# Micromagnetism

For this chapter we follow {cite}`abert2019`. The micromagnetic model is used at the micron scale, and the main assumption is that the magnetization $\mathbf{M} (\mathbf{x})$ is a continuous field $\Mbf: \Omega \to \Omega$ with constant magnitude and can be written as

$$
\mathbf{M} (\xbf) = \Ms \ \mbf (\xbf),
$$

where  $\Omega \subset \Rbb^3$ is the magnetic domain, the field $\mbf$ is the *unit magnetization* (also called slightly incorrectly *magnetization*) and satisfied $\| \mbf (\xbf) \| = 1$ on $\bar\Omega$, and $\Ms \in \Rbb$ is the *spontaneous magnetization*. This varies with temperature and at temperature $0 K$ it is called *saturation magnetization*.

Let us define $\Mcal$ the set of all possible unit magnetization profiles, i.e.

$$
\Mcal := \{ \mbf = (m_1, m_2, m_3) \ : \ m_i \in H^1(\Omega) \ \text{for} \ i=1,2,3 \}.
$$

In this notation, we can write the unit condition as

$$
m_1^2 (\xbf) + m_2^2 (\xbf) + m_3^2 (\xbf) = 1 \qquad \forall \xbf \in \bar\Omega = \Omega \cup \partial\Omega.
$$

Let us furthermore observe that this set together with the norm

$$
\| \mbf \|_\Mcal = \left[ \| m_1 \|_{H^1(\Omega)}^2 + \| m_2 \|_{H^1(\Omega)}^2 + \| m_3 \|_{H^1(\Omega)}^2 \right]^{1/2}
$$

is a Banach space.