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# A numerical and theoretical study of the multipactor appearance in the presence of a DC parallel electric field

Adrien Plaçais, Julien Hillairet, Mohamed Belhaj, Jérôme Puech

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Adrien Plaçais, Julien Hillairet, Mohamed Belhaj, Jérôme Puech. A numerical and theoretical study of the multipactor appearance in the presence of a DC parallel electric field. AIP Advances, 2022, 12 (2), pp.025022. 10.1063/5.0062499. hal-03584916

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# A numerical and theoretical study of the multipactor appearance in the presence of a DC parallel electric field

Cite as: AIP Advances **12**, 025022 (2022); https://doi.org/10.1063/5.0062499
Submitted: 06 July 2021 • Accepted: 01 February 2022 • Published Online: 18 February 2022

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A. Plaçais,

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J. Hillairet,

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M. Belhaj, et al.

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# A numerical and theoretical study of the multipactor appearance in the presence of a DC parallel electric field

Cite as: AIP Advances 12, 025022 (2022); doi: 10.1063/5.0062499

Submitted: 6 July 2021 • Accepted: 1 February 2022 •

Published Online: 18 February 2022

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A. Plaçais,<fnref n="1" />,ᵃ ID J. Hillairet,<fnref n="1" />,ᵇ ID M. Belhaj,<fnref n="2" />,ᶜ ID and J. Puech⁵,ᵈ ID

## AFFILIATIONS

## ABSTRACT

In this paper, we study the appearance of the multipactor in a simple system with a bottom plate holding a static potential $V_d$. We developed a numerical model to compute the evolution of the multipactor threshold as a function of the potential. We identify the nature of the resonant modes that appear at different $V_d$ using resonance theory. Then, we use it to compute the theoretical threshold of the structure. As we use a very simplistic electron emission model to identify the multipactor modes, we compute the threshold in a more realistic configuration in a second step.

© 2022 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0062499

## I. INTRODUCTION

The multipactor effect is a phenomenon appearing in vacuum radio-frequency (RF) devices when two requirements are met: (i) residual electrons in the system enter in resonance with the RF electric field and (ii) their energy is high enough to emit other electrons when impacting the walls. In practice, the multipactor appears when the RF electric field reaches a certain amplitude threshold $E_{th}$. This phenomenon can degrade the RF signal or partially dissipate the RF power. In the nuclear fusion and space telecommunications fields, the main issue with the multipactor is that it can cause overheating of the surfaces.<fnref n="1" /> When the heat flux is large enough, it can lead to outgassing of the surfaces; the ejected matter can ionize, potentially leading to destructive corona breakdowns.<fnref n="2" />

Electron emission phenomena are at the origin of the augmentation in electron population. When an electron called a Primary

Electron (PE) impacts a sample, it can emit other electrons. We distinguish three populations of emitted electrons: Secondary Electrons (SEs), which are electrons removed from the material, and Inelastically Backscattered Electrons and Elastically Backscattered Electrons (IBEs, EBEs), which are PEs reflected by the sample, respectively, with and without energy loss.

We characterize these emitted electron populations by three quantities: an energy distribution, an angular distribution, and an emission yield. The emission yield of a population is the ratio between the flux of this population and the flux of PEs. The secondary electron emission yield is noted as SEEY or $\delta$. The inelastically backscattered electron emission yield and the elastically backscattered electron emission yield are, respectively, noted IBEEY, $\eta_i$ and EBEEY, $\eta_e$. These quantities mainly rely on the nature of the sample and surface state and on the energy and direction of the flux of PEs. We define the Total Electron Emission Yield (TEEY) as

<footnote n="1">¹CEA, IRFM, F-13108 Saint-Paul-Lez-Durance, France</footnote>

<footnote n="2">²DPHY, ONERA, Université de Toulouse, F-31055 Toulouse, France</footnote>

<footnote n="3">³CNES, DSO/RF/HNO, F-31401 Toulouse, France</footnote>

<footnote>ᵃAuthor to whom correspondence should be addressed: adrien.placais@lpsc.in2p3.fr. Also at: DPHY, ONERA, Université de Toulouse F-31055 Toulouse, France and CNES, DSO/RF/HNO, F-31401 Toulouse, France.</footnote>

<footnote>ᵇElectronic mail: julien.hillairet@cea.fr</footnote>

<footnote>ᶜElectronic mail: mohamed.belhaj@onera.fr</footnote>

<footnote>ᵈElectronic mail: jerome.puech@cnes.fr</footnote>
$$ \sigma \equiv \delta + \eta_e + \eta_i. \qquad (1) $$

This quantity is of the utmost importance, as $\sigma > 1$ conveys an increase in the electron population and is one of the necessary conditions for multipacting. This is not a multipactor appearance sufficient condition as it does not guarantee that the electron cloud enters in resonance with the RF signal.

The multipactor appearance mechanisms are well understood in simple configurations, such as rectangular or coaxial metallic waveguides, for which the multipactor susceptibility chart is commonly employed. However, multipacting predictability becomes less reliable in several situations, including RF devices with complex geometries and RF devices involving a magnetic field.³,⁴ In particular, dielectric materials are of primordial interest in modern RF applications. Unlike metals, they can hold a net electric charge, which influences both their emission properties⁵ and the RF signal/electron resonance.⁶,⁷ Thus, these materials strongly complicate the appearance of the multipactor. In addition, the charge trapped at the surface of dielectric is not uniform⁸ and evolves with each electron collision. In this study, we focus on the influence of dielectrics on resonance mechanisms; we model charged dielectrics materials by a DC electric field.

