Effect of desorbed gas on microwave breakdown on vacuum side of dielectric window

Pengcheng ZHAO (赵朋程),Zhongyu LIU (刘忠玉),*,Rui WANG (王瑞),Panpan SHU (舒盼盼),Lixin GUO (郭立新) and Xiangxin CAO (曹祥鑫)

1 School of Physics,Xidian University,Xi’an 710071,People’s Republic of China

2 National Key Laboratory of Science and Technology on Space Microwave,Xi’an Institute of Space Radio Technology,Xi’an 710100,People’s Republic of China

3 School of Sciences,Xi’an University of Technology,Xi’an 710054,People’s Republic of China

Abstract The gas desorbed from the dielectric surface has a great influence on the characteristics of microwave breakdown on the vacuum side of the dielectric window.In this paper,the dielectric surface breakdown is described by using the electromagnetic particle-in-cell-Monte Carlo collision (PIC-MCC) model.The process of desorption of gas and its influence on the breakdown characteristics are studied.The simulation results show that,due to the accumulation of desorbed gas,the pressure near the dielectric surface increases in time,and the breakdown mechanism transitions from secondary electron multipactor to collision ionization.More and more electrons generated by collision ionization drift to the dielectric surface,so that the amplitude of self-organized normal electric field increases in time and sometimes points to the dielectric surface.Nevertheless,the number of secondary electrons emitted in each microwave cycle is approximately equal to the number of primary electrons.In the early and middle stages of breakdown,the attenuation of the microwave electric field near the dielectric surface is very small.However,the collision ionization causes a sharp increase in the number density of electrons,and the microwave electric field decays rapidly in the later stage of breakdown.Compared with the electromagnetic PIC-MCC simulation results,the mean energy and number of electrons obtained by the electrostatic PIC-MCC model are overestimated in the later stage of breakdown because it does not take into account the attenuation of microwave electric field.The pressure of the desorbed gas predicted by the electromagnetic PIC-MCC model is close to the measured value,when the number of gas atoms desorbed by an incident electron is taken as 0.4.

Keywords: electron multipactor,collision ionization,desorbed gas,electromagnetic particle-incell-Monte Carlo collision model

1.Introduction

In recent years,the peak power of high-power microwave has reached the GW level,which easily causes breakdown on the vacuum side or even the gas side of the dielectric window [1–11].The secondary electron multipactor is one of the dominant reactions of breakdown on the vacuum side of the dielectric window,i.e.the phenomenon that the number of electrons increases sharply due to frequent collisions between free electrons and the dielectric surface under the action of strong electromagnetic fields [12].In the multipactor process,the collision between electrons and the dielectric surface also causes the desorption of gas molecules or atoms from the dielectric surface [13].Due to the accumulation of desorbed gas molecules or atoms,the vacuum side of the dielectric window gradually changes from vacuum environment to the gas of low pressure.At low pressure,the number density of plasma becomes very high since a large number of collisional ionization events occur.In this case,the breakdown plasma has a significant impact on the transmission of high-power microwave.Therefore,it is important to study the effect of desorbed gas on the breakdown of dielectric window for understanding and controlling breakdown.

In view of the important effect of desorbed gas on dielectric window breakdown,researchers have carried out relevant experimental studies [13,14].The high-speed optical instrument was employed by Neuberet alto diagnose the breakdown of alumina dielectric window in vacuum [13].They found that the gas pressure near the dielectric window reached 1.5 Torr in the later stage of breakdown.The experiment of breakdown on the vacuum side of microwave window was created by Changet alto understand the phenomenon of gas desorption [14].This experiment showed that the local pressure of the desorbed gas rose to several Torr,which is in agreement with the results of Neuberet al[13].

