Thermodynamic properties
In the last chapter we laid the groundwork necessary for calculating the thermodynamic properties of ferromagnetic materials. These are, naturally, related to the internal energy of the crystal, which includes interactions of the various spins, given by Eq. (37.3). For the energy of the spontaneous magnetization below the Curie point, we can set $H=0$ in Eq. (37.3), and—noticing that $\tanh x=M/M_{\text{sat}}$—we find a mean energy proportional to $M^2$: \begin{equation} \label{Eq:II:37:5} \av{U} = -\frac{N\mu\lambda M^2} {2\epsO c^2M_{\text{sat}}}. \end{equation}  If we now plot the energy due to the magnetism as a function of temperature, we get a curve which is the negative of the square of the curve of Fig. 37–1, as drawn in Fig. 37–2(a). If we were to measure then the specific heat of such a material we would obtain a curve which is the derivative of 37–2(a). It is shown in Fig. 37–2(b). It rises slowly with increasing temperature, but falls suddenly to zero at $T=T_c$. The sharp drop is due to the change in slope of the magnetic energy and is reached right at the Curie point. So without any magnetic measurements at all we could have discovered that something was going on inside of iron or nickel by measuring this thermodynamic property. However, both experiment and improved theory (with fluctuations included) suggest that this simple curve is wrong and that the true situation is really more complicated. The curve goes higher at the peak and falls to zero somewhat slowly. Even if the temperature is high enough to randomize the spins on the average, there are still local regions where there is a certain amount of polarization, and in these regions the spins still have a little extra energy of interaction—which only dies out slowly as things get more and more random with further increases in temperature. So the actual curve looks like Fig. 37–2(c). One of the challenges of theoretical physics today is to find an exact theoretical description of the character of the specific heat near the Curie transition—an intriguing problem which has not yet been solved. Naturally, this problem is very closely related to the shape of the magnetization curve in the same region. Now we want to describe some experiments, other than thermodynamic ones, which show that there is something right about our interpretation of magnetism. When the material is magnetized to saturation at low enough temperatures, $M$ is very nearly equal to $M_{\text{sat}}$—nearly all the spins are parallel, as well as their magnetic moments. We can check this by an experiment. Suppose we suspend a bar magnet by a thin fiber and then surround it by a coil so that we can reverse the magnetic field without touching the magnet or putting any torque on it. This is a very difficult experiment because the magnetic forces are so enormous that any irregularities, any lopsidedness, or any lack of perfection in the iron will produce accidental torques. However, the experiment has been done under careful conditions in which such accidental torques are minimized. By means of the magnetic field from a coil that surrounds the bar, we turn all the atomic magnets over at once. When we do this we also change the angular momenta of all the spins from “up” to “down” (see Fig. 37–3). If angular momentum is to be conserved when the spins all turn over, the rest of the bar must have an opposite change in angular momentum. The whole magnet will start to spin. And sure enough, when we do the experiment, we find a slight turning of the magnet. We can measure the total angular momentum given to the whole magnet, and this is simply $N$ times $\hbar$, the change in the angular momentum of each spin. The ratio of angular momentum to magnetic moment measured this way comes out to within about $10$ percent of what we calculate. Actually, our calculations assume that the atomic magnets are due purely to the electron spin, but there is, in addition, some orbital motion also in most materials. The orbital motion is not completely free of the lattice and does not contribute much more than a few percent to the magnetism. As a matter of fact, the saturation magnetic field that one gets taking $M_{\text{sat}}=N\mu$ and using the density of iron of $7.9$ and the moment $\mu$ of the spinning electron is about $20{,}000$ gauss. But according to experiment, it is actually in the neighborhood of $21{,}500$ gauss. This is a typical magnitude of error—$5$ or $10$ percent—due to neglecting the contributions of the orbital moments that have not been included in making the analysis. Thus, a slight discrepancy with the gyromagnetic measurements is quite understandable.