Unperturbed-orbit integration and the 3D kinetic dispersion relation of the electron cyclotron drift instability

TL;DR

This paper derives a comprehensive 3D kinetic dispersion relation for electron cyclotron drift instabilities in crossed-field plasmas, emphasizing unperturbed orbit integration and finite-Larmor-radius effects to improve understanding of plasma instabilities.

Contribution

It provides a self-contained derivation of the electron density perturbation using unperturbed orbit integration, including finite-Larmor-radius effects, and clarifies the connection between different mathematical formulations.

Findings

Derived the missing step in the dispersion relation involving electron density perturbation.

Clarified the relationship between Poisson and dielectric form representations.

Discussed assumptions and potential extensions for more realistic plasma configurations.

Abstract

High-frequency instabilities in crossed-field ($\bm E\times\bm B$) plasmas are widely implicated in anomalous cross-field electron transport in Hall thrusters and related devices. Building on the fully kinetic 3D electrostatic dispersion relations reported by Ducrocq \emph{et al.} and later by Lafleur \emph{et al.}, we provide a concise, self-contained derivation of the key missing step: the magnetized-electron density perturbation $n_{e1}$ obtained from the linearized Vlasov equation via a retarded integration along unperturbed orbits, including finite-Larmor-radius effects and cyclotron harmonics. We collect the required mathematical identities in appendices and clarify the mapping between the Ducrocq Poisson-form and the Lafleur dielectric-form representations, including ion closures (cold-fluid versus kinetic Landau response). We conclude with a brief discussion of the assumptions and possible extensions toward more realistic configurations.