Mass-separation by standing waves near the cyclotron frequency

Standing waves near the ion cyclotron frequency push ions along the axial steady magnetic field by the nonlinear ponderomotive force. The direction of this axial force is determined by whether the wave frequency is larger or smaller than the ion cyclotron frequency. This dependence can be used for ion mass separation, a much-needed process for various applications. In traveling waves there is no axial pushing, but there is ion cyclotron heating, which is linear in the wave fields. We calculate single-particle trajectories under forces by standing waves, where there is an axial ponderomotive force, and compare them to the trajectories under traveling waves, where the axial ponderomotive force is absent, but there is cyclotron heating. The equations of motion are solved for the case that the particle displacement is much smaller than a quarter wavelength. In that case we present analytical solutions that show the particle pushing along the magnetic field with the mass-dependent direction as well the growth of particle perpendicular velocity. We find numerically the particle trajectories when the particle displacement is close to a quarter wavelength and show when the trajectory deviates from the analytical solution.