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Charge-exchange ions in an ion-thruster grid set: where they land and what they erode.

A gridded ion thruster accelerates xenon ions through a stack of slit electrodes, and the grids wear out from the inside: unionised propellant leaks through the apertures, a fast beam ion meets a slow xenon atom, and resonant charge exchange (Xe+ + Xe -> Xe + Xe+) leaves a fast neutral and a slow ion. The slow ion is born at the local potential and falls into the most negative electrode nearby, the accelerator grid, which it sputters. This is the classic lifetime limit of gridded thrusters (the “pits and grooves” worn into the accel grid’s downstream face, and the barrel erosion that widens its apertures).

The grid set is a three-grid slit geometry with the thicknesses and gaps of a 30 cm NSTAR-class thruster: a screen grid at +1100 V, an accel grid at -250 V and a grounded decel grid, each 0.38 mm thick, with 0.66 / 1.0 mm gaps and half-apertures of 0.95, 0.45 and 0.9 mm. The accel slit is narrower than NSTAR’s round accel holes (0.57 mm radius) because a slit lets the screen potential through much more than a round hole: at 0.57 mm and -180 V the potential on the axis of the accel aperture would be positive (about +16 V), and the grid would no longer stop beam-plasma electrons from streaming back. Each grid is a simgeometry Aperture2D.slit, panelled by :func:tree_d_3d.conductors_from_shape with panels graded toward the aperture tips. Two wide plates close the problem: one upstream at the discharge-plasma potential (1125 V) and a grounded one far downstream, standing in for the beam plasma; without them the net charge of the isolated grids would set the far field. :class:tree_d_3d.BoundaryIntegralElectrodes solves for the vacuum field once.

Model and assumptions, all in a 2D slice (per unit length of slit), lengths in mm, energies in eV and code units with q = m = 1 (so a kinetic energy is v^2 / 2):

  • Primary ions start 1 mm upstream of the screen grid with the energy they gain from the plasma potential and are pushed ballistically (no beam space charge: the potential saddle and the trajectories are the vacuum ones; a high-perveance beamlet would need the tree-code space charge of examples/ion_extraction.py). Each primary carries I' / e / N ions per second per metre of slit, with I' = 0.13 A/m, the beamlet current density of an NSTAR-class grid (about 70 A/m^2) times the screen aperture. * The neutral xenon density n = p / (k_B T) (T = 300 K) is uniform. Real neutral efflux peaks at the grids and thins out downstream, so the absolute rates here are indicative; the scaling and the impact pattern are the point. * Resonant charge exchange uses the fit of Miller et al., J. Appl. Phys. 91, 984 (2002), sigma = (87.3 - 13.6 log10(E / eV)) * 1e-20 m^2 at the primary’s lab energy E, tabulated as a plasmacoll CrossSection (used here from 1 eV to 1.3 keV). Along each primary step of length ds a charge exchange happens with probability n sigma(E) ds; the primary current is attenuated by the same optical depth (the fast neutral leaves). Charge-exchange events are rare (n sigma L is about 1e-3 at the reference pressure), so the sampling boosts the probability by a constant factor and gives each slow ion the inverse as its weight. * Each slow ion is born with the thermal velocity of the gas, pushed ballistically in the electrode field (velocity Verlet with a step limited to 0.02 mm), and absorbed at the first boundary crossing of a step: shape.crossings(start, end) tests the whole step, so no ion jumps through a plate, and gives the impact point and the surface normal, from which the impact energy and angle of incidence follow. Slow ions do not collide again. * Sputtering uses the Bohdansky formula, Nucl. Instrum. Methods B 2, 587 (1984), Y(E) = Q s_n(eps) (1 - (E_th / E)^(2/3)) (1 - E_th / E)^2, with the Kr-C reduced nuclear stopping s_n (Eckstein) and the Lindhard screening length, for xenon on molybdenum at normal incidence. The threshold E_th = 50 eV and Q = 4 are illustrative values that put the yield near one atom per ion at 1 keV and the threshold near those of published Xe -> Mo fits; replace them with a vetted fit before using the erosion numbers quantitatively. Angular effects are left out (the angles are reported).

The production of slow ions is checked against the expected number, the sum over primary steps of the boosted probability B n sigma(E) ds exp(-tau), within Poisson error.

Source: examples/cex_grid_erosion.py. Run it yourself with python examples/cex_grid_erosion.py [--output examples/output] [--quick] (needs pip install "tree-d-3d[examples]").

Figures

Section titled “Figures”

the potential of the three grids and the boundary plates, with the negative saddle in the accel aperture (about -100 V on the axis) that keeps beam-plasma electrons out.

the primary beamlet and a sample of charge-exchange ions, coloured by where they end: most fall back onto the accel grid, from the gap and from downstream.

the flux of charge-exchange ions onto each face of the accel grid (upstream face, aperture barrel, downstream face) against position along the face.

impact energy and angle of incidence of the ions that hit the accel grid, weighted by flux.

the estimated erosion rate of each face of the accel grid, in micrometres per thousand hours of operation (an estimate: see the sputter-yield caveat above). The barrel wears fastest, at its upstream edge, which widens the aperture.

charge-exchange current to the accel grid and the peak erosion rate against neutral pressure: linear at low pressure, falling below linear only once the beam’s optical depth n sigma L approaches one (by then slow ions would collide again too, which the model leaves out).

--quick uses fewer primaries, slow ions and scan points (well under a minute); --smoke is the tiny configuration the test suite runs. Needs Plotly and plasmacoll (pip install 'tree-d-3d[examples]' plasmacoll).