Gridless inductive output tubes (IOTs) offer compact size and high-power amplification at sub-GHz frequencies. Minimizing cavity dimensions in the interest of compactness leads to smaller gaps, which may cause multipactor discharge under high-power operating conditions. The uncontrolled electron growth resulting from multipactor breakdown can lead to undesired effects including surface damage and system failure. This paper performs a parallel-plate multipactor analysis for a high-Q, L-shaped, aluminum, 431 MHz cavity designed for a gridless IOT to be operated in the MW-power regime. The cavity gap is 27 mm, and diameter is 339 mm. Multipactor susceptibility regions are calculated for non-zero emission energy, half-cycle, and non-half-cycle multipactor using a semi-analytic approach and a standard aluminum secondary electron yield (SEY) curve. The analytical results are validated with particle-in-cell simulation in CST Studio. Simulation results show a voltage range of 6.4–19 kV, compared to the analytically calculated values of 8.2 and 18.3 kV for the lower and upper bounds, respectively. Fluorocarbon coating as a means to reduce secondary electron emission is simulated, which shows 46% reduction in peak particle population with an 11.2 nm PTFE coating, with further reduction as coating thickness increases. The results show that the L-shaped cavity is a suitable choice for this IOT design as it does not exhibit single-surface multipactor and will not develop two-surface multipactor at full-power operation.

## I. INTRODUCTION

Multipactor discharge, or simply multipactor, is a nonlinear resonant electron discharge, which occurs in radio frequency (RF) systems under low-pressure and high-vacuum conditions.^{1} It leads to disruption in device performance, such as the generation of harmonics, cavity detuning, localized heating, and device breakdown.^{2} Multipactor occurs when an electron, accelerated by an electric field, strikes a metallic or dielectric surface.^{3} The electron source may be thermionic discharge, photoemission, or—in the case of microwave tubes—electrons from the beam source itself.^{4} If the electron impact velocity is within a certain range, one or more secondary electrons may be generated depending on the material, known as the secondary electron yield (SEY). For SEY greater than unity, the secondary electrons may generate a third generation of electrons, and so on. If multipactor conditions are satisfied, this may lead to exponential growth, also known as electron avalanche.^{5} Multipactor has been observed in high-power microwave systems,^{6} including traveling wave tubes (TWTs)^{7} and klystrons.^{8} Multipactor is also a concern in space-based systems.^{9} Some issues in particle accelerators, such as the Large Hadron Collider (LHC), have also been attributed to multipactor.^{10}

Early multipactor models assumed that the ratio of primary electron impact energy to the secondary electron emission (SEE) velocity was constant. This is commonly referred to as a constant-*k* model.^{11} The model was improved by Vaughan, for parallel-plate geometries, which replaces the constant-*k* assumption with a non-zero emission velocity model.^{12} Since then, other techniques have been presented such as the Furman model,^{13} which is notably more complex. Another method is statistically computing multipactor using Monte Carlo simulation, which, while faster, cannot account for the electron cloud or space-charge effects.^{14} The Vaughan model is widespread due to its simplicity and accuracy and can be used in most simulation tools available today.^{15} Moreover, particle-in-cell (PIC) tools greatly aid in the visualization of multipactor and the verification of analytic approaches.

Recently, as part of our study on inductive output tubes (IOTs), we have presented a 431 MHz gridless IOT with annular beam and MW-class operation.^{16,17} With high mesh density and secondary electrons enabled, uncontrolled electron growth was not observed in PIC simulations. However, as multipactor is a localized phenomenon under very specific operating conditions, it requires a closer examination. For pulsed microwave devices, it has been noted in the literature that a shorter pulse length reduces the chances of multipactor.^{18} However, the IOT operates with longer pulse durations, up to a few milliseconds.^{19} Therefore, a multipaction analysis of the structure is warranted. In this paper, a semi-analytic approach from literature is applied. The gap distance and voltages are used to calculate these susceptibility regions for parallel-plate multipactor. The analytic expressions for this susceptible gap voltage are solved numerically for multiple electron emission scenarios. A very high-density mesh must be simulated with a tool capable of modeling electron trajectories, with support for secondary electron emissions using material-specific yield data. In this paper, the effects of dielectric coating are also simulated. Therefore, the tool should be capable of simulating the effects of the material on electron emission. CST Studio is used for the simulations in this paper as it offers the required features to simulate multipactor with a user-friendly interface. The paper includes a detailed simulation setup for the localized multipactor simulation.

