Numerical simulations of a low-pressure electrodeless ion source intended for air-breathing electric propulsion

Journal of Physics D: Applied Physics J. Phys. D: Appl. Phys. 57 (2024) 495203 (14pp) https://doi.org/10.1088/1361-6463/ad7471 Numerical simulations of a low-pressure electrodeless ion source intended for air-breathing electric propulsion Marek ˇ St′ astn´ y1,2,∗, Kryˇ stof Mrózek2,3,4, Karel Juˇ rík1, Lukᡠs Havlícˇek2, Michal Novotn´ y2 and Adam Obrusník3,4 1Department of Theoretical and Experimental Electrical Engineering, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 3058/10, 616 00 Brno, Czech Republic 2SpaceLabEU, Sukova 49/4, 602 00 Brno, Czech Republic 3PlasmaSolve s.r.o., Sukova 49/4, 602 00 Brno, Czech Republic 4Department of Plasma Physics and Technology, Faculty of Science, Masaryk University, Kotlᡠrská 267/2, 611 37 Brno, Czech Republic E-mail: [email protected] Received 15 April 2024, revised 10 July 2024 Accepted for publication 28 August 2024 Published 18 September 2024 Abstract Air breathing electric propulsion (ABEP) systems offer a promising solution to extend the lifetime of very low earth orbit (VLEO) missions by using residual atmospheric particles as propellants. Such systems would operate in very low-pressure environments where plasma ignition and confinement prove challenging. In this contribution, we present results of a global plasma model (GPM) of a plasma ignited in a very low-pressure air mixture. The results are validated against experimental measurements acquired using a laboratory electrodeless ion source utilizing a resonator for plasma ignition. The device is specifically designed to operate within low-pressure environments as it holds potential applications in ABEP systems for VLEO missions. Parametric studies are carried out via GPM to investigate the resonant behavior and its implications. The potential of the model serving as a predictive tool is assessed through experimental validation against measured data, mainly investigating the extracted ion current dependency on operational pressure and external magnetic field strength. The verified model is further utilized to extrapolate additional information about the resonant plasma such as ion composition or a degree of ionization. Keywords: very low earth orbit, air breathing electric propulsion, electron cyclotron resonance, low pressure ion sources, global plasma model, numerical simulations 1. Introduction In recent years, the growing demand for more efficient and sustainable satellite operation has led to an increasing interest in novel electric propulsion concepts. Among these, the ∗Author to whom any correspondence should be addressed. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. theoretical concept of an air breathing electric propulsion (ABEP) system promises a long-lasting operation of satellites in the lowest of Earth’s orbits [1]. Many proposed solutions rely on ion sources capable of efficient ion production which offer high exhaust velocities, long operational lifetimes, and reduced propellant consumption. The ion source presented in this article leverages a combination of electric and magnetic fields, displaying a resonant plasma behavior to achieve these desirable characteristics in low-pressure environments. As such, it is particularly suitable for ABEP applications. Introducing satellites to a very low earth orbit (VLEO) approximately below 400 km of altitude results in various 1 © 2024 The Author(s). Published by IOP Publishing Ltd J. Phys. D: Appl. Phys. 57 (2024) 495203 M ˇ St′ astn´ yet al benefits including faster communication, better weather forecasting, and improvements in Earth observation [2]. Moreover, low orbits are desirable due to the minimal concentration of space debris, therefore the risk of catastrophic collisions is reduced. However, residual atmospheric particles in low Earth orbits pose a significant challenge to satellite operation in the form of substantial atmospheric drag. ABEP systems present a theoretical solution by employing these residual particles as a fuel source, subsequently compressing, and ionizing them in a discharge chamber. The resulting plasma serves as a reservoir for expelling particles which in turn creates thrust large enough to offset the said drag force. Due to the harsh atmospheric conditions, the operational pressure in ABEP systems is expected to fall in the range of millipascals [3]. Therefore, a high frequency ion source with resonant behavior is considered as a suitable device capable of igniting and sustaining plasma at such low pressure. This is further corroborated by contemporary studies that emphasize the necessity for thrusters capable of operating at very low