1.1.3.2 Ferromagnetism. Weiss molecular field

Some materials present very strong magnetization, typically in the order of the saturation magnetization, also in absence of external field, i.e. they present spontaneous magnetization. This kind of materials are referred to as ferromagnetic materials (Fe, Co, Ni, Gd, alloys, etc.). Typical properties of some ferromagnetic materials can be found in Appendix A. The behavior of very small regions of ferromagnetic materials can be treated by following the same line of reasoning used for paramagnetism. With respect to the continuum model introduced in section 1.1.1, we are now dealing with phenomena occurring inside our elementary volume $dV_\textbf{r}$, which involve the interactions between single spins. Here we report the theory developed by Weiss which is very similar to the one used for paramagnetism. In fact, the main difference stays in the postulation of an additional magnetic field $H_$w whose non magnetic (Maxwellian) origin is not investigated. This field was called molecular field by Weiss [8]; by adding the field $H_$w$=N_$w$M$ ( $N_$w is characteristic of the material) to the external field in Eq. (1.24), one ends up with the following equation:

$$M=M_s \mathcal{L}\left(\frac{\mu_0 m_0 (H_\text{a}+N_\text{w} M)}{kT}\right) \quad.$$ (1.28)

The latter equation can be linearized for high temperatures, which corresponds to small $\beta$ as seen before. Then, one can find the well-known Curie-Weiss law that once again expresses the dependance of the susceptibility on the temperature

$$\chi\propto\frac{1}{T-T_c} \quad,\quad T_c=\frac{\mu_0 M_s m_0 N_\text{w}}{3k} \quad,$$ (1.29)

where $T_c$ is the Curie temperature, characteristic of the material. Thus, for temperatures $T>T_c$ the ferromagnetic materials behave like paramagnetic. For temperature $T<T_c$, one can use Eq. (1.28) to derive the relationship between the saturation magnetization $M_s$ and the temperature $T$. The resulting relationship $M_s=M_s(T)$ behaves like in Fig. 1.2. This behavior qualitatively matches with experimental observations [5].

Figure 1.2: Typical behavior of spontaneous magnetization as function of temperature.

In addition, the phenomenological approach of molecular field was theoretically justified when Heisenberg introduced the exchange interaction on the basis of quantum theory (1931). Nevertheless, the Weiss theory gives information about the magnitude of magnetization, but nothing can be said about the direction. In this respect micromagnetics has the purpose to find the direction of magnetization at every location within the magnetic body. In this respect, for constant temperature, the magnetization vector field $\textbf{M}(\textbf{r},t)$ can be written as

$$\textbf{M}(\textbf{r},t)=M_s \textbf{{m}}(\textbf{r},t) \quad,$$ (1.30)

where $\textbf{{m}}(\textbf{r},t)$ is the magnetization unit-vector field.