The rest of this paper is organized as follows: In Sec. II, we use the numerical tool POTOMAC to compute the multipactor threshold $E_{th}$ of a parallel-plate waveguide as a function of the potential of the bottom plate. The electron emission model is voluntarily kept very simple. As the chart $E_{th}(V_d)$ shows distinct regimes, we develop, in Sec. III, a resonance theory to study these regimes, which is solvable with the very simple electron emission model. In Sec. IV, we compare electrons emission phase calculated with POTOMAC with emission phases obtained with the resonance theory. In Sec. V, we compute the multipactor thresholds with the resonance theory in order to compare them with the thresholds calculated with POTOMAC. Finally, in Sec. VI, we make a second set of POTOMAC simulations with a more realistic electron emission model, to study its influence on the multipactor appearance.

## II. NUMERICAL STUDY

### A. Presentation of the model

The system under study, illustrated in Fig. 1, is composed of two parallel metallic plates, infinite in the $\hat{y}$ and $\hat{z}$ directions. The plates are separated by a gap of length $d$. The RF electric field $E_{RF}$ has an amplitude $E_0$, a frequency $f$, and is oriented along $\hat{x}$,

$$ E_{RF}(t) = E_0 \sin(\phi)\hat{x}, \qquad (2) $$

with the phase $\phi = 2\pi ft$, where $t$ is the time. The model is in 1D, and electrons move along $\hat{x}$. The origin of coordinates is set on the bottom plate. The motion of electrons is non-relativistic. The bottom plate is maintained at a potential $V_d$, whereas the top plate is grounded. The voltage difference between the plates creates a static and uniform electric field $E_d$, oriented along $\hat{x}$. Their charge is static, uniform, and does not affect electron emission phenomena.

As the electron emission model and its parameters have a tremendous influence on the calculated multipactor threshold,⁹ we use the same emission model for simulation and theory. Yet, the main assumption underlying resonance theory is that all electrons participating to the growth of the electron cloud follow the same

<figure><img src="image_14.png" /><figcaption>FIG. 1. Infinite parallel plates separated by a distance $d$. Potential of the bottom plate is $V_d$, the top plate being maintained at ground potential. A uniform RF electric field $E_{RF}$ propagates between the plates.</figcaption></figure>

<table><caption>TABLE I. Full electron emission model used in this study. There is no model for EBEs, as they are not considered here.</caption><thead><tr><th>Population</th><th>Emission yield</th><th>Angular distrib.</th><th>Energy distrib.</th></tr></thead><tbody><tr><td>SE</td><td>Dionne model</td><td>No (1D)</td><td>Monoenergetic</td></tr><tr><td>IBE</td><td>Constant</td><td>No (1D)</td><td>Monoenergetic</td></tr><tr><td>EBE</td><td></td><td></td><td></td></tr></tbody></table>

trajectory. This hypothesis forbids the use of energy and angular distributions for emitted electrons. Hence, we use a naive electron emission model; all electrons are emitted with the same constant energy and there is no EBE. SEs and IBEs are not discriminated from each other. The SEEY is computed from Dionne model as defined in Ref. 10. The model is summarized in Table I.

For the sake of simplification, we consider that the metallic plates have the same emission properties and that $V_d \ge 0$ V. For symmetry reasons, this study and all its conclusions are still valid with a negative potential. In addition, we can easily adapt it to configurations where the plates have different emission properties.

### B. Presentation of the numerical tool POTOMAC

POTOMAC (Physical simulatiOn TOol for Multipactor in Advanced Configurations) is a physical Electro-Static Particle-in-Cell (ES-PIC) code that models the multipactor appearance and, in particular, compute the multipactor threshold of a structure. It is possible to simulate configurations involving dielectric materials and/or magnetic fields.¹¹,¹² $N_e$ seed electrons are injected in the waveguide during the three first signal periods. The simulation is carried on for up to 60 000 periods. If the number of electrons goes higher than $5N_e$, the multipactor is found and the simulation stops. If it drops below $N_e/10$ or if the end of simulation is reached, simulation restarts with a higher $E_0$. We found this criterion to be more robust than the detection of an exponential growth of the electron
population. As a matter of a fact, this last one led to premature detection of the multipactor at very low electric fields, caused by temporary increase in electron population. As we are only interested in the multipactor threshold determination, space charge effects are neglected.¹ Such approximation would be, of course, too strong to study the multipactor evolution in an established regime.¹³,¹⁴

The electron positions and velocities are calculated with a leapfrog scheme. In PIC codes, the time step $\Delta t$ has a tremendous importance. In general, two conditions have to be matched,¹⁵

$$ \begin{cases} \Delta t \ll 1/f, \\ \Delta t < \Delta x/v_{\text{max}}. \end{cases} \qquad \begin{aligned} & (3a) \\ & (3b) \end{aligned} $$

$\Delta x$ is the spatial step. $v_{th}$ is the velocity of the most energetic electrons appearing in the system; its value depends on the geometry under study. The pseudo-random numbers generator is a Taus-worthe generator:¹⁶ it has a period and is uniform. At each collision, $\sigma$ is calculated. The number of electrons emitted is an integer calculated with a Poisson distribution.¹⁷ When only one electron is emitted, it has a probability $\eta_e/\sigma$ to be an EBE, a probability $\eta_i/\sigma$ to be an IBE, and a probability $\delta/\sigma$ to be a SE. When many electrons are emitted, they are always SEs.¹⁷,¹⁸

We implemented numerical diagnostic tools in POTOMAC. They make it possible to register several internal parameters, such as the nature (seed, SE, EBE, IBE) of every electron and of the electron that emitted them. In this paper, we utilize it to record the emission phase, the collision phase, the emission plate, and the collision plate of every simulated electron for every potential $V_d$ at the multipactor threshold $E_{th}$.

### C. POTOMAC simulation results

In this study, we consider $d = 0.1$ mm, and the RF electric field has a frequency of 11 GHz. We sweep the RF electric field amplitude between 4.3 and 15 kV cm⁻¹. The material is copper as received (without any surface treatment), which TEEY was measured at the ONERA. Experimental data can be found in Ref. 19 and details on the measurements in Ref. 10. The TEEY, as a function of the impact energy of PEs, is represented in Fig. 2. The first cross-over energy, defined as the first energy for which $\sigma = 1$, is $E_{c,1} = 29$ eV. The list of the electron emission model parameters is presented in Table II. All electrons are emitted with the same energy of 3 eV.