pressures, due to constraints imposed by in-orbit conditions [4]. Recent ABEP technologies present various theoretical solutions to spacecraft operation in VLEO. The feasibility of these distinct concepts is rarely tested in VLEO representative laboratory conditions, and mostly via numerical simulations [5]. Recent investigations offer a comprehensive review of the existing efforts in developing ABEP systems [6,7], discussing trade-offs between crucial parameters to achieve optimal performance and efficiency. A few of the existing approaches rely on high frequency sources for plasma ignition. According to publicly available sources, none of these concepts have undergone in-orbit testing. The Japan Aerospace Exploration Agency implemented electron cyclotron resonance (ECR) as one of their early concepts. Although research on an air breathing ion engine was not followed through, a study from 2012 discussed the validity of the design. The study concluded that the compression inside the discharge chamber should be high enough at 200 km of altitude for successful ECR ionization [6]. More recently, Institut für Raumfahrtsysteme reported a successful ignition of an RF helicon-based inductive plasma thruster (IPT) [8]. The concept features an electrodeless design to avoid electrode degradation frequently occurring in VLEO due to the presence of corrosive anomalous oxygen. Currently, the IPT undergoes thruster optimization based on advanced plasma diagnostics results and simulations [9]. This contribution investigates an electrodeless magnetized high frequency ion source intended for ABEP by means of ab-initio numerical simulations and laboratory measurements of the extracted ion current. This work compares theoretical results in measurements and demonstrates that the model correctly captures the resonant behavior of the plasma source and can be predictive across a broad range of conditions with only two fitting parameters. The resonant plasma was chosen especially as it can sustain plasma discharges in low pressure environments and can achieve a sufficient degree of ionization, overcoming the main obstacles in ABEP design. To simulate the ion source behavior for various settings, we use the so-called global plasma model (GPM). GPMs can be seen as a more efficient simulation technique compared to the 2D or 3D particle-in-cell codes [10] or fluid codes [11, 12]. It should be noted that GPMs do not offer spatial resolution because they rely on analytical solutions of spatial derivatives (rather than numerical), which makes them especially suitable for simple geometries (cylinder, box, etc). On the other hand, they offer an unparalleled efficiency and can capture complex plasma reaction schemes with hundreds or thousands of plasma-chemical reactions [13,14]. GPMs have also been employed in the research and development of electricpropulsion systems with various on-board propellants [15]. Furthermore, Lucken et al used a GPM in conjunction with the particle-in-cell (PIC) method to study magnetic confinement in plasma thrusters using xenon and iodine [16]. The first chapter of this paper addresses the GPM method of simulation in more detail and introduces the key features of global modeling while distinguishing the presented model from its predecessors. In this work, we improve the treatment of magnetic confinement and power absorption, which proves to be crucial for predictive simulation of electrodeless ion sources. Two distinct methods of magnetic confinement are compared while discussing the preference of one over the other. In the second chapter, we comment on the technical details of the ion source and describe the method that is used for ion extraction. In the resulting section, the output of the GPM equipped with a more suitable magnetic confinement method is compared to experimental measurements to verify the ability of the GPM to simulate resonant plasma behavior. Finally, the verified GPM is subjected to extrapolation of information regarding plasma composition, demonstrating its potential as a useful analytical tool. 2. GPM theory GPMs serve as a way of gaining insight into various plasma processes via computer simulation. GPMs are unique in their approach to the spatial distribution of plasmas as they approximate spatial derivatives analytically to drastically shorten computational times down to the order of tens of seconds per configuration. This is thanks to the fact that, mathematically, GPMs are a set of ordinary differential equations, instead of a set of partial differential equations. The speed and efficiency are counterbalanced by providing solely volume averaged quantities as their output. Thanks to the short computation time, extensive parametric