Theory of the anomalous momentum exchange from wave–particle interactions in Hall-effect ion accelerators and comparisons with measurements

Abstract

A first-principles model of the anomalous momentum-transfer collision frequency for electrons (vea) in E×B ion accelerators, also known as Hall-effect thrusters, is presented. The theory on which the model is based adopts a two-stage evolution of unstable waves. First, short-wavelength (k⊥ρe>1), high-frequency (ω∼ωce) modes that are driven by the cross-field drift υE=E×B/B2 grow and saturate at a level of turbulence too low to explain the observed measurements. Then, the wave energy is dominated by modes of longer wavelength (k⊥ρe<1) and in the range of the lower-hybrid frequency ωLH=ωpi/1+ωpe2/ωce2½. The lower-hybrid modes combine wave growth in the azimuthal direction that is driven by the diamagnetic drift υDe=∇pe×B/enB2, with growth parallel to B due to a higher effective mass of electrons. The latter has been typically identified as the modified two-stream instability. The diamagnetic-driven modes are found to be important in regions of the channel where ions begin to accelerate since υE ∼ υDe there. The theoretical model compares extremely well with a large set of empirical profiles of vea derived from laser-induced fluorescence measurements. Our model validation comparisons spanned thrusters with >10× range in discharge power, various sizes and operating conditions, in unshielded and shielded magnetic field topologies. The kinetic version of our closed-form expression yields the scaling vea∼ωceυTieτ¯/(υE+υDe), where τ¯ ∼ ωLH/vi, vi is the sum of the ionization and charge-exchange frequencies and υTi is the ion thermal speed. The latter must be determined by the appropriate integration of the ion velocity distribution function and include not only random changes of the drift velocity but also ion production.