Quasi-Analytical Derivation of Parallel-Plate
Multipactor Trajectories in the Presence of HigherOrder Mode Perturbations
Scott Rice and John Verboncoeur
Department of Electrical and Computer Engineering
Michigan State University
East Lansing, MI, USA
[email protected], [email protected]
Abstract— Numerical simulation of multipactor trajectories is
frequently accomplished by calculating the force on a particle as a
function of position and time, and employing a numerical
differential equation solver to step the particle trajectory through
time. In this work, a different approach using linear timedependent system theory is employed, allowing an analytical
treatment up until a final numerical integration step. An example
is provided as a demonstration of consistency between the quasianalytic results and the typical numerical differential equation
approach.
Let the RF excitation be composed of a fundamental and a
harmonic mode of order 𝑛, with the fields in this geometry
specified by:
⃗⃗⃗⃗
𝐸1 (𝑡) = 𝑥̂𝐸𝑜 cos(𝜔𝑡 + 𝜑) ,
(1)
⃗⃗⃗⃗
𝐸𝑛 (t)= 𝑥̂𝛽𝐸𝑜 cos( 𝑛 ∙ (𝜔𝑡 + 𝜑) + 𝜁),
(2)
𝛼 𝐸
⃗⃗⃗⃗
𝐵1 (t)= 𝑦̂ 1 𝑜 cos(𝜔𝑡 + 𝜑),
(3)
𝛼 𝛽𝐸
⃗⃗⃗⃗
𝐵𝑛 (t)= 𝑦̂ 𝑛 𝑜 cos( 𝑛 ∙ (𝜔𝑡 + 𝜑) + 𝜁),
(4)
𝑐
Keywords—multipactor, numerical simulation
𝑐
I. INTRODUCTION
Multipactor [1][2] is a resonant phenomenon in which an
electromagnetic field causes a free electron to impact a surface,
resulting in the surface emitting one or more secondary
electrons. If the surface geometry and electromagnetic fields are
appropriately arranged, the secondary electrons can then be
accelerated and again impact a surface in the bounding
geometry. If the net number of secondary electrons participating
in multipactor is non-decreasing, then the process can repeat
indefinitely. This phenomenon is of considerable practical
interest in the design and operation of radio frequency (RF)
resonant structures, windows, and supporting structures.
Multipactor is commonly studied either experimentally or
via numerical simulations. In the case of numerical simulations,
frequently the particle trajectories are determined by calculating
the force on a particle at any given step in time, and stepping the
particle through many time steps using a numerical differential
equation solver. In this work, a different approach using linear
time-dependent system theory is employed, allowing an
analytical treatment up until a final numerical integration step.
This mathematical treatment assumes that space-charge effects
are negligible and can be safely ignored, which is the case for
multipactor initiation in a vacuum environment.
II. GEOMETRY DESCRIPTION
Consider a parallel-plate geometry that is bounded in the xdimension by a gap distance d, infinite in the y- and zdimensions, and subject to RF excitation as shown in Fig 1.
where 𝐸𝑜 is the peak electric field strength (V/m), 𝛽 is the
relative field strength of the 𝑛th harmonic relative to the
fundamental mode (unitless), 𝛼1 is the relative magnetic field
strength of the fundamental mode (unitless), 𝛼𝑛 is the relative
magnetic field strength of the 𝑛th harmonic (unitless), 𝑐 is the
speed of light (m/s), 𝜔 is the fundamental mode radian
frequency (rad/s), 𝑡 is time (s), 𝜑 is the phase of the fundamental
mode excitation (rad), and 𝜁 is the relative phase of the 𝑛th
harmonic to the fundamental mode (rad).