studies can be conducted and examination of systems with complex chemistry and reaction schemes becomes possible. These GPM strengths facilitate the examination of plasma discharges ignited in the N2/O2mixture in the ion source investigated in this paper. The model requires specifications of the studied system despite the approximations of spatial derivatives. Firstly, a geometrical configuration must be provided as an input. GPMs 2 J. Phys. D: Appl. Phys. 57 (2024) 495203 M ˇ St′ astn´ yet al function most efficiently with a simple geometry: e.g. a cylindrical discharge chamber of an ion source. Secondly, a kinetic scheme incorporating a list of present species and the reactions among them is required. The reaction scheme grows more complex with an increasing number of different species, including excited and metastable states and positive or negative ions in the investigated plasma system. Moreover, the model must include the interaction of species with walls, thus increasing complexity. Lastly, the method of power transfer must be well established as it is crucial to simulate the resonant behavior of the investigated ion source. The following section introduces the previously published model by Mrozek et al which serves as a basis for the improved model discussed in later sections. 2.1. Base model and its limitations The core of the model presented in this paper is based on the efforts of Mrozek et al, who developed the equations and performed a validation of the model on a simple case [13]. The following paragraphs highlight the important parts of the model implementation as well as emphasize key differences made to better suit our circumstances. At the crux of the GPM lie two types of ordinary differential equations: •Power balance equations are solved for electron and gas temperature •Particle balance equations solved for number densities of all species separately, except for electrons The kinetic system contains 42 species: 21 neutrals, 12 positive ions, 8 negative ions and electrons. The particle balance equations are governed by source terms depicting the loss or gain channels and can be divided into: •inflow and outflow •wall reactions: neutral deexcitation, neutral recombination, ion neutralization •volume reactions The full list of 612 reactions is based on the kinetic scheme detailed in Obrusník et al [17] and can be seen fully implemented in the supplementary data. Readers are encouraged to refer to the supplementary material for further context, as it contains a machine-readable model configuration file with explanatory notes. Mrozek et al drew a conclusion based on the results of their numerical model regarding the feasibility of plasma ignition in magnetized high-frequency plasma sources at low pressures. Both the experimental and GPM results agree on a minimal pressure threshold for successful operation at approximately 5 mPa. Subsequent investigation using 3D DSMC simulation of gas compression in VLEO representative environments corroborates this conclusion [3]. However, the base model lacks sufficient experimental validation as the laboratory breadboard model had not yet been fully developed at the time. The main point of contention is the approach to simulating power deposition in the plasma and magnetic confinement while replicating the resonant behavior of the plasma. The base model exhibits fixed power deposition due to establishing absorbed power as a constant input parameter based on the known value of input power Pin delivered by the power supply. As we conduct experiments including an investigation of plasma process dependency on changing magnetic field, this assumption no longer remains valid and the approach to plasma deposition must change. Similarly, the magnetic plasma confinement model demonstrated in the base model omits crucial factors specific to the extremely low-pressure environment resulting in potential overestimation of the confinement effects. The GPM is a spatially approximative model that cannot resolve the spatial distribution of positive ions in a standalone fashion. The commonly used hfactors represent the ratio between the density of positive ions at the edge of the plasma sheath and the density of positive ions at the center of the discharge. Therefore, the value of his inversely proportional to the confinement strength. The h-factors explicitly appear in the particle balance equations for positive ions and influence certain source terms in the electron and gas temperature equations. In particular, the particle balance equation neutralization loss term for positive ions serves to illustrate how the system is closely intertwined as the same term appears as the gain channel in the neutral particle balance