We take $\Delta x = d/100$ and $N_e = 100$. We showed in another paper¹⁰ that, with POTOMAC and in this geometry, most energetic electrons had an energy of 300 eV. The $\Delta t$ value matching conditions (3a) and (3b) in these conditions were set to $\Delta t = T_{RF}/1000$, where $T_{RF}$ is the period of the signal. We verified the consistency of POTOMAC’s results with other simulation codes and multipactor theories, as well as its convergence. A consistency study in a simple configuration is presented in the Appendix, as well as a convergence study on the initial number of electrons.

Figure 3 represents the evolution of $E_{th}$ with $V_d$, as calculated with POTOMAC. Four regions of potentials are visible:

* zone 1 ($V_d = 0-19.3$ V), where $E_{th}$ increases with $V_d$ in a parabola-like shape.

* zone 2 ($V_d = 19.3-26.5$ V), where $E_{th}$ slowly decreases with $V_d$.

<figure><img src="image_15.png" /><figcaption>FIG. 2. Modeled TEEY as a function of the impact energy of PEs used in this study.</figcaption></figure>

<table><caption>TABLE II. Copper Dionne’s parameters as defined in Ref. 10.</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>A</td><td>12.48</td></tr><tr><td>n</td><td>1.00</td></tr><tr><td>d (nm)</td><td>4.80</td></tr><tr><td>S</td><td>0.15</td></tr><tr><td>Φ (eV)</td><td>4.65</td></tr><tr><td>η<sub>i</sub></td><td>0.25</td></tr></tbody></table>

<figure><img src="image_16.png" /><figcaption>FIG. 3. The evolution of the multipactor threshold with the potential at the surface of the bottom plate.</figcaption></figure>
* zone 3 ($V_d$ = 26.5–47.0 V), where $E_{th}$ drops by 39% and then linearly increases with $V_d$.

* zone 4 ($V_d$ = 47.0–54.5 V). At 47 V, $E_{th}$ is abruptly raised by 11%. Then, it linearly increases with $V_d$.

The isolated point at $V_d$ = 50 V is not reproducible, and it corresponds to a case where a significant number of seed electrons were injected in resonance with the RF electric field, leading to a very rapid increase in electron density. For $V_d$ > 54.5 V, the multipactor threshold dramatically increases. As RF systems have transmissible power limitations, we can then consider that the multipactor disappears.

### D. Comparison with previous studies

Sorolla et al.⁶ computed the multipactor threshold of an infinite parallel-plate dielectric-loaded system with a numerical model in 1D. They took into account the evolution of the dielectric slab charge with each electron collision, as well as the influence of this charge on the SEEY.⁵ They kept the charge relatively low, corresponding to our zone 1. In this zone, the shape of our threshold vs charge curves are similar.

Zhai et al.⁷ studied the evolution of $E_{th}$ with $V_d$ in a parallel-plate dielectric-loaded waveguide. Their simulation is 1D. They used Vaughan model to compute the SEEY,²⁰,²¹ and all electrons were emitted with an energy of 3.68 eV. Although the general shape of their $E_{th}(V_d)$ curve (Fig. 4) is similar to ours, they did not observe the strong discontinuities at 26.5 and 47.0 V. They are unphysical and are created by the lack of dispersion in electrons impact velocities, compelling all electrons to enter in the same multipactor mode. We believe that Zhai et al. did not observe these because they allowed the dielectric charge to vary, smoothing the transitions between modes. According to their hypothesis, the multipactor is in a Double-Sided (DS) mode at low potential and progressively switches to a Single-Sided (SS) mode when the potential difference between the plates increases. In a DS mode, the multipactor is created by electrons impacting a plate different from their emission plate. In a SS mode, it is created by electrons impacting their emission plate.

In order to verify these conclusions and validate POTOMAC simulation results, in Sec. III, we develop a multipactor resonance theory adapted to this configuration.

## III. DEVELOPMENT OF THE RESONANCE THEORY

### A. Motion equations and notations

In this section, we investigate the multipactor resonant trajectories and resonant phases, regardless of electrons impact energies. All electrons have a mass $m$ and a charge $-|e|$ [1]. They have a low energy (some keV at most); thus, their motion is non-relativistic, and we can neglect the RF magnetic field.²² We use the dimensionless set of equations proposed by Sorolla et al.,⁶

$$ \Lambda \equiv \frac{m\omega^2}{|e|E_0} \cdot x, \qquad (4a) $$

$$ V_{\omega} \equiv \frac{m\omega}{|e|E_0} \cdot v_x, \qquad (4b) $$

$$ \Omega \equiv \frac{E_d}{E_0}. \qquad (4c) $$

The dimensionless position is $\Lambda$, and the dimensioned position is $x$. The dimensionless velocity is $V_\omega$, and the dimensioned velocity is $v_x$.

We set the origin of positions on the bottom plate, which is therefore at the position $\Lambda = 0$. The top plate is at $\Lambda_d$. All electrons emitted from the bottom plate have an initial velocity $V_{\omega,0}$, and those emitted from the top plate have an initial velocity $-V_{\omega,0}$. The motion equations for an electron emitted at $\phi_1$ at the position $\Lambda_1$ with the velocity $V_{\omega,1}$ are as follows:⁶

$$ V_{\omega} = \cos \phi - \cos \phi_1 - \Omega(\phi - \phi_1) + V_{\omega,1}, \qquad (5a) $$

$$ \Lambda = (\phi - \phi_1)(V_{\omega,1} - \cos \phi_1) + \sin \phi - \sin \phi_1, \qquad (5b) $$

$$ -\frac{\Omega}{2}(\phi - \phi_1)^2 + \Lambda_1. $$

We call crossing electrons as electrons that are emitted on the bottom plate and impact the top plate (short form: $b \to t$) and electrons that are emitted on the top plate and impact the bottom plate (short form: $t \to b$). Similarly, we call recollected electrons as electrons emitted on the bottom plate and impact the bottom plate ($b \to b$), and electrons emitted on the top plate and return to it ($t \to t$). Due to the positive potential on the bottom plate, $t \to t$ electrons cannot enter in resonance with the RF electric field, though individual $t \to t$ electrons can exist. Without any loss of generality, we consider that the first resonant trajectory always starts on the bottom plate.

