Magnetization

Intuition

Inside a magnetic material, every atom carries a tiny magnetic moment — a microscopic compass needle, born from unpaired electron spins and orbital motion. Most of the time these compass needles point every which way and their effects cancel. But when they line up — spontaneously, as in a ferromagnet, or under an applied field — the material as a whole acquires a net magnetic character. The magnetization $\vec{M}$ is the field that captures this collective alignment: at every point in space, it tells you the density and direction of the local magnetic moments. It is the quantity that ultimately is the magnet.

Formal Definition

The magnetization $\vec{M}(\vec{r})$ is the magnetic moment per unit volume:

$$\vec{M}(\vec{r})=\underset{\Delta V\to 0}{\lim }\frac{\Delta \mathbf{\mu }}{\Delta V}[A/m],$$

where $\Delta \mathbf{\mu }={\sum }_{i\in \Delta V}{\mathbf{\mu }}_i$ is the sum of the atomic magnetic moments contained in a small volume $\Delta V$ around $\vec{r}$. The limit is taken in the continuum sense: $\Delta V$ is small on the macroscopic scale but large compared to the atomic lattice, so that $\vec{M}$ varies smoothly.

In a saturated ferromagnet, $|\vec{M}|=M_s$, the saturation magnetization — a material-specific constant. Micromagnetics works with the reduced (unit) magnetization

$$\vec{m}(\vec{r},t)=\frac{\vec{M}(\vec{r},t)}{M_s},|\vec{m}|=1,$$

which is the field whose dynamics are governed by the LLG equation.

Key Results

1. What $\vec{M}$ encodes

The magnetization vector field captures three pieces of information at once:

1. The density of magnetic dipoles in the material.
2. The degree of alignment of those dipoles.
3. The direction of the net alignment.

2. Bridging scales

While a single magnetic moment $\mathbf{\mu }$ is a property of an individual atom or ion (units: A·m²), the magnetization $\vec{M}$ is a macroscopic field (units: A/m). Magnetization is the continuum coarse-graining that lets us forget the lattice and treat the magnet as a continuous medium — the starting point of micromagnetism.

3. Role in Maxwell’s equations — B, H, and M

Magnetization enters macroscopic electromagnetism through the constitutive relation

$$\vec{B}={\mu }_0(\vec{H}+\vec{M}).$$

The three fields play distinct roles:

• $\vec{B}$ — the magnetic flux density: what is really present inside the matter, and what determines the Lorentz force on a moving charge.
• $\vec{H}$ — the magnetic field: tied to free currents (the currents the experimentalist controls), i.e. what is “imposed from outside.”
• $\vec{M}$ — the response of the material: the density of magnetic moments induced by, or pre-existing in, the medium.

So $\vec{M}$ is exactly the source of the difference between $\vec{H}$ and $\vec{B}$ in matter. Spatial variations of $\vec{M}$ act as bound currents and magnetic charges that source the stray (demagnetizing) field ${\vec{H}}^{\text{ms}}$ — the nonlocal ingredient of the magnetostatic energy.

4. Linear response: susceptibility and permeability

For most non-ferromagnetic materials, and for ferromagnets in weak fields away from saturation, the response is linear:

$$\vec{M}=\chi \vec{H},\vec{B}={\mu }_0(1+\chi )\vec{H}\equiv {\mu }_0{\mu }_r\vec{H},$$

with

• $\chi$ — the (dimensionless) magnetic susceptibility, which can be positive, negative, or strongly direction-dependent,
• ${\mu }_r=1+\chi$ — the relative permeability of the medium.

The sign and magnitude of $\chi$ are what classify materials as diamagnetic, paramagnetic, or ferromagnetic — see magnetic-materials for the full taxonomy.

5. Remanent magnetization

In a ferromagnetic material, $\vec{M}$ can remain non-zero after the external field is removed:

$${\vec{M}}_r=\vec{M}({\vec{H}}^{\text{ext}}=0)\text{(remanence)}.$$

This is the defining feature of permanent magnets — and a direct consequence of hysteresis in the domain structure.

Summary

Quantity	Symbol	Units	Scale
Magnetic moment	$\mathbf{\mu }$	A·m²	atomic (single dipole)
Magnetization	$\vec{M}$	A/m	continuum (density of moments)
Saturation magnetization	$M_s$	A/m	material constant
Reduced magnetization	$\vec{m}=\vec{M}/M_s$	dimensionless	$

Magnetization tells us how magnetized a material is, in what direction, and at every point in space — the foundational field of micromagnetism.

Connections

• magnetic-moment — the atomic-scale building block
• llg-equation — dynamics of the reduced field $\vec{m}$
• micromagnetic-energy — energy functional $W[\vec{m}]$
• magnetic-domains — spatial structure of $\vec{M}$ at equilibrium
• hysteresis — origin of remanent magnetization (stub)

References

• J. M. D. Coey, Magnetism and Magnetic Materials (Cambridge, 2010), Ch. 2.
• B. D. Cullity & C. D. Graham, Introduction to Magnetic Materials (Wiley, 2009).