Periodic electron structures in gases: a fluid model of the ‘‘window’’ phenomenon

Periodic electron spatial structures in gases occur within a window of voltages and pressures. Recent accurate solutions of Boltzmann’s equation portray this effect, but offer little physical insight into the causes of windowing. Here we show for the first time how such insight can be obtained using the fluid model established by Robson, White, and Petrović [Rev. Mod. Phys. 77, 1303 (2005)], with an appropriate generalization of the heat flux ansatz. Conversely, the success in portraying windowing itself becomes a stringent test of the integrity of this fluid model, which can then be applied to a wider range of problems

The "0.4 eV" Shape Resonance of Electron Scattering from Mercury in a Franck-Hertz Tube

The alternative version of the Franck-Hertz experiment with mercury, in which a two-grid tube is used as a combination of electron gun, equipotential collision space, and detection cell, was analyzed recently in considerable detail. In particular, it was inferred that, at optimal pressure, the formation of peaks in the anode current at inelastic thresholds is mediated inside the detection cell by the large variation, a maximum at 0.4 eV, in the cross section for elastic scattering. This variation is due to a shape resonance in the electron-mercury system and is observable persuasively at the onset of anode current as a sharp peak followed by a clear minimum. In the present paper, the passage of electrons through the second grid to anode region is analyzed in terms of kinetic theory. The discussion is based on a simplified expression for the electron current derivable from an approximate form of the Boltzmann transport equation that maintains the spatial density gradient but omits elastic energy losses. The estimated range of pressure underlying this kind of idealization is in good agreement with experiment. An explicit solution is obtained by constructing an analytic expression for the momentum transfer cross section of mercury using a recent theory of generalized Fano profiles for overlapping resonances. This solution is used in order to model successfully the formation of peaks at the threshold of anode current and at excitation potentials, and to explain the dependence of the observed profiles on the pressure and on the sign and magnitude of the potential across the detection cell

A fluid analysis of electron resonances and non-locality in gases

The origin of periodic electron spatial structures in gases subject to spatially uniform electric fields E0 has been recently analyzed through fluid equations [1] thereby providing greater physical understanding of the 'window' phenomenon in the Franck-Hertz experiment, and complementing the more accurate but purely numerical results provided by Boltzmann's equation [2]. Similar physical insight can be obtained for a spatially varying\ud electric field, which modulates electron properties substantially if the applied wavelength matches the natural, 'Franck-Hertz' wavelength, simultaneously producing large phase shifts ('non-local effects'). Such cases have been analysed extensively through solutions of\ud the Boltzmann equation [3] but never by physically tenable, benchmarked fluid modeling as described in [1, 4]

Fluid-model analysis of electron swarms in a space-varying field: nonlocality and resonance phenomena

The physically based, benchmarked fluid model developed by Robson et al. [R. E. Robson, R. D. White and Z. Lj. Petrovic Rev. Mod. Phys. 77 1303 (2005)] and extended to analyze electron swarms in a spatially homogeneous electric field under conditions corresponding to the Franck-Hertz experiment [P. Nicoletopoulos and R. E. Robson Phys. Rev. Lett. 100 124502 (2008)] is generalized to investigate the nonlocal response and resonance phenomena associated with electrons subject to an externally prescribed, spatially varying electrostatic field. Analytic expressions are first derived for the mean velocity, energy, and heat flux of electrons in a harmonically varying field, and expressions are then given for fields with more general spatial dependences. Numerical examples are given for both benchmark model cross sections and a real gas

Key Factors in Fluid Modelling of Plasmas and Swarms

In this paper we start with general fluid equations for both ions and electrons in neutral gases, obtained as velocity moments of Boltzmann's cquation. Two distinct approximations are required for these exact equations to be of allY practical use:\ud 1. The collision transfer terms (right hand side of the fluid equations) must be approximated, and\ud II. Some closure ansatz (hypothesis) is required for the "streaming terms" (left hand side of the fluid equations), to ensure that the number of equations corresponds to the number of unknowns.\ud For step I, swarm (frce diffusion) limit results using, e.g., momentum transfer theory, may be taken over directly to low temperature plasmas, but step II remains problematic, with little guide from swarm physics, and serious doubts about existing assumptions in the plasma literature. We focus on electron fluid equations with closure at the level of momentum and energy balance, which requires an accurate heat flux ansatz in order to produce physically meaningful results. The crucial nature of this ansatz is illustrated using a simple benchmark calculation for infinite plane parallel geometry, where we show for the first time how periodic spatial structures (Franck-Hertz oscillations) may be generated from fluid equations

Kinetic theoretical and fluid modelling of plasmas and swarms: the big picture

Since the 1950s there has been great progress in the fundamental kinetic theory of charged particles (electrons, positrons, muons and ions) in gases, but many of the ideas and results have still to find their way into modern low temperature plasma physics. This paper stresses the bigger picture, in the context of the kinetic theory of gases and fluid modelling, with a view to reconciling the plasma and swarm literature. We focus especially on the importance of a unified approach to transport analysis, appropriate to all types of charged particles in all situations. We discuss both plasmas and swarms in general, and make recommendations for 'best practice' in both kinetic theoretical and fluid modelling

Fundamental issues in fluid modeling: direct substitution and aliasing methods

It is shown how the accuracy of fluid models of charged particles in gases can be improved significantly by direct substitution of swarm transport coefficient data, rather than cross sections, into the average collision terms. This direct substitution method emerges in a natural way for fluid formulations in which the role of the mean energy is transparent, whatever the mass of the charged particles in equation (ions or electrons), and requires no further approximations. The procedure is illustrated by numerical examples for electrons, including the operational window of E/N for an idealized Franck-Hertz experiment. Using the same fluid formulation, we develop an aliasing method to estimate otherwise unknown mobility data for one type of particle, from known mobility data for another type of particle. The method is illustrated for muons in hydrogen, using tabulated data for protons in the same gas