Extraordinary magnetic materials
We would now like to discuss some of the more exotic magnetic materials. There are many elements in the periodic table which have incomplete inner electron shells and hence have atomic magnetic moments. For instance, right next to the ferromagnetic elements iron, nickel, and cobalt you will find chromium and manganese. Why aren’t they ferromagnetic? The answer is that the $\lambda$ term in Eq. (37.1) has the opposite sign for these elements. In the chromium lattice, for example, the spins of the chromium atoms alternate atom by atom, as shown in Fig. 37–13(b). So chromium is “magnetic” from its own point of view, but it is not technically interesting because there are no external magnetic effects. Chromium, then, is an example of a material in which quantum mechanical effects make the spins alternate. Such a material is called antiferromagnetic. The alignment in antiferromagnetic materials is also temperature dependent. Below a critical temperature, all the spins are lined up in the alternating array, but when the material is heated above a certain temperature—which is again called the Curie temperature—the spins suddenly become random. There is, internally, a sudden transition. This transition can be seen in the specific heat curve. Also it shows up in some special “magnetic” effects. For instance, the existence of the alternating spins can be verified by scattering neutrons from a crystal of chromium. Because a neutron itself has a spin (and a magnetic moment), it has a different amplitude to be scattered, depending on whether its spin is parallel or opposite to the spin of the scatterer. Thus, we get a different interference pattern when the spins in a crystal are alternating than we do when they have a random distribution. There is another kind of substance in which quantum mechanical effects make the electron spins alternate, but which is nevertheless ferromagnetic—that is, the crystal has a permanent net magnetization. The idea behind such materials is shown in Fig. 37–14. The figure shows the crystal structure of spinel, a magnesium-aluminum oxide, which—as it is shown—is not magnetic. The oxide has two kinds of metal atoms: magnesium and aluminum. Now if we replace the magnesium and the aluminum by two magnetic elements like iron and zinc, or by zinc and manganese—in other words, if we put in magnetic atoms instead of the nonmagnetic ones—an interesting thing happens. Let’s call one kind of metal atom $a$ and the other kind of metal atom $b$; then the following combination of forces must be considered. There is an $a$-$b$ interaction which tries to make the $a$ atoms and the $b$ atoms have opposite spins—because quantum mechanics always gives the opposite sign (except for the mysterious crystals of iron, nickel, and cobalt). Then, there is a direct $a$-$a$ interaction which tries to make the $a$’s opposite, and also a $b$-$b$ interaction which tries to make the $b$’s opposite. Now, of course we cannot have everything opposite everything else—$a$ opposite $b$, $a$ opposite $a$, and $b$ opposite $b$. Presumably because of the distances between the $a$’s and the presence of the oxygen (although we really don’t know why), it turns out that the $a$-$b$ interaction is stronger than the $a$-$a$ or the $b$-$b$. So the solution that nature uses in this case is to make all the $a$’s parallel to each other, and all the $b$’s parallel to each other, but the two systems opposite. That gives the lowest energy because of the stronger $a$-$b$ interaction. The result: all the $a$’s are spinning up and all the $b$’s are spinning down—or vice versa, of course. But if the magnetic moments of the $a$-type atom and the $b$-type atom are not equal, we can get the situation shown in Fig. 37–13(c), and there can be a net magnetization in the material. The material will then be ferromagnetic—although somewhat weak. Such materials are called ferrites. They do not have as high a saturation magnetization as iron—for obvious reasons—so they are only useful for smaller fields. But they have a very important difference—they are insulators; the ferrites are ferromagnetic insulators. In high-frequency fields, they will have very small eddy currents and so can be used, for example, in microwave systems. The microwave fields will be able to get inside such an insulating material, whereas they would be kept out by the eddy currents in a conductor like iron. There is another class of magnetic materials which has only recently been discovered—members of the family of the orthosilicates called garnets. They are again crystals in which the lattice contains two kinds of metallic atoms, and we have again a situation in which two kinds of atoms can be substituted almost at will. Among the many compounds of interest there is one which is completely ferromagnetic. It has yttrium and iron in the garnet structure, and the reason it is ferromagnetic is very curious. Here again quantum mechanics is making the neighboring spins opposite, so that there is a locked-in system of spins with the electron spins of the iron one way and the electron spins of the yttrium the opposite way. But the yttrium atom is complicated. It is