6.6.4 Molecular Field Theory

Now the question arises: What leads to this ordering and what is the effect of the ordering of magnetic moment? This ordering or shall we say a neighbour effect gives rise to a modified internal field experience by these magnetic moments (often denoted as magnetic dipoles) in a sort of a similar manner as we witnessed in the derivation of Claussis Mossotti relation in dielectrics.

In 1907, Weiss postulated that mutual interaction between the magnetic moments keeps them parallel and aligned together  (wrongly!) subscribing this to an internal field (not wrong!). This extra field or called as Weiss molecular field, H_w plays a crucial role in keeping the moments aligned in one direction.
H_w is expressed as a product of magnetization with a molecular field constant γ a characteristic of the material. Hence, this internal field can be expressed as

(6.32)

Now, for a ferromagnetic material, we need to replace H with H_i in equation (6.23 and 6.24) which leads to

(6.33)

OR

(6.34)

(6.35)

where T_C = γC and it has a unit of temperature and is called as Curie temperature. This is Curie-Weiss Law!

The figure above shows that while M_s  start dropping at T_C and then dies off slowly as there is a divergence in 1/χ at θ in the paramagnetic region and is a signature of a phase transition to a spontaneously ordered phase. A positive value of T_C indicates that molecular field is acting in the same direction as of applied field and acts to align the magnetic moments parallel to each other, as should be the case with a ferromagnetic material.

However, in real picture as later physicists pointed out, the origin of Weiss molecular field is the exchange interaction as a consequence of the Pauli’s exclusion principle and the Coulomb interaction between electrons. For instance, in case of two electrons, electrons can arrange themselves parallel or antiparallel. If parallel, Pauli’s exclusion principle requires them to remain far apart and if antiparallel, the electrons may come closer together and their wave functions overlap considerably. Hence, the electrostatic energy of an electron system is governed by the relative orientation of the spins and the difference in energy defines the exchange energy.  For transition element materials like iron, nickel and cobalt, the exchange energy is minimum for parallel spin configuration while for certain other materials like MnO  and various other ferrites, exchange energy (actually super-exchange energy) is minimum when the spins are arranged in antiparallel fashion and these materials are called antiferromagnetic or ferrimagnetic materials which we will discuss in the following sections.

The exchange interaction is of short range, thus only nearest neighbor atoms produce the molecular field. The magnitude of this exchange or molecular field can be worked out by equating the exchange energy (μ_B H_W) with thermal energy, kT at T_C, i.e. H_W = (kT_C/μ_B) which turns to be of the order of about 1 kilo-tesla which is quite large a field, and not even found in the best of machines.