Compared to experimental research,more theoretical studies were devoted to exploring the effect of desorbed gas on microwave window breakdown in vacuum [12,15–19].Wanget alused the electrostatic particle-in-cell-Monte Carlo collision (PIC-MCC) model to numerically simulate the window breakdown involving desorbed gas [15].The effect of desorbed gas yield on the evolution of electron number density and mean electron energy was analyzed.Chenget alinvestigated numerically the suppression of window breakdown by an external magnetic field when considering the influence of desorbed gas [16].Under different yields and speeds of desorbed gas molecular,the electrostatic PIC-MCC model was employed by Donget alto simulate the changes in the number and mean energy of electrons over time [17–19].This study also discussed the influence of desorbed gas on the spatial distribution of charged particle density and normal electric field.A dynamic model was proposed by Zhanget alto analyze the effects of gas desorption and diffusion on the microwave window breakdown [12].The accuracy of the dynamic model was verified by comparing its breakdown prediction with the results of the electrostatic PIC-MCC model.

When a large number of gas molecules or atoms are desorbed from the dielectric surface,the number density of electrons generated in low pressure breakdown is much higher than that in vacuum.The breakdown plasma absorbs and reflects the microwave,and leads to changes in the microwave fields.Nevertheless,we still do not have a sufficient understanding of this phenomenon because the electrostatic PIC-MCC model was usually used in previous studies[12,15,17–19].In this paper,the electromagnetic PIC-MCC model is used to investigate the evolution of dielectric surface breakdown involving desorbed gas.We focus on the changes in microwave fields caused by the surface breakdown.The breakdown characteristics obtained from the electromagnetic PIC-MCC model,such as the evolution of electron number,are compared with the results from the electrostatic PIC-MCC model in order to illustrate the limitations of the latter.Finally,the gas desorption yield is approximately determined by comparing the simulated values of desorbed gas pressure with the experimental data.

2.Model

Figure 1 is the schematic of microwave dielectric surface breakdown involving desorbed gas in vacuum.In the initial stage,electrons only come from secondary electron multipactor,so they exist near the dielectric surface.As time increases,the gas pressure near the surface rises because the electrons impact the surface and cause gas molecules or atoms to desorb from the surface.The collision ionization between electron and desorbed gas molecule produces an additional electron and ion.Therefore,there are a large number of electrons,ions,and desorbed gas molecules or atoms near the surface in figure 1.The normal electric fieldEnis generated by charged particles on the surface and in space.TheEnhas a significant impact on the motion of electrons and ions.In addition,there is a significant attenuation of the amplitude of the microwave electric field when the plasma density increases to a sufficiently high level.

Figure 1.Schematic diagram of microwave dielectric surface breakdown involving desorbed gas in vacuum.

In order to fully consider these reaction processes mentioned above,the one-dimensional spatial distribution and three-dimensional velocity (1D3V) distribution electromagnetic PIC-MCC model is employed to describe the breakdown on the dielectric surface in vacuum.We use this model to focus on the variation of microwave field amplitude and its feedback effect on the dielectric surface breakdown.This section is divided into three subsections,including Maxwell equations,PIC-MCC algorithm and the method for calculating desorbed gas.

2.1.Maxwell equations

The breakdown on the dielectric surface caused by an uniform plane microwave is considered in this work.The microwaves polarized along thex-axis propagate in a negativez-axis direction,as can be seen in figure 1.The electric field of microwave is parallel to the dielectric surface.In this case,the breakdown characteristics can be described by the 1D 3V electromagnetic PIC-MCC model [20].The one-dimensional Maxwell equations in this model are as follows:

In equations (1) and (2),Exis the electric field of microwave inx-axis direction,Hyis the magnetic field of microwave iny-axis direction,Jxis the current density generated by the motion of electrons and ions,and ϵ0and μ0are the dielectric constant and magnetic permeability in vacuum.It is worth noting that the self-organized normal electric field is formed due to the spatial separation of electrons and ions.The normal electric field,in turn,affects the density and velocity of charged particles,and thus causes the corresponding current density to change with time.In equation (1),the current densityJxis obtained by computing the sum of the products of charge,density,and velocity of charged particles under the action of self-organized electric field.In other words,the influence of self-organized normal electric field on electromagnetic wave propagation can be considered using equations (1) and (2).The calculation of this self-organized electric field can be found in subsection 2.2.The portion of computational domain is set in the dielectric.When calculating the microwave field in the dielectric using equations (1) and (2),it is necessary to replace ϵ0and μ0with the dielectric constant and magnetic permeability of the dielectric.

