First-principles Model of the Anomalous Momentum-transfer Collision Frequency in Hall Thrusters and its Validation

A first-principles model of the anomalous electron momentum-exchange collision frequency (v_ea) in Hall-effect thrusters (HET) is presented. The theory on which the model is based assumes that the Electron Cyclotron Drift Instability grows in the acceleration channel, pre-heats electrons and ions to comparably warm values, and then saturates before it can affect significantly the anomalous transport. Modes in the range of the lower-hybrid (LH) frequency ω_LH (m_e/m_i )^"½" ω_ce also grow and it is their saturation that ultimately drives the quasi-steady state of v_ea. The closed-form model compares remarkably well with a large set of empirical profiles of v_ea along the channel centerline of several HETs. These profiles were derived from a combination of laser-induced fluorescence measurements and steady-state simulations using the axial-radial fluid-electron/kinetic-ion code Hall2De. Our model validation comparisons have spanned thrusters with operation range of 1-12.5 kW in discharge power and 200-600 V in discharge voltage, a factor of ~3 in size, and have included both unshielded and magnetically shielded topologies. The simplest form of the two expressions we derived yields the scaling v_ea~ω_ce υ_Ti e^τ ̅ /υ_ye where the azimuthal drift velocity is υ_ye=υ_E+υ_De with the cross-field and diamagnetic drifts being υ_E=E×B/B^2 and υ_De=∇p_e×B/enB^2, respectively. The characteristic dimensionless exponent τ ̅ is proportional to ω_LH/v_i with v_i denoting the ion production frequency, and υ_Ti is the ion thermal speed. The latter must be determined by the appropriate integration of the ion velocity distribution function and account for both random changes of the drift velocity and ion generation. In addition to the growth of different modes spanning both short and long wavelengths relative to electron Larmor radius, two other findings are noteworthy. First, to the point that v_ea is inversely proportional to υ_ye, in the region of the channel upstream of the location of maximum E×B drift, we found υ_De to be comparable to υ_E in magnitude and pointing in the opposite direction. Therefore, υ_De appears to offer the explanation behind the well-observed non-monotonic behavior of v_ea in this region of the channel. This non-monotonic behavior has been found in most empirically-derived profiles of v_ea, in multiple HETs, and has proven to be critical in our ability to simulate accurately ion beam divergence and front pole erosion. The mechanism that caused this behavior, however, had remained elusive for several years. Second, to the point that v_ea is proportional to υ_Ti e^τ ̅ , we concluded that ionization and change exchange must play a critical role in the saturation of the instabilities and, in turn, on the anomalous transport of electrons.