In the previous chapters, I have developed a theory on the form of non-hydrodynamic effects in time of flight experiments. The obvious extension to this work is to develop such a theory for the other main classes of swarm experiments, namely the Steady State Townsend experiments and the Pulsed Townsend experiment. In this chapter, I develop a theory of the parallel plane experiment. The generalization to the Townsend-Huxley experiment, involving as it does elements of multivariable complex analysis, is not straightforward, and so little will be said in this case.
In the conventional theory of steady state experiments, the
diffusion equation (
) is
solved with the time
derivative set to zero. The homogeneous equation gives rise to
solutions in terms of modified Bessel functions of half integer
order, which are merely polynomials in the inverse distance from
the source, multiplied by an exponential of this distance. At
large distances, the dominant term from a compact source is
, where 
and 
.
For a planar source, as in the parallel plane experiment, the
dominant term is 
, which as noted in the previous section, is seen
experimentally. The remaining terms, in the case of a compact source, arise
from the structure of the source. These decay polynomially with respect to the
dominant mode,
away from the source.
Since these effects are clustered near the source, a full theory
should also include a description of the relaxation to local
thermal equilibrium of the particles as they leave the source.
This requires a theory based on the Boltzmann equation.
The usual technique for analysing the parallel plane experiment by means of the Boltzmann equation was first developed by Thomas [1969], and subsequently used by many authors. This involves assuming the solution has an exponential dependence on distance. The Boltzmann equation with one spatial dimension is solved by solutions of the form
where the 
are the roots of
,
where r indexes the roots of 
.
It might be supposed that only solutions () are needed for the
general solution:
Since it is known experimentally that the distribution function
varies exponentially as a function of the distance from the
source, one might assume that there is a root whose
real part is larger than all the others, and so contributes
dominantly to the distribution function at large distances from
the source. If this is the case, then we can identify this root
with 
.
For the one dimensional Klein-Kramers model, one can readily
solve 
from the one dimensional version of ().
In the following, we will use dimensionless units, in which 
and 
. In these units, the mean free time of the charged
particle is 
and the mean free path is 
.
From the usual quadratic formula, the roots of 
are easily found to be
The first Townsend ionization coefficient
can be identified with the largest root of the negative branch 
.
[Blevin and Fletcher
[1984]]
There are an infinite number of positive branch roots that one would
expect to dominate over the 
term in (), and so ()
does not in this case agree with what one expects to see physically. This
raises the question of why only the negative branch roots contribute to
the solution, and of what role the positive branch roots
play. To get some insight into the problem, the number density was
computed numerically for the Klein-Kramers model. This work has been
reported as Standish [1989].
