The magnetization vector

Consider a ferromagnet with aligned magnetic moments

The magnetization is the density of magnetic moments $\bm{\mu}$ in a small volume $\Delta V$,

$$ \mathbf{M} = \frac{\Delta\bm{\mu}}{\Delta V}. $$

(Left) Magnetic domains. (Right) Saturated magnet.

The magnetisation may be different in different regions of the magnet. The volumes were the magnetization is roughly constant are called magnetic domains.

Consider that, by applying a strong magnetic field, we align all magnetic moments (''saturate'' the magnetisation) along the field direction and then we can measure the saturation magnetisation $M_s$.

A disc-shaped magnetic particle. (Left) Single domain. (Right) Two domains (bidomain). [Moutafis et al, PRB, 2007.]

Magnetization configurations. [Tonomura et al.]

The Bloch sphere

The magnetization vector takes values on the Bloch sphere

The following assumtion is made.

A ferromagnet is described by the magnetisation vector $\mathbf{M}=\mathbf{M}(x,t)$ with

$$ |\mathbf{M}|=M_s (=\text{const.}). $$

The magnetization is made up of many spins. Therefore, it is a strong assumption to constraint $\mathbf{M}$ to have a fixed length.

[See, Landau, Lifshitz, Pitaevskii, ``Statistical Physics'' Vol II.]