Larmor Precession

Intuition

Place a magnetic moment $\mathbf{\mu }$ in a field $\vec{B}$ and your first guess is that it snaps into alignment, like a compass needle. It doesn’t. Instead it precesses — its tip traces out a cone around the field direction, at constant angle, like a spinning top under gravity. The reason is that $\mathbf{\mu }$ carries angular momentum: a torque can’t simply pull it in, it has to turn the angular momentum vector. The resulting circular motion is Larmor precession, and it is the conservative (energy-preserving) heart of the LLG equation: the first term in LLG describes nothing else. In real magnets, damping eventually closes the cone — but for the duration of the precession, $|\vec{m}|$ and the angle between $\vec{m}$ and ${\vec{H}}^{\text{eff}}$ are both conserved.

Formal Definition

A magnetic moment in a field experiences a torque

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

Because $\mathbf{\mu }$ is proportional to the carrier’s angular momentum, Newton’s law for angular momentum ($\dot{\vec{L}}=\vec{\tau }$) becomes an equation of motion for $\mathbf{\mu }$ itself. In micromagnetic notation, applied to the reduced magnetization $\vec{m}$ in the effective field ${\vec{H}}^{\text{eff}}$, this is the Landau–Lifshitz (precessional) term:

$$\frac{\partial \vec{m}}{\partial t}=-{\gamma }_0\vec{m}\times {\vec{H}}^{\text{eff}},{\gamma }_0\equiv {\mu }_0{\gamma }_G.$$

The cross product on the right is always perpendicular to $\vec{m}$, so $|\vec{m}|$ is preserved. Likewise the angle between $\vec{m}$ and ${\vec{H}}^{\text{eff}}$ is preserved — only the azimuthal angle around ${\vec{H}}^{\text{eff}}$ evolves.

Key Results

1. The Larmor frequency

For a static, uniform ${\vec{H}}^{\text{eff}}$, the equation of motion is linear and solved by uniform precession at the Larmor angular frequency

$${\omega }_L={\mu }_0{\gamma }_G{H}^{\text{eff}}={\gamma }_0{H}^{\text{eff}},$$

with Larmor period

$$T_L=\frac{2\pi }{{\omega }_L}.$$

The frequency scales linearly with the field magnitude — strong fields mean fast precession.

For an electron, ${\gamma }_G/2\pi \approx 28GHz/T$, so a field of $1\mathrm{T}$ produces precession at $\sim 28\mathrm{GHz}$ — squarely in the microwave range, which is why ferromagnetic resonance (FMR) lives there. The numerical value follows directly from the electron gyromagnetic ratio ${\gamma }_e=g_s{\mu }_B/\hslash \approx 1.76\times 10^{11}\mathrm{rad}{\mathrm{s}}^{-1}{\mathrm{T}}^{-1}$.

1bis. The same physics, three resonances

Larmor precession is the single mechanism behind a whole family of experimental techniques. Only the carrier — and therefore $\gamma$ — changes:

Technique	Carrier	Used for
EPR / ESR	unpaired electron spin in a paramagnet	identifying defects, radicals, transition-metal sites
FMR	the collective magnetization $\vec{M}$ of a ferromagnet	damping ${\alpha }_G$, anisotropy fields, spin-wave spectra
NMR / MRI	nuclear spin (mostly ${}^1$H) — ${\gamma }_p\approx 2.7\times 10^8$ rad/s/T, $\sim 660\times$ smaller	medical imaging, chemistry, neuroscience

In all three cases, a static field ${\vec{B}}_0$ sets the Larmor frequency ${\omega }_L=\gamma B_0$, and a transverse microwave (or radiofrequency) field tuned to ${\omega }_L$ tips the magnetization. The huge spread of $\gamma$ across electrons/nuclei is why EPR sits in the GHz band and NMR in the MHz band at the same applied field.

2. Sense of precession

The minus sign in the equation of motion (inherited from the negative electron gyromagnetic ratio) makes the precession clockwise when viewed along ${\vec{H}}^{\text{eff}}$. The motion is purely kinematic: no energy is gained or lost, and the cone angle is fixed by the initial condition.

3. With damping — the cone closes

In a real ferromagnet, the Gilbert term of the LLG equation adds a viscous torque that pulls $\vec{m}$ toward ${\vec{H}}^{\text{eff}}$. The motion becomes a shrinking spiral at frequency ${\omega }_L$, with the cone angle decaying on the timescale $\tau \sim 1/({\alpha }_G{\omega }_L)$ until $\vec{m}\parallel {\vec{H}}^{\text{eff}}$.

Summary

Quantity	Expression	Meaning
Torque	$\mathbf{\mu }\times \vec{B}$	drives precession, not alignment
Equation of motion	${\partial }_t\vec{m}=-{\gamma }_0\vec{m}\times {\vec{H}}^{\text{eff}}$	LL (precessional) term of LLG
Larmor frequency	${\omega }_L={\gamma }_0{H}^{\text{eff}}$	rate of precession
Period	$T_L=2\pi /{\omega }_L$	one orbit
Conserved	$	\vec{m}

Larmor precession is what gives magnetization its clock: every dynamical phenomenon in micromagnetism — FMR, spin waves, vortex gyration, STT-driven switching — beats at, or is detuned from, the local Larmor rhythm.

Connections

• llg-equation — Larmor precession is the LL term; add Gilbert damping and STT to get the full equation
• effective-field — what plays the role of ${\vec{H}}^{\text{eff}}$ in a real micromagnet
• magnetic-moment — the carrier whose angular momentum makes precession happen
• magnetization — the field that precesses in the continuum theory
• thiele-equation — gyrotropic vortex motion is a collective-coordinate cousin of Larmor precession

References

• J. Larmor, Phil. Mag. 44, 503 (1897) — original derivation.
• C. Kittel, Introduction to Solid State Physics, ch. on magnetic resonance.
• D. D. Stancil & A. Prabhakar, Spin Waves: Theory and Applications (Springer, 2009).