Spinning tops in a magnetic field
Protons, many other nuclei and electrons behave like tiny magnets that also spin. In a magnetic field B₀ they do not simply line up: like a spinning top tilted in gravity, their magnetic moment precesses around the field direction. The rate of that precession is the Larmor frequency, named after the Irish physicist Joseph Larmor. A radio wave at exactly this frequency can flip the spins, which is the resonance used in NMR spectroscopy and MRI.
The formula
- Larmor frequency: f = γ̄ × B₀, where γ̄ = γ ÷ 2π is the reduced gyromagnetic ratio of the particle
- Angular frequency: ω = 2π × f = γ × B₀
- Period: T = 1 ÷ f
Values of γ̄ used here, in MHz per tesla: hydrogen-1 42.577, fluorine-19 40.078, phosphorus-31 17.235, sodium-23 11.262, carbon-13 10.708, nitrogen-15 −4.316, and the electron −28,025. The sign only tells the direction of precession; the frequency itself is positive.
Example: a 1.5 T MRI scanner
- Proton frequency: 42.577 × 1.5 = 63.87 MHz
- Angular frequency: 4.01 × 10⁸ rad/s
- Period of one precession: 15.7 ns
- Wavelength of that radio wave in vacuum: 4.69 m
At 3 T the frequency doubles to 127.7 MHz. The wavelength in the body is much shorter than in vacuum, roughly 25 to 30 cm at 3 T because water slows radio waves, which is one cause of uneven brightness in high-field images.
Carbon-13 in the same 1.5 T magnet resonates at only 16.06 MHz, which is why a spectrometer needs a separate channel for each nucleus.
Why it matters
- MRI encodes position by adding small field gradients: each slice of the body sees a slightly different B₀ and so answers at a slightly different frequency.
- In NMR, the electrons around a nucleus shield it a little, shifting its frequency by a few parts per million. This chemical shift reveals the structure of molecules.
- Electron spin resonance uses the much larger electron ratio, about 28 GHz per tesla, so it works with microwaves.
Limits
The gyromagnetic ratios are those of bare nuclei. In a real sample the shielding changes them by parts per million, too little to matter for the field strength but essential for spectroscopy.