We consider an electron emitted at the phase $\phi_k$, with $k$ an integer different from zero. It impacts a plate (the same or another one) at the phase $\phi_{k+1}$. The time between the collision of a PE and the emission of the electrons is in the range $10^{-11}-10^{-12}$ s,²³ which is at least two orders of magnitude lower than the period of a GHz signal. Thus, we consider that the electron emission phenomena are instantaneous. As a consequence, $\phi_k$ is also the collision phase of the precedent electron and $\phi_{k+1}$ the emission phase of the next electron. Riyopoulos et al.²⁴ showed that making this hypothesis had little influence on the resonance mechanisms.

The DS multipactor is created by crossing electrons. The classic DS multipactor resonance condition reads¹,²⁵

$$ \phi_2 = \phi_1 + N\pi, \quad N = 1, 3, 5, \dots, \qquad (6) $$

with $N$ the multipactor order. The solid lines in Fig. 5 represent the trajectories of two electrons between two grounded plates. The first one checks the resonance condition (6). As $V_d = 0$ V, the second electron also checks this condition and resonance is ensured. The dashed lines in Fig. 5 represent the trajectories of two electrons, the bottom plate being set to $V_d = 10$ V. The first electron checks the resonance condition (6). The positive potential on the bottom plate breaks the symmetry of the problem, and the second electron is out of resonance.

### B. Asymmetric multipactor modes

In the asymmetric multipactor modes, the multipactor involves $n_\phi = 1, 2, 3, \dots$ different resonant trajectories. The $k$th electron is emitted at the phase $\phi_k$ on the plate whose position is $\Lambda_k$, with an initial velocity $V_{\omega,k}$. It impacts the plate of position $\Lambda_{k+1}$ at the phase $\phi_{k+1}$. We define the multipactor period $\Pi$,

$$ \Pi = \phi_{n_{\phi}+1} - \phi_1. \qquad (7) $$
<figure><img src="image_17.png" /><figcaption>FIG. 4. The multipactor threshold calculated by POTOMAC as a function of V<sub>d</sub> (left axis, —). The evolution of the electrons emission phase distribution with V<sub>d</sub>, extracted from POTOMAC simulations (right axis, ▲). Electrons are sorted depending on their trajectories: (a) all electrons, (b) b → t electrons, (c) t → b electrons, and (d) b → b electrons.</figcaption></figure>

We define the resonant or fixed phases,

$$ \varphi_k \equiv \mathrm{mod}(\varphi_k, \Pi), \quad k = 1, 2, 3, \dots \qquad (8) $$

The sequence of resonant phases ($\varphi_1, \varphi_2, \dots, \varphi_{n_\varphi}$) is repeated every $\Pi$.

We name asymmetric DS<sup>26</sup> and SS modes by the subset ($\Pi, n_\varphi$). For example, the symmetric multipactor resonance mechanism

presented in Fig. 5 is called DS ($\Pi = \pi, n_\varphi = 1$). We represented in Fig. 6 the seven resonant trajectories of the SS ($\Pi = 12\pi, n_\varphi = 7$).

The ping-pong modes involve both crossing and recollected electrons.<sup>27,28</sup> In order to completely characterize these modes, we should provide the list and ordering of the impacted plates. We use the notation $x_1 \rightarrow x_2 \rightarrow \dots \rightarrow x_{n_\varphi}$, where $x_k$ is “b” if the kth electron is emitted on the bottom plate and “t” if it is emitted on the top plate. We present in Fig. 7 the trajectories of the PP ($\Pi = 4\pi, n_\varphi = 3$) b → t → b.
<figure><img src="image_18.png" /><figcaption>FIG. 5. Solid lines: resonant trajectory of the two first electrons in the symmetric first order DS mode, with $V_d = 0$ V, $\phi_1 = 2.73$ rad, and $E_0 = 7.00$ kV cm$^{-1}$. Dashed lines: trajectory of two electrons, the first one checks (6), with $V_d = 10$ V, $\phi_1 = 2.98$ rad, and $E_0 = 7.00$ kV cm$^{-1}$.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>FIG. 6. Trajectory of the electrons in the SS ($\Pi = 12\pi, n_\phi = 7$). $V_d = 33.0$ V, $\phi_1 = 2.650$ rad, and $E_0 = 6.03$ kV cm$^{-1}$.</figcaption></figure>

### C. Method of resolution

In a fashion similar to the symmetric case and for a given $\Omega$, we solve the $n_\phi$ position equations, the unknowns being the phases $\phi_k$,

$$ \Lambda_{k+1} = (\phi_{k+1} - \phi_k)(V_{\omega,k} - \cos \phi_k) + \sin \phi_{k+1} - \sin \phi_k - \frac{\Omega}{2}(\phi_{k+1} - \phi_k)^2 + \Lambda_k, \quad k = 1, 2, \dots, n_\phi. \quad (9) $$

The number of phases $n_\phi$, the values of $\Lambda_k$, $V_{\omega,k}$, and of $\phi_{n_\phi+1} - \phi_1$ are set by the multipactor mode under study. We solve this system numerically; we used a least-squares algorithm.$^{29,30}$

We set up some additional constraints to ensure the convergence and physicality of the results. We impose $\phi_{k+1} - \phi_k > 2\pi/100$ to avoid electrons being recollected at the same time they are emitted. We also force $\phi_1 \in [0, 2\pi]$. Finally, we impose $\Lambda(\phi) \in [0; \Lambda_d]$ to forbid the premature impact of an electron on the wrong plate, such as a $b \to t$ electron going through negative values of position before impacting the top plate.

