s is a time constant, and f is the operating frequency.

_{a}= 2.5 × 10

^{4}J/m

^{3}, see [17]). Initially, ipMS almost linearly increases with grain size, while the opMS is effectively zero. After reaching the maximum value, the ipMS acutely decreases to SSD susceptibilities. The peak ipMS decreases with increasing frequency and shifts towards the smaller particle volumes. The opMS increases, creating a bell-like curve and subsequently dropping to effectively zero. The peak opMS also decreases with increasing frequency and shifts towards the smaller particle volumes. Consequently, whereas the ipMS is non-zero in the entire interval of particle volumes considered and therefore affected by the particles of relatively wide grain-size interval, the opMS is non-zero and dominantly affected by the particles of much narrower interval.

#### 3.2. Electrical Eddy Currents

_{0}is permeability of surrounding medium (air or rather matrix, in the context of measurement of magnetic susceptibility), and ω is angular frequency of the applied field.

^{7}S/m, ipMS is only slightly negative at frequencies less than 2 × 10

^{3}Hz; thereafter, it decreases substantially with increasing frequency. opMS is positive and increases with increasing frequency until 10.5 kHz, and then decreases but remains positive. In smaller and less conductive spheres, the basic trends are retained, but are less expressive and shifted to higher frequencies. Figure 6b shows the phase angle (δ) of the same sphere. At low frequencies (f~100 Hz), the phase angle δ = 90°, then increases with increasing frequency, reaching δ = 180° at very high frequencies when eddy currents are confined to a thin surface layer of the sphere and create a magnetic moment that is antiparallel to the applied field.

#### 3.3. Weak Field Hysteresis

^{2}

## 4. Examples of Geological Applications

#### 4.1. Instrumentation and Data Processing

_{m}= (K

_{1}+K

_{2}+ K

_{3})/3

_{1}/K

_{3}

_{2}− η

_{1}− η

_{3})/(η

_{1}− η

_{3}) = 2lnF/lnP − 1

_{1}≥ K

_{2}≥ K

_{3}are the principal susceptibilities, η

_{1}= lnK

_{1}, η

_{2}= lnK

_{2}, η

_{3}= lnK

_{3}, and F = K

_{2}/K

_{3}. The parameter K

_{m}is called the mean susceptibility and characterizes the qualitative and quantitative content of magnetic minerals in a rock. The parameter P, called the degree of AMS, indicates the intensity of the preferred orientation of magnetic minerals in a rock. The parameter T, called the shape parameter, characterizes the symmetry or shape of the AMS ellipsoid. If 0 < T < +1, the AMS ellipsoid is oblate (the magnetic fabric is planar); T = +1 means that the AMS ellipsoid is rotationally symmetric (uniaxial oblate). If −1 < T < 0, the AMS ellipsoid is prolate (the magnetic fabric is linear); T = −1 means that the AMS ellipsoid is uniaxial prolate.

#### 4.2. Fabric of Ultrafine Magnetic Particles in Loess/Palaeosol Sequences—Viscous Relaxation

^{−4}, and the opMS is about an order lower, with both showing virtually no variations with the measuring field. The ipMS possesses relatively high frequency dependence; ipMS at 15,616 Hz is up to 10% lower than at 976 Hz. In addition, the percentage loss of susceptibility between 976 and 15,616 Hz increases with ipMS (Figure 9a). This increase can be traditionally (since [40]) interpreted as resulting from creation of new SP particles in soil layers during pedogenesis. Consequently, the opMS is evidently due to viscous relaxation and indicates preferably oriented viscous particles in the transition between SP and SSD states (cf. [13]). The ipAMS, by comparison, is controlled not only by these grains, by also by the MD, SSD, and SP grains, and by paramagnetic grains.

#### 4.3. Preferred Orientation of Graphite in Graphite-Bearing Rocks and Graphite Ores—Eddy Currents

^{–5}, compared to the ipMS, which is an order-of-magnitude higher. Both opMS and ipMS are independent with respect to the intensity of the magnetizing field (Figure 10a,b). The ipMS shows virtually no dependence on the operating frequency, whereas the opMS is strongly frequency dependent (not shown in the figure).