To determine the microwave electric and magnetic fields that vary with time and space,the finite-difference timedomain (FDTD) method is employed to solve numerically equations (1) and (2) [21].For example,the difference scheme of equation (1) has the following form:

where Δtand Δzare temporal and spatial steps,respectively,nis zero or a positive integer that denotes the timet=nΔt,andkis also zero or a positive integer that denotes the positionz=kΔz.The spatial step in the difference scheme is taken as a very small value,i.e. Δz=0.5 μm,because the gradient of electron density near the dielectric window is large.The time step and space step in equation (3) must obey the stability conditioncΔt≤Δzin whichcis the light speed.As a result,Δtis taken as Δz/1.1c.

2.2.PIC-MCC algorithm

We use the 1D3V PIC-MCC method to deal with the number,velocity,and position of charged particles.Under the action of electromagnetic fields and self-organizing normal electric field,the velocity υ and positionzof an electron or ion satisfy the following equations:

where υzdenotes the component of velocity on thez-axis,qis the charge of an electron or ion,mis the mass of an electron or ion,Enrepresents the normal electric field produced by charged particles on and near the dielectric surface,andEmwandHmwrepresent the electric and magnetic field components of the microwave,respectively.

The calculation of microwave fields is described in subsection 2.1.The normal electric fieldEnis determined by using Poisson equation

In equation (6),φ denotes the electric potential,Edc=and ρ denotes the charge density near the dielectric surface.The particle cloud method is used to distribute the charged particles to the corresponding grids[22,23].The charge density on the grid can be deduced from superposition of the charge amount on the grid distributed by all charged particles.

Electrons and ions can achieve great velocities in a strong electric field of microwave,leading to frequent collisions between them and desorbed gas atoms.The elastic collision,excitation collision and ionization collision between electrons and desorbed gas atoms are involved in the simulation.The threshold energies for excitation and ionization of argon are 11.5 and 15.8 eV,respectively.We also consider the elastic collision and charge exchange between ions and desorbed gas atoms.The collision cross sections of argon can be found in the work of Peterson and Allen [24].The Monte Carlo collision method is used to compute the random collision process between particles[25].When the generated random numberRis less than or equal to the collision probabilitypc,a collision occurs between two particles.On the contrary,there was no collision between these two particles.pcdepends on the collision cross-section and gas number density.For example,the collision probability between thei-th electron and desorbed gas atom is

where σTis the total collision cross-section,εiand υiare the energy and velocity of thei-th electron,respectively,ziis the position of thei-th electron,andng(zi) is the number density of desorbed gas atoms atzi.

The impact of electrons on the dielectric surface may induce the emission of secondary electrons.If the secondary electron emission yield δ ＞1,the electron number increases dramatically in time.Conversely,the electron multipactor cannot occur.The secondary electron emission yield is a function of the primary electron energy εeand incidence angle θ.Here the empirical formula of Vaughan is used to estimate the secondary electron emission yield [26].

In order to ensure the stability and correctness of the PIC algorithm,the space step cannot be greater than the Debye length [27],that is,

whereKBTeis the electron temperature andqeis the electronic charge.The time step should meetin which ωpeis the plasma frequency.In addition,because the number of collisions between an electron and neutral particles is required to be less than 1 in each time step,the time step must be less than the average collision time,that is,ΔtPIC≤τ=where υ is mean velocity of electrons,and σTis the total collision cross section.Therefore,the time step must meet [27]

In this work,the peak value of the plasma number density is about 1×1021m-3,the peak value of the gas number density is about 5×1023m-3,and the plasma temperature is around 50 eV.In that case,Δz=0.5 μm and Δt=Δz/1.1c=1.65×10-15s used here satisfy the relationships shown in equations (8) and (9),respectively.Because Δt=Δz/1.1candc≫υ,Δt＜Δz/υ,that is,the relationship between Δzand Δtobeys the Courant criterion.To sum up,the Δzand Δtused in this work meet the requirements of stability and accuracy of PIC algorithm.The calculation efficiency of electromagnetic PIC-MCC model is significantly lower than that of electrostatic PIC-MCC model because the time step of the latter can take a larger value.