## II. MODEL

The schematic quarter view of the IOT is given in Fig. 1(a), and the manufactured cavity is shown in Fig. 1(b). The system is comprised of the following: an input cavity, which consists of an annular cathode and a modulating electrode, a resonant cavity, a beam collector, and the RF output line. From previous simulations,^{16} the IOT operating parameters are as follows. The average output power is 2.24 MW, with beam voltage of 70 kV, a gain of 16.5 dB, and an average efficiency of 53.9%. The space-charge induced beam spread is controlled using a constant 0.7 T magnetic field along the length of the IOT. For the gridless IOT, the beam is not controlled by cathode-grid gap spacing.^{20} Instead, the beam is controlled via emission gating. The RF input is applied to the “modulating anode,” which controls the beam switching via E-field. Achieving resonance in the modulation cavity is, therefore, not needed. The frequency of operation is, instead, determined by the beam switching frequency and the resonance frequency of the output cavity. Moreover, due to this emission gating, the gridless IOT operates during the half-cycle of the applied RF period, and the beam remains off for the remaining half period.

Due to the half-cycle operation of this IOT, the electric field in the input cavity, drift tunnel, and beam collector remains unidirectional, as the beam is switched off for the remaining half-cycle. For a one-sided E-field polarity, there is no reverse field to sustain multipactor. The output cavity is only the resonant structure in the gridless IOT. Therefore, the cavity and the output RF window^{21} are the only two locations in this design where multipactor may occur. The primary focus of this paper is the output cavity. As seen in Fig. 1(a), the L-shaped cavity consists of a uniform gap that may be subject to multipactor during resonance, as will be discussed in Sec. II A. When the beam crosses the gap, electrons may enter the cavity and lead to secondary emission. In the following, we will present the two-surface multipactor model as it applies to the field distribution inside this cavity.

### A. Multipactor model

Multipactor theory is based on a resonant condition between an electric field and an electron.^{22} For an alternating RF field, e.g., during resonance, the field causes the electron to accelerate and strike a surface with a certain velocity and angle. The impacted surface may be the same as the emitting electrode, known as single-surface multipactor, or it may be the opposing surface in which case it is parallel-plate or two-surface multipactor.^{23} The multipactor model used is as follows. From Fig. 2, we see that the E-field of the operating mode inside the L-shaped cavity, during beam operation, is normal to the surface, which accelerates the electron to the opposing surface. The magnetic field is applied in the z direction, along the direction of the beam. The external magnetic field has been used as a means to inhibit multipactor in some literature.^{24} This is a functional use for cases where the electric and magnetic fields are perpendicular,^{25} which causes the *E* × *B* vector to point away from the surface. This leads to an increase in the flashover voltage and can also suppress multipactor.^{26} In our case, however, the E-field of the operating mode is parallel to the magnetic field in the cavity, and this suppression does not apply.

Given an RF electric field of frequency $f=\omega /2\pi =1/T$ across a constant gap *d*, we can write the expression of the E-field as $E=Eg\u2009sin(\omega t)$. The magnitude of the field can be expressed in terms of voltage as $Eg=(Vg/d)$, where *V _{g}* is the voltage across the gap. The electron must impact the parallel surface within an odd number of RF half-cycles to be in resonant condition with the RF field.

