Spatially periodic structures in electron swarms: Ionization, NDC effects and multi-terms analysis

In reference (Robson R E, Li B and White R D 2000 J. Phys. B: At. Mol. Opt. Phys. 33 507), we revisited the Franck-Hertz experiment, and gave solutions of the Boltzmann equation describing the spatially-resolved relaxation profiles of a non-hydrodynamic swarm of electrons streaming at a steady rate from a plane source into mercury vapour. In this paper, we extend this study to other cases and develop a formalism for both ions and electrons and consider situations where both conservative and non-conservative collisions may take place. As in Robson et al (2000), we employ a 'two-temperature' Burnett function representation of operators in velocity space in the Boltzmann equation. Configuration space is represented by a finite mesh of points and a finite difference technique is developed accordingly. Boundary conditions are specified for the general problem and techniques for solving the resulting large system of algebraic equations are discussed. The importance of a 'multi-term' analysis and the existence of negative differential conductivity (NDC) under non-hydrodynamic conditions is displayed by considering electrons in methane. The explicit effect of ionization on the spatial relaxation profiles is considered along with a study on the importance of treating ionization as a true on-conservative process as opposed to another inelastic process. The spatial relaxation profiles are compared with predictions from eigenvalue theory.