_{1}and K

_{3}axes are relatively tightly grouped. The ipAMS foliations dip moderately, being similarly oriented as the metamorphic schistosity (Figure 11d). The ipAMS lineations plunge and lie near the mesoscopic lineation. The degree of opAMS is very high and the opAMS fabric ranges from neutral to linear (Figure 11a). K

_{1}and K

_{3}axes are switched between ipAMS and opAMS (Figure 11c).

#### 4.4. Preferred Orientation of Pyrrhotite in Eclogite with Complex Magnetic Mineralogy—Weak Field Hysteresis

#### 4.5. Ferromagnetic Mineral Fabric Masked by Paramagnetic Fraction in Whole-Rock AMS in Sedimentary Rocks

## 5. Discussion

_{a}≤ N

_{b}≤ N

_{c}(e.g., [48,49]). The apparent complex susceptibility of such a grain measured in a weak field is a second rank tensor controlled by the internal susceptibility and demagnetizing factor (e.g., [36,50,51,52]).

_{j}is the measured (apparent) principal susceptibility, N

_{j}is the principal value of the tensor of the demagnetizing factor, and κ is the isotropic internal susceptibility. The degree of AMS is then:

^{2}. This approach is probably applicable to titanomagnetite, whose phase angle is very low; the maximum phase angle of the high-Ti magnetic fraction extracted from a volcanic rock is 3.6°. If there was a mineral with isotropic internal susceptibility and much higher phase angle, one would have to apply the full Equations (18) and (19) with no simplification.

## 6. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Definition of in-phase and out-of-phase components of AC susceptibility: (

**a**) magnetization delayed behind the field, and (

**b**) delayed magnetization resolved into in-phase and out-of-phase components. For the sake of better illustration, susceptibility is considered larger than 1, although it is much less than 1 in common rocks.

**Figure 2.**Model of the effect of the fraction with only in-phase response (having positive K

_{ip}) on the whole rock phase angle. δ

_{w}is the whole rock phase angle, δ

_{mix}is the phase angle of the fraction with non-zero opMS, r = K

_{ip}/K′

_{mix}. Adapted from [9].

**Figure 3.**Relative errors in ipMS (ϑ′, red symbols) and opMS (ϑ″, blue symbols) according to the rock phase angle (δ) and relative error in measuring directional susceptibility ϑ. Adapted from Hrouda et al. [1].

**Figure 4.**Theoretical (

**a**,

**b**) and experimental (

**c**,

**d**) variations of ipMS and opMS with magnetic particle volume and operating frequency. (

**a**) Theoretical variations of ipMS (κ′) and opMS (κ″) for a single grain (in brackets are frequencies rounded to integers in kHz). (

**b**) Theoretical variations of ipMS (κ′) and opMS (κ″) with operating frequency for a population of magnetic particles showing lognormal distribution of particle volumes (for details see the text). Two full circles on the κ′ curve correspond to 976 and 15,616 Hz frequencies. Experimental field and frequency dependence of total (not corrected for volume) opMS (

**c**) and ipMS (

**d**) in a ferromagnetic liquid consisting of ultrafine magnetic particles in mineral oil. (

**a**,

**b**) adapted from [9]; (

**c**,

**d**) original measurements for this paper.

**Figure 5.**Schematic orientations of maximum and minimum susceptibilities with respect to grain shape or crystal lattice: (

**a**) magnetite, titanomagnetite, and maghemite grains with shape anisotropy; (

**b**) graphite with magnetocrystalline anisotropy, copper with shape anisotropy; (

**c**) titanomagnetite with shape anisotropy, pyrrhotite and hematite with magnetocrystalline anisotropy.

**Figure 6.**(

**a**) Model of the ipMS (κ′, in blue) and opMS (κ″, in red) variation of a sphere with zero DC susceptibility according to operating frequency, electrical conductivity (σ), and sphere diameter (r); (

**b**) phase angle in the same sphere. Legend: solid lines r = 1 mm, σ = 6 × 10

^{7}S/m, dash lines r = 0.1 mm, σ = 6 × 10

^{7}S/m, dash-and-dot lines r = 1 mm, σ = 6 × 10

^{6}S/m. Calculated using Equation (7). (

**c**) opMS vs. field and frequency and (

**d**) ipMS vs. field and frequency in copper (for more details see [25]); opMS and ipMS are in terms of total susceptibility (not normalized against volume).