^{27}This is commonly known as half-cycle multipactor. The equation of motion for the electron through this field can be used to determine the frequency, RF field intensity, and gap spacing required to sustain multipactor discharge. It should be noted, however, that simply modeling the electron trajectory with a given RF field (or voltage) is not sufficient to determine the existence of multipactor. Electron impact velocity and the secondary electron yield (SEY) are also required to determine multipactor. The SEY is a material-specific property and may vary significantly with certain modifications to the material, such as baking and oxidation.

^{28}Moreover, higher-order multipactor and non-half-cycle multipactor can also exist in tandem with the half-cycle case.

^{29}In the former case, the electron takes more than one half-cycle to reach the opposite electrode. In the latter, the first and second half-cycle trajectories are not equal. Analysis of these cases is also addressed for the IOT cavity.

*y*=

*d*is

^{30}

*m*is electron mass, $u0=2eE0/m$ is the initial velocity of the electron at emission energy

*E*

_{0},

*e*is the charge of the electron, and $\varphi $ is the phase of the electric field at which the electron is emitted, i.e., at $(Vg/d)\u2009sin(\omega t+\varphi )$. At this instant, the impact velocity at

*y*=

*d*is given as

For the following voltage and velocity calculations, (1) and (2) are solved numerically using MATLAB. This particle trajectory for the cavity under study is given in Fig. 3(a). The minimum and maximum gap voltages, *V _{gmin}* and

*V*, require the material-specific properties of SEY. To this end, the SEY curve for aluminum is plotted in Fig. 3(b). The dataset is sourced from Dennison.

_{gmax}^{31}The first crossover point is the lowest energy at which the $SEY=1$, which, in this case, is 51 eV. The maximum SEY is $\delta max=2.35$ at 320 eV. The impact velocity of (2) vs phase and the velocity of the first crossover point are shown in Fig. 3(c). The phase at which the electron impact velocity is equal to the velocity of the crossover point is termed the crossover phase ( $\varphi x$). By plotting (1) vs phase, two other phase values of interest are determined. These are the phase at which the voltage is zero ( $\varphi 0$) and the phase at which the voltage is at its lowest value ( $\varphi min$). The values of the phases that correspond to the highest and lowest voltage are then selected. These highest and lowest voltage values are

*V*and

_{gmax}*V*, respectively. The set of parameters that will be used for the following simulations is given in Table I.

_{gmin}Quantity . | Symbol . | Value (unit) . |
---|---|---|

Cavity gap | d | 27 mm |

Cavity radius | r | 169.5 mm |

Energy of first crossover point | E_{1} | 51 eV |

Impact velocity of first crossover point | u_{1} | $4.235\xd7106\u2009m/s$ |

Cavity operating frequency | f | 431.5 MHz |

RF time period | T | 2.317 ns |

Quantity . | Symbol . | Value (unit) . |
---|---|---|

Cavity gap | d | 27 mm |

Cavity radius | r | 169.5 mm |

Energy of first crossover point | E_{1} | 51 eV |

Impact velocity of first crossover point | u_{1} | $4.235\xd7106\u2009m/s$ |

Cavity operating frequency | f | 431.5 MHz |

RF time period | T | 2.317 ns |

Here, it should be noted that one-sided multipactor may also exist in tandem with the parallel-plate type. However, single-surface multipactor is generally a concern in the presence of dielectrics,^{32} where an electron emitted from the surface creates a positive charge on the dielectric surface resulting in a dc electric field normal to the surface.^{33} An electron emitted at an angle *θ* is then forced back to the emitted surface, leading to secondary generation. It may be observed on a metallic surface,^{34} but requires specific conditions to exist. It necessitates a magnetic field tangential to the electrode surface,^{35} in addition to the RF field which accelerates the electron to be tangential to the surface.^{36} These conditions are not satisfied for this cavity. Two-surface multipactor is generally of primary concern in microwave tubes and is the focus of this paper.