equation. Furthermore, the electron density neis calculated in the model from the quasineutrality assumption ne=n+−n−, featuring the difference between positive n+and negative ion densities n−, respectively. Consequently, confinement hfactors are ubiquitous throughout the entire model and their implementation may fundamentally change the modeled plasma characteristics. Interested readers are encouraged to refer to [13] for the exact implementation of hfactors in the system of ODEs solved by the GPM. They are commonly divided into a longitudinal hL, and radial hRin the case of cylindrical domains based on the two main directions of ion loss. The method chosen by Godyak et al leverages the use of Godyak’s cylindrical formula [18] coupled with an expression accounting for the plasma electronegativity [19] due to the presence of oxygen in the mixture. The added magnetic field strengthens the confinement in the radial direction (perpendicular to the external magnetic field). The hLfactor is considered to be mostly unaffected by the presence of magnetic field. The work of Mrozek et al relies on the following modifier for the radial hRfactor [13]: fB=(1+ωci νin )−2 ,(1) where ωci denotes the ion cyclotron frequency and νin represents the mean ion-neutral collision frequency. In other words, fBis based on the assumption that heavier ions govern the confinement effect, especially in low pressure environments. However, using this approximation almost perfectly confines all ions in the plasma even at low magnetic fields. This is demonstrated in the following figure 1. 3 J. Phys. D: Appl. Phys. 57 (2024) 495203 M ˇ St′ astn´ yet al Figure 1. Magnetic confinement strength and extracted ion current Iext dependency on external magnetic field strength at 35 mPa at different constant input powers Pin using the magnetic confinement method from the base model [13]. Figure 1shows almost total confinement due to fBreaching 10−4order of magnitude on average. In this case, virtually no ion loss is observed in the radial direction and as a result, all ions reach the collection surface and the Iext thus remains constant. It stands to reason that such overestimation of the confinement strength, i.e. underestimation of the hRfactor by a nonmarginal amount would negatively influence all plasma processes in the same way as Iext. Therefore, a change in the confinement method is in order. The following section depicts the improvements made to the base model and discusses the methods of plasma confinement and power deposition in further detail. 2.2. Improving magnetized plasma treatment In contrast to the previously published model, we make improvements to the following key parts of the global model to acknowledge the resonant nature of the ion source of interest. To begin with, the description of plasma confinement must change to reflect the extremely low pressure in the discharge chamber and strong external magnetic field. Next, the approach to plasma power absorption is changed in the manner described below. 2.2.1. Plasma confinement. Due to the presence of a strong external magnetic field, the investigated plasma exhibits a nonMaxwellian shape of the electron energy distribution function (EEDF). This is reflected in the model by employing a BOLSIG+Boltzmann equation solver [20] to calculate the EEDF and to more accurately estimate rate coefficients of certain electron-impact reactions. However, for the remainder of the model, the Maxwellian EEDF approximation is used due to the lack of a better alternative. To the author’s knowledge, there are currently no published works dealing with global plasma modeling at comparably low pressures and high magnetic fields. Under these conditions, Coulomb collisions between electrons are expected to be the dominant randomization mechanism for electron velocity, so the hfactors must be modified to reflect the physics. The most common hfactors are purely based on the geometrical configurations of the simulated domain. The previous work of Godyak, and later Liebermann and Lee [21] culminated in a semi-empirical hfactor expressions suitable for electropositive or electronegative plasma in all pressure regions in a cylindrical domain of length Land radius R: hR= 1+(3αavg γ) 1+αavg 0.8 √4+R λi+(0.8RuB χ01J1(χ01)Da)2,(2) hL= 1+(3αavg γ) 1+αavg 0.86 √3+L 2λi+(0.8LuB πDa)2,(3) where λirepresents the ion mean free path in neutral gas, J1 stands for the first order Bessel function of the first kind and χ01 ≈2.405 represents the first zero (sometimes called the first root) of the zeroth order Bessel function of the first kind. The ambipolar diffusion coefficient is denoted by Da. Finally, the Bohm velocity uBrepresents the velocity of ions at the sheath