Atomic Nuclei: Larmor Precession Frequency

The earth's gravitational field produces a 'twisting force' perpendicular to the angular momentum of a spinning mass (such as a spinning top) that causes the mass to 'wobble' around the gravitational field axis in a phenomenon called precession. Similarly, the magnetic moment (μ) of a spinning nucleus precesses due to an external magnetic field directed along the z-axis. The precession of the magnetic moment vector about the magnetic field is called Larmor precession, and the angular frequency of precession (ω) is called the precessional or Larmor frequency, which is directly proportional to the applied magnetic field (B₀). The proportionality constant is the gyromagnetic ratio (γ), which is fixed for each nucleus. This is because each nucleus has a unique ratio of magnetic moment to angular momentum on account of its charge and mass. For a hydrogen nucleus (a proton), the value of γ is 2.675 × 10⁸ T⁻¹ s⁻¹. Expressing ω in terms of the operating frequency (ν) of the NMR instrument, rearranging the terms, and multiplying both sides of the equation by Planck's constant yields the relationship between the Larmor frequency and the energy difference between the spin states (ΔE).