Equivalent equilibrium ion distributions in ion sources and beams
Ion sources and ion beams often contain ions of more than one charge-to-mass ratio, due to the presence of multiple ionization states and/or multiple atomic and molecular ion species in the source. We show that, under certain conditions, analysis of these systems can be simplified using the notion of “equivalent equilibrium ion distributions” (EEIDs). These distributions have identical ion trajectories, space-charge distributions, and electric-field distributions, but can be composed of ion species with arbitrary charge-to-mass ratios. We use this result to obtain multi-species generalizations for the perveance of an ion diode, which relates the diode voltage to the extracted beam current, and for the perveance of a drifting ion beam, which determines the axial length-scale for beam expansion due to the self-electric field. Applied to a quasi-neutral plasma ion source containing cold ions and hot electrons, we find that the location of the beam-plasma interface (meniscus), where ions are extracted from the plasma, depends on the electron density and ion energy per unit charge at the interface, independent of the ion composition. For certain types of numerical simulations, EEIDs allow one to calculate the beam optics of an intense multi-species ion source by doing a simulation for a single ion species with an artificial charge-to-mass ratio chosen to minimize computation time. We illustrate these results using the 3D particle-in-cell, finite-element code Aleph. The simulations show how ions in a plasma source self-consistently accelerate to the supersonic speed needed for steady, radially expanding flow. Multi-species equilibria are subject to streaming instabilities because, while the different species have the same kinetic energy per unit charge, their velocities scale with the square root of the charge-to-mass ratio. The simulations show that a strong ion streaming instability can occur in a multi-species plasma ion source.