Kinetic simulation of electron cyclotron resonance assisted gas breakdown in split-biased waveguides for ITER collective Thomson scattering diagnostic

For the measurement of the dynamics of fusion-born alpha particles $E_\alpha \leq 3.5$ MeV in ITER using collective Thomson scattering (CTS), safe transmission of a gyrotron beam at mm-wavelength (1 MW, 60 GHz) passing the electron cyclotron resonance (ECR) in the in-vessel tokamak `port plug' vacuum is a prerequisite. Depending on neutral gas pressure and composition, ECR-assisted gas breakdown may occur at the location of the resonance, which must be mitigated for diagnostic performance and safety reasons. The concept of a split electrically biased waveguide (SBWG) has been previously demonstrated in [C.P. Moeller, U.S. Patent 4,687,616 (1987)]. The waveguide is longitudinally split and a kV bias voltage applied between the two halves. Electrons are rapidly removed from the central region of high radio frequency electric field strength, mitigating breakdown. As a full scale experimental investigation of gas and electromagnetic field conditions inside the ITER equatorial port plugs is currently unattainable, a corresponding Monte Carlo simulation study is presented. Validity of the Monte Carlo electron model is demonstrated with a prediction of ECR breakdown and the mitigation pressure limits for the above quoted reference case with $^1$H$_2$ (and pollutant high $Z$ elements). For the proposed ITER CTS design with a 88.9 mm inner diameter SBWG, ECR breakdown is predicted to occur down to a pure $^1$H$_2$ pressure of 0.3 Pa, while mitigation is shown to be effective at least up to 10 Pa using a bias voltage of 1 kV. The analysis is complemented by results for relevant electric/magnetic field arrangements and limitations of the SBWG mitigation concept are addressed.


I. INTRODUCTION
ITER will be the first fusion reactor to achieve a fusion power gain of Q ≥ 1 (goal Q = 10).
Collective Thomson scattering (CTS) has been proposed as a primary diagnostic for the measurement of fusion-born alpha particles (E α < 3.5 MeV). 1,2 It relies on scattering of an intense gyrotron beam at mm-wavelength (1 MW, 60 GHz) by fluctuations in the plasma, which are captured by receiver mirrors. The fast ions' velocity distribution is inferred from the measured spectrum. The gyrotron beam must pass the electron cyclotron resonance (ECR, where ω rf = ω ce ) within the in-vessel waveguide of the diagnostic, subject to the tokamak 'port plug' vacuum. However, the intense radio frequency (RF) radiation of the beam may cause an ECR-assisted electric gas breakdown within the waveguide, which may lead to failure and damage of the diagnostic. This phenomenon is qualitatively depicted by the Paschen law. 3,4 Gas breakdown and the development of a discharge occurs when the gas pressure is (i) high enough so that ionizing collisions of electrons (which are accelerated by the driving electric field) with the neutral gas atoms are frequent enough to compensate for diffusion and losses to the walls, but (ii) low enough that the electrons gain sufficient energy for ionization (from the electric field) in between subsequent collisions. A positive net balance between electron count gain and loss results in an exponential growth in the free electron population and a corresponding ionization avalanche, ∂n e ∂t = n e (ν iz − ν loss ), (1) n e (t) = n e,0 exp{(ν iz − ν loss ) t}.
The minimum voltage to maintain a discharge is correspondingly specified at break even.
On breakdown, the resulting transition from an under-to an over-dense regime, where a RF electric field is substantially affected by the plasma, corresponds to an angular frequency ω rf approximately equal to the electron plasma frequency, ω rf ≈ ω pe (n e ). It is marked by the critical electron density n e,crit = ω 2 rf 0 m e /e 2 . 5 e is the elementary charge, m e is the electron mass, and 0 is the vacuum permittivity. While static and RF breakdown are conceptually related, they differ quantitatively. [3][4][5][6] Two contributions govern breakdown. On the one hand, the electron count gain is determined by the mean ionization frequency which scales proportional to gas pressure p, electron velocity v e , and velocity dependent ionization collision cross section σ iz . Therein, parentheses . denote the mean value, averaged over the electron energy distribution, T is the neutral gas temperature, and k B is Boltzmann constant. On the other hand, the electron count loss is determined by electron diffusion loss. It is governed by the rate ν loss = v e /L with which electrons are lost from the volume, where breakdown is investigated, at a typical distance L (e.g., waveguide diameter 2L = D = 88.9 mm). Assuming an energy independent ionziation collision cross section, the balance of the processes is approximately determined by the growth ratio ν iz /ν loss ∝ pL.
Consequently, with ν iz /ν loss < 1, diffusion loss typically dominates at low pressures (below 1 Pa), resulting in a negative net electron balance and thus no breakdown. Due to the factual dependence of the ionization collision cross section on energy (cf. Section III C), an optimum electron energy (respectively velocity) may be determined. It maximizes σ iz (v e )v e and therefore, above this energy, electrons are subject to overheating (i.e., are increasingly unlikely to cause ionization).
The presence of a magnetic field (e.g., of a fusion device) significantly alters the gas breakdown dynamics in two respects: (i) diffusive transport is inhibited across magnetic field lines, reducing wall losses; and (ii) the RF beam may need to be transmitted through the ECR region. The first mechanism predominantly influences the gain/loss balance, enabling gas breakdown at much lower pressures. Regarding the second aspect, when electrons are accelerated subject to the ECR (if present) this brings about a substantial increase in electron heating (but little influence on the particle balance for electron energies above the ionization threshold). Resonance occurs when the frequency ω rf equals an integer multiple k of the electron cyclotron frequency kω ce = keB ⊥ /m * e = ω rf (where B ⊥ is the magnetic field component transverse to the RF electric field and m * e = γm e is the relativistic electron mass; γ → 1 is a safe approximation for ITER CTS, but should be treated with caution in ECR heating schemes). Electrons are accelerated in phase with their gyration about a magnetic field line and rapidly gain energy from the RF electric field component perpendicular to the magnetic field, E rf ⊥ B. The related phenomenon of ECR-assisted gas breakdown has been studied theoretically, based on analytical model formulations, since the 1950s. 7,8 The approaches utilize global particle and energy balance relations, paired with considerations on the high frequency electron kinetics. This phenomenon is greatly exploited in heating of fusion plasmas. 9,10 While the afore-mentioned considerations and models approximately capture the global phenomena, they do not provide any detailed spatio-temporal information. Spatially resolved kinetic models may accurately predict these dynamics, limited by their computational requirements. To the best of our knowledge, no corresponding simulation study has been previously conducted investigating ECR-assisted gas breakdown. Conceptually similar previous studies have been mostely concerned with the decisively different DC gas breakdown and streamer regime. 11,12 In addition, non-magnetized microwave air breakdown was recently investigated by means of Monte Carlo simulations, taking into account electronsurface interaction. 13 In particular, in the frame of RF diagnostics for fusion devices or heating schemes, such as ITER CTS or ECR heating, no theoretical study has been concerned with the analysis of mitigation schemes to prevent undesired ECR-assisted breakdown within the diagnostic's waveguide. This aspect has been addressed experimentally in previous works by Moeller et al., [14][15][16][17][18] who proposed a longitudinally-split electrically-biased waveguide (SBWG) design to avoid in-waveguide ECR breakdown by promoting the removal of electrons from the central region of high RF electric field strength and amplifying wall loss (detailed later).
To address the aspects of ECR breakdown and SBWG mitigation in the context of the ITER CTS SBWG design, 19 the onset of a gyrotron pulse and the corresponding ECR breakdown dynamics are simulated and analyzed in this work, using a spatially resolved Monte Carlo electron model. Initially, a reference setup described by Moeller et al. [14][15][16][17][18] and the ITER CTS configuration 19 are introduced in Section II. Thereafter, the simulation approach and the fundamental prerequisites and model inputs are presented in Section III. This is followed by simulation results for the 'Moeller' reference case, which are established for verification and validation in Section IV A. The ITER CTS setup is addressed subsequently in Section IV B, and the effectiveness of SBWG breakdown mitigation is demonstrated for this setup. After a discussion of the results, the work is concluded.

