Understanding ferromagnetism
In this chapter we will discuss the behavior and peculiarities of ferromagnetic materials and of other strange magnetic materials. Before proceeding to study magnetic materials, however, we will review very quickly some of the things about the general theory of magnets that we learned in the last chapter. First, we imagine the atomic currents inside the material that are responsible for the magnetism, and then describe them in terms of a volume current density $\FLPj_{\text{mag}}=\FLPcurl{\FLPM}$. We emphasize that this is not supposed to represent the actual currents. When the magnetization is uniform the currents do not really cancel out precisely; that is, the whirling currents of one electron in one atom and the whirling currents of an electron in another atom do not overlap in such a way that the sum is exactly zero. Even within a single atom the distribution of magnetism is not smooth. For instance, in an iron atom the magnetization is distributed in a more or less spherical shell, not too close to the nucleus and not too far away. Thus, magnetism in matter is quite a complicated thing in its details; it is very irregular. However, we are obliged now to ignore this detailed complexity and discuss phenomena from a gross, average point of view. Then it is true that the average current in the interior region, over any finite area that is big compared with an atom, is zero when $\FLPM$ is uniform. So, what we mean by magnetization per unit volume and $\FLPj_{\text{mag}}$ and so on, at the level we are now considering, is an average over regions that are large compared with the space occupied by a single atom. In the last chapter, we also discovered that a ferromagnetic material has the following interesting property: above a certain temperature it is not strongly magnetic, whereas below this temperature it becomes magnetic. This fact is easily demonstrated. A piece of nickel wire at room temperature is attracted by a magnet. However, if we heat it above its Curie temperature with a gas flame, it becomes nonmagnetic and is not attracted toward the magnet—even when brought quite close to the magnet. If we let it lie near the magnet while it cools off, at the instant its temperature falls below the critical temperature it is suddenly attracted again by the magnet! The general theory of ferromagnetism that we will use supposes that the spin of the electron is responsible for the magnetization. The electron has spin one-half and carries one Bohr magneton of magnetic moment $\mu=$ $\mu_B=$ $q_e\hbar/2m$. The electron spin can be pointed either “up” or “down.” Because the electron has a negative charge, when its spin is “up” it has a negative moment, and when its spin is “down” it has a positive moment. With our usual conventions, the moment $\FLPmu$ of the electron is opposite its spin. We have found that the energy of orientation of a magnetic dipole in a given applied field $\FLPB$ is $-\FLPmu\cdot\FLPB$, but the energy of the spinning electrons depends on the neighboring spin alignments as well. In iron, if the moment of a nearby atom is “up,” there is a very strong tendency that the moment of the one next to it will also be “up.” That is what makes iron, cobalt, and nickel so strongly magnetic—the moments all want to be parallel. The first question we have to discuss is why. Soon after the development of quantum mechanics, it was noticed that there is a very strong apparent force—not a magnetic force or any other kind of actual force, but only an apparent force—trying to line the spins of nearby electrons opposite to one another. These forces are closely related to chemical valence forces. There is a principle in quantum mechanics—called the exclusion principle—that two electrons cannot occupy exactly the same state, that they cannot be in exactly the same condition as to location and spin orientation.1 For example, if they are at the same point, the only alternative is to have their spins opposite. So, if there is a region of space between atoms where electrons like to congregate (as in a chemical bond) and we want to put another electron on top of one already there, the only way to do it is to have the spin of the second one pointed opposite to the spin of the first one. To have the spins parallel is against the law, unless the electrons stay away from each other. This has the effect that a pair of parallel-spin electrons near to each other have much more energy than a pair of opposite-spin electrons; the net effect is as though there were a force trying to turn the spin over. Sometimes this spin-turning force is called the exchange force, but that only makes it more mysterious—it is not a very good term. It is just because of the exclusion principle that electrons have a tendency to make their spins opposite. In fact, that is the explanation of the lack of magnetism in almost all substances! The spins of the free electrons on the outside of the atoms have tremendous tendency to balance in opposite directions. The problem is to explain why for materials like iron it is just the reverse of what we should expect. We have summarized the supposed alignment effect by adding a suitable term in the energy equation, by saying that if the electron magnets in the neighborhood have a mean magnetization $M$, then the moment of an electron has a strong tendency to be in the same direction as the average magnetization of the atoms in the neighborhood. Thus, we may write for the two possible spin orientations,2 \begin{equation} \begin{aligned} \text{Spin “up” energy} &=+\mu\biggl(H+\frac{\lambda M}{\epsO c^2}\biggr),\\[1ex] \text{Spin “down” energy} &=-\mu\biggl(H+\frac{\lambda M}{\epsO c^2}\biggr). \end{aligned} \label{Eq:II:37:1} \end{equation}  When it was clear that quantum mechanics could supply a tremendous spin-orientating force—even if, apparently, of the wrong sign—it was suggested that ferromagnetism might have its origin in this same force, that due to the complexities of iron and the large number of electrons involved, the sign of the interaction energy would come out the other way around. Since the time this was thought of—in about 1927 when quantum mechanics was first being understood—many people have been making various estimates and semicalculations, trying to get a theoretical prediction for $\lambda$. The most recent calculations of the energy between the two electron spins in iron—assuming that the interaction is a direct one between the two electrons in neighboring atoms—still give the wrong sign. The present understanding of this is again to assume that the complexity of the situation is somehow responsible and to hope that the next man who makes the calculation with a more complicated situation will get the right answer! It is believed that the up-spin of one of the electrons