The multipactor can only appear, thanks to the phase-focusing effect, when a small deviation in the electron emission phase leads to a smaller deviation of its collision phase. Then, the mode is said to be stable. This condition can be written as$^{26}$

<figure><img src="image_20.png" /><figcaption>FIG. 7. Trajectory of the electrons in the PP ($\Pi = 4\pi, n_\phi = 3$) $b \to t \to b$. $V_d = 25.00$ V, $\phi_1 = 3.52$ rad, and $E_0 = 7.21$ kV cm$^{-1}$.</figcaption></figure>

$$ \left| \frac{\delta \phi_{n_\phi}}{\delta \phi_1} \right| = \left| \frac{\delta \phi_2}{\delta \phi_1} \cdot \frac{\delta \phi_3}{\delta \phi_2} \cdots \frac{\delta \phi_{n_\phi}}{\delta \phi_{n_\phi-1}} \right| < 1. \quad (10) $$

### IV. EMISSION PHASE STUDY

For every potential $V_d$ and at the multipactor threshold, we recorded the emission and collision phase as well as the emission and collision plate of every simulated electron (cf. Subsection II B). Emission phases are represented as heatmaps to show their evolution with $V_d$ and are represented in the range $[0, 2\pi]$. In Figs. 7(b)-7(d), we represented the distribution of all the emission phases of all electrons. In Figs. 7(b)-7(d), we represented the emission phases of electrons according to their trajectory: $b \to t, t \to b$, and $b \to b$. We did not represent $t \to t$ electrons as their number is always negligible or null.

Each emission phase distribution was normalized between 0 and 100, and color bars were chosen so as to discern three types of emission phases:

* phases where a negligible number of electrons were emitted (noise);

* phases where a relatively small number of electrons were emitted; and

* phases where a relatively high number of electrons were emitted.

In order to compare these heatmaps to the resonance theory, we calculated the resonant phases of every relevant multipactor mode for every potential $V_d$ (cf. Subsection III C). $\Omega = E_{th}/V_d$ is calculated with the POTOMAC thresholds presented in Subsection II C (Fig. 3 results).

#### A. Zone 1

In this zone, there are no $b \to b$ electrons; hence, the multipactor is created by DS dynamics. Here, the only relevant mode is the DS ($\Pi = 2\pi, n_\phi = 2$), and the emission phases are represented in Fig. 8. We observed that equations of higher order and/or higher period multipactor did not have any solution in this configuration. The DS ($\Pi = 2\pi, n_\phi = 2$) admits two solutions: the first one is perfectly matching POTOMAC data, and the second solution is unstable.$^1$ Around $V_d = 18$ V, POTOMAC emission phases start to scatter, even if they remain close to the theoretical phases.
<figure><img src="image_21.png" /><figcaption>**FIG. 8.** The multipactor threshold calculated by POTOMAC as a function of $V_d$ (left axis, ●—). The evolution of the crossing electrons emission phase distribution with $V_d$, extracted from POTOMAC simulations (right axis, ▲) and for various modes. Electrons are sorted depending on their trajectories: (a) b → t electrons and (b) t → b electrons.</figcaption></figure>

## B. Zone 2

Emission phases are scattered in zone 2; thus, the multipactor is created by non-resonant dynamics. As there is no finite suite of fixed phases ($\varphi_1, \varphi_2, \dots, \varphi_{n_\varphi}$), this multipactor cannot be described by resonant theory.

<figure><img src="image_22.png" /><figcaption>**FIG. 9.** The multipactor threshold calculated by POTOMAC as a function of $V_d$ (left axis, ●—). The evolution of the b → b electrons emission phase distribution with $V_d$, extracted from POTOMAC simulations (right axis, ▲) and for various modes.</figcaption></figure>

## C. Zone 3

$E_{th}$ drops by 39% at the transition between 2 and 3, which is correlated with a sharp change in the emission phases. It translates the passage of a $V_d$ threshold after which SS dynamics can appear. We represented in Fig. 9 this zone relevant modes. Between 26.5 and 27 V, the SS ($\Pi = 2\pi, n_\varphi = 1$) theoretical emission phases correspond to emission phases calculated by POTOMAC. From 27 V and up to 31.5 V, they scatter, and we cannot discern delimited phases. Increasing the number of initial electrons would solve this issue as it would increase the resolution of the heatmap. It would, however, drastically increase the computation time.

Around 31.5 V and above, we observe separated emission phases corresponding to various SS modes, summarized in Table III. The highest emission phase of all these modes (around 3.6 rad) is barely visible on the heatmap. As a matter of fact, a multipactor mode with several resonant trajectories is likely to involve collisions with a TEEY inferior to unity, resulting in a temporary decrease in electron density. As an example, Table IV shows the emission phases and TEEYs of the SS ($\Pi = 12\pi, n_\varphi = 7$) collisions, for $V_d = 33$ V. We write $\sigma_k$ the TEEY of the kth collision occurring at $\phi_{k+1}$. We can observe that $\sigma_2$ is particularly low. Thus, very few electrons are emitted at the phase $\phi_3$, and it is nearly invisible in the heatmap. Each mode transition appears chaotic as the emission phases scatter.