II. SETUP
In the following, the two relevant configurations are presented. The main parameters and features are collected in Table I. Schematics of the configurations are depicted in Figure 1.
When not specified otherwise, a homogeneous magnetic field magnitude B = 2.14 T is imposed in both cases (note the different magnetic field directions for the 'Moeller' and ITER CTS cases).

A. 'Moeller' configuration
The results due to Moeller et al. [14][15][16][17][18] were obtained considering a circular smooth waveguide with D = 19.1 mm inner diameter and TE 11 microwave mode propagation. 21 The waveguide surface material was stainless steel. 18 The magnetic field was created by a solenoid magnet, therefore, pointing predominantly in the axial direction (cf. Figure 1a). A two dimensional model well represents the system inside the solenoid. It takes the invariant axial direction as z and assumes a homogeneous axial magnetic field B z = 2.14 T. The TE 11 polarization is taken in the y direction. The bias electric field E bias (consistently calculated numerically) points predominantly in the x direction. Therewith, effective ECR heating, as well as breakdown mitigation mainly based on an E bias × B drift are realized. For simplifi-  cation, the configuration is simulated in a two dimensional transversal cross section of the waveguide. Moeller reports on gas breakdown and mitigation in hydrogen 1 H 2 . However, as noted outgassing from the walls may have had an important contribution. 18

B. ITER CTS configuration
The ITER CTS configuration differs from the 'Moeller' case in several respects. Firstly, the proposed corrugated waveguide with inner diameter D = 88.9 mm supports LP 01 (HE 11 respectively) mode propagation. 22 Moreover, the waveguide surface material is most certainly CuCrZr (or Cu coating), which among other aspects suppresses outgassing from the walls (stainless steel retains much H 2 O). 17,18 Secondly, the magnetic field structure within the waveguide is neither axial nor transversal, but entails contributions in both directions. The relevant section of the waveguide is subject to a magnetic field strength of B ≈ 2.14 T to be affected by ECR heating. This is schematically depicted in Figures 1b) and c) and Figure 2, whereas the waveguide axis is aligned with the z axis. The linearly polarized RF electric field E rf points in the y direction, whereas the magnetic field B has components in the x and z direction. The bias electric field E bias points predominantly in the x direction. Two model representations are investigated both assuming a hydrogen 1 H 2 neutral gas background: (i) A two dimensional setup which neglects the axial magnetic field component B z , but maintains the magnetic field magnitude B = 2.14 T and includes all governing mechanisms concerning ECR breakdown as well as mitigation (cf. Figure 1b).
(ii) A three dimensional setup depicting a waveguide section (L = 80 mm) and which includes a more realistic magnetic field (including relevant B ≈ 2.14 T; cf. Figures 1b and   2). While the axial magnetic field is B z = 0.735 T, the transverse magnetic field B x varies with dB x /dz = 1.674 T/m. Magnetic field lines are oblique at an angle with the z axis. The 3D magnetic field structure is approximated from the complicated 3D structure of the ITER tokamak baseline scenario. 23 Due to computational limitations, only a limited number of representative cases were simulated for this setup.