2.3.Calculation of desorbed gas density

There is an adsorbed gas layer on the dielectric surface in vacuum.The adsorbed gas molecules become desorbed gas during the collision between electrons and the dielectric surface because gas molecules break away from the binding of dielectric surface.The mass spectrometry has been used to identify the components of desorbed gases [28].Hydrogen,steam,nitrogen,carbon monoxide,argon and other gases may be produced in the dielectric surface discharge.The average gas desorption yield has also been measured experimentally,which is between 0 and 1 [29].The desorption yield is defined as the number of gas molecules released by an incident electron.

In view of the diversity of desorbed gas components and the complexity of desorption process,it is very difficult to directly simulate the interaction between electrons and adsorbed gas on the dielectric surface.For this reason,people usually assumed that the desorbed gas species is a single gas,such as argon or hydrogen,when simulating dielectric surface discharge [12,15-17].The gas desorption yield is taken as a constant between 0 and 1 according to the previous experimental results.The energy of desorbed gas molecules is assumed to obey Maxwell distribution,and the average energy of molecules is taken as the energy corresponding to room temperature of 298 K.The direction angle of the gas molecular emission is randomly sampled from the dielectric surface towards the half space of the vacuum.

The desorbed gas density can be obtained by tracking the drift of gas atoms and the collisions between them.The average collision frequency between gas atoms is

wheredandrepresent the diameter and average speed of a gas atom,respectively.The pressure of the desorbed gas can increase to the order of 1 Torr [14].If the pressure and temperature of the desorbed gas are taken as 1 Torr and 300 K,respectively,the～106s-1.Here,we only focus on the breakdown process within a few nanoseconds after the desorbed gas pressure reaches 1 Torr.The number of collisions between an atom and other atoms is much less than unity during this process.Therefore,collisions between gas atoms are ignored.The desorbed gas density continuously changes over time and is non-uniform in space.The gas density on each grid point is continuously updated when calculating the collision between charged particles and desorbed gas atoms (see subsection 2.2).In this case,the effect of the desorbed gas on the surface breakdown is considered.

3.Simulation results

We use the model in section 2 to compute and analyze the microwave breakdown on the dielectric surface in vacuum involving desorbed gas.There are a small number of seed electrons near the surface to cause the occurrence of secondary electron multipactor.A plane microwave is incident perpendicular to the dielectric surface,as shown in figure 1.The electric field of microwave is parallel to the surface.The wave frequency is 9.4 GHz.The microwave field amplitudeEmwnear the dielectric surface is 3 MV m-1before the breakdown occurs.It is worth noting thatEmwis different from the amplitudeEmw,iof the incident electric field,andEmwandEmw,isatisfy the following relationship:Emw=(1+Γ)Emw,i.The reflection coefficient Γ=-0.23 when the dielectric material is taken as polyethylene.The secondary electron emission parameters of polyethylene are δmax0=2.6 and εmax0=280 eV.The dielectric material and parameters of incident wave are similar to the experimental conditions of Changet al[14].It is assumed that the species of desorbed gas is argon and the desorption yield is 0.4.The simulation domain is in the area of 0 ≤x≤9000 μm.The interface between the dielectric and vacuum is set atz=1 μm.

Figure 2 shows the time evolution of the number of electrons and ions.In the initial stage of breakdown,the dielectric surface is in a vacuum or extremely low pressure environment,where the change in the number of electrons is mainly due to secondary electron multipactor.Therefore,the number of electrons first rapidly increases,and then oscillates at twice the microwave frequency after approximately 1.1 ns.The impact of primary electrons on the dielectric surface not only generates secondary electrons,but also causes gas atoms to desorb from the surface.As a result,the number of desorbed gas atoms gradually increases with time in figure 3.More and more electrons and ions are produced since ionization events between electrons and desorbed gas atoms occur frequently.This also leads to an increase in the frequency of electron impact on the dielectric surface (see figure 3),as well as more gas atoms desorbed from the surface per unit time.As the pressure near the surface of dielectric window increases,the dominant mechanism of breakdown transitions from electron multipactor to collision ionization.Therefore,the number of electrons and ions increases sharply in time,and the difference between the two is small after 15 ns.