## D. Zone 4

At the transition between zones 3 and 4, $E_{th}$ significantly increases by 11%. As in zone 2, the emission phases computed by
<table><caption>TABLE III. The single-sided modes found to re-appear in zone 3 with the range of potentials on which they match POTOMAC data.</caption><thead><tr><th>SS mode</th><th>V<sub>d</sub> range (V)</th></tr></thead><tbody><tr><td>(Π = 2π, n<sub>φ</sub> = 1)</td><td>[26.5; 27.0]</td></tr><tr><td>(Π = 14π, n<sub>φ</sub> = 8)</td><td>[31.5; 32.3]</td></tr><tr><td>(Π = 12π, n<sub>φ</sub> = 7)</td><td>[32.5; 33.5]</td></tr><tr><td>(Π = 10π, n<sub>φ</sub> = 6)</td><td>[33.7; 35.5]</td></tr><tr><td>(Π = 8π, n<sub>φ</sub> = 5)</td><td>[35.7; 37.8]</td></tr><tr><td>(Π = 6π, n<sub>φ</sub> = 4)</td><td>[39.7; 43.0]</td></tr><tr><td>(Π = 10π, n<sub>φ</sub> = 7)</td><td>[44.5; 46.8]</td></tr></tbody></table>

<table><caption>TABLE IV. TEEY of the collisions in the SS (Π = 12π, nφ = 7). Vd = 33.0 V and E0 = 6.03 kV cm⁻¹. These parameters are identical to those used in Fig. 5.</caption><thead><tr><th>k</th><th>mod(φ<sub>k</sub>, 2π) (rad)</th><th>σ<sub>k</sub></th></tr></thead><tbody><tr><td>1</td><td>2.650</td><td>1.10</td></tr><tr><td>2</td><td>2.238</td><td>0.28</td></tr><tr><td>3</td><td>3.661</td><td>1.31</td></tr><tr><td>4</td><td>3.250</td><td>1.33</td></tr><tr><td>5</td><td>3.121</td><td>1.33</td></tr><tr><td>6</td><td>3.011</td><td>1.31</td></tr><tr><td>7</td><td>2.882</td><td>1.27</td></tr></tbody></table>

POTOMAC are scattered. The resonance theory cannot describe this configuration, and we did not find any solution matching POTOMAC data.

# V. THEORETICAL MULTIPACTOR THRESHOLD

## A. Methodology

In Sec. IV, we used the resonance theory to find which multipactor mode appeared at each potential. We injected Ω calculated with POTOMAC in the position equations. In this section, the theoretical multipactor threshold of the structure is calculated without using any data calculated by POTOMAC.

For a period-Π multipactor with nφ fixed phases, the multipactor will appear if

$$ \prod_{k=1}^{n_{\varphi}} \sigma_k > 1. \qquad (11) $$

$E_0$ is incremented from 4 to 17 kV m⁻¹ with 121 steps, and $V_d$ is swept from 0 to 60 V with 101 steps. For a couple ($E_0, V_d$), the resonant phases of the mode under study (cf. Subsection III C) are calculated. We inject these phases in Eq. (5a) to compute the TEEY of each collision and verify if (11) is fulfilled. If it is not, $E_0$ is incremented. Otherwise, the threshold is found, and the next potential is studied.

The convergence of every solution was checked. Their stability was verified [Eq. (10)], with the numerical value $\delta\phi_1 = 10^{-2}$ rad. As the stability results can rely on the value of $\delta\phi_1$,³¹,³² we tried various values ranging from $10^{-3}$ to $10^{-1}$ rad, with no appreciable effect. The multipactor modes presented in Sec. IV, as well as the

SS (Π = 2π, nφ = 2) and the PP (Π = 4π, nφ = 3) b → t → b, were studied. At a given $V_d$, several modes can have a solution, and each mode can have several solutions. In this case, we consider that the solution with the lowest threshold is the most favored or fundamental, as it would be the first to appear when increasing the RF electric field amplitude.

## B. Calculated theoretical multipactor thresholds

In Fig. 10, we present the evolution of $E_{th}$ as calculated by POTOMAC with $V_d$. We also show the theoretical multipactor thresholds. In general, there is a good matching between these two curves. This is particularly true in the zones 1 and 3, where the modes at the origin of the multipactor were clearly identified in Sec. IV.

In zone 1, the theoretical threshold of the DS (Π = 2π, nφ = 2) matches the threshold calculated by POTOMAC. This mode is stable, and the relative error between the thresholds is under 2% at 0–16.8 V. At 16.8–19.3 V, where POTOMAC emission phases scatter, this modes becomes unstable and the error grows up to 9%. This instability does not prevent the multipactor from appearing; it leads to an important loss of electrons, compensated by higher impact energies.³³

In zone 2, the DS (Π = 2π, nφ = 2) is the most favored between 19.3 and 22.2 V. It is unstable on this range of potentials. The threshold calculated by POTOMAC is underestimated at 19.3–21 V; in this range, the loss of electrons is too important to be compensated by the higher impact velocities. Thus, the non-resonant multipactor occurs, with a slightly higher threshold. At 22.2–26.5 V, the PP (Π = 4π, nφ = 3) b → t → b has the lowest threshold. It is stable on this whole range. Even if the multipactor is chaotic, we believe

<figure><img src="image_23.png" /><figcaption>FIG. 10. The evolution of the multipactor threshold calculated by POTOMAC (—) with $V_d$. The evolution of the theoretical threshold of the most favored mode, for each potential.</figcaption></figure>
that the resonant trajectories of the DS ($\Pi = 2\pi, n_\phi = 2$) and the PP ($\Pi = 4\pi, n_\phi = 3$) $b \to t \to b$ are good approximations of the real electrons trajectories.

In zone 3, the nature and threshold of the modes are consistent with that discussed in Sec. IV. Their error stays under 5%, and they are all unstable. According to Semenov et al.,³⁴ for SS modes, the increase in deviation $\delta\phi_k$ saturates and turns into finite oscillations.