C. Electron cyclotron resonance and gyrotron excitation
The RF excitation frequency is f = 60 GHz for both cases discussed. The corresponding RF period is T = 1/f ≈ 16.67 ps. The ECR frequency ω ce defines the intrinsic timescale of the electron dynamics which needs to be resolved. At the gyrotron beam frequency ω rf ≈ ω ce , the magnetic field magnitude of the fundamental resonant mode is B ≈ 2.14 T, whereas the second harmonic resonant mode is excited when ω rf ≈ 2ω ce at B ≈ 1.07 T.
Moeller reports pulses of τ = 5 ms, while ITER CTS design specifies typical pulse lengths of τ = 10 ms (between τ = 1 ms to 1 s depending on operating conditions). The pulse rise time is typically on the order of τ rise ≈ 100 µs, 24 which is in line with the design specifications for ITER CTS. Moeller does not specify the pulse rise time.
Both the pulse length and the rise time are orders of magnitude larger than the intrinsic timescale of the ECR heating dynamics (see above paragraph). Moreover, while the electron heating timescale needs to be resolved, the dominant timescale for gas breakdown is governed by electron impact ionization collisions with the gas background. The mean collision time for ionization provides the relevant timescale. It represents an intermediate timescale between the ECR heating time and the pulse rise time and duration. As tabulated in Table II, it is below τ c 5 µs for the relevant pressure range.
The excitation pulse envelope is schematically depicted in Figure 3 for a simplified gyrotron pulse. It is approximated by a Gaussian ramp up phase defined by the rise time τ rise = 100 µs, a plateau defined by the pulse duration τ = 10 ms, and an analogue decay phase τ decay = τ rise . Notably, the pulse magnitude is approximately constant on the timescale of gas breakdown (10 µs depicted in Figure 3b). Hence, the RF modulated pulse waveform can be considered a continuous wave (CW) concerning the intrinsic electron heating dynamics, as well as the collisional processes leading to gas breakdown. The time interval which  is most relevant for gas breakdown should be considered. The latter is dictated by the RF electric field magnitude E 0 which is required to heat electrons to sufficient energies -this is detailed in Section III A.
The field pattern within the waveguide is defined by the propagation modes TE 11 for 'Moeller' and LP 01 for ITER CTS. In the latter case, a corrugated waveguide is used. Corrugations on the waveguide surface are located, where the field magnitude is minimum (i.e., close to zero). Consequently, their effect on gas breakdown is negligible and corrugations have been omitted in the geometry of the simulation. The electric field patterns of the relevant modes are depicted in Figure 4 for both configurations. While the TE 11 mode contains two transverse electric field contributions E x and E y , LP 01 entails only an E y contribution. In both cases, the y component dominates and is peaked in the center of the waveguide.
The region of high RF electric field strength is more distributed for TE 11 compared to LP 01 , resulting in a smaller peak electric field at equal intensity and waveguide inner diameter.
See references 21,22 for more details.

D. Longitudinally-split electrically-biased waveguide
The SBWG mitigation scheme has been originally proposed and detailed by Moeller. 14-18 The fundamental reasoning is to enhance drift-diffusion of electrons from the central resonant region of high RF electric field strength, before these electrons initiate or participate in a gas breakdown avalanche. This is achieved by longitudinally splitting the waveguide in two halfcylinders and applying a DC bias voltage, V bias , between them. (Note that the symbol V bias is used to denote the electric potential difference V bias = φ anode − φ cathode .) For 'Moeller' a bias voltage in the range V bias = 1 to 2.3 kV has been reported, while ITER CTS is specified with V bias = 1 to 2 kV. Depending on the geometry and the resulting bias electric field E bias , two mechanisms drive electrons out of the central waveguide region and to the walls: (i) Electron acceleration and transport along magnetic field lines may proceed freely due to E bias B. Assuming a uniform bias electric field E bias = V bias /D and neglecting collisions, the maximum sweep time can be approximated by This assumes an electron acceleration and trajectory from one side of the waveguide to the opposite side and provides an upper bound. For ITER CTS with 1 kV bias voltage this evaluates to τ sweep ≈ 10 ns, for 2 kV it corresponds to τ sweep ≈ 6.7 ns. Electron transport is principally inhibited by collisions with the gas background. By comparison with the approximate mean collision times from to slowdown but also to multiplication of electrons, due to ionization. Hence, at higher pres-sures especially mechanism (i) may additionally contribute to the gas breakdown dynamics.
The influence of the latter, as well as nonuniform bias electric fields cannot straightforwardly be included in the presented approximations. This stresses the need for accurate numerical simulation predictions.

