Larmor Precession

The core idea

Place a tiny magnet — an atom, a nucleus, an electron — into a uniform magnetic field B. Naively you might expect it to swing into alignment with the field like a compass needle. But these microscopic magnets are not inert needles: they carry angular momentum. The magnetic moment μ and the angular momentum L point along the same axis, locked together by the gyromagnetic ratio:

μ = γ L

The field exerts a torque τ = μ × B. Crucially this torque is perpendicular to both μ and B, so it does not pull the moment toward B — it pushes the tip sideways. And torque is the rate of change of angular momentum (dL/dt = τ), so the angular momentum vector swings in the direction of the torque. The result: the moment circles around B at a fixed tilt angle, tracing a cone. That circular wobble is Larmor precession.

The equation of motion

Combine μ = γL with dL/dt = μ × B to get the governing equation directly in terms of the moment:

dμ/dt = γ (μ × B)

For a field along the z-axis, B = B ẑ, this is the equation of a vector rotating about z at constant rate. Writing it out component by component:

dμ_x/dt = γ B μ_y
dμ_y/dt = −γ B μ_x
dμ_z/dt = 0

The z-component never changes — the cone angle is fixed. The x and y components rotate at angular frequency

ω_L = γ B        (Larmor angular frequency)
f_L = γ B / 2π   (Larmor frequency in Hz)

Three facts fall straight out of this and surprise newcomers:

• The rate is independent of the tilt angle. A moment tipped 5° from B precesses at exactly the same frequency as one tipped 85°.
• The rate is independent of the moment's magnitude. Doubling μ also doubles L, and the two cancel.
• The rate is proportional to the field. Double B and you double the precession frequency. This linearity is what makes magnetic-resonance imaging spatially resolvable.

The gyromagnetic ratio

The single number γ controls everything. It is the ratio of magnetic moment to angular momentum and is an intrinsic property of each particle. For a classical orbiting charge q of mass m, γ = q/2m. For real particles the value is modified by a dimensionless g-factor: γ = g · q/2m. The electron's g ≈ 2.0023 (the famous "g minus 2"), nearly twice the classical orbital value.

Particle	γ (rad·s⁻¹·T⁻¹)	γ/2π	Sense
Electron	−1.761 × 10¹¹	28.0 GHz/T	opposite to B
Proton (¹H)	2.675 × 10⁸	42.58 MHz/T	same as B
Neutron	−1.832 × 10⁸	29.16 MHz/T	opposite to B
Carbon-13	6.728 × 10⁷	10.71 MHz/T	same as B
Fluorine-19	2.518 × 10⁸	40.05 MHz/T	same as B

Because the electron's γ is about 658 times the proton's, an electron in the same field precesses 658 times faster — gigahertz versus megahertz. That is why electron spin resonance (ESR/EPR) lives in the microwave band while NMR sits in the radio band. A negative γ simply means the moment precesses in the opposite rotational sense (the moment points opposite to L).

Numerical examples

Scenario	Larmor frequency
Proton in Earth's field (≈ 50 µT)	f_L ≈ 42.58 × 0.00005 ≈ 2.1 kHz
Proton in 1.5 T clinical MRI	f_L ≈ 63.9 MHz
Proton in 3 T clinical MRI	f_L ≈ 127.7 MHz
Proton in 23.5 T research NMR (1 GHz spectrometer)	f_L ≈ 1.0 GHz
Electron in 0.35 T (X-band ESR)	f_L ≈ 9.8 GHz
Electron in Earth's field (≈ 50 µT)	f_L ≈ 1.4 MHz

The proton example in Earth's field is the basis of the proton precession magnetometer — polarize the protons in a water sample, release them, and measure their ~2 kHz precession to read the local field to nanotesla accuracy. Surveyors and geophysicists use exactly this device to map magnetic anomalies.

