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

Since the equation has the same form as Eq. (3.1), it will also have the
same solutions and as the initial conditions are identical, the formulas
for the resonant field amplitude and the impact velocity will be identical.
However, it should be noted that instead of νc one will have ν and v0
should be replaced by u0. An important difference is that the velocity
in the previous model was only directed in the x-direction, but now
there is also a thermal velocity component, vt, i.e. the total velocity
is v = u + vt. With these new designations, the expressions for the
resonant field amplitude and the impact velocity become:
E0 =
m
e (ω2 + ν2 )(d + u0
ν (Φ − 1))
(1 + Φ)sin α + ((1 − Φ) ω
ν
2ν + ω )cos α
(3.11)
uimpact = u0Φ + Λ(1 + Φ)(ω cos α − ν sinα) (3.12)
where Φ = exp (−Nπν/ω) has been introduced for simplicity. Λ is given
by Eq. (3.4) as before, but νc should be replaced by ν.
In order to construct the multipactor boundaries, the same approach
as in the simple model case is taken and an expression for the resonant
phase is obtained by combining Eqs. (3.11) and (3.12), which yields,
tan α = ω2 [ρΦ + χ] + 2ν 2 (Φu0 − uimpact)
(dν + uimpact − u0)νω(1 + Φ)
(3.13)
where ρ = dν + uimpact + u0 and χ = dν − uimpact − u0 have been
used for convenience. The reason why the expression looks somewhat
different from Eq. (3.7) is that the constant initial velocity approach has
been used instead of the assumption of a constant ratio between impact
and initial velocities. This will also affect the expression for the phase
stability factor, which in this case becomes
G = (Φ − 1)(ν2 + ω 2 )sin αΛ − Φνu0
ν((1 + Φ)(ν sin α − ω cos α)Λ − Φu0)
(3.14)
So far, the differences between the simple and the more advanced
model are fairly trivial. However, the parameters used (νc and νiz)
are not constants, they depend to a great extent on the total electron
velocity, which is the vector sum of the drift and thermal velocities. The
thermal velocity will have a random direction and therefore the average
total velocity will be equal to the drift velocity. However, the total
(average) energy, ɛ, will still depend on both velocities and it becomes,
ɛ = mv2
2
= m
2 (u2 + 〈vt 2 〉) (3.15)
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