Larmor Frequency

Larmor frequency: ω = γB

The Larmor frequency tells you how fast a magnetic moment precesses around an external field: ω_L = γ·B, where γ is the gyromagnetic ratio (which depends on the particle) and B is the magnetic field in tesla. For the proton, γ/2π = 42.58 MHz/T. That works out to ≈63.87 MHz in a clinical 1.5 T MRI, ≈127.7 MHz at 3 T, and around 298 MHz on a 7 T research scanner. The relation is named after Joseph Larmor (1897), and it fixes the radio-frequency tuning of every NMR and MRI experiment. Example: a proton in a 1.5 T field has ω_L = 2.675·10⁸ · 1.5 ≈ 4.01·10⁸ rad/s (≈63.87 MHz).

Applications: MRI, NMR spectroscopy, atomic clocks

The Larmor relation sets the resonant frequency of clinical MRI scanners. It also drives the NMR spectroscopy that chemists use to identify molecules through chemical shifts, makes MRSI possible by combining a spectrum with an image, tunes atomic clocks built on hyperfine transitions, and underlies the magnetometers that pick up tiny magnetic fields.

FAQ

What is the gyromagnetic ratio? It's a particle's magnetic moment divided by its angular momentum. The proton has γ = 2.675·10⁸ rad/s/T, and for the electron γ is about 658× larger.

Why does MRI use higher fields? A stronger B pushes ω_L up, and that buys you a better signal-to-noise ratio and finer spatial resolution. The trade-off is higher cost and tighter safety constraints.

What is chemical shift? Electrons shield the nucleus to some degree, which nudges the local B and therefore ω_L. NMR spectroscopy reads those tiny shifts, measured in ppm, to map out molecular structure.

Is the formula relativistic? No. ω_L = γB is non-relativistic. At very strong fields, QED corrections and relativistic effects start to change it, but the simple form holds up fine for ordinary lab work and clinical MRI.