boundary while accounting for the electronegativity of the discharge, which takes the form described in [19]: uB=√kBTe(1+αavg) mi(1+αavgγ),(4) with Boltzmann constant kB, electron temperature Tein Kelvin, and mean ion mass mi. The electronegativity is described via the averaged electronegativity of the plasma αavg =n−/neand by γ=Te/Tiwhich denotes the ratio between the temperature of electrons and negative ions, respectively. The ion temperature is assumed to be equal to the gas temperature, as the studied plasma is considered to be a low-temperature magnetized high-frequency plasma. The Bohm velocity formula (4) was derived with a Maxwellian EEDF in mind. Therefore, applying it to a system with nonMaxwellian EEDF lessens its validity; however, it remains a reasonable approximation in systems with low electron temperatures. Furthermore, an electronegative factor is derived 4 J. Phys. D: Appl. Phys. 57 (2024) 495203 M ˇ St′ astn´ yet al to describe a singular type of ion present in the simulated plasma. Therefore, it must be treated with a degree of skepticism as a mere approximation meant to offer sufficient insight into the confinement of electronegative plasmas. Furthermore, the main strength of GPM lies in its computational efficiency, which would be lost by implementing overly complicated electronegativity models. The ion source of interest operates in conjunction with a strong applied magnetic field. The magnetic field is generated using a pair of Helmholtz coils, ensuring a nearly homogeneous magnetic field during experimental measurements. Therefore, the magnetic field intensity is assumed to be constant over the modeled plasma domain. Whereas the previously published model presents a factor fBmultiplied by an already derived hRfactor, we opt to leverage the ambipolar diffusion coefficient Dain the equations (2) and (3). In the absence of a magnetic field, the extremely low pressure renders the plasma effectively collisionless, resulting in a large Da. Excluding electronegativity, both equations would reduce to the original Godyak’s version describing plasma in an infinitely long cylinder. However, in the presence of a strong magnetic field, the Dadecreases in the radial (perpendicular) direction described by the component D⊥a. The longitudinal (parallel) component of Dais believed to be virtually unaffected by the magnetic field. Combining the previously gained insight, we arrive at the following expressions: hRB = 1+(3αavg γ) 1+αavg 0.8 √4+R λi+(0.8RuB χ01J1(χ01)D⊥a)2(5) hLB = 1+(3αavg γ) 1+αavg 0.86 √3+L 2λi .(6) The D⊥ais implemented in the model based on several assumptions. The magnetic field is assumed to mainly affect diffusion in the perpendicular direction. According to Liebermann and Lichtenberg [22], D⊥acan be expressed as: D⊥a=D⊥eµ⊥i+D⊥iµ⊥e µ⊥e+µ⊥i ,(7) consisting of ion µ⊥iand electron µ⊥emobility, ion D⊥iand electron D⊥ediffusion coefficients acting in the perpendicular direction. The plasma in the discharge chamber can be described as a low pressure highly non-isothermal plasma due to its resonant nature. The assumption that µe≫µiand De≫ Diholds true even for their perpendicular components for magnetic fields of the order of mT. The perpendicular components of mobility and diffusion coefficients can be expressed as: µ⊥i,⊥e=µi,e 1+(ωci,ce νin,eff )2,(8) D⊥i,⊥e=Di,e 1+(ωci,ce νin,eff )2,(9) for ions and electrons, respectively. Both expressions are modified by the ratio between the corresponding cyclotron frequency and the mean collision frequency of ions/electrons with neutrals. The mean ion-neutral collision frequency is calculated as the ratio between the mean thermal velocity of ions and λi. The cyclotron frequencies can be written as: ωci,ce =eB0 mi,e ,(10) for the external magnetic field B0and electron mass meor ion mass mi. Whereas the movement of ions is mainly driven by the mean collision frequency of ions and neutrals, the motion of electrons is influenced by additional factors. There are numerous processes in low pressure resonant magnetoplasmas which influence particle behavior. In a low-pressure environment, electron–neutral collisions fail to accurately describe the electron movement on their own. Moreover, electron trajectories appear to be further influenced by plasma oscillations, instabilities, fluctuations, and electron–electron interactions. It is therefore reasonable to replace the original electron–neutral collision frequency νen with an effective frequency νeff while describing electron motion in radial direction in equations (8) and (9). The necessity for introducing νeff as well as its definition is addressed in the following paragraphs. Recent research indicates that plasmas with perpendicular electric