1.1.1 Continuum Hypothesis

Let us consider a region $\Omega$ occupied by a magnetic body. Let us now focus on a `small' region $dV_\textbf{r}$ within the body, denoted by the position vector $\textbf{r}\in \Omega$. The word `small' here indicates that the volume $dV_\textbf{r}$ is large enough to contain a huge number $N$ of elementary magnetic moments $\mu$$_j$, $j=1,\ldots,N$, but small enough in order that the average magnetic moment varies smoothly. In this respect, we define the magnetization vector field $\textbf{M}(\textbf{r})$, such that the product $\textbf{M}(\textbf{r}) dV_\textbf{r}$ represent the net magnetic moment of the elementary volume $dV_\textbf{r}$:

$$\textbf{M}(\textbf{r})=\frac{\sum_j^N \mbox{\boldmath$\mu$}_j}{dV_\textbf{r}} \quad.$$ (1.1)

Moreover, we assume that the magnetization is also a function of time $t$:

$$\textbf{M}=\textbf{M}(\textbf{r},t) \quad.$$ (1.2)

First of all, it is important to recall that the micromagnetic model [3,4,5] is interested in magnetic phenomena which arise in a wide spatial scale, going from few nanometers (nm) to few microns ($\mu m$).

Figure 1.1: Different kinds of magnetic interactions depending on the distance between dipoles.

The micromagnetic framework includes short and long-range (maxwellian) interactions between magnetic moments. In this respect, we shall start the discussion from the short-range exchange and anisotropy interactions introduced with phenomenological approach. Finally, we will introduce the long-range magnetostatic interactions due to `maxwellian' magnetic fields. All the these interactions can be described in terms of the free energy of the body. In the next section a brief overview of basic thermodynamic laws and definitions is reported before each contribution to the micromagnetic free energy is analyzed in some details.