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Magnetic susceptibility of natural rocks and ores is important in many applications. In a few rock types magnetic susceptibility is independent of the direction in which a weak magnetic field is applied. Such rocks are magnetically isotropic. In most rock types, however, the magnitude of magnetic susceptibility in a constant weak field depends on the orientation of the magnetic field applied. Such rocks are magnetically anisotropic and such directional variation in magnetic susceptibility with these rocks is termed as anisotropy of magnetic susceptibility (AMS). Although attempts have been made on describing AMS using mathematical models, there is still a need to present a more consistent and united mathematical process for AMS. This paper presents a united AMS model by rationalizing the existing pieces of different AMS models through a consistent approach. A few examples of AMS from some types of natural rocks and ores are also presented to substantiate this united AMS model.

Magnetic susceptibility of natural rocks and ores plays important roles either directly or indirectly in many applications, such as oil and mineral explorations [

AMS is a reflection of uneven and directed distribution of ferromagnetic minerals in some rocks and ores during their formation or later deformation by various means. Effect of AMS can be either positive, such as identifying historic structural and tectonic events through magnetic fabrics in the rocks [

Some attempts have been made on describing AMS using mathematical models [

Magnetic susceptibility

where

It is important to distinguish extrinsic or measured susceptibility

where N is the demagnetization factor. When

In a few rock types the induced magnetization in symmetrically shaped specimens is independent of the direction in which a weak magnetic field is applied. Such rocks are magnetically isotropic. In most rock types, however, the strength of the induced magnetization in a constant weak field depends on the orientation of a rock sample within the field. Such rocks are magnetically anisotropic. The variation of susceptibility with orientation can be described mathematically in terms of a symmetric second-rank tensor as,

Or simply visualized as a susceptibility ellipsoid (

tropy of magnetic susceptibility (AMS). A more detailed mathematical description of AMS follows in the next section.

Susceptibility is a second-rank tensor expressed as Equation (3). Because it is symmetric, it can be converted to another specialized orthogonal coordinate system by rotating the coordinates,

where

By Equations (1) and (4), in the new coordinates, the populations of induced magnetization are expressed as

or

This indicates that the three principal susceptibilities are parallel to their corresponding populations of magnetic field, and no interactions occur among the three orthogonal populations.

We do not normally measure each component of the susceptibility tensor. Instead, we measure the directional susceptibility along the applied magnetic field. For an anisotropic material, assuming there is an angle

If the direction cosines of

Then the directional susceptibility

As

In practice, the susceptibility tensor of a sample can be calculated by means of the least-squares method from a set of

Assuming there is a vector

Then the direction cosines are

From Equations (10)-(12), the following equation can be determined

This is a ternary quadratic polynomial that can be converted to an ellipsoid using the three eigenvectors coincident with the three principal susceptibilities:

This is called the magnitude susceptibility ellipsoid or simply the susceptibility ellipsoid. It is a standard ellipsoid with the half-axis length of

The output of AMS measurements is the susceptibility ellipsoid defined by the length and orientation of its three principal axes. The parameters usually presented are the bulk susceptibility

And the magnetic anisotropy A, lineation L and foliation F as given below

The magnetic foliation plane contains the maximum and intermediate susceptibility axes. The magnetic lineation is parallel to the maximum susceptibility axis, so it lies within the foliation plane. The minimum susceptibility axis is normal to the magnetic foliation plane so it can be regarded as the pole to the foliation plane (

The magnitude of AMS depends on two factors: the anisotropy of individual magnetic particles; and the degree of their alignment. The anisotropy of the individual particles comprises two populations-crystalline and shape anisotropy. The preferred orientation of crystallographic axes commonly controls grain shape and determines the AMS for the vast majority of minerals. However, for a few special rock types, such as banded iron formations (BIFs), ferromagnetic grains are concentrated along some bands. In such cases the magnetic interactions among the ferromagnetic grains can generate an overall magnetic behavior completely different from the behavior of individual grains. This type of anisotropy is often called textural anisotropy [

High-grade iron ores derived from BIFs have a weak AMS and the average degree of anisotropy is below 1.05 (

Basalt and dolomite are weak in magnetism and generally isotropic (

This paper presents a united AMS model by rationalizing the existing pieces of different AMS models through a consistent approach. A few examples of AMS from some types of natural rocks and ores are also presented to substantiate this united AMS model. This unified mathematical model serves as a consistent guide for any studies involving AMS.

William W. Guo, (2015) Mathematical Model of Anisotropy of Magnetic Susceptibility (AMS). Journal of Applied Mathematics and Physics,03,399-404. doi: 10.4236/jamp.2015.34050