and magnetic fields are prone to developing instabilities, leading to cross-field electron transport in E×Bdirection. This anomalous diffusion of electrons across magnetic field lines naturally decreases the effectivity of magnetic confinement and is observed frequently in several types of E×B plasmas in cylindrical geometries (e.g. in Hall thrusters [23]). Due to anomalous diffusion, D⊥ascaling with ∼B−2(7) is sometimes discarded in favor of Bohm diffusion DBohm ∼B−1 which better corroborates experimental measurements [24]. Nevertheless, the description of charged particle diffusion in a magnetic field remains a highly contentious topic. Therefore, we choose to implement equation (7) in hope of accounting for the anomalous diffusion by introducing νeff. The realization of νeff in our model is also based on the following research; Popov ruminated on the existence of νeff as a frequency including collisional and non-collisional electron effects, possibly represented as a function of neor p,Er, Teand B0[25]. This is further supported by Lucken et al, who employed extensive 2D PIC simulations to account for instabilities in the magnetoplasma and arrived at a formula for νeff as a function of the above-mentioned parameters [16]. The effective frequency is also introduced by Kim et al as a collisional electron frequency enhanced by instabilities leading to cross-field transport [26]. However, their proposed expressions of νeff are tailored to their specific type of plasma and application. Obtaining and implementing the exact formula for νeff of our studied system would render the GPM too complex to investigate the plasma processes efficiently. Therefore, we opt to introduce a multiplicative factor ξas a fitting parameter: νeff =ξ νen. According to Popov, experiments confirm that νeff eventually equals to νen at pressures higher than approximately 5 J. Phys. D: Appl. Phys. 57 (2024) 495203 M ˇ St′ astn´ yet al 1 Pa and a condition νeff ≫νen applies to lower pressures. This also holds true for the expressions of νeff proposed by Lucken and Kim. Additionally, the exact values of νeff reported in pertinent studies span a large interval from νeff ≈ωce/2 [25] down to νeff ≈ωce/16 [23], depending on which effects are being considered and which are being neglected. It must be stated that the resulting formulas (5) and (6) come with many caveats. Regardless, in conjunction with a novel approach to plasma power absorption, the simulation of resonant behavior can be achieved. 2.2.2. Power deposition. This GPM differs from its predecessors in the approach to power deposition into plasma, which is an integral part of the power balance equations. In our case, the power is transferred directly from the power supply to the electron population via resonance. Electrons are subjected to helical trajectories and experience gyromotion with cyclotron frequency ωce based on the applied external magnetic field B0. If the resonator is tuned to frequency ωof the same magnitude, a resonance condition ωce =ωis satisfied, and electrons gain a substantial amount of energy. A simple formula for the average power absorbed by electrons proposed by Kelly et al was modified and employed [27]: Pabs =e2neVplE2 r 4me[νeff ν2 eff + (ω−ωce)2+νeff ν2 eff + (ω+ωce)2], (11) where Vpl stands for the plasma volume. Pabs depends on the magnitude of the magnetic field (through ωce). Naturally, it must also depend on the amount of power provided by the power supply Pin.Pin and Pabs are connected through the parameter Er. The physical interpretation of Erlies in its representation as the magnitude of the radial component of the electric field (i.e. direction perpendicular to the magnetic field). However, since we are unable to measure this quantity experimentally and since the radial electric field is likely not constant over the plasma diameter, we use it as a tuning parameter of the model. We set its value so that the Pabs (value in the model) is equal to 80% of Pin (the nominal value as read out from the power supply), when the ECR condition is satisfied (ωce =ω)—see figure 2. A factor of 80% was estimated experimentally, by measuring the forward and reflected power of the transmission line. Lastly, νeff stands for an effective frequency and represents a deviation from the original formula. Kelly et al proposed expression (11) for ECR plasmas, neglecting plasma instabilities. Due to the reasons described above, we opt to replace the original νen for νeff. Kelly et al also concluded that at resonance conditions, the collision frequency in expression (11) represents the half-width of the resonance peak. This insight becomes apparent while discussing the results of our simulations, making the use of this formula intuitive. In addition, its simplicity complements the core strengths of GPM: computational efficiency and flexibility. Figure 2. Illustration of the dependence of the absorbed power on the magnetic field. The usage of νeff is further supported by the conclusion drawn by Kelly et al, who deemed the usage of νen not suitable for pressures lower than approximately 1 Pa. Instead, experiments conducted under this pressure threshold revealed a discrepancy between calculated νen and the experimentally obtained half-width of the resonance peak. This deviation increases with decrease in pressure, which corresponds to the insights into ξand νeff mentioned in the plasma confinement section 2.2.1. 