Larmor precession

If an electron is constrained somehow to move in an orbit, it has an associated magnetic moment m =γ, where γ is the gyromagnetic ratio. The effect of the magnetic field is to exert a torque

Γ = m × B .......(1)

on the current loop. Newton’s law for angular momentum Γ = d/dt gives

dm/dt = γm × B. .........(2)

When B is along the z-axis, the vector product in Cartesian coordinates gives

........(3)

The z component m_z = mcos θ is independent of time, but the x and y components oscillate. The solution is m(t ) =(msin θ sin ω_Lt , msin θ cos ω_Lt , mcos θ),where ω_L = γB. The magnetic momentmtherefore precesses around the applied field direction at the Larmor frequency f_L = ω_L/2π; here

.....(4)

The precession continues indefinitely if there is no way for the system to dissipate energy, and the angular momentum remains constant. Note that the Larmor precession frequency for an orbital moment (γ = −e/2m_e) is just half the cyclotron frequency, 28 GHz T⁻¹, whereas it is equal to the cyclotron frequency for a spin moment (γ = −e/m_e). The precession of the spin angular momentum around Oz occurs at the Larmor frequency.