Normal modes and resonant confinement of charged particles in oscillating electric and magnetic fields

Abstract

The classical motion of a charged particle in oscillating magnetic and electric fields is investigated. Introducing a rotating coordinate system frame greatly simplifies the equations of motion and allows a formal analogy with the problem of three coupled oscillators with anisotropic damping. The solutions are calculated analytically. It is shown that when the resonance condition is achieved, that is, the frequency of the oscillating field equals the cyclotron frequency of the particle (ω c ), the particle is confined to a region of space of volume given approximately by 8v 3 0 /3ω c ω 2 1 , where v 0 is the modulus of the initial velocity of the particle. ω 1 = qB 1 /m is the frequency of the particle about B 1 , the magnitude of the oscillating field. The orbit of the particle is shown to be a closed curve in the rotating frame, or a closed surface in the laboratory system. Off resonance, the particle drifts away. We simulate the motion of particles with charge-to-mass ratios differing by about 1%, and show that they can be resonantly separated. The results suggest a resonant method for charged particle confinement and isotope separation.