Theory of a stable strong electrostatic double layer generated in a   two-grid Franck-Hertz tube
  There is a classic alternative to the Franck-Hertz experiment designed to show more than a recurrence of the first excited state. Instead of being subjected to a rising potential between source and accelerating grid, electrons are now accelerated in less than one excitation mean-free-path by an extra grid, and then drift towards the second grid across a large equipotential region. In this arrangement one must face the difficulty that the space potential between the grids is strongly modified by space charges. A recent analysis of this experiment with mercury showed that there is a particular form of discharge that generates and sustains the ideal design dynamically. The inevitable variations of potential inside the intergrid volume are then confined within a narrow sheath, a free double layer, joining two field-free plasma regions. The position of the double layer can be controlled so as to optimize the experiment. In the present paper, those phenomena are studied theoretically in steady-state conditions. The essence of the method is to specify velocity distributions for electrons and ions and use them to solve the Poisson equation. Vortex-type functions are used for trapped particles and mono-energetic beams for free particles. The model explains the change of position of the double layer and provides a natural explanation of a threshold condition for its amplitude that governs the transition between two critical configurations, in accordance with experiment. Stability and minimum field-energy considerations lead to a unique solution in a four-dimensional parameter space where the Langmuir ratio is equal to one. The method is adaptable to other experiments on double layer formation under discharge conditions and correctly predicts the values of several plasma parameters. 
Peter Nicoletopoulos