in the inside shell, which is making the magnetism, tends to make the conduction electrons which fly around the outside have the opposite spin. One might expect this to happen because the conduction electrons come into the same region as the “magnetic” electrons. Since they move around, they can carry their prejudice for being upside down over to the next atom; that is, one “magnetic” electron tries to force the conduction electrons to be opposite, and the conduction electron then makes the next “magnetic” electron opposite to it. The double interaction is equivalent to an interaction which tries to line up the two “magnetic” electrons. In other words, the tendency to make parallel spins is the result of an intermediary that tends to some extent to be opposite to both. This mechanism does not require that the conduction electrons be completely “upside down.” They could just have a slight prejudice to be down, just enough to load the “magnetic” odds the other way. This is the mechanism that the people who have calculated such things now believe is responsible for ferromagnetism. But we must emphasize that to this day nobody can calculate the magnitude of $\lambda$ simply by knowing that the material is number $26$ in the periodic table. In short, we don’t thoroughly understand it. Now let us continue with the theory, and then come back later to discuss a certain error involved in the way we have set it up. If the magnetic moment of a certain electron is “up,” energy comes both from the external field and also from the tendency of the spins to be parallel. Since the energy is lower when the spins are parallel, the effect is sometimes thought of as due to an “effective internal field.” But remember, it is not due to a true magnetic force; it is an interaction that is more complicated. In any case, we take Eqs. (37.1) as the formulas for the energies of the two spin states of a “magnetic” electron. At a temperature $T$, the relative probability of these two states is proportional to $e^{-\text{energy}/kT}$, which we can write as $e^{\pm x}$, with $x=\mu(H+\lambda M/\epsO c^2)/kT$. Then, if we calculate the mean value of the magnetic moment, we find (as in the last chapter) that it is \begin{equation} \label{Eq:II:37:2} M=N\mu\tanh x. \end{equation}  Now we would like to calculate the internal energy of the material. We note that the energy of an electron is exactly proportional to the magnetic moment, so that the calculation of the mean moment and the calculation of the mean energy are the same—except that in place of $\mu$ in Eq. (37.2) we would write $-\mu B$, which is $-\mu(H+\lambda M/\epsO c^2)$. The mean energy is then \begin{equation*} \av{U} = -N\mu\biggl(H+ \frac{\lambda M}{\epsO c^2}\biggr)\tanh x. \end{equation*}  Now this is not quite correct. The term $\lambda M/\epsO c^2$ represents interactions of all possible pairs of atoms, and we must remember to count each pair only once. (When we consider the energy of one electron in the field of the rest and then the energy of a second electron in the field of the rest, we have counted part of the first energy once more.) Thus, we must divide the mutual interaction term by two, and our formula for the energy then turns out to be \begin{equation} \label{Eq:II:37:3} \av{U} = -N\mu\biggl(H+ \frac{\lambda M}{2\epsO c^2}\biggr)\tanh x. \end{equation}  In the last chapter we discovered an interesting thing—that below a certain temperature the material finds a solution to the equations in which the magnetic moment is not zero, even with no external magnetizing field. When we set $H=0$ in Eq. (37.2), we found that \begin{equation} \label{Eq:II:37:4} \frac{M}{M_{\text{sat}}}=\tanh\biggl(\frac{T_c}{T}\, \frac{M}{M_{\text{sat}}}\biggr), \end{equation}  where $M_{\text{sat}}=N\mu$, and $T_c=\mu\lambda M_{\text{sat}}/k\epsO c^2$. When we solve this equation (graphically or otherwise), we find that the ratio $M/M_{\text{sat}}$ as a function of $T/T_c$ is a curve like that labeled “quantum theory” in Fig. 37–1. The dashed curve marked “cobalt, nickel” shows the experimental results for crystals of these elements. The theory and experiment are in reasonably good agreement. The figure also shows the result of the classical theory in which the calculation is carried out assuming that the atomic magnets can have all possible orientations in space. You can see that this assumption gives a prediction that is not even close to the experimental facts. Even the quantum theory deviates from the observed behavior at both high and low temperatures. The reason for the deviations is that we have made a rather sloppy approximation in the theory: We have assumed that the energy of an atom depends upon the mean magnetization of its neighboring atoms. In other words, for each one that is “up” in the neighborhood of a given atom, there will be a contribution of energy due to that quantum mechanical alignment effect.  But how many are there pointed “up”? On the average, that is measured by the magnetization $M$—but only on the average. A particular atom somewhere might find all its neighbors “up.” Then its energy will be larger than the average. Another one might find some up and some down, perhaps averaging to zero, and it would have no energy from that term, and so on. What we ought to do is to use some more complicated kind of average, because the atoms in different places have different environments, and the numbers up and down are different for different ones. Instead of just taking one atom subjected to the average influence, we should take each one in its actual situation, compute its energy, and find the average energy. But how do we find out how many are “up” and how many are “down” in the neighborhood?  That is, of course, just what we are trying to calculate—the number “up” and “down”—so we have a very complicated interconnected problem of correlations, a problem which has never been solved. It is an intriguing and exciting one which has existed for years and on which some of the greatest names in physics have written papers, but even they have not completely solved it. It turns out that at low temperatures, when almost all the atomic magnets are “up” and only a few are “down,” it is easy to solve; and at high temperatures, far above the Curie temperature $T_c$ when they are almost all random, it is again easy. It is often easy to calculate small departures from some simple, idealized situation, so it is fairly well understood why there are deviations from the simple theory at low temperature. It is also understood physically that for statistical reasons the magnetization should deviate at high temperatures. But the exact behavior near the Curie point has never been thoroughly figured out. That’s an interesting problem to work out some day if you want a problem that has never been solved.