III. MODEL
A. Kinetic electron model The utilized Monte Carlo electron model was developed within the OpenFOAM framework. 25,26 The underlying particle in cell particle (PIC) code has been used for pure simulation studies, 27-29 but has also been validated with experiments. [30][31][32] In addition, the code has been validated against the benchmarked reference PIC code yapic. 33 Electron collisions with the gas background are included in a Monte Carlo collision scheme. The neutral gas is assumed as a stationary Maxwell-Boltzmann distributed background with temperature T and adjustable gas composition. Collisional processes are incorporated using a modified no-time counter. 28,37 The selection of different collision processes further uses a null-collision approach. 38 The magnetic field and the RF electric field are imposed within the domain, whereas the static bias electric field is calculated using the finite volume method. 39 Feedback of charged species onto the electromagnetic fields is neglected.
This approach is valid in the underdense regime with n e n e,crit . Initially, a homogeneous electron density of n e (t = 0 s) = 10 10 m −3 is imposed. The previous assumption is well justified for the subsequent evolution, since simulations starting with this low density are conducted only until gas breakdown is detected from a noticeable rise in n e . The electric field due to local space charge effects can be estimated from Poisson's equation. In 1D Cartesian coordinates, the maximum electric field of a uniform charge density ρ ≈ e · 10 11 m −3 over the length of a waveguide diameter D = 88.9 mm is estimated to 160 V/m. This is several orders of magnitude smaller than the bias electric field and the RF electric field imposed in this work, and consequently negligible. Moreover, due to the low charge carrier density, collective effects such as quasi-neutrality and Debye shielding occur on the length scale of the geometric configuration, and are negligible as well.
The pseudo-particle weight was chosen to maintain sufficient statistics ( 80 electrons per mesh cell 1). The time step was set to ∆t = 200 fs, to capture the gyration of electrons around magnetic field lines ( 80 time steps per gyration 1).
The computational effort of the simulation and the divergent physical timescales render an evaluation of the complete pulse waveform infeasible. As schematically depicted in Figure 3, for a time interval less than approximately 10 µs the pulse magnitude is nearly time invariant.
This is inherently the case for the intrinsic electron dynamics. Hence, a modified CW waveform is considered in the simulation without loss of generality. The rise of the pulse is not evaluated to scale, but an initial start-up phase using an RF modulated Gaussian with σ = 10 ns and a subsequent CW signal specifies the RF electric field waveform. The maximum electric field strength E 0 is chosen such that a gas breakdown avalanche is initiated most effectively, taking into account electron impact ionization in the volume as well as electron-induced secondary electron emission and reflection at the surface.
Electrons need to be heated to sufficient energies, but not overheated. The latter may occur once the product of σ iz (v e )v e -which defines the collision probability -attains a negative slope and decreases with increasing kinetic energy. For energies in the range E ≈ 70 to 700 eV, σ iz (v e )v e varies less than 10 % from the maximum at E ≈ 285 eV (cf. Sections III C). Consequently, strongest initiation of an ionization cascade is expected for the mentioned energy range. The exact energy, however, is of subordinate relevance, as long as a minimum energy of E 70 eV is maintained.
In addition to electron impact ionization in the volume, however, an electron count balance more realistic than Equation (1) also depends on surface processes in a realistic scenario.
That is, electron-induced secondary electron emission and reflection contribute respectively.
Due to markedly different energy dependencies (e.g., maximum emission at 400 to 600 eV; cf. Section III B), these alter the optimum RF electric field strength E 0 , which most effectively causes gas breakdown. For the 'Moeller' scenario an RF electric field magnitude, Following this reasoning, the proposed simulations operate at a relatively low RF electric field strength, corresponding to the start-up phase of the gyrotron pulse. This circumstance does not impose any limitation on the validity for the case with higher RF electric field strengths encountered later during the 'experimental realization'. The RF electric field strength and the corresponding energy window were selected in the simulations to provide a conservative gas breakdown estimate. Higher RF electric field strengths cause electron overheating, suggesting raised pressure limits for gas breakdown. The same reasoning applies to the situation when the pressure increases during operation and becomes critically high only after the RF modulated pulse magnitude has reached its maximum.
Compared to an experimental realization, the simulation procedure differs in two aspects: (i) The simulated RF modulated pulse increases within τ rise = σ = 10 ns to the CW electric field magnitude. This rise imposes a rapid excitation of the system of electrons (which are initially Maxwell-Boltzmann distributed). As apparent from the results, this is associated with a noticeable 'ringing' in the average electron energy, due to a slower time response of the system. The rise time τ rise = 10 ns is chosen as a compromise, minimizing these 'ringing' oscillations, but also the computational load (i.e., the time duration to be simulated).
(ii) In the proposed ITER CTS realization, the mitigation bias is designed to be constantly active. In contrast, the simulations are initially evolved until a noticeable gas breakdown occurs (exponential rise in n e ), and thereafter the mitigation bias voltage is switched on. On the one hand, this procedure is used to demonstrate the effectiveness of the SBWG mitigation scheme. On the other hand, it is also used because the onset of the gyrotron pulse cannot feasibly be simulated due to the simulation run-time. This and the inclusion of background ionization processes (instead of a pre-specified initial electron density) would, however, be required to capture the early phase of ECR breakdown and mitigation appropriately.
Both of the above raised aspects signify limitations of the simulation approach. However, as the conditions for ECR breakdown depicted by the procedure are more severe than in an experimental realization, the above points do not seem to entail any implications regarding the validity of the conclusions. Hence, the limits determined by the approach can be regarded as conservative bounds. This reasoning is supported by the agreement between experimental and simulation results, as elaborated for the 'Moeller' reference case in Section IV A.

