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

where 〈vt 2 〉 represents the average of the square of the magnitude of
the thermal velocity. In paper C, a differential equation for the thermal
velocity is derived, viz.
d〈vt 2 〉
dt + (νcδ + νiz)〈vt 2 〉 = u 2 (νc(2 − δ) + νiz) (3.16)
where δ is the energy loss coefficient. By assuming that νc, νiz and δ are
constants, like before, Eq. (3.16) can be solved explicitly and with the
initial condition 〈vt 2 (t = α/ω)〉 = 0, the thermal impact velocity, when
ωt = Nπ + α, can be found and thus the total impact velocity can be
determined. However, the expression is very complicated and will not
be reproduced here.
The total impact velocity will determine the secondary electron emission
yield. For vacuum multipactor as well as in the previous simple
model for low pressure multipactor, the impact velocity was perpendicular
to the electrodes. In such a case, the secondary yield depends only
on the impact velocity. However, for angular incidence, which will be
the case now with the random three dimensional thermal velocity component,
the yield will be a function not only of the impact energy but
also of the angle of incidence. To account for the angular incidence the
expressions given in Ref. [22] have been used and for ease of reference
they are reproduced here,
ɛmax(θ) = ɛmax(0)(1 + θ 2 /π) (3.17)
σse,max(θ) = σse,max(0)(1 + θ 2 /2π) (3.18)
η = ɛimpact − ɛ0
ɛmax(θ) − ɛ0
σse = σse,max(θ)(η exp 1 − η) k
(3.19)
(3.20)
where θ is the impact angle with respect to the surface normal. ɛmax is
the impact energy when the secondary emission reaches its maximum,
σse,max. ɛimpact is the total impact energy and ɛ0 is the energy limit
for non-zero σse. The formulas are valid for “a typical dull surface”,
according to Ref. [22]. The coefficient k is given by k = 0.62 for η < 1
and k = 0.25 for η > 1.
In vacuum multipactor, the only source of new electrons is secondary
yield from each impact. When the phenomenon takes place in a gas,
another potential source of new electrons is impact ionisation of the
gas molecules. The ionisation threshold of most gases of interest is
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