<figure><img src="image_24.png" /><figcaption>FIG. 11. Infinite metallic rectangular waveguide with a gap $d$ and a width $W$. Insulating inserts (in blue) isolate the bottom plate from the top plate. They do not influence the TE$_{10}$ propagation of $E_{RF}$. The potential of the bottom plate is $V_d$, the top plate being maintained at ground potential.</figcaption></figure>

<table><caption>TABLE V. Full electron emission model used in this study.</caption><thead><tr><th>Population</th><th>Emission yield</th><th>Angular distrib.</th><th>Energy distrib.</th></tr></thead><tbody><tr><td>SE</td><td>Dionne<sup>10</sup></td><td>Cosine law<sup>36</sup></td><td>Chung and Everhart<sup>37</sup></td></tr><tr><td>EBE</td><td>SLAB<sup>38</sup></td><td>SLAB<sup>38</sup></td><td>Monoenergetic</td></tr><tr><td>IBE</td><td>Constant</td><td>Cosine law<sup>36</sup></td><td>Uniform</td></tr></tbody></table>

<table><caption>TABLE VI. Copper Dionne's parameters as defined in Ref. 10 for the more realistic study.</caption><thead><tr><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>A</td><td>6.67</td></tr><tr><td>n</td><td>1.00</td></tr><tr><td>d (nm)</td><td>9.47</td></tr><tr><td>S</td><td>0.14</td></tr><tr><td>k<sub>a</sub></td><td>0.40</td></tr><tr><td>k<sub>b</sub> (°)</td><td>0.14</td></tr><tr><td>Φ (eV)</td><td>4.65</td></tr><tr><td>η<sub>i</sub></td><td>0.25</td></tr></tbody></table>

In zone 4, none of the modes that we studied matches the emission phases calculated by POTOMAC. In particular, the sudden augmentation in $E_{th}$ at 47.0 V is not reproduced by any of the theoretical thresholds.

## VI. POTOMAC SIMULATIONS IN 3D, CONSIDERING ENERGY AND ANGULAR DISTRIBUTIONS AS WELL AS EBES

### A. Presentation of a more realistic model

We made strong assumptions in the precedent simulations: lack of energy and angular distributions, 1D, and no EBEs. Thus, there was no dispersion in electrons velocity, which created the $E_{th}$ drops

<figure><img src="image_25.png" /><figcaption>FIG. 12. Modeled TEEY as a function of the impact energy of PEs used in this study. TEEY is represented for various incidence angle $\phi_0$, defined prior to the normal of the sample. The TEEY used in precedent studies is also shown.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>FIG. 13. The multipactor threshold calculated by POTOMAC as a function of the potential on the bottom plate $V_d$, with the simple and more realistic models.</figcaption></figure>
at 2–3 and 3–4 transitions. In this section, we make POTOMAC simulations in a more realistic configuration to study the influence of these hypotheses.

We study in 3D a rectangular waveguide of width $W \gg d$, represented in Fig. 11. The bottom plate is maintained at the potential $V_d$, while the top plate is grounded. In practice, the insulation can be realized by placing insulating inserts in the waveguide lateral walls. We consider that they do not influence the propagation of the fundamental TE$_{10}$ mode,

$$E_{\text{RF}}(t, y) = E_0 \sin(\phi) \sin\left(\frac{\pi y}{W}\right) \dot{x}. \quad (12)$$

We neglect the RF magnetic field, which is justified when $W \gg d$.35 We also neglect the $z$ propagation term of the RF electric field, which comes back to study the $(x, y)$ plane where the electric field amplitude is maximum. The full electron emission model is resumed in Table V.10 In the remaining part of the paper, we call this new configuration the more realistic model, in opposition to the first that was simplistic.

For this simulation, we took the closest dimensions possible to the previous dimensions: $d = 0.1$ mm, $W = 19.05$ mm, and $f = 11$ GHz. It corresponds to a WR75 waveguide, with a gap reduced from 9.525 to 0.1 mm. As it is very narrow, the waveguide cannot transmit a significant power and has no industrial

<figure><img src="image_27.png" /><figcaption>FIG. 14. Heatmaps of electron emission phases in the more realistic configuration. Electrons are sorted depending on their trajectories: (a) all electrons, (b) $b \to t$ electrons, (c) $t \to b$ electrons, and (d) $b \to t$ electrons.</figcaption></figure>
application. However, its very low threshold eases the multipactor measurements. We performed measurements on this system in collaboration with a private company; results will be published in a dedicated article.

In this section, we use a slightly different electron emission model that is in 3D and takes EBEs into account. We fitted the TEEY on the same experimental data as for the first simulation; we reported the model parameters in Table VI. We present in Fig. 12 the TEEY as a function of impact energy; for comparison, we also present the TEEY used in the previous study.

Both fits are similar, except the low-energy peak due to EBEs. New TEEY curve is slightly higher than previous at high energies; we do not expect these differences to be of importance, as the majority of PEs impact walls with a low energy. Here, $E_{c,1}(0^{\circ}) = 29$ V.

## B. Simulation results

We represented in Fig. 13 the evolution of the multipactor threshold with $V_d$, as calculated by POTOMAC in the two configurations. As a whole, the $E_{th}(V_d)$ curves are similar in terms of absolute value and general shape. The most notable difference is that the introduction of velocity dispersion erased the strong discontinuities at 26.5 and 47.0 V. In addition, the *more realistic* configuration exhibits a local minimum in threshold at 10 V.

As in Sec. IV, we represented in Fig. 14 electrons emission phase distributions and their evolution with $V_d$. Each distribution is normalized between 0 and 100. We did not represent $t \to t'$, as there is very few of them to contribute to the multipactor. The color bars were chosen so as to discern four main types of emission phases:

* phases where a negligible number of electrons were emitted (noise);

* phases where a relatively small number of electrons were emitted;

* phases where a relatively high number of electrons were emitted; and

* phases where the majority of electrons were emitted, allegedly matching emission phases of *simplistic* model.

As expected, the emission phases are significantly more scattered than in precedent study. Thus, we cannot identify resonant modes as we did earlier in this study. In contrast with the *simplistic* study, there are $b \to b$ electrons in zone 1 and crossing electrons in zone 3.

## VII. DISCUSSION

We observed that the threshold was always minimum for $V_d = 0$ V. It is to be noted that its position depends on the geometry of the system as well as on the RF frequency. For example, we observed that if we set $d = 0.15$ mm, the minimum was at $V_d = \pm 11$ V. Thus, the results presented in this study should not be generalized. In particular, the addition of a potential difference between the plates does not necessarily raise the threshold.

