• hollow cathodes
• Langmuir probe
• hall thrusters
• electric propulsion
• numerical model
• plasma
• plume

For many years research on hollow cathodes for Hall thrusters was focused only on the internal part of the cathode (the emitter and orifice region), giving less importance to the cathode-to-keeper gap and the external region where the plasma plume expands. Only later, more detailed studies on the plasma plume have been carried out, increasing the interest in understanding phenomena outside the cathode due to plasma instabilities and the presence of high energy ions which lead to strong erosion of the keeper and consequently to a limited cathode life. At the same time, there has been a development of diagnostic techniques for the investigation of the plasma, with the use of intrusive probes and non-intrusive diagnostics. With this regard, a previously developed plume model for hollow cathodes, based on fluid equations for different species (electrons, ions and neutrals), has been revised and improved through the reduction of uncertainty on the input parameters. The model is based on nine equations (continuity, momentum and energy equations for each species) which are manipulated to obtain a suitable set of ODEs and introduced in the Matlab ODE solver to obtain the values of the unknown parameters of the plasma at different positions from the keeper exit. An approximation of the properties of the plasma to the keeper with the values obtained at the orifice is not possible since a strong variation of the main plasma parameters is present in the orifice-to-keeper gap. Thus, to get a better calculation of the input values, a further time-independent model that describes the region between the orifice and the keeper has been built, allowing for a connection between the internal part of the cathode and the external one. The model, based on balance equations, has been developed to determine the properties of the plasma at the keeper exit without calculating the trends in the region, requiring as input at the orifice experimental data or theoretical values obtained through the internal cathode model developed at SITAEL. Due to the different nature of the models (the plume model is based on a 1D spherical reference system, while the orifice-keeper model is not), the point where the output values of the orifice-keeper model are calculated is different from the application point of the initial conditions of the plume model and this causes an error on the values. As a consequence, a calibration process is carried out on the calculated values. The calibration is achieved by considering the gap between the keeper exit and the starting point of the integration for the plume model and by knowing the variation of the parameters in the gap for some of the cathodes examined (NSTAR and NEXIS). For the cathodes of which there are no experimental data, the calibration is still subject to error since the variation of the plasma properties depends on numerous factors and is certainly not the same as the variations considered for the NEXIS and NSTAR cathodes. However, this calibration process, which is used to obtain better estimates of the plasma properties, involves an additional step compared to a simpler system where the variation of the parameters in the initial gap is neglected. Simulations for the different types of cathodes are carried out, and a comparison with the experimental data has been made whenever possible. Knowing the values of the plasma properties at different operating points, a model is developed to design Langmuir probes (single and triple Langmuir probes) with the aim of reducing the intrusive effects. In fact, the different couples of plasma parameters (plasma density and electron temperature) are used to calculate the Debye length which, in turn, is used to find the probe dimensions. The theoretical values obtained are the minimum for which the thin sheath theory can be used. The most important dimensions are the tungsten wire diameter and length and the distance between the electrodes in the triple Langmuir probes: the different tungsten wires shall be at such a distance that the sheath of an electrode does not influence the others. At the same time, however, if the distance between the wires is too large, the measurement becomes sensitive to the gradients in the plasma parameters. Finally, considering a triple Langmuir probe that works in Voltage mode, the model calculates the minimum voltage to be applied between the two electrodes. Also in this case, simulations for the different types of cathodes are carried out, and a comparison with the dimensions obtained from the experimental data has been made (probe diameters).