III. SOLUTION DERIVATION
Without loss of generality, assume that the electron
trajectory occurs in the x-z plane. The equations of motion for
the particle are then given as:
𝑥̈ (𝑡) =
𝑞
(𝐸1 + 𝐸𝑛 ) − 𝑧̇
𝑚
𝑞
𝑧̈ (𝑡) = 𝑥̇
𝑚
𝑞
𝑚
(𝐵1 + 𝐵𝑛 ),
(𝐵1 + 𝐵𝑛 ),
(5)
(6)
where the quantities 𝐸1 , 𝐸𝑛 , 𝐵1 , and 𝐵𝑛 are the time-dependent
magnitudes of the respective vector-fields defined in (1)-(4), 𝑞
is particle charge (C), and 𝑚 is particle mass (kg). For notational
brevity, let's define the following quantities:
𝑓(𝑡) =
𝑔(𝑡) =
𝑞𝐸𝑜
𝑚
𝑞𝐸𝑜
𝑚𝑐
[cos( 𝜔𝑡 + 𝜑) + 𝛽 cos( 𝑛(𝜔𝑡 + 𝜑) + 𝜁)],
(7)
[𝛼1 cos(𝜔𝑡 + 𝜑) + 𝛼𝑛 𝛽 cos( 𝑛(𝜔𝑡 + 𝜑) + 𝜁)]. (8)
The equations of motion can thus be expressed as:
𝑓(𝑡)
𝑥̈
𝑥̇
[ ] = 𝐴(𝑡) [ ] + [
],
𝑧̈
𝑧̇
0
where
Fig. 1. System geometry
This work supported by an MSU Foundation Strategic Partnership Grant on Accelerator Technology.
(9)
0
𝐴(𝑡) = [
𝑔(𝑡)
−𝑔(𝑡)
].
0
(10)
Next define
The normal solution approach at this point would be to use a
numerical differential equation solver to solve for 𝑥(𝑡) and
𝑧(𝑡). Let's instead apply linear time-varying system theory [3]
[4] to solve for the trajectories.
The homogenous solution for the first derivatives is
computed via:
𝑡
𝐺(𝑡, 𝜏) ≡ ∫𝜏 𝑔(𝜎) 𝑑𝜎 ,
which we can evaluate analytically. Doing so, we arrive at:
𝐺(𝑡, 𝜏) =
𝛼𝑛 𝛽
𝑥̇ (𝑡)
𝑥̇ (𝑡 )
[
] = 𝛷(𝑡, 𝑡0 ) [ 0 ].
𝑧̇ (𝑡)
𝑧̇ (𝑡0 )
(11)
Since 𝐴(𝑡1 )𝐴(𝑡2 ) = 𝐴(𝑡2 )𝐴(𝑡1 ) we have (see [3] for proof):
𝑡
𝛷(𝑡, 𝑡0 ) = 𝑒𝑥𝑝 (∫𝑡 𝐴(𝜏) 𝑑𝜏)
= 𝑒𝑥𝑝 ([
𝑡
− ∫𝑡 𝑔(𝜏) 𝑑𝜏
∫𝑡 𝑔(𝜏) 𝑑𝜏
0
0
𝑡
0
]).
(12)
Noting that:
cos(𝑘)
𝑘
]) = [
sin(𝑘)
0
0
𝑒𝑥𝑝 ([
𝑘
−sin(𝑘)
]
cos(𝑘)
(13)
for real-valued constant k, we thus have:
𝑡
0
𝑡
−𝑠𝑖𝑛 (∫𝑡 𝑔(𝜏) 𝑑𝜏)
0
𝑡
𝑐𝑜𝑠 (∫𝑡 𝑔(𝜏)
0
𝑑𝜏 )
]). (14)
The complete (non-homogenous) solution is given by:
[
𝑥̇ (𝑡)
𝑥̇ (𝑡 )
𝑡
𝑓(𝜏)
] = 𝛷(𝑡, 𝑡0 ) [ 0 ] + ∫𝑡 𝛷(𝑡, 𝜏) [
] 𝑑𝜏
0
𝑧̇ (𝑡)
𝑧̇(𝑡0 )
0
(15)
For a particle starting from rest, 𝑥̇ (𝑡0 )=0 and 𝑧̇ (𝑡0 )=0. In this
situation we thus have:
𝑡
𝑡
∫𝑡 𝑓(𝜏)𝑐𝑜𝑠 (∫𝜏 𝑔(𝜎)𝑑𝜎) 𝑑𝜏
𝑥̇ (𝑡)
[
] = [ 𝑡0
].