B. Surface Coefficients
Particles interacting with bounding walls are subject to several interaction mechanisms.
In the present context, in particular electron-induced secondary electron emission (e-SEE, denoted by δ) and electron reflection (denoted by η) are important. 40 E m defines the energy of maximal electron emission, whereas δ m specifies its value. x m and k are dimensionless fitting coefficients. All listed parameters are material dependent.
The process of electron reflection has been found to predominantly depend on atomic number Z. 42 Table III. Parameters used for the evaluation of surface coefficients using Equations (7) and (8).
Cu parameters from 40,44  Whereas the dependence m(Z) governs an energy power law, c(Z) scales the total electron reflection. Both are dimensionless functions of the atomic number Z.
While data for Cu has been proposed, 42,46,47 the uncertainty of the available data suggests that atomic fractions 70% Fe, 20% Cr, 10% Ni provide a reasonable estimate for SS. These were used accordingly in the following. Surface coefficients for Cu and SS are depicted in Figure 5. The parameters used to evaluate Equations (7) and (8) are listed in Table III.
While the e-SEE coefficient δ initially increases to δ m with incident energy, it steadily drops for energies E > E m . In contrast, the electron reflection coefficient η is nearly constant for the relevant energies.  (7) and (8) using the values from Table III.
The process of ion-induced electron emission (i-SEE) is substantially weaker than electron-induced electron emission at ion bombardment energies E i in the eV to keV range. 42,48 For the present investigation, two cases need to be distinguished: (i) Ions created in volume ionization processes are not significantly heated by the RF electric field, due to their large mass and inertia. Without a bias electric field and in the absence of a fully established plasma (and corresponding boundary sheaths), ions are close to thermal equilibrium with the gas background. Their approximate average energy is correspondingly low, With a bias electric field, the ion energy is on the order of the bias voltage, Ions are removed from the waveguide volume within approximately the ion sweep time τ sweep ≈ 570 ns, much longer than the fast ECR heating dynamics, as estimated for 1 H 2 following Equation (5). In both cases (i) and (ii), the ion-induced electron emission process is in the potential and kinetic emission transition regime (eV to keV range) and can be consequently neglected, as reasoned by an emission coefficient of γ 0.1. [49][50][51][52] Moreover, to incorporate i-SEE into the model and assess its influence on gas breakdown would require to also simulate the ion dynamics. The expected small influence does not justify these additional, significant computational costs.
Electrons emitted from the surface in the simulation are assumed to have a Maxwell-Boltzmann distribution with k B T e = 2.6 eV (approximating fractional energy input from incident electrons). In the absence of more reliable data, electrons reflected from the surface are divided into 90% diffuse and 10% specularly reflected contributions, approximating measured emitted electron energy spectra. 40,42 The diffuse fraction is re-emitted identical to the primary emitted secondaries.

C. Collision Processes
Collisional interactions included in the calculation are reduced to elastic scattering and direct electron impact ionization to the singly ionized state. The cross sections have been obtained using LXCat from the IST Lisbon and Biagi database. 53? -56 Argon, as well as molecular hydrogen, nitrogen, and oxygen were obtained and implemented. [54][55][56][57][58][59][60][61][62][63] In the absence of reliable cross section data for hydrogen isotopes (deuterium and tritium), 64 their cross sections are approximated using hydrogen 1 H 2 cross sections (which are correspondingly used throughout).  Table II. Given no precise gas temperature specifications, in all cases T = 400 K is assumed (both for estimates and simulations).
One of the constituents of the ITER fuel is Tritium ( 3 H 2 ). Tritium is subject to radioactive β-decay with a lifetime of approximately 12.32 years. It correspondingly acts as a constant electron source (average electron energy of 5.7 keV), with a source rate on the order of 10 11 m −3 s −1 (assuming a pressure of 1 Pa and a tritium fraction of 0.5). Although these βelectrons have a substantial chance of subsequently undergoing an electron-impact ionization collision (σ iz (v e )v e ≈ 2.5 · 10 −14 m 3 /s), the total source rate is estimated to be much smaller than a conservatively approximated electron source rate due to thermal seed electrons and electron-impact ionization on the order of 10 17 m −3 s −1 (using an initial electron density n e = 10 10 m −3 and σ iz (v e )v e ≈ 5 · 10 −14 m 3 /s). Consequently, even if tritium was used in the modeling, this process would be of subordinate importance for the present study. Note, however, that Tritium β-decay will be a constant source of initial seed electrons for the initiation of the breakdown process. This also means that it is not possible to completely deplete the resonant volume free electrons.

A. Validation with results of Moeller
Moeller reports on observations of ECR breakdown for a number of specific cases as summarized in Table IV. In the following, simulation results for these cases are presented.
The simulated configuration is distinct in the sense that the axial magnetic field allows for transport in the axial direction (not resolved in the simulation), but effectively inhibits Breakdown has also been reported for an increased pressure of p = 0.08 Pa and successful mitigation has been observed using a bias voltage of V bias = 1 kV. This result is reproduced by the simulation, using the bias electric field presented in Figure 8. The latter has been consistently calculated in the simulation domain, imposing the respective boundary electric potential. While the bias electric potential and electric field are symmetric with respect to the x axis, the circular geometry enforces a corresponding curvature of the electric field lines. This effect is of relevance for electron removal, due to an E bias × B drift, which scales in magnitude proportional to the local fields and is directed in the local perpendicular direction. The bias electric field in the center of the waveguide is systematically larger than the one dimensional approximation, E bias ≈ 67 kV/m > V bias /D ≈ 52 kV/m. Figure 9 are For the mitigation case, depicted by the black lines in Figure 9, a rise in electron energy is noticeable within approximately 10 ns after switching on the bias electric field, associated with an initial distortion of the electron dynamics. The decay of the number of electrons     (Figure 11a), electron removal following the E bias × B drift proceeds more than twice as fast, τ sweep ≈ 350 ns, compared to the case with V bias = 1 kV and τ sweep ≈ 800 ns.