The resonance theory is based on the strong assumption that all electrons follow the same trajectory; it was shown to be unreasonable when the gap is too big or when the dispersion in emission velocities is too wide.³³,³⁹ In this study, the waveguide is extremely tight, which should not invalid the theory. The main difference between the

<figure><img src="image_28.png" /><figcaption>FIG. 15. The multipactor threshold calculated by POTOMAC as a function of the potential on the bottom plate $V_d$. Threshold calculated with the *more realistic* model and with the *more realistic* model without EBEs.</figcaption></figure>

*simplistic* and the *more realistic* configurations is that we included emission velocities dispersion in the latter. It results on a smoothing of the $E_{th}(V_d)$ curve, but does not fundamentally change its shape. However, it would be incorrect to assume that, in this configuration, velocity dispersion should be neglected.

As a matter of a fact, this notion also encompasses the treatment of EBEs that have an energy distribution very different from SEs. We realized another simulation in the *more realistic* configuration, with an EBEEY set to 0. The evolution of the multipactor threshold calculated by POTOMAC is represented in Fig. 15. Removing EBEs increases the multipactor threshold of 5%–27% between 0 and 40 V. For potentials higher than 40 V, it cancels the multipactor. It is due to the fact that, for high potentials, the multipactor appearance involves a lot of very low energy collisions for which the EBEEY is the highest. These results highlight an important point: even in small gap geometries, the velocity spread plays an important role.

We also neglected the influence of the $H_{RF}$ field. It tends to push electrons toward the lateral walls of the waveguide (Miller force),³⁵ creating a spatial instability. As the RF electric field amplitude is lower at the sides of the waveguide, the electrons pushed away from the center go under a weaker acceleration, and their energy is too low to ensure $\sigma > 1$. When the time between the emission and the collision of electrons, called the flight time, is relatively small, neglecting $H_{RF}$ is an acceptable hypothesis.³⁵ This is typically the case when $W \gg d$. However, in our configuration, some multipactor modes involve electrons with an important flight time. We expect that considering $H_{RF}$ would slightly raise the multipactor threshold, without fundamentally change the dynamics at stake. A new set of simulations would be required to precisely assert the importance of this hypothesis.

## VIII. CONCLUSIONS

In this paper, we led a detailed study of the multipactor appearance in two RF systems holding a potential difference $V_d$ between their plates. The first one was an infinite parallel-plate system, studied in 1D, with a simplified electron emission model. The second
system was a rectangular waveguide, studied in 3D, with a more real-
istic electron emission model. We calculated the evolution of these
systems' multipactor threshold with $V_d$, with the numerical tool
POTOMAC. In the first configuration, we used resonance theory to
determine which mode appeared at different potentials and the theo-
retical threshold of the system. For the two RF systems, POTOMAC
simulations outlined an important dependence of the multipactor
threshold $E_{th}$ on the potential difference between the plates $V_d$.

In the parallel-plate system, there was no dispersion in the
impact velocities of the electrons, which caused significant and
unphysical threshold variations at $V_d$ = 26.5 and 47.0 V. We
observed that, when increasing $V_d$, the multipactor progressively
switches from DS (Π = 2π, $n_φ$ = 2) dynamics to chaotic multipactor.
Then, there is a smooth transition between various asymmetric SS
modes. For very high potentials, the multipactor becomes chaotic
again. In this same geometry, we computed the theoretical thresh-
old, which matched POTOMAC. Hence, we can validate both
POTOMAC and our implementation of resonance theory.

In the rectangular waveguide, the dispersion in electrons
impact velocities makes the notion of modes irrelevant. However,
we confirmed the assertion of other authors<fnref n="7" /> according to whom
multipactor progressively switches from DS to SS dynamics. The
general shape of the $E_{th}(V_d)$ curve was similar to that of the par-
allel plate, at the exception of the unphysical variations of $E_{th}$. We
observed that omitting the EBEs in this configuration led to a pre-
mature disappearance of the multipactor (40 V instead of 55 V). It
outlines the important dependence of $E_{th}$ not only on the electron
emission properties of the walls of the system wall but also on its
modeling.

ACKNOWLEDGMENTS

This work was carried out within the framework of the
EUROfusion Consortium and received funding from the Euratom
Research and Training Programme 2014–2018 and 2019–2020
under Grant Agreement No. 633053. The views and opinions
expressed herein do not necessarily reflect those of the European
Commission.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

DATA AVAILABILITY

Some of the data that support the findings of this study
are openly available in Zenodo at http://dx.doi.org/10.5281/
zenodo.3648704.¹⁹ The remaining data that support the findings
of this study are available from the corresponding author upon
reasonable request.

APPENDIX: POTOMAC CONVERGENCE AND
CONSISTENCY STUDIES

In their study,⁴⁰ Anza et al. determined the multipactor thresh-
old of an infinite parallel-plate system. Their study has the merit
to compute the threshold of the structure with different methods:
statistical theories, PIC simulations, and an experimental study. We

<figure><img src="image_29.png" /><figcaption>FIG. 16. The evolution of the multipactor threshold with $V_d$, in the geometry presented in Sec. II and with different values of $N_e$.</figcaption></figure>

led a set of POTOMAC simulations in their configuration: the gap
d = 1 mm, and the frequency f = 1.64 GHz. The material is sil-
ver, and TEEY is calculated with modified Vaughan model.²⁰,²¹,⁴⁰,⁴¹
For this fd product, we can refer to the study of Anza et al.
(Fig. 11), which states that the voltage threshold is around 100 V:
it is consistent with POTOMAC, whose threshold is 104 V in this
configuration.

We represented in Fig. 16 the evolution of the multipactor
threshold with $V_d$ as calculated by POTOMAC for different num-
bers of initial electrons. The configuration is the same as in Sec. II,
where 100 electrons are enough to ensure convergence. $N_e$ = 100 is
also adequate in the configuration discussed in Sec. VI.

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