Particle Motion in the Presence of Three-Dimensional Finite Amplitude Harmonic Cyclotron Waves

The motion of charged particles in six-dimensional phase space is investigated when the particles are influenced by a strong uniform magnetic field and finite amplitude electric fields with frequencies at or near a harmonic of the cyclotron frequency. In general, it is found that the particles move in a vortex filament pattern in this six-dimensional phase space. Using the constraint of this motion the amount of energy available to go into electric field energy is estimated for a loss-cone distribution function with several assumed simple types of three-dimensional wave forms. It is found that only a small fraction of the particle kinetic free energy can go into waves.