Depicted in
The ratio of the sweep times is consistent with the inverse ratio of the corresponding bias voltages and can clearly be attributed to the E bias × B transport mechanism.
Using a reduced bias voltage of V bias = 1 kV (Figure 11b) reveals that even in this case -in contrast to the observation by Moeller -mitigation is effective and gas breakdown is disrupted. The principle dynamics and transport mechanisms are identical (breakdown and ECR electron heating without mitigation; removal of electrons from the high RF field region with mitigation). Electron removal takes place with the previously reported sweep time of τ sweep ≈ 800 ns. This appears to be sufficiently fast in comparison with a mean collision time τ c ≈ 425 ns: A ratio τ sweep /τ c ≈ 1 means that an electron experiences approximately a single collision encounter during sweep out. In contrast, to initiate a substantial ionization avalanche, τ sweep /τ c > 1 would be required.
A number of aspects can be raised to explain the apparent discrepancy between simulations and experimental observations (in presumed order of importance): (i) Moeller has mentioned that conditioning of the waveguide surfaces had a significant impact on the gas breakdown behavior. 18 In particular, he has mentioned the influence of outgassing from the SS waveguide walls, which probably assisted the breakdown phenomenon (cf. subsequent paragraph). (ii) The simplifying assumption of a two dimensional geometry with a homogeneous magnetic field is not an exact representation of the setup which has been used experimentally. 14,15 Possible improvements of the model are uncertain, however, as only insufficient details on the geometry and the magnetic field setup are documented. (iii) The To obtain an estimate of the influence of outgassing, simulations were performed assuming a fraction of 10% of synthetic air (78% N 2 , 21% O 2 , 1% Ar) as an impurity and otherwise unaltered parameters (i.e., a maintained total pressure p = 0.133 Pa). The corresponding number of electrons and the average energy per electron are presented in Figure 12. It is found that, after an initial relaxation of the system, ECR heating and breakdown evolution are decisively different. The lower ionization thresholds and larger cross sections for the introduced gas impurity enhance breakdown. This is reasoned by the decreased mean collision time and is observed despite the impurity's minor concentration. At t = 500 ns, the relative increase in the number of electrons is more than 10% larger compared to the case without impurity. Notably, breakdown follows the expected exponential dependence leading to an even more pronounced effect on the later evolution. Electron heating also appears to proceed more efficiently due to the more local energy conversion. Due to a shorter mean free path and an inhibited transport, electrons remain in the high RF electric field region for a longer period of time. It should be noted that outgassing from the walls in the experiments is merely an addition to the present gas, not a substitute (varied total pressure).
Consequently, a synergistic effect of a lower ionization threshold paired with an increased gas pressure would be expected for the situation reported by Moeller.

B. Simulation prediction for ITER CTS
The ITER CTS scenario differs in three main aspects from the 'Moeller' case: (i) waveguide inner diameter D = 88.9 mm (i.e., lower peak RF electric field strength E 0 ); (ii) axially varying magnetic field with only about 35% axial component in the resonant region, with B ≈ 2.14 T; (iii) CuCrZr ITER-grade alloy waveguide surfaces (approximated by Cu).
In the following, two and three dimensional simulation results are presented, each depicting a specific aspect of the gas breakdown phenomenon. Initially, two dimensional simulation results are discussed, focusing on gas breakdown with a magnetic field solely in the transversal waveguide plane (i.e., directed toward the bounding waveguide walls). Subsequently, three dimensional simulation results highlight the influence of an axial magnetic field contribution. Finally, two particular situations are analyzed related to variations in the magnetic field. The cases to be considered are summarized in Table V.

