Modelling Ionisation by Helicon Waves
A. W. Degeling, R. W. Boswell
Space Plasma Group, Plasma Research Laboratory, RSPhysSE,
The Australian National University, Canberra ACT 0200, Australia
The response of the electron distribution function in one dimension to a travelling
wave electric field is modelled for parameters relevant to a low pressure Helicon
wave plasma source, and the resulting change in the ionisation rate calculated. The
simulation shows pulses in the ionisation rate which move away at the phase velocity of
the wave, demonstrating the effect of resonant electrons trapped in the wave's frame
of reference. We find that the ionisation rate is highest when the phase velocity
of the wave is between 2 and 3 x106 m/s, where the electron's interacting strongly with the wave (ie. electrons with velocities
inside the wave's 'trapping width') have initial energies just below the ionisation
threshold. Results from the model are compared with experimental data taken from the WOMBAT helicon source, and show reasonable qualitative agreement.
1. Introduction
Recent experiments in low pressure Argon helicon plasmas indicate that anomalous ionisation
occurs downstream from the antenna when the axial phase velocity (vf) approaches the threshold velocity for ionisation (vionise = 2.35x106 m/s for Argon).
This suggests that the helicon wave is interacting with the electron distribution
function for velocities near vf.
The mechanism by which this occurs is generally thought to be electron trapping in
the axial component of the wave's electric field. A trapped electron is confined
between consecutive electric field wavefronts as it is accelerated periodically above
and blow vf. The size of the perturbation v' depends on the wave parameters, the electron's initial
velocity, and the electron's timing with respect to the wave phase.
2. Modelling the Perturbed f(v)
The effect of the strong interaction with electrons about vf on the distribution function f(v) and the resulting ionisation rate is investigated
by numerically calculating the electron trajectories in 1D between the antenna position
(z = 0) and varying final positions (zf) and times (tf).
Photographs of the distribution function in space (constant tf, varying zf) are shown in figure 1. The greylevel represents the value of f as a function of
position and velocity, with low dark areas representing few electrons and light areas
representing many electrons.
The developing circular patterns in both diagrams (each one wavelength across) are
due to electron trapping. If these diagrams were animated in time the patterns would
propagate to the right at the phase velocity. Note that at the lower electric field
of 10 V/m, the velocity range for trapping is smaller, and the circular pattern takes
a longer distance to form.
3. Results: The ionisation Rate
The ionisation rate is calculated by considering the ionisation cross section for
Argon and electron flux as a function of position and time.
Averaging over time and varying vf gives figure 2, which shows average ionisation rate as a function of z and vf. In this diagram high and low ionisation rates are represented by light and dark
greylevels respectively. This diagram shows that the maximum average ionisation rate
occurs at vf = 2.35x106 m/s and z = 0.35 m, and corresponds to the distance taken for the most
trapped electrons to be accelerated from below vionise to above vionise. The second and third peaks with increasing z are due to the second and third oscillations
of the trapped electrons (the dashed lines following the troughs correspond to the
position at which the bulk of the trapped electrons return to their initial velocities). The peaks are smaller in amplitude because the period of oscillation of the
trapped electrons differs with velocity, so the motion of the trapped electrons gradually
becomes incoherent with increasing z, reducing the average ionisation rate. The peaks in ionisation rate also move to higher vf as z is increased (while the magnitude of the peak decreases), so that at z = 1.0
m for example, the first peak occurs at vf = 3x106 m/s. This is because the period of oscillation for the trapped electrons increases
with phase velocity.
4. Comparison with Experiment
Figure 3 shows a comparison of the model average ionisation rate as a function of
vf at z = 1.0 m for various Eo values with experimental density measurments taken in the centre of a large cylindrical
helicon plasma 1.0 m from the antenna as a function of measured vf.
This diagram shows that the experiment and model agree qualitatively where the ionisation
rate is relatively insensitive to changes in Eo over the range shown (the peak at vf = 2.8x106 m/s). The sensitivity of the peak in ionisation rate at 2.35x106 m/s to changes in Eo suggests that this region will not be stable in an experiment, and may explain the
lack of data here, given that the phase velocity is linked to the density via the
helicon dispersion relation.
5. Conclusion
The modelled ionisation rate downstream is peaked in vf due to electron trapping when the most electrons are accelerated above vionise by the wave. This qualitatively supports the existing experimental data in helicon
systems.