𝑡
𝑧̇ (𝑡)
∫𝑡 𝑓(𝜏)𝑠𝑖𝑛 (∫𝜏 𝑔(𝜎)𝑑𝜎 ) 𝑑𝜏
0
𝑚𝑐
[𝛼1 cos(𝜔𝑡 + 𝜑) − 𝛼1 cos(𝜔𝜏 + 𝜑) +
cos( 𝑛(𝜔𝑡 + 𝜑) + 𝜁) −
𝛼𝑛 𝛽
𝑛
cos( 𝑛(𝜔𝜏 + 𝜑) + 𝜁)] (18)
Then with a slight abuse of integral notation for integration
bounds, we have:
𝑡
𝑡
𝑡0
𝑡0
(19)
Thus, given 𝑥(𝑡0), 𝑧(𝑡0 ), 𝐸𝑜 , 𝜔, 𝛽, 𝛼1 , and 𝛼𝑛 , we can
uniquely determine the trajectory x(t) and z(t), although the final
resulting integrations will need to be handled numerically. This
time-varying linear systems approach provides an alternative
way to compute multipactor trajectories in the regime of
negligible space-charge, in lieu of the common approach of
using a numerical differential equation solver on the equations
of motion.
IV. NUMERICAL EXAMPLE
𝛷(𝑡, 𝑡0 )
𝑐𝑜𝑠 (∫𝑡 𝑔(𝜏) 𝑑𝜏)
0
= 𝑒𝑥𝑝 ([
𝑡
𝑠𝑖𝑛 (∫𝑡 𝑔(𝜏) 𝑑𝜏 )
𝑛
𝑞𝐸𝑜
∫𝑡 (∫𝑡 𝑓(𝜏)𝑐𝑜𝑠( 𝐺(𝑡, 𝜏)) 𝑑𝜏) 𝑑𝑡
𝑥(𝑡)
[
] = [ 𝑡0 𝑡0
].
𝑧(𝑡)
∫ (∫ 𝑓(𝜏)𝑠𝑖𝑛( 𝐺(𝑡, 𝜏)) 𝑑𝜏) 𝑑𝑡
0
0
(17)
(16)
To demonstrate the numerical equivalence of this quasianalytical approach with a numerical differential equation solver
for the equations of motion, in Fig. 2 below we show the x- and
z-positions vs. time for the present method and a solution based
on Matlab's ODE45 differential equation numerical solver, for a
particle of electron mass (9.109E-31 kg) and elementary charge
(1.602E-19 C) starting at rest from x=0 in a system with
parameters 𝐸𝑜 = 3000 V/m, 𝜔 = (2π)·100 MHz, 𝑛=3, 𝛽 = 1.75,
𝜑=0, 𝜁=-π/3, 𝛼1 =1, and 𝛼3 = 1. A plate separation distance
(parameter d in Fig. 1) of 0.1 m was used, such that the particle
will strike the opposite bounding wall and re-start with zero
velocity at approximately 230 ns into this simulation. Matlab’s
default integral() function was used for the numerical
integrations required in the present method. These simulation
parameters were chosen to demonstrate numerical agreement for
a relatively complicated trajectory over approximately 36
periods of the fundamental mode. A more sophisticated error
analysis would depend on the specific algorithms used in the
numerical integration and differentiation schemes, and is
beyond the scope of this present paper.
V. CONCLUSION
An alternative formulation for calculating multipactor
trajectories has been presented, in lieu of the standard approach
of using numerical differential equation solver on the equations
of motion. This formulation assumes that space-charge effects
are negligible. The benefit of such an approach is that it avoids
numerical computation until the final numerical integration,
which can allow for greater understanding of system dynamics,
and can more easily facilitate perturbation and sensitivity
analyses within a given multipactor scenario.
REFERENCES
[1]
[2]
[3]
[4]
Fig 2. Comparison of particle trajectory in x-direction (top) and
z-direction (bottom), using the new quasi-analytic method and a
typical numerical differential equation method.
Vaughan, “Multipactor”, IEEE Transactions on Electron Devices, Vol.
35, No. 7, July 1988, pp. 1172-1180
Hasan Padamsee, Jens Knobloch, and Tomas Hays, RF Superconductivity
for Accelerators, 2nd ed, Wiley-VCH, 2008, Chapter 10: Multipacting,
pp. 182-197
Michael Baake and Ulrike-Schlägel, The Peano-Baker Series,
arXiv:1011.1775v2
[math.CA],
available
online
at
http://arxiv.org/abs/1011.1775v2
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