To compute 
for the Klein-Kramers model, we
need to make some assumptions about the initial phase space
distribution. The swarm is collected in a potential well formed
by the electric shutter, before being released into the drift
tube. For calculational simplicity, we assume that the inital
velocity distribution is a Gaussian. Thus, we have
Substituting this into (
), we get
England and Elford [1987] have measured the mobility of
Li
ions in helium at various distances, and fitted the
experimental data to equation (), truncating the series
at 
. Their fitted values for 
ranged from
0.5 to 2 mm, as a function of electric field strength. It is of
interest to compare this value of 
, which is the
combined error due to all end effects, with the numerical value
for 
obtained by substituting the
experimental parameters into equation (). Even
though the Klein-Kramers model is not really applicable to
lithium in helium, where the ions and neutrals have nearly the
same mass, the result should indicate whether non-hydrodynamic
effects are a significant proportion of the total end effect.
The experiment was performed with a neutral gas pressure of 50
Pa and a temperature of 300 K. Using the ideal gas equation of
state, this corresponds to a number density of 
particles per m
. The measured drift velocity was
. From Viehland [1982], the
cross section at this kinetic energy (
) is about 
, where 
is the Bohr radius. This figure is very similar to 
obtained by considering the collision of two spheres of radius
. From this, we can conclude that the collision frequency
will be of the order of 
, or 
. Upon substituting this into equation
(), the value of 
is found to be of
the order of 1 mm, to be compared with the typical drift distance of 100mm.
While experimental corrections must include other end effects, as discussed by England and Elford [1987], we can conclude that non-hydrodynamic effects are a significant contribution to experimental end effects.
