The system of Eqs. (1.39), (1.42) and (1.43) div B = 0

which describes the magnetic field in a condensed matter is clearly incomplete, because we still do not know the relationship between M and H (or between B and H) inside a body. This relationship depends heavily on the material from which the body under study is made. Fortunately, for an overwhelming majority of physical substances, the required relationship is very simple:

where scalar quantities m and w are called correspondingly magnetic permeability and susceptibility (so here m is not the magnitude of the total magnetic moment!). From Eqs. (1.43) and (1.48), the relationship between m and w is w = (m _ 1)/4ft. In some cases (e.g., for solid monocrystalline samples), m and w appearing in the proportionality relationships (Eq. 1.48) are tensors of a corresponding rank. For ferro-magnets, the situation is even more complicated - the relationship between B and H is nonlinear in general case and depends on the history of the sample.

The magnetic susceptibility is the most important quantity characterizing the magnetic properties of a material (Landau and Lifshitz, 1975a; Kittel, 1986). Namely, it enables us to calculate the magnetization (and hence the magnetic moment) of the body in an external field. If w < 0, then, as can be seen from Eq. (1.48), the magnetic moment induced by an external field is directed opposite to this field. Such materials are called diamagnets (e.g., inert gases, organic liquids, graphite, bismuth). According to our discussion of the force acting on a body in a nonhomogeneous magnetic field (see text following Eq. 1.38), diamagnetic bodies are repelled from the magnet.

For substances with w > 0, the induced magnetic moment caused by the external field, points in the same direction as the external field. Materials with positive but very small (for most materials w @ 10~6) susceptibility are called paramagnets (some gases, organic free radicals, most metals).

Finally, there exists a narrow class of materials for which magnetic susceptibility defined by Eq. (1.48) - when possible - is huge (w @ 103, but for some specially prepared materials w @ 106 can be achieved). Such substances are known as ferro-magnets (iron, cobalt, nickel and their alloys, some iron and chromium oxides, etc.). It is evident that these materials are most interesting, for both theoretical studies and practical applications. We shall consider corresponding problems in the final paragraph of this section and again in Section 1.2.4. Here, it should only be mentioned that the relationships (1.48) for ferromagnets are, generally speaking, not valid - the induced magnetic moment is not simply proportional to the external field.

Diamagnetism and paramagnetism can be explained in terms of classical physics (to be more precise, in these terms we can provide explanations which appear reasonable). Let us begin with diamagnetism. The molecules of diamagnetic substances do not have their own magnetic moments; that is, they do not possess a spontaneous moment - a magnetic moment in the absence of an external field. From basic electromagnetism we are familiar with Lenz's law: when we try to change a magnetic flux through a conducting contour, then an electric current in this contour is induced in such a way, that the magnetic field created by this current opposes the change of the external magnetic flux. In other words, if we try to increase a magnetic field inside a closed contour, the magnetic moment associated with the current induced by this external field will be directed opposite to it.

From the classical point of view, electrons moving in atoms or molecules can be considered as currents. Hence, by applying a magnetic field to a body, we try to increase magnetic flux through contours formed by these currents (electron orbits).

According to Lenz?s law, this increase leads to changes in corresponding currents, with the result that the magnetic moment of the body induced by this change is directed opposite to the external field, which means diamagnetism.

Molecules of paramagnetic substances already possess their own dipole moments. When we apply an external field, these moments tend to align themselves along this field, because it would minimize their magnetic energy according to Eq. (1.37). The chaotic thermal motion tries to prevent such an alignment, but an average magnetic moment nevertheless appears and points in the field direction, leading to paramagnetism (w > 0).

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