a rare-earth element and gets a large contribution to its magnetic moment from orbital motion of the electrons. For yttrium, the orbital motion contribution is opposite that of the spin and also is bigger. Thus, although quantum mechanics, working through the exclusion principle, makes the spins of the yttrium opposite those of the iron, it makes the total magnetic moment of the yttrium atom parallel to the iron because of the orbital effect—as sketched in Fig. 37–13(d). The compound is therefore a regular ferromagnet. Another interesting example of ferromagnetism occurs in some of the rare-earth elements. It has to do with a still more peculiar arrangement of the spins. The material is not ferromagnetic in the sense that the spins are all parallel, nor is it antiferromagnetic in the sense that every atom is opposite. In these crystals all of the spins in one layer are parallel and lie in the plane of the layer. In the next layer all spins are again parallel to each other, but point in a somewhat different direction. In the following layer they are in still another direction, and so on. The result is that the local magnetization vector varies in the form of a spiral—the magnetic moments of the successive layers rotate as we proceed along a line perpendicular to the layers. It is interesting to try to analyze what happens when a field is applied to such a spiral—all the twistings and turnings that must go on in all those atomic magnets. (Some people like to amuse themselves with the theory of these things!) Not only are there cases of “flat” spirals, but there are also cases in which the directions the magnetic moments of successive layers map out a cone, so that it has a spiral component and also a uniform ferromagnetic component in one direction! The magnetic properties of materials, worked out on a more advanced level than we have been able to do here, have fascinated physicists of all kinds. In the first place, there are those practical people who love to work out ways of making things in a better way—they love to design better and more interesting magnetic materials. The discovery of things like ferrites, or their application, immediately delights people who like to see clever new ways of doing things. Besides this, there are those who find a fascination in the terrible complexity that nature can produce using a few basic laws. Starting with one and the same general idea, nature goes from the ferromagnetism of iron and its domains, to the antiferromagnetism of chromium, to the magnetism of ferrites and garnets, to the spiral structure of the rare earth elements, and on, and on. It is fascinating to discover experimentally all the strange things that go on in these special substances. Then, to the theoretical physicists, ferromagnetism presents a number of very interesting, unsolved, and beautiful challenges. One challenge is to understand why it exists at all. Another is to predict the statistics of the interacting spins in an ideal lattice. Even neglecting any possible extraneous complications, this problem has, so far, defied full understanding. The reason that it is so interesting is that it is such an easily stated problem: Given a lot of electron spins in a regular lattice, interacting with such-and-such a law, what do they do? It is simply stated, but it has defied complete analysis for years. Although it has been analyzed rather carefully for temperatures not too close to the Curie point, the theory of the sudden transition at the Curie point still needs to be completed.  Finally, the whole subject of the system of spinning atomic magnets—in ferromagnetic, or in paramagnetic materials and in nuclear magnetism, has also been a fascinating thing to advanced students in physics. The system of spins can be pushed on and pulled on with external magnetic fields, so one can do many tricks with resonances, with relaxation effects, with spin-echoes, and with other effects. It serves as a prototype of many complicated thermodynamic systems. But in paramagnetic materials the situation is often fairly simple, and people have been delighted both to do experiments and to explain the phenomena theoretically. We now close our study of electricity and magnetism. In the first chapter, we spoke of the great strides that have been made since the early Greek observation of the strange behaviors of amber and of lodestone. Yet in all our long and involved discussion we have never explained why it is that when we rub a piece of amber we get a charge on it, nor have we explained why a lodestone is magnetized!  You may say, “Oh, we just didn’t get the right sign.”  No, it is worse than that. Even if we did get the right sign, we would still have the question: Why is the piece of lodestone in the ground magnetized?  There is the earth’s magnetic field, of course, but where does the earth’s field come from?  Nobody really knows—there have only been some good guesses. So you see, this physics of ours is a lot of fakery—we start out with the phenomena of lodestone and amber, and we end up not understanding either of them very well. But we have learned a tremendous amount of very exciting and very practical information in the process!