  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.