2.3. Magnetic confinement model comparison During experiments with resonant plasma, the effects of magnetic field on confinement and resonant energy transfer cannot be separated. However, numerical simulations allow us to conduct independent investigations of these effects and directly compare both confinement models at constant input powers. The new confinement model is examined in the following figure 3, sharing the same initial conditions as figure 1. Both figures 1and 3feature simulation results from studies conducted at different input powers while keeping the same pressure in the discharge chamber. Although both confinement models display a decrease in hRwith respect to an increase in magnetic field strength, their absolute values differ by multiple orders of magnitude (10−4vs 10−1). Not only is such strong confinement seen in figure 1unrecognized in preliminary experimental measurements but research concerned with magnetic confinement reports hRvalues comparable to the results seen in figure 3. This is due to the aforementioned additional effects experienced by electrons such as plasma instabilities which disrupt the magnetic confinement via a cross-field electron transport, causing electron–neutral collisions to occur further from the axis, thus changing the ion density profile. Moreover, recent results by Kim et al show an increase in magnetic confinement with increasing externally applied magnetic field B0, which is also apparent in figure 3[26]. Despite both figures displaying an increase in the extracted ion current with increasing input power, the measured Iext 6 J. Phys. D: Appl. Phys. 57 (2024) 495203 M ˇ St′ astn´ yet al Figure 3. Magnetic confinement strength and extracted ion current Iext dependency on external magnetic field strength at 35 mPa at different constant input powers Pin using the novel magnetic confinement method. does not remain constant while increasing the magnetic field strength. Additionally, the increase in Iext observed in figure 3 corresponds well to the gradual decrease in hRresulting in limiting the ion loss to walls. In summary, the improved magnetic confinement method conforms more to theoretical expectations and experimental observations. The main difference between the two applied models is the dependence on increasing strength of the external magnetic field. It stands to reason that, eventually, increasing B0governs the magnetic confinement as it overwhelms effects stemming from resonant behavior, plasma instabilities and fluctuations. 2.4. Model implementation The model was implemented using proprietary software MatSight Global Model by PlasmaSolve. The software enables computing large non-linear systems of non-linear ODEs and conducting multi-parametric and optimization studies. The advantage of this solution is the separation of software and physics input, which enables us to attach a humanreadable configuration file containing all the equations and source terms as supplementary data to this publication. 3. Experimental The experimental layout of the investigated setup has already been described in Mrozek et al [13]. It consists of a glass tube with a metal electrode wrapped around it. The electrode is driven by a high-frequency alternating voltage in the frequency range of 100–900 MHz. Furthermore, there is an external source of magnetic field which can be either a set of two solenoids or an array of permanent magnets. For the purposes of source development and for the purposes of this paper, we use exclusively magnetic coils to generate the magnetic field. The fluidic interface consists of the Pfeiffer EVR 116 electronics-driven needle valve and the Inficon MPG500 widerange pressure gauge used for regulating and measuring pressure in the discharge channel. The chemical composition of the N2/O2mixture comprises 80% nitrogen and 20% oxygen. The plasma source used in this work is very similar to the one described in [13]. The main difference lies in the current extraction stage, which was simplified for the