Two dimensional
A bias voltage of V bias = 1 kV is initially specified for ITER CTS and was correspondingly used in the proceeding analysis once mitigation was switched on. It was later increased to 2 kV, following the findings in this study. The corresponding bias electric potential and field are shown in Figure 13. The bias electric field in the center of the waveguide is again Table V. Collection of ITER CTS cases considered in the proceeding discussion. systematically larger (≈ 28 %) compared to a one dimensional approximation due to the circular geometry, E bias ≈ 14 kV/m > V bias /D ≈ 11 kV/m. Consistent with the scaling in waveguide diameter, the maximum bias electric field is approximately 5 times smaller than for the 'Moeller' case (cf. Figure 8). In the E bias × B configuration, this results in a correspondingly slowed down removal of electrons. However, by aligning the SBWG halves appropriately with the magnetic field in the resonant region (i.e., E bias B), electron removal can proceed along magnetic field lines, resulting in short electron sweep times on the order of τ sweep ≈ 10 ns (cf. Section II D).
As a reference case for ITER CTS gas breakdown and mitigation simulations, a hydrogen pressure of p = 1 Pa was used. In the two dimensional representation, the magnetic field points solely in the x direction. The axial magnetic field component (along the waveguide) is set to zero. A transversal magnetic field magnitude of B = 2.14 T is used to maintain ECR conditions. In Figure 14 the evolution of the number of electrons and the average energy per electron is shown. Without mitigation, breakdown is observed at the timescale of the mean collision time τ c ≈ 55 ns, following the initial onset of the RF modulated pulse.
ECR heating proceeds accordingly, increasing above E ≈ 500 eV for t 40 ns. When a bias voltage V bias = 1 kV is applied after t = 100 ns, the expected behavior of a rapid depletion of electrons within 15 ns is observed. This is accompanied by a steep increase in electron energy after switching on the bias voltage. It is reasoned by the acceleration of electrons due to the bias electric field. For the case with mitigation (black line), the average energy per electron after t ≈ 110 ns is subject to substantial statistical fluctuations, due to the low number of electrons involved and should be considered with caution.
The dynamics of electron removal can be understood from the spatial distributions of the electron density right before the bias voltage is switched on and 10 ns after. Figure 15 shows corresponding electron density profiles plotted over the simulated domain. Notably, even before active mitigation the electrons diffusively distribute along the magnetic field lines. The density maxima close to the wall stem from the electrons' wall interactions and subsequent reflection or secondary electron emission. This effect is correspondingly enhanced with a bias electric field which promotes the electron flux toward the wall. That is, electrons are rapidly removed from the central region of high RF electric field strength (indicated by white isocurves), diminishing ECR heating, but at the same time may accumulate close to the wall until breakdown is finally disrupted. Consequently, wall processes have a noticeable influence on the total electron count (slow absorption), but little influence on the removal of electrons from the high RF electric field region and hence the mitigation dynamics.
A different characteristic can be observed in the two dimensional simulations with a  Indicated by contours is the LP 01 /HE 11 RF electric field magnitude (cf. Figure 4). Mitigation requires an increased bias voltage for pressures above p 10 Pa. To illustrate  Figures 4 and 15). In contrast, however, they accumulate temporarily close to the absorbing (and partially emitting) walls until they are finally removed due to the continued drag toward the wall (see left wall in Figure 15b). The mitigation scheme in the 'Moeller' case, in contrast, relies on the significantly slower E bias × B drift, which does not lead to emphasized accumulation in front of the wall (cf. Figure 10b).

Three dimensional
To verify the mechanisms involved in gas breakdown for ITER CTS, also three dimensional simulations were performed for a section of the waveguide of length L = 80 mm.
The main difference compared to the two dimensional case is posed by an axial magnetic field component, which is expected to reduce the efficacy of ECR heating, due to a smaller The slow breakdown dynamics -compared to the two dimensional case of Figure 14 may be attributed to less efficient ECR heating at an identical RF electric field magnitude.
Electrons are heated in the central high RF electric field region of the waveguide. This is depicted in Figure 19, where a) the instantaneous electron density and b) their kinetic energy per electron is plotted over a cross sectional cut through the x−z plane at t = 350 ns.
While electrons distribute along the magnetic field lines, ECR heating is effective only in the narrow resonant zone where B ≈ 2.14 T. As electron transport is bound to the oblique magnetic field lines, however, electrons diffuse out of the resonant zone at the respective angle. Subsequently, they gain kinetic energy only when passing through the resonant region and remain off-resonance along their remaining trajectory along B. This leads to a large discrepancy between the local and the average energy per electron: local 3 keV, global average 300 eV (cf. Figures 18b and 19). It further appears to influence the electron   An additional asymmetry is found in the electron density below (z < 0 mm) and above (z > 0 mm) the ECR location. Despite the statistical fluctuations apparent in Figure 19, clearly more electrons populate the volume below the resonance condition, where the magnetic field is smaller and the RF beam originates from. This is a known phenomenon which can be understood from the gradient in B approximated by the given field structure. 5 Electrons experience a net drift toward regions of smaller magnetic field, while generally being constrained to their respective field lines. Note that to some extent, the electron and gas breakdown dynamics may be affected by the limited statistics of the three dimensional simulation (i.e., fewer particles per computational cell ≈ 20).
The mechanism of electron removal subsequent to switching on the bias voltage V bias = 1 kV, at time t = 350 ns, is illustrated in Figures 18 and 20. Transport and removal of electrons proceeds mainly in the direction of the magnetic field, whereas an E bias × B drift contribution is estimated to be orders of magnitude smaller. With an angle ∠( E bias , B) ≈ 20 • , the effective bias electric field component parallel to B is slightly reduced to approximately 94 % of the 2D case. Irrespective of this reduction, also in this case electrons are rapidly removed from the high RF electric field region within a sweep time τ sweep ≈ 10 ns. By following the oblique magnetic field lines, electrons are additionally drawn out of the resonant zone with B ≈ 2.14 T, further reducing energy input from the RF electric field. The number of electrons decays quickly after a short phase of accumulation at the wall (cf. Figure 20).
To obtain further insight into the ECR heating and breakdown dynamics in the three dimensional situation, an altered scenario was simulated, where also the RF electric field points in the x direction, E rf E bias B trans. . ECR heating in this case is due only to the RF electric field contribution E rf perpendicular to B, which is reduced to approximately 34 % of the 2D case, as estimated from the axial magnetic field contribution. For otherwise unaltered parameters the results are shown in Figure 21. Gas breakdown is observed after  Figure 13). In the other half of the waveguide, E bias · B/B has opposite sign, pinching electrons into the volume of this waveguide half. This pinching is associated with a corresponding continuous energy gain due to the fundamental bias electric field acceleration (see Figure 22b). In addition, the constricted electrons are concentrated close to the waveguide center with a high RF electric field. This leads to a continued gas breakdown in this half of the waveguide after t 380 ns. The proceeding longtime breakdown dynamics for times t 400 ns are governed by a balance between ionization processes in the pinched waveguide half and transport out of the resonant region following an E bias × B drift. As this drift is orders of magnitude slower than the E bias B removal (cf. Section IV A), the subsequent dynamics cannot be feasibly resolved with the present three dimensional simulation. In addition, a simulation of the referenced effect is problematic for the considered three dimensional setup, because the corresponding drift is directed to the wall only for a sufficiently long waveguide section. As the currently simulated waveguide section is comparably short (L = 8 cm), and the simulation assumes periodic boundary conditions at the top and bottom, this transport mechanism can only insufficiently be reproduced by the model. It is, however, expected to be effective in a 'non-simplified' ITER CTS setup.
A final three dimensional study concerns a high pressure case p = 5 Pa with E rf ⊥ E bias B trans. . As seen from Figure 23, gas breakdown proceeds significantly faster compared to the p = 1 Pa case, on the order of the mean collision time τ c ≈ 11 ns. The principle ECR heating and breakdown dynamics remain similar, however. It can be seen from the case with a bias voltage V bias = 1 kV, switched on after t = 120 ns (black line), that the SBWG mitigation scheme is also effectively interrupting gas breakdown at this pressure. The mitigation timescale is consistently on the order of the electron sweep time τ sweep ≈ 10 ns.
From the preceding analysis it is argued that despite the different model assumptions for the two and three dimensional ITER CTS setup, the principle ECR heating dynamics, gas breakdown, as well as a conceptually similar electron removal mechanism is involved.
While the two dimensional situation systematically does not account for any processes re-  lated to an axial magnetic field component, it provides a meaningful approximation for the prediction of gas breakdown and mitigation effectiveness. At the same time -due to the reduced computational cost -it allows for more detailed studies concerning the specifying parameters, providing a reliable understanding of the inherent processes. In fact, the two dimensional situation merely represents a worst case scenario of the three dimensional case concerning ECR heating, since the RF electric field and the magnetic field are aligned most ideally concerning the ECR condition. Notably, however, the three dimensional scenario also shows physical aspects, which are not present in the simplified two dimensional situation.
This especially relates to the peculiar combination of electron transport to the walls, their interaction with the respective surfaces, and their accumulation due to the magnetic field structure and strength. Most apparently this manifests in an altered optimum RF electric field determined to be almost an order of magnitude smaller in the three dimensional case.