The second crossover point in Fig. 3(b) occurs at very high energy. In this case, at $\u223c2200\u2009eV$. Empirically, it is known that the energy of most emitted electrons falls between 0 and 20 eV.^{37,38} This high energy is not achievable by the collision electrons in practice. Therefore, the second crossover energy of the SEY curve is generally ignored and it is assumed that any electron over the first crossover point energy can fall into the multipactor range.

Furthermore, for the multipactor simulations in this paper, the space-charge effects are neglected. The simulations while considering the space-charge effect are more computationally intensive and may not always be necessary. Notably, the space-charge debunching is relevant to the later stages of multipactor and is not the cause of the multipactor itself. Initially, when the electron population is low, the RF field is primarily responsible for focusing the electron trajectories or moving electrons between the surfaces. As the electron population grows, the collective electron effects start to dominate.^{39} It primarily affects the electron saturation state. Generally, reaching the multipactor saturation state can take up to 300 RF cycles^{40} which would cause an unreasonably large simulation time for a 3D model.

During sustained multipactor, the electron cloud grows in tandem with the avalanche. The electron (avalanche) bunch experiences two effects. First, the electrons may strike the surface during an opposing phase of the RF field, which causes zero secondary emission regardless of the impact energy. Second, the outward space-charge debunching force opposes the RF field, which can cause the electrons to be driven back into the emitting surface resulting in net zero secondaries.^{39} It has been shown that this leads to a kind of multipactor “equilibrium” state where the electron energies tend to SEY = 1.^{41} Additionally, the space-charge saturation effects have a minimal impact on the steady-state multipactor behavior for cavities with larger Q-factor (*Q* > 10),^{42} whereas, for this L-shaped cavity, we have measured the Q-factor to be *Q* = 1812.^{16}

## III. SIMULATION AND RESULTS

### A. Zero and non-zero emission velocity cases

The simplest case for a parallel surface multipactor is one with half-cycle electron trajectory, zero initial electron energy ( $u0=0$), and first-order (*N* = 1). Using Eq. (1), the values of *V _{gmin}* and

*V*are plotted against a frequency range in Fig. 4. This is the traditional susceptibility chart, which signifies the region between the curves at which multipactor can occur.

_{gmax}Figure 5 shows the change in susceptibility region between zero and non-zero initial energy for first-order two-surface multipactor. While there is a minor change in the lower bound, it is observed that the upper voltage shifts upward causing an increase in the susceptibility range. It should also be noted that the early values are omitted in the plot where the electron impact velocity is lower than the electron velocity at the first crossover point, i.e., $ui(\varphi )<ui(\varphi x)$, where multipactor cannot occur. At 431 MHz, we find $Vgmin=9.2\u2009kV$ and $Vgmax=16.1\u2009kV$.

### B. Non-half-cycle and higher-order multipactor

^{29}i.e., the primary electron may reach the opposite surface at a time instant before or after an odd multiple of the half-cycle. Correspondingly, the return trajectory of the emitted secondary can take a longer or shorter time to reach the opposite surface. This can occur due to shifts or curvatures in the E-field, or by variation in electron emission angle. For these electrons to cause multipactor discharge, it is still necessary to have an energy or velocity for which the electrode's SEY is greater than unity. This time duration difference is here written as $\Delta \xi \omega $. For instance, in the case of

*N*= 1, the electron may complete the positive trajectory to reach

*y*=

*d*in a shorter time ( $t=\varphi \omega +T2\u2212\Delta \xi \omega $), than the negative cycle ( $t=\varphi \omega +T2+\Delta \xi \omega $), or vice versa. This is visualized in Fig. 6. In this case, the gap voltage and impact velocity are instead written as,

^{43}

The change in susceptibility curve for the case of $\Delta \xi =15\xb0$ and $E0=10\u2009eV$ is shown in Fig. 7, compared to the half-cycle case $\Delta \xi =0\xb0$ and $E0=10\u2009eV$. For this case, we find $Vgmin=8.2\u2009kV$ and $Vgmax=18.3\u2009kV$ at 431 MHz.

