On the Theory of Ferromagnetism¹)

R. Becker, Göttingen

The concept of spontaneous magnetization introduced by Weiss has proved extraordinarily fruitful in the description of ferromagnetic phenomena. But here, as happens very often in physics, it may turn out that a newer concept makes the simple and natural interpretation of many phenomena more difficult, if this concept is taken too literally. The phenomena near the Curie point, considered by Gerlach, serve as a typical example of this kind. Below I shall give a description of ferromagnetic phenomena which, almost everywhere, is in practical agreement with Weiss’s theory wherever that theory is confirmed by experiment; at the same time, however, this description clearly shows how, on approaching the Curie point, the concept of spontaneous magnetization gradually loses its meaning. The conclusions of the new theory concerning behavior near the Curie point differ from the conclusions of Weiss’s theory precisely in such a way that they agree with experiment.

I shall call this new description the theory of “ordering” (Ausscheidungstheorie) of ferromagnetism. It proceeds from the following assumptions. As quantum theory requires, a ferromagnet may be described schematically as follows: in each cubic centimeter there are \(n\) spins, each of them with magnetic moment \(\mu\). Of these, \(r\) spins are directed to the right, and \(l\) to the left \((r + n = l)\). The resulting magnetization is equal to

\[ J = \mu(r - l) = \mu(2r - n). \tag{1} \]

If we further assume that the exchange energy \(I\) has an appreciable value only for nearest neighbors, then it can be determined by the following schematic reaction:

\[ \uparrow\uparrow + I = \uparrow\downarrow, \tag{2} \]

i.e., in other words, the energy \(I\) must be added to two parallel spins in order to transform them into antiparallel ones.

¹) Contribution to the discussion occasioned by W. Gerlach’s report, printed in this issue (p. 368).

The magnetic state will be completely determined if we know that part of the energy of the body \(E = E(J,T)\) which depends on the temperature and the magnetization. In the limiting case of very high temperatures this calculation is easy to carry out. If \(r\) right and \(l\) left spins are distributed in complete disorder, then any randomly chosen right spin will have \(Z\cdot \dfrac{r}{n}\) right and \(Z\cdot \dfrac{l}{n} = Z\cdot \dfrac{(n-r)}{n}\) left neighbors (\(Z\) is the number of nearest neighbors). The total number of “right-left” neighborhoods is, obviously, \(Zr\dfrac{n-r}{n}\), and the energy is equal to \(E = Zn\dfrac{r}{n}\cdot\dfrac{-r}{n} I\). But from (1) it follows that

\[ \begin{aligned} \frac{r}{n} &= \frac{1}{2}\left(1+\frac{J}{J_\infty}\right),\\ \frac{l}{n} &= \frac{1}{2}\left(1-\frac{J}{J_\infty}\right), \end{aligned} \tag{3} \]

where \(J_\infty = n\mu\). Thus,

\[ E=\frac{1}{4}ZnI\left(1-\frac{J^2}{J_\infty^2}\right) \quad \text{for } T\to\infty, \tag{4} \]

or, if we introduce the abbreviated notation:

\[ \left. \begin{aligned} \frac{1}{2}\frac{ZnI}{J_\infty^2} &= W,\\[4pt] E &= \frac{1}{2}W\left(J_\infty^2-J^2\right) \quad \text{for } T\to\infty, \end{aligned} \right\} \tag{5} \]

\(E=0\) at saturation (\(J=J_\infty\)). For complete destruction of the magnetization it is necessary to expend the energy \(\dfrac{1}{2}WJ_\infty^2\).

Together with (5), which gives the limiting value of \(E\), and with known \(E(J,T)\), the magnetic properties can be obtained from the thermodynamic relation

\[ \frac{\partial}{\partial T}\left(\frac{H}{T}\right)_J = -\frac{1}{T^2}\frac{\partial E}{\partial J}. \]

Namely,

\[ \frac{H}{T} = \int_T^\infty \frac{1}{T^2}\frac{\partial E}{\partial J}\,dT + \frac{k}{\mu}\Phi\left(\frac{J}{J_\infty}\right), \]

where

\[ \Phi(x)=\frac{1}{2}\ln\frac{1+x}{1-x}. \]

Weiss’s theory differs from the one set forth here in that it considers expression (5) for \(E(J,T)\) to be exact everywhere (although it clearly has force only in the limiting case of high temperatures) and, consequently, \(E\) does not depend explicitly on the temperature. This assumption would be correct if one were to suppose (as Weiss in fact did) that each given spin interacts with every other spin of the substance in the same way. But this assumes that the radius of action

of the orienting forces embraces many atomic distances. As we know from Heisenberg’s theory, the exchange forces act in practice only between nearest neighbors, and this necessarily leads to the fact that at low temperatures definite spin groups will be formed; namely, each right spin will be surrounded by right neighbors, and a left one by left neighbors. Thus, for given values of \(r\) and \(l\), the number of right-left neighborhoods and, with it, the energy will be much smaller than in a statistically disordered distribution. Therefore the energy \(E\), for a given \(J\), at low temperatures will be substantially smaller than is required by (5). A strictly numerical solution of this problem has not yet been obtained. The principal difficulty is the following (see P. Becker, Metallwirtsch., 16, 573, 1937).