V. CONCLUSIONS
The scope of this work was to investigate ECR heating and the proposed SBWG mitigation scheme for the ITER CTS diagnostic. The results were divided in two main sections: (i) In Section IV A, the simulation scheme was compared against experimental reference data available from Moeller et al. [14][15][16][17][18] It was shown that ECR-assisted gas breakdown dynamics reported from experiments could be reproduced with our model for breakdown and the Monte Carlo electron simulations. The effectiveness of the SBWG mitigation schemeas also reported by Moeller -was subsequently shown by the simulations. A critical assessment was provided regarding the discrepancy of the lower mitigation bias voltage limit for a pressure of p = 0.133 Pa. An analysis of the underlying cause was conducted. It was argued that the uncertainty in outgassing from the SS waveguide walls in the experiments is the most probable contribution to which the deviations are attributed. This is also corroborated by the commissioning phase at the tokamak DITE, where a gyrotron 'conditioning' of the waveguides was necessary, before significant power could be transmitted through the ECR for significantly lengths of time. 18 The proposed simulation procedure is finally argued to be sufficiently accurate for a reliable prediction of the phenomena expected for the ITER CTS diagnostic system.
(ii) In Section IV B, the ITER CTS scenario was initially investigated with respect to the fundamental physical processes taking place during gas breakdown and mitigation. Pressure limits (cf. following paragraph) were determined for a reduced two dimensional situation, whereas they were hypothesized to provide a reliable measure also for the more realistic three dimensional situation. This three dimensional scenario was subsequently investigated for a number of representative cases to manifest the following aspects: 1. The similarity in ECR breakdown dynamics for the simulations for 2D and 3D ITER CTS scenarios was initially elaborated.
2. The similarity of SBWG mitigation for 2D and 3D ITER CTS scenarios, with a bias electric field predominantly parallel to the transversal magnetic field component, was laid out thereafter. Notably, for both relevant three dimensional cases ( E rf ⊥ E bias B trans. and E rf E bias B trans. ), the effectiveness of SBWG mitigation could be demonstrated.
3. The physical processes and the mitigation effectiveness was investigated for a less ideal 3D case with E bias ⊥ B trans. , which is conceptually related to the 'Moeller' scenario.
The attention was drawn to the peculiar effect of electron 'pinching' and continued ECR heating in one half of the SBWG.
To conclude, the pressure limits for gas breakdown and mitigation for ITER CTS -as It is worth noting that -given the corresponding parameters -the procedure and the model used in this work can in principle be extended to accommodate various gas mixtures

ACKNOWLEDGMENT
The work leading to this publication has been funded partially by Fusion for Energy under the Framework Partnership Agreement F4E-FPA-393. This publication reflects the views only of the authors, and Fusion for Energy cannot be held responsible for any use, which may be made of the information contained therein. The authors would also like to thank Charles Moeller for considerable input regarding the integration of SBWGs into existing tokamaks.

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