Higher-order multipactor, for $N=3,5,7\u2026$, may also exist in tandem with first-order. Increasing the multipactor order raises the minimum frequency required to sustain that order, i.e., a higher frequency is needed to satisfy the $NT/2$ condition such that the RF period aligns with electron impact given a fixed distance. Moreover, increasing *N* increases the impact velocity of the first crossover point in (1) and (2). When the crossover point is increased, more of the velocity curve of Fig. 3(c) falls below the crossover point, below which SEY < 1 and multipactor cannot sustain. This essentially increases the lower threshold for the higher-order multipactor to occur. For our case, the third-order susceptibility region is shown in Fig. 8 for $E0=10\u2009eV$ for *N* = 1 and $E0=0\u2009eV$ for *N* = 3.

### C. Multipactor PIC simulation

The details of the simulations are as follows. Multipactor simulation requires a very high-density localized mesh, for accurate modeling of electron trajectories. The rapid growth of electrons during avalanche is highly computationally intensive, especially in a 3D simulation. One technique to circumvent this is to simulate a duration where multipactor clearly occurs even if it does not reach a steady state. For instance, in Ref. 15, the authors performed the simulation for up to 40 ns. This approach has an acceptable computational load and can be used to verify the thresholds for the susceptibility regions. For this paper, the cavity model is first designed in the eigenmode solver to generate the field distribution of the desired operating mode. The hexahedral-type mesh is used, as it provides faster computation and lower cell count, with high compatibility for imported data.^{44}

It is possible to generate the field data directly in the PIC simulation. However, this eigenmode approach is utilized for two main reasons. First, the eigenmode solver is not encumbered with particle trajectories, which results in faster simulation. Second, this approach allows us to control the field intensity as follows. CST allows the imported field data to be scaled with a desired factor, which can be set to a desired value in the simulation setup window. Since the electric field across the constant gap represents the gap voltage, this scaling factor can be used to perform the PIC simulation across a range of voltages to determine the existence of multipactor and the corresponding susceptibility thresholds. A region of 1600 mm^{2} is defined with a localized high-density mesh on the cavity wall where multipactor is to be observed. For the L-shaped cavity, this region can be defined across the gap at any point along its length as the field is uniformly distributed, as shown in Fig. 2.

Furthermore, simulation details are as follows. A Gaussian point electron source emitting $ne\u22481000$ electrons is placed in this dense mesh region. This results in a density of ∼15 × 10^{6} mesh cells for PIC. Since the number of electrons grows exponentially,^{18} the time duration must be set such that multipactor can be observed without causing large computational overhead. The maximum simulation duration is set to 50 ns, which encompasses ∼21 cycles of the RF field (42 impacts for *N* = 1), which is sufficient to observe any multipactor growth. The field data in the eigenmode solver can be used to find the voltage and peak electric field across the gap. The simulations are performed on a desktop computer, with a 12-core AMD Ryzen™ 9 CPU, 32 GB RAM, and an NVIDIA^{®} GeForce^{®} RTX 3080 GPU. Each iteration takes ∼1 h to complete.

From the PIC simulation, it is seen that the multipactor grows between the ranges $\u223c6.4$ to $\u223c19\u2009kV$. The particle population vs the voltage is plotted in Fig. 9(a). It is observed that for voltages higher than 19 kV, the number of electrons decays over time, and the multipactor is not sustained.

It is noted that the PIC simulation shows a wider multipactor range compared to the ranges obtained from (1) and (3) on both the lower end and upper end. This may be attributed to electrons with $E0>10\u2009eV$, a non-half-cycle angle of $\Delta \xi >15\xb0$, or a combination thereof. The maximum number of particles at $t=25\u2009ns$ are shown in Fig. 9(b).

