Theory of the Anomalous Momentum Exchange from Wave-particle Interactions in Hall-effect Ion Accelerators and Comparisons with Measurements

A first-principles model is presented of the anomalous momentum-transfer collision frequency for electrons (v_ea) in E×B ion accelerators also known as Hall-effect thrusters. The theory on which the model is based adopts a two-stage evolution of unstable waves. First, short-wavelength (k_ ρ_e>1), high-frequency (|ω|~ω_ce) modes that are driven by the cross-field drift υ_E=E×B/B^2 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_ ρ_e<1) and frequencies in the lower-hybrid range ω_LH=ω_pi/(1+ω_pe^2/ω_ce^2 )^"½" . The lower-hybrid modes combine wave growth in the azimuthal direction that is driven by the diamagnetic drift υ_De=∇p_e×B/enB^2, with growth parallel to B. The latter has been typically called the modified two-stream instability. The gradient-driven modes are found to be important in regions of the channel where ions begin to accelerate since |υ_E |~|υ_De | there. The theoretical model compares extremely well with a large set of empirical profiles of v_ea derived from laser-induced fluorescence measurements. Our 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 v_ea~ω_ce υ_Ti e^τ ̅ /(υ_E+υ_De) where τ ̅~ω_LH/v_i, v_i is the sum of the ionization and charge-exchange frequencies and υ_Ti 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.