Figure 2.The number of electrons and ions as a function of time.The amplitude of microwave electric field is 3 MV m -1 near the surface of dielectric window before the breakdown occurs,and its frequency is 9.4 GHz.The gas desorption yield is 0.4.

Figure 3.The number of desorbed argon atoms and collision frequency between electrons and dielectric surface.The amplitude of microwave electric field is 3 MV m -1 near the surface of dielectric window before the breakdown occurs,and its frequency is 9.4 GHz.The gas desorption yield is 0.4.

When the gas desorption phenomenon is ignored,secondary electron multipactor is the main physical mechanism of dielectric surface breakdown in vacuum.The number of electrons first increases sharply,and then oscillates at twice the microwave frequency,that is,it reaches the saturation state [30].The time evolution of electron number is very different from figure 2 where the gas desorption phenomenon is considered.

Figure 4 shows the time evolution of mean electron energy and average secondary electron emission yield.The mean electron energy oscillates between 100 and 400 eV before 1 ns,and the secondary electron emission yield is higher than 1.In this case,the number of electrons increases rapidly in time (figure 2).The self-organized normal electric field becomes very high after 1 ns,which causes the transit time of electrons to decrease.As a result,the mean electron energy first decreases and then oscillates around the critical energy at which the secondary electron emission yield is equal to 1.Although the multipactor has reached saturation between 5 and 15 ns,an increase in the amplitude of electron energy occurs.This is because the impact of electrons on gas atoms changes the normal velocity of electrons,and results in an increase in the transit time of electrons.However,a decrease in the electron energy amplitude occurs after 15 ns,which is due to the following two factors.The first factor is that as the pressure of the desorbed gas increases,frequent collisions between electrons and neutral particles result in significant energy loss of electrons.The second factor is that the number density and thickness of electrons become large after 15 ns,and the microwave field amplitude decays.

Figure 4.Time evolution of mean energy of electrons and average secondary electron emission yield.The amplitude of microwave electric field is 3 MV m -1 near the surface of dielectric window before the breakdown occurs,and its frequency is 9.4 GHz.The gas desorption yield is 0.4.

After the multipactor reaches saturation in vacuum,the amplitude of the mean electron energy remains almost unchanged [5,20].However,due to the influence of the desorbed gas,the amplitude of mean electron energy changes in time,as can be seen in figure 4.The average secondary electron emission yield is still close to 1.This phenomenon suggests that there is no significant change in the energy of the electrons impacting the dielectric surface.In other words,the mean electron energy in space is different from the mean energy of electrons impacting the surface of dielectric window.

Figure 5 shows the spatial distributions of electron and ion densities at 3.2,11.2,and 21.8 ns.The surface of dielectric window is at z =1 μm.The spatial distributions of the desorbed gas density at the three different times are shown in figure 6.The density of the desorbed gas atoms is very low at t =3.2 ns (figure 6),and t he number of charged particles produced by collision ionization is very small.Only electrons are produced in the secondary electron multipactor.Therefore,the density of electrons is much larger than the density of ions,as can be seen at t =3.2 ns in figure 5(a).The electron number density monotonically decreases with the increase of z.It is because the normal velocity of electrons emitted from the surface is low and they have a small transit distance.Electrons impact on the surface of dielectric window hardly lead to any change in the number of electrons between 3.2 and 11.2 ns since the average secondary electron emission yield is close to 1 (figure 4).However,more collision ionization events occur due to the increase in the density of the desorbed gas atoms (figure 6),and result in higher densities of electrons and ions at time t=11.2 ns compared to them at time t =3.2 ns.The density and thickness of the desorbed gas further increase between 11.2 and 21.8 ns (figure 6).The dominant mechanism of breakdown changes from multipactor to collision ionization.Therefore,the number density of electrons is close to the number density of ions at time t=21.8 ns,as shown in figure 5(c).Furthermore,the peak density and thickness of electrons in figure 5(c) are much greater than those in figure 5(a) where the electron multipactor plays a dominant role.