It is necessary to determine the number of ways of placing \(r\) right and \(l\) left spins on the lattice sites so as to obtain a prescribed number of right-left neighborhoods. A certain substitute for solving this problem (which is still impossible) is, in our opinion, the consideration of simple alloys of two components \(A\) and \(B\), which at high temperatures are unlimitedly soluble, while at low temperatures they can form mixtures only of a definite composition, as shown in Fig. 1. Qualitatively this may be described by saying that each atom \(A\) of the alloy is more strongly bound to its \(A\)-neighbor than to a \(B\)-neighbor. Quantitatively, this difference \(V\) in energies can be defined by the following reaction:

\[ \frac{1}{2}[AA + BB] + V = AB. \tag{6} \]

Fig. 1.

Here it is likewise impossible to calculate the equilibrium distribution, for a given concentration of \(a\)-atoms \(A\), just as it is to calculate the spin distribution for a given \(r\). But one can obtain a first approximation if one assumes that within a homogeneous phase the atoms are distributed statistically. Then it is easy to calculate that, at a given temperature \(T_1\), an alloy of composition lying within the limits between \(a_1\) and \(1-a_1\) decomposes into crystallites with compositions \(a_1\) and \(1-a_1\). The temperature dependence of \(a_1\) is given by the boundary curve of the phase diagram. In (6) the energy \(V\) plays the same role as the exchange energy \(I\) in equation (2); therefore one may try to adapt the phase diagram (Fig. 1) to the spins of a ferromagnet. In this case each of the abscissae will correspond to a given number of right spins \(r\), i.e. to a given value of the relative magnetization \(\eta\):

\[ \eta=\frac{J}{J_{\infty}}=\frac{2r-n}{n}=2a-1 \quad \text{or} \quad a=\frac{1}{2}(\eta+1). \]

As a result, material with a resulting magnetization equal to zero decomposes at \(T_1\) into regions with magnetization \(+\eta_1\) and \(-\eta_1\).

Unlike alloys, for which an overall composition is established, in the magnetic case an external magnetic field can reverse the region of magnetization \(-\eta_1\) and transform it into a region of type \(+\eta_1\). Thus the whole material as a whole will possess the total magnetization \(\overline{DF}\). The branch \(ODF\) of our boundary curve therefore proves to be identical with the Weiss curve \(J_s = J_s(T)\) for spontaneous magnetization. As for the energy \(E(J,T)\), we arrive at the conclusion that (5) is replaced by the following formulae:

\[ \left. \begin{aligned} E&=\frac{1}{2}W\left(J_\infty^2-J_s^2\right)\quad &&\text{for } |J|<J_s,\\ E&=\frac{1}{2}W\left(J_\infty^2-J^2\right)\quad &&\text{for } |J|\geq J_s. \end{aligned} \right\} \tag{7} \]

Until the regions of homogeneous magnetization, determined on the boundary curve by the points \(D\) and \(G\), contain an enormous number of atoms, the description of the ferromagnet obtained from the state diagram (Fig. 2) coincides with the Weiss theory; but it is to be expected that the sizes of these regions become ever smaller as we approach the Curie point \(\theta\). The regions corresponding to the points \(E\) and \(H\) differ little in their state; their mutual surface energy is relatively negligible, which facilitates their division into small islands. Owing to the smallness of these regions, their orientation depends much more strongly on thermal motion; therefore stronger fields are required for their orientation than for regions of type \(G\). But in stronger fields there also occurs an increase of \(J_s\), i.e. the point \(E\) moves in the direction toward \(K\). It therefore loses all meaning to speak of a “spontaneous” magnetization determined by the points \(E\) or \(H\). On the other hand, immediately above \(\theta\) the distribution of spins is still far from statistical; therefore even above \(\theta\) there exist in equilibrium states “swarms” of parallel spins, which disappear (dissolve) only with a further rise in temperature, and before that manifest themselves, for example, in anomalies of the heat capacity. The description of a ferromagnet as an ordering problem makes it possible to clarify qualitatively in what sense the expression for the energy (7) must be modified in order to obtain a non-sharp character of the Curie point. For details I refer the reader to my book on ferromagnetism, written jointly with W. Döring (R. Becker und Döring, “Ferromagnetismus,” J. Springer, Berlin, 1939).

Fig. 2