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

on the approximate solution for the electron trajectory, Eq. (5.3). The
analytical expression is accurate when the oscillations are small and a
consequence of this is that accurate stable phase multipactor regions are
only found for N ≥ 2, i.e. when the duration of the trajectory is at least
2 RF-periods.
By analysing the resonance and stability conditions, one can show
that single-sided breakdown will have not only one region of stable resonant
phase, but rather two stable regions can be found. One somewhat
wider region with resonance close to zero and another, which is resonant
close to π/4. It can be shown that these regions, in the case when
v0 = 0, are approximately given by:
and
0 < αR < α1
α2 < αR < α3
(5.13)
(5.14)
where
α1 ≈ 4
(5.15)
Nπ
α2 ≈ π 1
− (5.16)
4 Nπ
α3 ≈ π 1
+ (5.17)
4 Nπ
For increasing N, the second region converges to αR = π/4, which according
to Eq. (5.7) corresponds to Rmin ≈ Ro/ √ 2. In Fig. 5.5 the
resonant stable phase, αR, has been plotted as a function of N. Except
for the lowest order resonance (N = 1), the regions of phase stability for
the numerically and analytically obtained phases agree very well.
To obtain the multipactor threshold, it is necessary to know the
impact velocity, which is given by
vimpact ≈ 2Vω,o cos α + v0
(5.18)
The lower boundary shown in Fig. 5.6 is obtained for the maximum
impact velocity for each mode, i.e. vimpact = 2Vω,o when v0 = 0. In the
same figure one can also identify a second set of regions with a higher
breakdown threshold. These zones correspond to the second stable phase
region, Eq. (5.14). Since the phases in this region are close to π/4,
the impact velocity is vimpact ≈ √ 2Vω,o. This value is also indicated in
Fig. 5.6, but it should not be expected to serve as an exact envelope of the
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