Figure 5. Spatial distributions of electron and ion densities at (a) 3.2 ns,(b) 11.2 ns,and (c) 21.8 ns.The surface of dielectric window is at z =1 μm.The amplitude of microwave electric field is 3 MV m -1 near the surface of dielectric window before the breakdown occurs,and its frequency is 9.4 GHz.The gas desorp tion yield is 0.4.

Figure 6. Spatial distributions of densities of desorbed argon atoms at 3.2,11.2,and 21.8 ns.The amplitude of microwave electric field is 3 MV m -1 near the surface of dielectric window before the breakdown occurs,and its frequency is 9.4 GHz.The gas desorp tion yield is 0.4.

It can also be found from figures 5(c) and 6 that electrons are distributed in the region of 0 -200 μm,while neutral gas atoms are only distributed in the region of 0 -20 μm,that is,the distribution region of electrons is much larger than that of gas atoms.The reasons for this phenomenon are described below.The neutral gas atoms are distributed in the region of 0 -20 μm,where collision ionization occurs.However,the velocity of electrons is much faster than that of gas atoms,so that the former can drift to the region farther than 20 μm.

Figure 7 shows the time evolution of the self-organizing normal electric fieldEn,wallon the dielectric surface.In the early stage of breakdown,En,wallfirst increases sharply and then oscillates with a roughly fixed amplitude,similar to the case of multipactor in vacuum.The amplitude ofEn,wallis about 2.2 MV m-1,which is on the same order of magnitude as the microwave electric field amplitude (3 MV m-1).Although the microwave electric field is along thex-direction,that is,parallel to the dielectric surface,electrons can obtain high energy from the microwave electric field.Due to the collision between energetic electrons and the dielectric surface,the secondary electron multipactor occurs,so that a large number of electrons are generated near the dielectric surface.The whole discharge system is electrically neutral.In that case,the number of positive charges deposited on the dielectric surface is equal to the number of electrons in space.Therefore,the surface density σwallof positive charges on the dielectric surface is very high.TheEn,wallis proportional to σwall,so the former can reach the order of microwave electric field.A similar phenomenon can also be found in the work of Kim and Verboncoeur [30].

Figure 7. Time evolution of self-organized normal electric field on the dielectric surface.The surface of dielectric window is at z= 1 μm.The amplitude of microwave electric field is 3 MV m -1 near the surface of dielectric window before the breakdown occurs,and its frequency is 9.4 GHz.The gas desorption yield is 0.4.

As time increases,the electron density near the dielectric surface grows rapidly because of the collision ionization in the desorbed gas.In this case,more and more electrons collide with the surface.Therefore,the amplitude ofEn,wallincreases with time.Moreover,En,wallis no longer constant greater than zero,which leads to the inability to maintain the steady state of secondary electron multipactor.After 20 ns,the amplitude ofEn,wallon the negative half axis is greater than its amplitude on the positive half axis.This is because the average secondary electron emission yield is slightly lower than 1 (figure 4),and the primary electrons deposited on the surface are more than the secondary electrons emitted from the surface.The amplitude ofEn,wallreaches tens of megavolts per meter after 20 ns,which is much higher than the amplitude of microwave electric field.

