Larmor Precession

Equivalent forms

f_L = \frac{\gamma B}{2\pi}

\frac{d\boldsymbol{\mu}}{dt} = \gamma\,\boldsymbol{\mu}\times\mathbf{B}

The precession rate is independent of the tilt angle — every proton in a 1.5 T magnet hums at exactly the same 63.9 MHz.

Symbol	Name	SI unit	Dimension	Range
$omega_L$	Larmor angular frequencyoutput Angular rate of precession about the field axis	rad/s	T^-1	0 – 400000000000
$gamma$	Gyromagnetic ratio Magnetic moment per unit angular momentum (electron: 1.7609e11, proton: 2.6752e8)	rad/(s*T)	M^-1TI	100000000 – 200000000000
$B$	Magnetic flux density Strength of the applied static magnetic field	T	MT^-2I^-1	0 – 10

Unit systems

Symbol	SI	CGS	Imperial
$omega_L$	rad/s	rad/s	rad/s
$gamma$	rad/(s*T)	rad/(s*G)	rad/(s*T)
$B$	T	G	T

Where it holds

Exact for an isolated magnetic moment in a uniform static field. Chemical-shift screening, field gradients, and spin-spin couplings shift or broaden the frequency — which is precisely what NMR spectroscopy measures.

Dimensional analysis

omega_L -> T^-1 (rad/s)

gamma*B ->

[M^-1*T*I] * [M*T^-2*I^-1] = T^-1

Both sides are inverse time.

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Discovery

Joseph Larmor · 1897

Larmor derived the precession classically for orbiting charges decades before spin was known. Rabi's 1937 resonance method exploited it to measure nuclear moments, and Bloch and Purcell's 1946 NMR work — built entirely on this one frequency — became the foundation of MRI.

Try this

How does an MRI machine make the protons in your body sing at 64 MHz?

Put a magnetic moment in a field and it doesn't snap into alignment — it precesses around the field like a tilted gyroscope, at a frequency set only by the field strength and a constant of the particle. That frequency is the radio signal MRI listens to.

Research status: stable

Real-world applications

MRI: hydrogen protons at 63.9 MHz in a 1.5 T scanner
NMR spectroscopy for molecular structure determination
Atomic magnetometers and the proton precession magnetometer used in geophysics
Muon g-2 precision tests of the Standard Model

Common misconceptions

Precession frequency does NOT depend on tilt angle — only the signal amplitude does
The spin is not 'spinning faster' in a stronger field — its precession about the field axis is
An aligned spin does not precess at all; precession requires a transverse component (a superposition)

Experimental verification

NMR resonance lines sit at gamma*B/(2*pi) to parts per billion — the basis of chemical-shift spectroscopy. Muon spin rotation (muSR) tracks precessing muons inside materials. The g-2 experiments at Fermilab measure tiny anomalies in the muon's precession rate against this baseline.

Derivation

Quantum: the Hamiltonian

H = -\gamma *B*S_z

splits spin states by

\delta E = \gamma

*hbar*B.

A superposition of up and down acquires relative phase exp(-i*delta E*t/hbar), so the transverse spin expectation rotates at

\omega = \delta E

/hbar

= \gamma *B

.

Classical: torque mu x B on angular momentum

S = \mu /\gamma

gives

dS/dt = \gamma *S x B

— same frequency, a rare exact quantum-classical agreement (Ehrenfest at work).

Limiting cases

B = 0

⟶

\omega _L = 0

No field, no precession — the spin direction is frozen.

Tilt angle -> 0 (spin along B)⟶ energy eigenstate, stationaryAligned spin states are stationary; precession is visible only for superpositions (tilted spins).

Add resonant transverse drive at

\omega = \omega _L

⟶ Rabi oscillationsDriving at the Larmor frequency tips the spin coherently — the basis of NMR pulses.

What if…

What if the field is doubled?

The precession frequency doubles exactly — that linearity lets MRI use field gradients to encode position into frequency.

What if you watch from a frame rotating at omega_L?

The static field vanishes from the dynamics — only the small RF drive remains. This rotating-frame trick makes NMR pulse design tractable.

What if neighboring spins disturb each other?

Local field variations dephase the precession — the decay times T1 and T2 that result are exactly the contrast mechanisms of MRI images.

1

Proton in a 1.5 T MRI scanner

Given ·

gamma:: 267522187.44
B:: 1.5

Find · f_L

Steps

$\omega _L = \gamma *B = 2.6752218744e_{8} * 1.5 = 4.013e_{8} rad/s$
$f_L = \omega _L/(2*pi) = 4.013e_{8} / 6.2832$
$\sim 6.39e_{7} Hz = 63.9\,\mathrm{MHz}$

Result ·

f_L = \gamma *B/(2*pi) \sim 63.9\,\mathrm{MHz}

— the scanner's RF coils transmit and receive at this frequency.

2

Electron in a 1 T laboratory field

Given ·

gamma:: 176085963000
B:: 1

Find · f_L

Steps

$\omega _L = 1.76085963e11 * 1\,\mathrm{rad/s}$
$f_L = 1.76085963e11/(2*pi)$
$\sim 2.80e10\,\mathrm{Hz} = 28.0\,\mathrm{GHz}$

Result ·

f_L \sim 28.0\,\mathrm{GHz}

— microwave range, which is why ESR uses microwave cavities.

Stern–Gerlach Deflection→Spin-1/2 Operators→Rabi Oscillations→Quantization of Angular Momentum→