JavaScript — simulating precession

// Larmor frequency from field and gyromagnetic ratio
function larmorFreqHz(gammaOver2pi_HzPerT, B_tesla) {
  return gammaOver2pi_HzPerT * B_tesla; // Hz
}

const PROTON = 42.577e6; // Hz/T
console.log(`1.5 T MRI proton: ${(larmorFreqHz(PROTON, 1.5)/1e6).toFixed(1)} MHz`); // 63.9
console.log(`3.0 T MRI proton: ${(larmorFreqHz(PROTON, 3.0)/1e6).toFixed(1)} MHz`); // 127.7

const ELECTRON = 28.025e9; // Hz/T
console.log(`X-band ESR field for 9.5 GHz: ${(9.5e9/ELECTRON*1000).toFixed(1)} mT`); // 339 mT

// Integrate dμ/dt = γ μ × B for a field along z (analytic cone)
function precess(mu0, gamma, Bz, t) {
  const omega = gamma * Bz;             // signed angular frequency (rad/s)
  const ct = Math.cos(omega * t), st = Math.sin(omega * t);
  return {                              // rotate (x,y) about z, leave z fixed
    x: mu0.x * ct + mu0.y * st,
    y: -mu0.x * st + mu0.y * ct,
    z: mu0.z
  };
}

const gammaP = 2.675e8;                 // rad/s/T for proton
const mu0 = { x: 0.6, y: 0, z: 0.8 };   // tilted ~37 degrees from field
const period = 2 * Math.PI / (gammaP * 1.5);
console.log(`Period in 1.5 T: ${(period * 1e9).toFixed(2)} ns`); // ~15.6 ns
console.log(precess(mu0, gammaP, 1.5, period / 4)); // quarter turn

// Cone angle is conserved: |z| component never changes
const after = precess(mu0, gammaP, 1.5, 1.234e-9);
console.log(`mu_z constant? ${after.z === mu0.z}`); // true

Quantum view: it is the same picture

Quantum mechanically a spin-½ particle in a field has two energy levels split by the Zeeman energy ΔE = ħγB. A spin prepared in a superposition of those levels has an expectation value ⟨μ⟩ that — by Ehrenfest's theorem — obeys exactly the classical equation d⟨μ⟩/dt = γ⟨μ⟩ × B. So the average spin vector precesses at the Larmor frequency, and the level splitting corresponds to the same energy:

ΔE = ħ ω_L = ħ γ B

This is why a photon of frequency f_L resonantly drives transitions between the two spin states — the resonance condition of NMR and ESR is identical to the classical precession frequency. Larmor precession is one of the cleanest places where the classical and quantum descriptions line up exactly.

Where Larmor precession shows up

• NMR spectroscopy. Chemists read precession frequencies of nuclei (¹H, ¹³C, ¹⁹F); tiny shifts from the local electron cloud — the chemical shift — reveal molecular structure.
• MRI. Field gradients make ω_L vary with position, so frequency encodes location. A radio pulse at f_L tips the magnetization; the precessing signal reconstructs an image.
• Electron spin resonance (ESR/EPR). Probes unpaired electrons in radicals, transition-metal complexes, and defects, in the microwave band.
• Atomic clocks and magnetometers. Optically pumped atoms and proton-precession magnetometers measure fields by their precession frequency to extreme precision.
• Astrophysics and plasmas. Charged particles gyrate around field lines; the electron gyrofrequency sets cyclotron and synchrotron emission.
• Spintronics and quantum computing. Single-spin qubits are manipulated by tuning B and driving at the Larmor frequency (Rabi control).

Common mistakes

• Thinking the moment aligns with the field. Without dissipation it never does — it precesses at a fixed angle forever. Alignment requires relaxation (energy loss to the surroundings).
• Believing the rate depends on the tilt angle. It does not. ω_L = γB regardless of how far the moment is tipped.
• Confusing it with cyclotron motion. Cyclotron motion is a charge's spatial circling in a field (ω_c = qB/m); Larmor precession is the orientation of a moment rotating. They share the q/m scaling but describe different things.
• Dropping the sign of γ. A negative γ (electron, neutron) reverses the precession sense; this matters for phase conventions in NMR and for which circular polarization drives resonance.
• Assuming the torque does work. The torque is always perpendicular to μ, so ideal precession conserves energy — the cone angle is constant, no work is done.
• Forgetting the factor of 2 from the g-factor. Electron spin γ uses g ≈ 2, not the classical orbital g = 1; using the wrong value gives a frequency off by roughly a factor of two.