The peak values of electron number density at 3.2 and 11.2 ns are about 2×1019and 3×1019m-3,respectively[figures 5(a) and (b)],which are higher than the cutoff number densitync=(ϵ0me/q2e)ω2of plasma in vacuum(nc≈1.1×1018m-3at 9.4 GHz) [31].Nevertheless,the microwave electric field amplitude atz＜100 μm is only slightly less than that before breakdown (3 MV m-1),as shown in figure 8(a).Figure 8 shows the spatial profiles of microwave electric field amplitudes at 3.2,11.2,and 21.8 ns.This is because the distribution thickness of electrons is about tens of micrometers,which is much smaller than the skin depth of the medium.It is worth noting that the mismatch between the wave impedance of the dielectric and the vacuum leads to the reflection of the microwave.The superposition of the incident field and the reflected field causes the microwave electric field to be spatially nonuniform at 3.2 and 11.2 ns.The peak density and thickness of the plasma at 21.8 ns are significantly greater than those at 3.2 and 11.2 ns.As a result,the microwave electric field amplitude at 21.8 ns is significantly lower than that before the breakdown since part of the microwave power is absorbed and reflected by the plasma.In addition,the reflection of microwaves by the dielectric surface and breakdown plasma results in a standing wave distribution in space.However,the distribution thickness of breakdown is on the order of 100μm,while the wavelength of the 9.4 GHz microwave is about 3.19×104μm.Therefore,the microwave electric field amplitude is roughly evenly distributed in the breakdown region as shown at 3.2,11.2,and 21.8 ns in figure 8(a).

Figure 8.(a) Spatial profiles of microwave electric field amplitudes at three different times.(b) Time evolution of microwave electric field amplitude at z=2 μm.The surface of dielectric window is at z =1 μm.The amplitude of microwave electric field is 3 MV m -1 near the surface of dielectric window before the breakdown occurs,and its frequency is 9.4 GHz.The gas desorption yield is 0.4.

Figure 9.Comparison between the mean energy and number of electrons obtained by the electrostatic PIC-MCC model and those obtained by the electromagnetic PIC-MCC model.The surface of dielectric window is at z =1 μm.The amplitude of microwave electric field is 3 MV m -1 near the surface of dielectric window before the breakdown occurs,and its frequency is 9.4 GHz.The gas desorption yield is 0.4.

In addition to observing the spatial profiles of the microwave electric field amplitudeEmwas shown in figure 8(a),we also show the time evolution of field amplitudeEmw,2atz=2 μ m in figure 8(b).Emw,2is approximately equal to 3 MV m-1before the breakdown occurs.During the period of 1 -1.4 ns,the secondary electron multipactor rapidly develops and then reaches saturation state,butEmw,2only decreases from 3 to 2.994 MV m-1.This phenomenon indicates that the multipactor has little effect on microwave transmission.As time increases,the desorbed gas atoms continuously accumulate near the surface of dielectric window,and the thickness and density of plasma produced by collision ionization gradually increase.This causesEmw,2to slowly decay between 1.4 and 15 ns.After the thickness and density of the plasma increase to a sufficiently large level,it strongly absorbs and reflects microwaves.Therefore,Emw,2rapidly decays after 15 ns.

The amplitude of microwave electric field shows a small change in the early and middle stages of breakdown,as shown in figure 8.In this case,the breakdown characteristics predicted by the electrostatic PIC-MCC model are close to the results of the electromagnetic PIC-MCC model.However,there is a significant attenuation in the amplitude of the microwave electric field in the later stage of breakdown.The electrostatic PIC-MCC model does not take into account the change in the amplitude of the microwave electric field,and results in the breakdown characteristics different from the results of the electromagnetic PIC-MCC model,as shown in figure 9.This figure shows the comparison of the mean energy and number of electrons obtained from the electrostatic PIC-MCC model with the results of the electromagnetic PIC-MCC model.The electrostatic PIC-MCC model overestimates the mean energy and number of electrons in the later stage of breakdown compared to the results of the electromagnetic PIC-MCC model because the former does not take into account the reduction of microwave electric field.This is also the reason why this work uses the electromagnetic PIC-MCC model rather than the electrostatic PIC-MCC model.

