Magnetic Moment

Intuition

A magnetic moment is the vector that says “this object behaves like a tiny bar magnet, of this strength, pointing this way.” It is the microscopic seed of all magnetism. At the atomic scale it comes from two sources — the orbital motion of electrons around the nucleus, and their intrinsic spin. Add the contributions of all the electrons in an atom and you get the atomic magnetic moment; sum atomic moments over a piece of material and you get the body’s magnetization. The entire ladder of magnetism — from a single electron to a permanent magnet — is built by adding magnetic moments.

Formal Definition

The magnetic moment $\mathbf{\mu }$ is the vector that relates the torque an object feels in an external magnetic field $\vec{B}$ to the field itself:

$$\vec{\tau }=\mathbf{\mu }\times \vec{B}.$$

For a continuous magnetized body of volume $V$, the total magnetic moment is the volume integral of the magnetization:

$${\vec{m}}_{\text{tot}}={\int }_V\vec{M}(\vec{r})dV[\mathrm{A}\cdot {\mathrm{m}}^2].$$

A magnetic moment $\mathbf{\mu }$ also generates a dipolar magnetic field in its surroundings — the field of a “point magnet.”

Key Results

1. Two sources at the atomic scale

In an atom, electrons contribute a magnetic moment through:

1. Orbital motion — an electron circulating around the nucleus is a tiny current loop, generating an orbital moment ${\mathbf{\mu }}_L=-\frac{{\mu }_B}{\hslash }\vec{L}$.
2. Intrinsic spin — every electron carries an irreducible spin angular momentum $\vec{S}$, with associated moment ${\mathbf{\mu }}_S=-g_s\frac{{\mu }_B}{\hslash }\vec{S}$, where $g_s\approx 2$ is the electron spin g-factor.

Spin therefore contributes twice as much magnetic moment per unit of angular momentum as the orbital motion — an irreducibly quantum relativistic effect, predicted by the Dirac equation. The natural unit of either contribution is the Bohr magneton

$${\mu }_B=\frac{e\hslash }{2m_e}\approx 9.274\times 10^{-24}\mathrm{A}\cdot {\mathrm{m}}^2.$$

The total atomic moment is the sum over all electrons; in many cases only the unpaired electrons contribute (paired spins cancel). The combined orbital + spin contribution of a one-electron atom is written compactly as

$$\mathbf{\mu }={\mathbf{\mu }}_L+{\mathbf{\mu }}_S=-{\mu }_B(\vec{L}+2\vec{S})/\hslash ,$$

with the conspicuous factor 2 in front of the spin — Landé’s $g_s$ factor, a strictly relativistic effect that drops out of the Dirac equation. Spin contributes twice as much moment as the equivalent orbital angular momentum, and that factor is what makes spin-dominated elements (Fe, Co, Ni) magnetically loud.

3. The gyromagnetic ratio

Because both the magnetic moment and the angular momentum scale linearly with the same electron, they are rigidly proportional:

$$\mathbf{\mu }=-{\gamma }_e\vec{L},{\gamma }_e=\frac{g_s{\mu }_B}{\hslash }\approx 1.76\times 10^{11}\mathrm{rad}{\mathrm{s}}^{-1}{\mathrm{T}}^{-1}.$$

The proportionality constant ${\gamma }_e$ is the gyromagnetic ratio of the electron. Its physical importance is enormous: in an external field $\vec{B}$ the moment does not line up instantaneously but precesses at the Larmor frequency ${\omega }_L={\gamma }_{e}B$ — exactly like a spinning top under gravity. This single equation is the foundation of:

• EPR (electron paramagnetic resonance),
• FMR (ferromagnetic resonance),
• NMR / MRI (with ${\gamma }_p\ll {\gamma }_e$ for the proton),
• the dynamical LLG equation of micromagnetism.

See larmor-precession for the geometry.

4. From atomic moments to macroscopic magnetization

In a ferromagnet, the exchange interaction locks neighboring atomic moments parallel inside domains. The domain magnetic moment is the vector sum

$${\vec{m}}_{\text{domain}}=\sum _{i\in \text{domain}}{\mathbf{\mu }}_i,$$

and the macroscopic magnetization is the density of such moments: $\vec{M}=d{\vec{m}}_{\text{tot}}/dV$. This is the conceptual ladder from quantum mechanics to a fridge magnet.

Notation Warning

Several conventions are in active use. In this wiki:

Symbol	Meaning	Units
$\mathbf{\mu }$	Microscopic / atomic magnetic moment (“quantum” notation)	A·m²
${\vec{m}}_{\text{tot}}$	Total magnetic moment of a body, ${\int }_V\vec{M}dV$	A·m²
$\vec{M}$	magnetization (moment per unit volume)	A/m
$\vec{m}=\vec{M}/M_s$	Reduced magnetization (everywhere else in this wiki)	dimensionless

Many textbooks write $\vec{m}$ for the magnetic moment instead of $\mathbf{\mu }$ — the two are equivalent there, but inside this wiki the symbol $\vec{m}$ is reserved for the reduced magnetization $\vec{M}/M_s$.

Summary

A magnetic moment is the vector that quantifies the strength and direction of the magnetic field produced by — and the torque felt by — a microscopic source. At the atomic scale it originates from electron spin and orbital motion. At the macroscopic scale, the collective alignment of many atomic moments produces the magnetization of the material.

Connections

• magnetic-atom — which atoms carry a non-zero moment, and why
• magnetization — the continuum field built from atomic moments
• spin — intrinsic source of half the atomic moment (stub)
• exchange-interaction — what aligns neighboring atomic moments (stub)
• magnetic-domains — regions of parallel moments

References

• J. M. D. Coey, Magnetism and Magnetic Materials (Cambridge, 2010), Ch. 3.
• D. J. Griffiths, Introduction to Electrodynamics, §6.1 — magnetic dipoles.
• B. D. Cullity & C. D. Graham, Introduction to Magnetic Materials (Wiley, 2009).