## Mean Field Theory of Ferromagnetism

The existence of ferromagnetism is one of (not very many) the macroscopic phenomena which, in principle, cannot be explained in terms of classical physics. To demonstrate this (Kittel, 1986), it is sufficient to estimate the magnitude of interactions between atomic magnetic moments which are responsible for the ferromagnetic phenomena using the following arguments. The main manifestation of ferromagnetism is the existence of the spontaneous magnetization - that is, a ferromagnetic sample can possess a spontaneous macroscopic magnetic moment. This means that there exists some strong interaction which results in the parallel alignment of all atomic magnetic moments inside the body. The magnitude of this spontaneous magnetic moment decreases if the sample temperature increases, because the thermal movement (thermal fluctuations) acts against any order trying to destroy it. At some temperature, Tc, which depends on the material and is termed the ''critical temperature'' or ''Curie point'', the spontaneous magnetization vanishes, and for temperatures T > Tc our body behaves like a paramagnet.

The interaction energy, Efm, for the interaction type responsible for the ferro-magnetism should be of the same order of magnitude as the thermal energy at the Curie point: Efm @ kTc. The only interaction known in classical physics which could cause the alignment of magnetic moments is the magneto dipole interaction between them. The interaction energy of two magnetic dipoles Edip can be estimated according to Eq. (1.37) as Edip @ mHdip, where the order of magnitude of the dipole field is (see Eq. 1.34) Hdip @ m/r3, so that Edip @ m2/r3. Substituting in this expression typical values of the atomic magnetic moment m @ mB a 10~20 erg/Gauss (mB is a so-called Bohr magneton which is a very convenient unit for measuring atomic magnetic moments) and the interatomic distance (@ lattice constant in a typical crystal) r @ (2... 3) ■ 10~8 cm, we obtain for the interaction energy Efm = Edip @ 10~17 erg. The value of the Boltzmann constant is k a 1.4 ■ 10~16 erg/K, so the critical temperature for a typical Ferro magnet should be Tc = Efm/k @ 0.1 K. This value has nothing in common with the experimentally measured Curie points which, for most ferromagnets, are of the order Tc @ 103 K (e.g., for iron, Tc = 1043 K). Hence, ferromagnetism cannot be explained by the magneto dipolar interaction, and in classical physics we have nothing else at our disposal.

For many decades, all attempts to develop a reasonable theory of ferromagnetism failed. The first phenomenological theory which succeeded in explaining some aspects of this phenomenon was suggested by Weiss (1907). Weiss postulated that: (1) there exists some (unknown) effective interaction field HE which tends to align atomic magnetic moments parallel to each other; and (2) the magnitude of this effective field is proportional to the average magnetization: HE = 1<M>. These assumptions, together with the well-known expression (the so-called Langevin function) for the average magnetization of a system of noninteracting magnetic moments in an external field as a function of the temperature T and field H (Kittel, 1986) (which in Weiss' theory should be set to the sum of the external H0 and effective HE fields), allowed Weiss to deduce the temperature dependence of the spontaneous magnetization. The result demonstrated a remarkable agreement with experimental data, which was more than acceptable for such a simple theory. However, as mentioned earlier, the existence of a ferromagnetic interaction itself was postulated by Weiss, so the nature of this interaction still required an explanation.

Such an explanation could be provided only after the appearance of quantum mechanics (for an excellent historical review, see Mattis, 1965). Here, an attempt will be made to provide a brief description of how ferromagnetism follows from its basic postulates. (Note: Should the reader feel uncomfortable when confronted with words such as "quantum", the following explanation may be missed out by simply accepting that permanent magnets do exist.)

Ferromagnetism occurs due to the collective behavior of electrons in some materials. Every electron possesses its own angular momentum S (called spin) which, being expressed in units of the so-called Planck constant (Feynmann et al., 1963; Landau and Lifshitz, 1971) is exactly S = 1/2. According to one of the basic principles of quantum mechanics - the Pauli principle - two particles with the spin 1/2 cannot occupy one and same quantum state (Feynmann et al., 1963; Landau and Lifshitz, 1971) which, for our purposes, can be reformulated as ''two particles having the same spin direction cannot occupy one and same space region''. In other words, if the spins of two electrons do not have the same direction, then the distance between them can, in principle, be very small, but electrons with parallel spins must be ''far away'' from each other.

This means, in turn, that the energy of a system of two electrons with different spin directions can be very large, because two close electrons exhibit a huge electrostatic repulsion as two charges of the same sign. Moreover, the electrostatic energy of two electrons with parallel spins should be quite small because such electrons must avoid each other due to the Pauli principle (please don't ask when have the electrons read any textbook on quantum mechanics!). For this reason, the state where spins of two electrons - and their magnetic moments! - are parallel is strongly preferred from the energy point of view, because the (average) electrostatic energy in this state is much lower! And this preferred state with all electron spins parallel is exactly what we want - the ferromagnetic state, where all electron magnetic moments are aligned and hence the body possesses macroscopic spontaneous magnetization. The phenomenon just described is called the ''exchange in teraction'', because its quantitative description is based on the so-called exchange integrals (Landau and Lifshitz, 1971). We realize that this reference does not make the things clearer, but the discussion on what these integrals are and why are they called ''exchange integrals'' is far too complicated to be presented here.

The explanation given above indeed accounts for the Curie temperatures observed experimentally. In this physical picture it is the strong Coulomb (electrostatic) interaction which is responsible for the appearance of ferromagnetism -not the weak magnetodipole forces. If we estimate Tc using the arguments given above, we simply determine the correct order of magnitude.

Of course, this is a very long way from our brief description of this basic idea to a real theory of the ferromagnetic phenomena (to see this, it is sufficient to note that if our arguments would represent the whole truth in all cases, then all substances would be ferromagnetic because there are some electrons in all materials!). But at least we have shown the beginning of the way that can lead to an explanation of ferromagnetism.