The voltage in the cavity from the beam-operated PIC simulation of the IOT is shown in Fig. 10, and the susceptibility range obtained from the multipactor simulation is highlighted. It can be seen that the device at maximum power operates beyond the susceptibility range, and the cavity is not at risk of electron avalanche.

In case, the operating power of the cavity falls within the susceptibility range, and one method of multipactor suppression is with the use of coatings. Diamond-like coatings (DLC) and^{46} fluorocarbon (FC) coating, such as PTFE on aluminum have been shown to significantly reduce SEE.^{45} CST has built-in support for coatings under the section of material properties. The SEY curve for an 11.2 nm PTFE coating is shown in Fig. 11. It is observed that a PTFE coating results in a significantly lower SEY of 1.82 vs an uncoated *δ _{max}* of 2.35.

The PIC simulation is repeated with the coating on the aluminum surface, and the results are shown in Fig. 12(a). It is observed that there is a notable decrease in the maximum number of particles at the same time instant, where the peak number of particles is nearly halved. This shows that coating can be a viable method of multipactor suppression if the system is not at risk of electrical breakdown with the inclusion of dielectrics. Furthermore, it is of interest to observe the effect of coating thickness on the electron emission. The simulation is repeated while varying the coating thickness from 0 to 113 nm and the maximum number of generated particles is recorded. The results are shown in Fig. 12(b). From this, it is noted that adding a coating notably reduces *n _{e}*. However, increasing the coating thickness beyond 60 nm results in diminishing returns.

## IV. CONCLUSION

Multipactor discharge is an undesired phenomenon in vacuum tubes. Oscillating electric fields can lead to uncontrolled electron growth under certain conditions. This paper has reported on multipactor conditions for an aluminum L-shaped cavity designed for use in a sub-GHz, MW-power gridless IOT. Two-surface susceptibility charts are presented to determine the likelihood of electron avalanche during device operation. Four main cases of electron emission are studied, namely, zero initial energy, non-zero initial energy, the non-half-cycle case, and higher-order multipactor. The largest susceptible voltage range is calculated to be between 8.2 and 18.3 kV, which corresponds to the conditions of 10 eV initial electron energy, first-order, and non-half-cycle multipactor. PIC simulation with a localized high-density mesh is used to verify these values. The simulation shows a minimum and maximum voltages of 6.4 and 19 kV, respectively. The PIC simulation shows a larger susceptible region compared to numerical calculation. This is to be expected, as the numerical calculation was performed for up to 10 eV emission energy and $\Delta \xi =15\xb0$, while the particle simulation considers a larger range of operating conditions. Meanwhile, the cavity gap voltage amplitude is up to 58 kV as shown by the PIC simulation data. Finally, the effect of PTFE coatings is demonstrated in simulation, which shows a $\u223c46%$ reduction in peak electron population in the susceptibility region. Coating thickness up to 113 nm is simulated, which shows a further decrease in the number of maximum particles. However, the coating effectiveness does not increase linearly with its thickness. The 3D multipactor simulations indicate that the cavity is not at risk of multipactor breakdown at full-power operation of this IOT.

## ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China under Grant Nos. 92163204, 61921002, and 62171098, and the Natural Science Foundation of Sichuan, China, under Grant No. 2022NSFSC1695.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Muhammad Khawar Nadeem:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Shaomeng Wang:** Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Visualization (equal). **Atif Jameel:** Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Software (supporting); Writing – review & editing (supporting). **Bilawal Ali:** Conceptualization (supporting); Formal analysis (supporting); Methodology (supporting); Software (supporting). **Jibran Latif:** Data curation (supporting); Resources (supporting); Software (supporting); Visualization (supporting); Writing – review & editing (supporting). **Yubin Gong:** Funding acquisition (lead); Project administration (lead); Resources (lead); Supervision (supporting).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Multipactor in Accelerating Cavities*

*Multipactor in Accelerating Cavities*

*Vacuum Tubes, McGraw-Hill Electrical and Electronic Engineering Series*