Multipactor in Low Pressure Gas and in ... - of Richard Udiljak

where d stands for the gap width, Vω = eEω/mω is the amplitude of
the electron velocity oscillations in the spatially uniform RF field, Eω is
the RF field amplitude inside the gap, and U is the voltage between the
conductors. In addition, for an electron avalanche to start, the impact
velocity should be between the first and the second cross-over points
(cf. Eq. (2.12)), which for zero initial velocity is given by,
v1 < 2Vω < v2 . (5.10)
Due to the asymmetry in the electron motion, the simple analytical
analysis that is feasible in the parallel plate case is not applicable for
the coaxial line, despite the fact that the approximate electron position
and velocity are known explicitly (Eqs. (5.3) and (5.4)). One way of
finding the resonance zones in the parameter space is to compute a series
of successive electron trajectories, searching for the conditions when it
converges to a periodically repeated sequence [69]. In Fig. 5.3 one can
see the result of such a method, where stable resonance zones have been
found numerically. To allow simple comparison with the parallel-plate
case, the normalised gap size has been used and in terms of the coaxial
line it becomes,
λ = m(ωd)2
eUc
= G(Ro − Ri) 2
R2 , (5.11)
o ln (Ro/Ri)
which coincides with Eq. (5.9) when Ro/Ri is close to unity. The convenient
parameter, G = ωRo/Vω,o, has been introduced as representing
a normalised outer radius and Vω,o = qEo/mω.
When Ro/Ri is close to unity the parallel-plate and coaxial models
give similar results. When the ratio becomes larger, the zones deviate
from the straight lines predicted by Eq. (5.8) and the deviation is more
pronounced for the higher modes. When the ratio becomes too large,
all two sided resonances disappear. This is a consequence of the Miller
force and for the higher order modes, where the approximate analytical
solution is very accurate, the prediction (Eq. (5.7)) is that the double
sided resonances should disappear roughly at Ro/Ri = √ 3 ≈ 1.73. Figure
5.3 shows that this is indeed true. The first order mode, however, is
not accurately described by the analytical expression and the numerical
calculations show that this mode can exist for values of Ro/Ri as high
as 4.
In Fig. 5.4 the double-sided discharge regions have been computed
numerically for Ro/Ri = 1.4. The classical multipactor zones have upper
and lower thresholds that satisfy Eq. (5.10) in the following sense: since
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