The seminal experiment of Franck and Hertz, which helped lay the foundations of quantum and atomic physics, is investigated through solution of the Boltzmann equation, using both eigenfunction expansion methods and spatial finite-difference techniques. We consider electrons in both a model gas and in mercury, the gas originally used by Franck and Hertz, and focus upon the effects of both inelastic and elastic collisions. It is pointed out that the periodic spatial structures encountered in the Franck-Hertz experiment have a physical origin similar to the oscillatory phenomena observed by Fletcher and others previously in low-pressure, low-current discharges, and by Winkler et al, and other contemporary authors in simulations of low-temperature plasmas.
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 non-conservative process as opposed to another inelastic process. The spatial relaxation profiles are compared with predictions from eigenvalue theory.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.