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Multipaction
Imagine a setup of two surfaces and in between these an oscillating electromagnetic field.
This could be for instance a parallel place capacitor connected to an AC power supply, or a
waveguide in which an electromagnetic field mode propagates. If free electrons exist on one
of the surfaces, and the phase of the field is just right, the field can accelerate these towards
the opposite surface as illustrated in Figure 1. If the mean free paths of the electrons are long
enough, and the field doesn’t alter direction and decelerate the electrons too much, they will
hit the other surface. If the energy is sufficiently high, secondary electrons can be knocked
off.
Figure 1: The first half-cycle of multipaction.
Now, if the period of the field is just right, these secondary electrons can be accelerated
back to the first surface repeating the process and generating more secondary electrons. Thus,
under the right conditions, an avalanche like generation of oscillating electrons can occur. This
effect is known as multipaction.
In space-borne equipment multipaction represents a severe problem that can generate RF
noise, alter impedance, generate heat and cause hardware damage. It is thus highly desirable
to understand the effect of multipaction and to investigate under what conditions it can occur.
From the description above, it is clear that the conditions for multipaction are that 1)
the mean free path length is sufficient to reach the other surface, 2) the energy is sufficient
to cause secondary emission and 3) the oscillation is close to resonance with the field. As a
result, multipaction is highly dependent on the frequency of the field, the dimensions of the
structure and the power of the field.
In order to examine multipaction, we thus need to know the physical relations describing
a) the propagation of electrons and b) the generation of secondary electrons. The following
two subsections introduc these.
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1.1
Equation of motion
A particle of charge q and mass m in an electromagnetic field is subject to the Lorenz force.
Disregarding other forces the equation of motion for the particle is thus
mr̈ = q (E + ṙ × B)
(1.1)
Knowing the EM-field we can from the above equation get both position and velocity of the
electrons as a function of time. Furthermore, each electron exerts a repulsive force on every
other electrons. So, to be exact, we need to include not only the oscillating electromagnetic
field, but also the fields of the individual electrons. When the density of electrons is low,
however, the field is dominated by the oscillating electromagnetic field, and the fields of the
electrons can be omitted. But as multipaction evolves the density of electrons grows and
so-called space charge effects will appear, owing to the fields of the electrons.
If no space-charge effects were present, and the conditions of multipaction were met, the
number of electrons would increase exponentially. However, the effects of space-charge cause
the oscillating electron cloud to disperse, and due to the resultant mismatch of phase the
number of electrons saturates.
1.2
Secondary emission
When an electron impacts a surface, either secondary electrons can be generated, or the
inbound electron can simply be reflected (backscattered) either elastically or inelastically.
The number of secondary electrons, their energy and direction, will of course depend on the
impact angle φ and energy Ep .
In this project, I have relied on a simple empirical model, known as the Vaughan model,
describing the average number of generated secondary electrons as a function of impact energy
and angle. The model is purely empirical and has been established by repeated measurements.
The relevant material constants can be found in industry standardization documents, such as
the ECSS documents published by The European Space Agency (ESA).
The Vaughan model
We define the secondary emission yield (SEY) δ as the number of secondary electrons (or
reflected primaries) generated per inbound electron. An example of an SEY curve is given in
Figure 2 where Emax is the energy that maximizes δ and δmax is the maximum value of δ.
In Vaughan’s model Emax and δmax are computed from two empirical relationships:
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Emax (φ) = Emax (0) 1 + kE2πφ
k φ2
(ii) δmax (φ) = δmax (0) 1 + φ2π
(i)
(1.2)
where Emax (0) and δmax (0) are values measured at normal incidence, and kE and kδ are
parameters indicating the smoothness of the surface from 0 to 2, often set to 1. Then,
depending on the ratio
v=
Ep − 12.5eV
Emax − 12.5eV
2
(1.3)
Figure 2: Secondary emission yield for a molybdenum surface at different angles.
the secondary emission yield is found as

 (ve1−v )0.56 , for v < 1
δ(Ep , φ) 
=
(ve1−v )0.25 , for 1 < v ≤ 3.6

δmax

1.125/v 0.35 , for v > 3.6
(1.4)
The concept of secondary emission yield is related to the average over many electron impacts. It gives the mean value of the number of secondary electrons for a given energy and
angle. However, the Vaughan model gives no expression for the probability distribution function of the energy nor the direction of the secondary electrons. So, these distributions have
to be obtained from other sources, or assumptions must be made.
Knowing a) the equation of motion for the electrons in a hollow structure and b) the average
number of secondary electrons generated upon impact with the structure surfaces, we have
the appropriate tools to create a model for simulating the behavior of individual electrons. In
my presentation I will explain this model, and how it is used to determine the conditions for
multipaction.
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