Cyclotron resonance condition (${\omega }_{\mathrm{RF}}={\omega }_c=|q|B/m$)

Take an ion of mass $m$ and electric charge $q\ne 0$, with velocity perpendicular to a uniform magnetic field of magnitude $B>0$. In the classical mechanics limit, the Lorentz force provides the centripetal acceleration:Here $r$ is the local radius of curvature and ${\omega }_c$ is the cyclotron frequency, expressed as an angular frequency. A half-turn takesIncreasing speed increases the path length in exactly the same proportion. Drive the gap at angular frequency ${\omega }_{\mathrm{RF}}={\omega }_c$, or ordinary frequency $f_{\mathrm{RF}}=|q|B/(2\pi m)$. In each half-turn the gap electric field reverses once, just as the particle reverses its direction of passage through the gap. The particle therefore receives two accelerating kicks per complete turn, using one oscillation per complete turn. This is the fundamental resonance; transit time through the gap and departures from a uniform magnetic field have been neglected.

Define $\Delta V={\Phi }_{\mathrm{entry}}-{\Phi }_{\mathrm{exit}}$ as the electric potential drop along the particle's passage. If the electric field changes negligibly during that short crossing, the work givesThe sign matters: a positive ion gains kinetic energy while going to lower electric potential; a negative ion needs the opposite drop. A fixed-sign field would accelerate on one crossing and decelerate on the next. The alternating field must reverse before the particle comes back.

For a fixed electric charge sign and circulation direction, absorb this alternating crossing direction into the definition of the arrival phase ${\varphi }_n$. If $V_0$ is the peak electric potential difference across the whole gap, rather than the amplitude of either electrode relative to ground, a thin-gap model isArrival at ${\varphi }_n=0$ gives the largest accelerating kick; $|{\varphi }_n|<\pi /2$ gives a positive kick. Arrival at ${\varphi }_n=\pi$ gives deceleration. For an ideal classical particle initially on crest, the cyclotron resonance condition keeps successive arrival phases equal. It does not automatically bring an initially mistimed particle onto crest.