**Figure 7.**Relationships between opMS or ipMS and the magnetizing field and frequency in a specimen of high-Ti titanomagnetite (

**a**,

**b**) and relationships between opMS or ipMS and the magnetizing field in several specimens of rocks with pyrrhotite (

**c**,

**d**). Adapted from [1].

**Figure 8.**Simplified geological map of Central Europe with study localities indicated. Compiled using EGDI maps.

**Figure 9.**opAMS and ipAMS in loess/palaeosol sequence in Bulhary locality: (

**a**) percentage loss of susceptibility (K

_{FD}= 100 (k

_{976}− k

_{15,616})/k

_{976}) vs. ipMS, (

**b**) P-T plot, (

**c**) opAMS lineations (Max) and opAMS foliation poles (Min), (

**d**) ipAMS lineations (Max) and ipAMS foliation poles (Min) in equal-area projection on lower hemisphere. Adapted from [19].

**Figure 10.**Variation in opMS and ipMS in a few specimens of graphite-bearing rock from Městský vrch locality according to the intensity of magnetizing field (

**a**,

**b**) and variation of ipMS with temperature (

**c**,

**d**). Adapted from [26].

**Figure 11.**The opAMS and ipAMS fabrics in the locality of Městský vrch. (

**a**) P-T plot for opAMS, (

**b**) P-T plot for ipAMS, (

**c**) directional elements for opAMS, (

**d**) directional elements for ipAMS. Symbols: Max—magnetic lineation, Min—magnetic foliation pole, S—metamorphic schistosity, A—mesoscopic lineation. Equal area on lower hemisphere. Adapted from [26].

**Figure 12.**Variation of opMS with magnetizing field (

**a**), variation of ipMS with field (

**b**), variation of ipMS with temperature (

**c**) (in red—heating curve, in blue—cooling curve) in the locality of Hutě.

**Figure 13.**Magnetic anisotropy P-T plot (

**a**) and orientations of magnetic foliation poles and magnetic lineations (

**b**) in eclogite of the locality of Hutě. Yellow symbols denote the opAMS fabric, other symbols belong to ipAMS. In (

**b**), equal-area projection on lower hemisphere.

**Figure 14.**Orientations of magnetic foliation poles (Min) and magnetic lineations (Max) in Skorušina Hills sedimentary rocks: (

**a**) ipAMS, (

**b**) opAMS, (

**c**) AMR. Tilt corrected coordinate system, equal area projection on lower hemisphere. Adapted from [46].

**Figure 15.**Variation in grain degree of ipAMS with magnetic particle volume and operating frequency in ultrafine particles.

**Figure 16.**Variations in the K

_{xy}/K

_{z}ratio of the hematite single crystals (denoted A, B, C, D) with magnetizing low-field: (

**a**) opMS, (

**b**) ipMS.

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**MDPI and ACS Style**

Hrouda, F.; Chadima, M.; Ježek, J.
Anisotropy of Out-of-Phase Magnetic Susceptibility and Its Potential for Rock Fabric Studies: A Review. *Geosciences* **2022**, *12*, 234.
https://doi.org/10.3390/geosciences12060234

**AMA Style**

Hrouda F, Chadima M, Ježek J.
Anisotropy of Out-of-Phase Magnetic Susceptibility and Its Potential for Rock Fabric Studies: A Review. *Geosciences*. 2022; 12(6):234.
https://doi.org/10.3390/geosciences12060234

**Chicago/Turabian Style**

Hrouda, František, Martin Chadima, and Josef Ježek.
2022. "Anisotropy of Out-of-Phase Magnetic Susceptibility and Its Potential for Rock Fabric Studies: A Review" *Geosciences* 12, no. 6: 234.
https://doi.org/10.3390/geosciences12060234