The classic sheath model for unmagnetized positive ion extraction codes was developed in the seventies. It assumes that the extracted ions originate from a positive plasma potential and experience a monotonic acceleration through the plasma sheath towards the extraction at negative potential. The model assumes that the compensating electron density can be stated as an analytic function of the potential. The classic model has been shown to predict the properties of extracted beams well.
Later, the use of the successful model extended to negative ion extraction systems, especially H$^-$, by changing the assumed potential distribution and by adding e$^-$, H$^+$, H$_2^+$, H$_3^+$ and possibly other particle populations in addition to the H$^-$ either by tracking the particles through the E-field and the geometry or by assuming that the particle densities are analytic functions of the potential. This type of codes have shown decent accuracy predicting the extracted H$^-$ and e$^-$ properties despite the fact that the effect of the filter magnetic field on the electrons in the plasma is not taken in account properly and that the particle flux emitted by the plasma is assumed to be uniform. Especially in the case of surface-enhanced production of H$^-$ it is believed that the
particle fluxes near the plasma electrode aperture are not uniform. In these cases methods are needed to model such cases as it is of importance to understand the effect of the nonuniformity on the beam properties.
Similar situation exists also in electron cyclotron resonance ion source (ECRIS) beam formation. The magnetic trap consisting of an axial magnetic mirror and a hexapole component is used to provide the resonance condition for heating the plasma and the confinement necessary for the creation of the high charge states.The beam is formed from the particle flux escaping the trap through the extraction aperture and due to the fact that electrons are trapped to the magnetic field lines the particle flux has a nonuniform triangular shape. Cylindrically symmetric uniform emission extraction simulations are often used to model ECRIS beam formation as sufficient information is not often available for use in more detailed simulations.
This paper describes the consequences of the approximations made by the extraction models and presents methods for modelling the formation of nonuniform beams.