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

est achievable radial position for an electron emitted from the outer
conductor, provided that the oscillations are not too large. The expression
for the minimum of R(t) will be a function of the field amplitude,
the frequency, the initial electron velocity as well as the initial phase,
α = ωt0:
Rmin = ΛRo(Λ 2 + 2Λ 2 cos α 2 + 4Λcos αv0ωRo + 2v 2 0ω 2 R 2 o) −1/2
(5.6)
However, for v0 = 0 a much more compact expression, which is independent
of field amplitude and frequency, is obtained,
Rmin ≈
Ro
� 1 + 2(cos α) 2
≥ Ro
√3 . (5.7)
This means that if the radius of the inner conductor, Ri, is smaller than
58% of the outer radius, Ro, then two sided multipactor is not possible
when the initial velocity is low and the oscillations are small.
5.1.2 Multipactor resonance theory
In a coaxial line, both double-sided and single-sided multipactor (on the
outer conductor) are possible. First, double-sided discharge will be considered
and typical for this is that the one way transit time corresponds
to an odd integer of half RF field periods. However, in a coaxial line,
the transit time is normally longer for electrons emitted from the outer
conductor than for electrons emitted from the inner conductor. Thus,
the sum of two transits must be considered and the condition for this is
that it should be an integer number of RF periods. This is the resonance
criterion and in addition to this the phase-focusing effect should be active,
which for the parallel-plate case is given by Eqs. (2.18)- (2.21). It
is instructive to compare the coaxial case with the parallel-plate case,
since in the limit when the Ri ≈ Ro the coaxial and parallel-plate models
should give the same results. For the parallel-plate case, when the
initial velocity is neglected (v0 = 0), the phase stability range is given
by the following inequalities [39],
πk < λ < � (πk) 2 + 4, (5.8)
where k is an odd positive integer. The normalised gap width, λ, is
defined by
λ = ωd/Vω = m(ωd) 2 /eU, (5.9)
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