Exercise 3.5 Graphically describe the potential and attraction of a uniform, thick-walled shell (inner radius a\ and outer radius CI2) along a line
5.3 Magnetic Permeability and Susceptibility
There is little to be gained from the distinction between B and H in most geophysical measurements of the magnetic field of the earth. Mea- surements of the earth's magnetic field, whether from aircraft, ship, or satellite, are made in environments very nearly free of magnetic material (e.g., Lowes [169]). Electrical currents can pose problems in some situ- ations, such as problems associated with ionospheric currents at satel- lite altitudes, but this issue is not relevant to our choice of B versus H. Indeed, the geophysical literature dealing with the interpretation of magnetic anomalies uses both B and H interchangeably. Here we will attempt to use B whenever possible.
5.3 Magnetic Permeability and Susceptibility
Materials can acquire a component of magnetization in the presence of an external magnetic field. For low-amplitude magnetic fields, say on the order of the earth's magnetic field, this induced magnetization is proportional in magnitude and is parallel (or antiparallel) in direction to the external field, that is,
M = XH . (5.8)
The proportionality constant \ is called the magnetic susceptibility.
Equation 5.8 is the same in both the SI and emu systems. Suscepti- bility is dimensionless in both systems but differs in magnitude by 4?r:
Susceptibility in emu equals 4TT times susceptibility in SI units.
A related quantity, the magnetic permeability /i, differs slightly be- tween the two systems, and separate derivations are necessary. Starting with equations 5.6 and 5.7, respectively, the derivations are as follows.
In the emu system,
B = H + 4TTM - H + 4TTXH
= (1 + 4TTX)H
(5.9)
whereas in SI units,
B = /xo(H + M)
(5-10) Kinds of Magnetization
Although x a nd M a r e derived in a simplistic mathematical way, they are in fact complex products of the atomic and macroscopic proper- ties of the magnetic material. The relationship between M and H is not necessarily linear as implied by equation 5.8; x m a v vary with field inten- sity, may be negative, and may be represented more accurately in some materials as a tensor. This section provides a very cursory description of the magnetization of solid materials. Indeed, this subject is worthy of its own textbook, and the interested reader is referred to books by Chikazumi [57] and Morrish [187] for information on magnetic materials in general, and to the book by Butler [47] for applications related to paleomagnetic and geomagnetic problems specifically. The implications of rock magnetism for magnetic-anomaly studies have been reviewed concisely and comprehensively by Reynolds et al. [243].
There are many kinds of magnetization. Diamagnetism, for example, is an inherent property of all matter. In diamagnetism, an applied mag- netic field disturbs the orbital motion of electrons in such a way as to induce a small magnetization in the opposite sense to the applied field.
Consequently, diamagnetic susceptibility is negative. Paramagnetism is a property of those solids that have atomic magnetic moments. Appli- cation of a magnetic field causes the atomic moments to partially align parallel to the applied field thereby producing a net magnetization in the direction of the applied field. Thermal effects tend to oppose this alignment, and paramagnetism vanishes in the absence of applied fields because thermal effects act to randomly orient the atomic moments. All minerals are diamagnetic and some are paramagnetic, but in either case these magnetizations are insignificant contributors to the geomagnetic field.
There is, however, a class of magnetism of great importance to geo- magnetic studies. Certain materials not only have atomic moments, but
5.3 Magnetic Permeability and Susceptibility 89 neighboring moments interact strongly with each other. This interaction is a result of a quantum mechanical effect called exchange energy, which is beyond the scope of this book. Suffice it to say that the exchange energy causes a spontaneous magnetization that is many times greater than paramagnetic or diamagnetic effects. Such materials are said to be ferromagnetic. There are various kinds of ferromagnetic materials too, depending on the way that the atomic moments align. These include ferromagnetism proper, in which atomic moments are aligned parallel to one another; antiferromagnetism, where atomic moments are aligned an- tiparallel and cancel one another; and ferrimagnetism, in which atomic moments are antiparallel but do not cancel.
At the scale of individual mineral grains, spontaneous magnetization of a ferromagnetic material can be very large. At the outcrop scale, how- ever, the magnetic moments of individual ferromagnetic grains may be randomly oriented, and the net magnetization may be negligible. The magnetization of individual grains is affected, however, by the applica- tion of a magnetic field, similar to but far greater in magnitude than for paramagnetism. Hence, rocks containing ferromagnetic minerals will acquire a net magnetization, called induced magnetization and denoted by Mi, in the direction of an applied field H, where
Of course the earth's magnetic field produces the same response in such materials, and the material is magnetic in its natural state. In small fields, with magnitudes comparable to the earth's magnetic field, the re- lationship between induced magnetization and applied field is essentially linear, and the susceptibility \ is constant.
Induced magnetization falls to zero if the rock is placed in a field-free environment. However, ferromagnetic materials also have the ability to retain a magnetization even in the absence of external magnetic fields.
This permanent magnetization is called remanent magnetization, which we denote here by Mr. In crustal materials, remanent magnetization is a function not only of the atomic, crystallographic, and chemical make-up of the rocks, but also of their geologic, tectonic, and thermal history. In geophysical studies, it is customary to consider the total magnetization M of a rock as the vector sum of its induced and remanent magnetiza- tions, that is,
M = Mi + Mr
The relative importance of remanent magnetization to induced magne- tization is expressed by the Koenigsberger ratio
In subsequent discussions regarding magnetic fields of crustal materials, we will consider just two kinds of magnetization: induced and remanent.
It is well to keep in mind, however, that both of these magnetizations arise from spontaneous magnetization, a complex property of the ferro- magnetic minerals in the earth's crust.
The spontaneous magnetization is dependent on temperature. As a material is heated, the spacing between neighboring atomic moments increases until a point is reached where the spontaneous magnetiza- tion falls to zero. This temperature is called the Curie temperature.
Hence, both induced and remanent magnetizations vanish at temper- atures greater than the Curie temperature. Paramagnetic and diamag- netic effects persist at these temperatures, but from the perspective of magnetic-anomaly studies we may consider rocks above the Curie tem- perature to be nonmagnetic.
Magnetite (FesO4) and its solid solutions with ulvospinel (Fe2TiC>4) are the most important magnetic minerals to geophysical studies of crustal rocks. Other minerals, such as hematite, pyrrhotite, and alloys of iron and nickel, are important in certain geologic situations, but the volume percentage, size, shape, and history of magnetite grains are of greatest importance in most magnetic surveys. Magnetite is a ferrimag- netic material with a Curie temperature of about 580° C.
Typical values of |Mr| and % f°r representative rock types are pro- vided by Lindsley, Andreasen, and Balsley [165] and Carmichael [54].
Generally speaking, mafic rocks are more magnetic than silicic rocks.
Hence, basalts are usually more magnetic than rhyolites, and gabbros are more magnetic than granites. Also, extrusive rocks generally have a higher remanent magnetization and lower susceptibility than intru- sive rocks with the same chemical composition. Sedimentary and meta- morphic rocks often have low remanent magnetizations and susceptibil- ities. These statements and the compilations referenced are only statis- tical guidelines with many exceptions. Values of Mr and \ often vary by several orders of magnitude within the same outcrop, for example.
Whenever feasible, interpretation of a magnetic survey should include