Figure 10 shows the time evolution of desorbed gas pressurePwallclose to the dielectric surface when the gas desorption yields are taken as 0.1,0.2,and 0.4,respectively.Pwallis the highest in space since the desorbed gas pressure decreases significantly as it moves away from the dielectric surface (figure 6).Changet almeasured the desorbed gas pressure in polyethylene surface breakdown at a microwave frequency of 9.4 GHz,amplitude of 3 MV m-1,and duration of 20 ns [14].In order to directly compare the simulation results with experimental data,the microwave in figure 10 is taken as a rectangular pulse with the duration of 20 ns,which is different from the condition of unrestricted microwave duration in figures 2 -9.It can be seen from figure 10 thatPwallincreases with time before 20 ns,but significantly decreases with time after 20 ns. This phenomenon is mainly caused by the following factors.The energy of electrons becomes very small after 20 ns due to the lack of acceleration by the microwave electric field.The secondary electron emission yield is much less than unity,and leads to a sharp decrease in the number of electrons close to the surface of dielectric window.In this case,the collision frequency between electrons and the dielectric surface is very small,while the diffusion of gas atoms leads to a decrease inPwall.

Figure 10.The time evolution of desorbed gas pressure near the dielectric surface at three different gas desorption yields.The surface of dielectric window is at z =1 μm.The amplitude of microwave electric field is 3 MV m -1 near the surface of dielectric window before the breakdown occurs,and its frequency and duration are 9.4 GHz and 20 ns,respectively.

It can also be observed from figure 10 that as the gas desorption yield increases,Pwallincreases more rapidly until it reaches a maximum value around 20 ns.The maximum desorbed gas pressure is 0.21,0.56,and 3.55 Torr at gas desorption yields of 0.1,0.2,and 0.4,respectively.The experiments of Changet alindicated that the desorbed gas pressure on the dielectric surface can reach 3.2 Torr [14].Therefore,thePwallat the gas desorption yield of 0.4 is closer to the experimental data compared to the other two different gas desorption yields.This is also the reason why the gas desorption yield is set to 0.4 in this work.

4.Conclusion

In summary,the microwave breakdown on the vacuum side of dielectric window has been investigated using the electromagnetic PIC-MCC model.Emphasis is placed on the effect of gas desorbed from the dielectric surface on the breakdown.This model not only considers the temporal and spatial evolution characteristics of charged particles and desorbed gas atoms,but also involves the effect of breakdown plasma on the microwave fields.The simulation results indicate that the change of electron number mainly depends on the secondary electron multipactor in the early stage of surface breakdown.The distribution thickness of electrons is only tens of micrometers after the multipactor reaches saturation,which causes a small change in the amplitude of the microwave electric field.The gas pressure near the dielectric surface increases in time since the impact of electrons on the surface causes a large number of gas atoms to desorb from the surface.Therefore,the dominant mechanism of breakdown transitions from secondary electron multipactor to collision ionization.Although the amplitude of mean electron energy first increases and then decreases in time,the secondary electron emission yield is still close to 1.The amplitude of self-organized normal electric field on the dielectric surface increases with time,and it is no longer constant greater than zero in the later stage of breakdown.This is due to the interaction between a large number of electrons produced in collision ionization and the surface of dielectric window.The number density and distribution thickness of electrons are significantly greater in the later stage of breakdown than in the early stage of breakdown.As a result,the amplitude of the microwave electric field rapidly decays in the later stage of breakdown.

The simulation results also indicate that the electrostatic PIC-MCC model overestimates the mean energy and number density of electrons compared to the electromagnetic PICMCC model.This is because the effect of breakdown plasma on the microwave field is neglected in the electrostatic PICMCC model.Under different gas desorption yields,the time evolution of desorbed gas pressure on the dielectric surface is also studied using the electromagnetic PIC-MCC model.The desorbed gas pressure predicted by this model is close to the measured value when the gas desorption yield is taken as 0.4.In the actual breakdown problem,the common wave mode is TE wave or TM wave.The electric field of TE wave or TM wave is non-uniform in both normal and tangential directions of the dielectric surface.In that case,we will use the the 2D3V PIC-MCC model to analyze the surface breakdown characteristics in the future work.

Acknowledgments

This work was supported by the National Key Laboratory Foundation 2021-JCJQ-LB-006,China (No. 6142411 132116),the Natural Science Basic Research Program of Shaanxi Province,China (Nos.2023-JC-YB-512 and 2023-JC-YB-042),the Fundamental Research Funds for the Central Universities,China (No.ZYTS23075),and the China Postdoctoral Science Foundation (No.2019M653545).