purposes of plasma source characterization. The schematic view of the setup is shown in figure 4below. The RF electrode circuit is completely isolated from the current collection circuit. The coils (in blue) generate an external magnetostatic field and are oriented co-axially (in Helmholtz configuration). The current in the coils is driven in the same direction by Rohde & Schwarz HMP4040 and is adjusted in such a way that the produced magnetic field on the coil axis is close to the resonant magnetic field (see equation (12) below). Since the plasma is electrodeless, the plasma potential is defined by a relatively sparse screen grid while the cylindrical collector electrode is connected to the negative terminal of the DC power supply. The extraction voltage is constant and limited to negative 200 V to prevent plasma breakdown between the screen grid and the collector electrode. The positive terminal of the Aim TTi PLH250 DC power supply is grounded through an ammeter, on which the extracted ion current is measured. The cylindrical shape of the collector electrode is driven by practical concerns—it is easier to assemble and does not obstruct side view of the plasma by other diagnostic methods (e.g. optical emission spectroscopy). In a possible ABEP setup, the collector electrode would be replaced by a planar acceleration grid and possibly a planar deceleration grid behind it. 4. Results This chapter addresses the outcomes of our study on the validation of the GPM model using experimental data. A comprehensive assessment is presented, describing various trends in measured quantities which display the resonant behavior. Lastly, the validation of the GPM output against experimental results is leveraged to extrapolate additional information about the studied system. 4.1. Model validation The following section addresses the validation of our GPM against the experimental data obtained via measurements 7 J. Phys. D: Appl. Phys. 57 (2024) 495203 M ˇ St′ astn´ yet al Figure 4. Schematic view of the experimental layout. using the laboratory ion source described in the previous chapter. The experimental data contains the dependencies of the extracted ion current on various input powers Pin, operational pressures p, and normalized applied magnetic fields Bnorm defined as: Bnorm =B0 Becr =qB0 meω,(12) allowing us to present the obtained data in a more organized manner. In practice, the resonant condition ω=ωce matches the value of Bnorm =1. Firstly, we compared all available datasets to determine whether the GPM equipped with a novel approach to magnetic confinement captures the resonant nature of the investigated plasma. We utilize both the previous approach to power deposition described by Mrozek et al (GPM-1) and the modification described in this paper (GPM-2) to observe the potential improvement directly. Conclusions about the resonant nature of the system can be drawn by investigating the profile of the extracted ion current dependency on the normalized magnetic field. The following figures 5and 6feature 2D graphs of Iext dependencies on Bnorm and Pin for simulated data gathered at an operational pressure of 52 mPa and experimental measurements. We expect the Iext (Bnorm)dependency to reach a maximum at the resonant condition: Bnorm =1. All figures display an area corresponding to the value Iext =0 as certain combinations of Bnorm and Pin extinguish the discharge. The measurement accuracy in the range up to 1 mA of the extracted current is 0.05% of reading and 0.006% of range. Figure 5 features a set of 2D graphs which clearly reinforces the need for a change in the power deposition approach as GPM-1 displays no sign of resonant behavior. On the contrary, GPM-2 follows the trends in experimental data well. The same level of agreement was reached for data gathered at pressures of 35 and 90 mPa. Two sets of experimental measurements were also conducted at higher pressures, 550 and 1000 mPa, respectively. The former is displayed in the subsequent figures 7and 8 featuring a comparison of experimental measurements to results obtained from GPM-1 and GPM-2, respectively. Our results show that resonant trends can be observed and simulated via GPM-2 even in an increased pressure environment. Moreover, all available datasets display an increase in Iext corresponding to an increase in Pin in the vicinity of the resonant condition. GPM-2 exhibits very similar trends and profiles as the experimental data. Nevertheless, Iext absolute values differ by an order of magnitude between the experimental and simulated data on the graphs shown above. This notion is addressed in the following section. 4.2. Model fitting A closer examination was carried out to quantify the discrepancy in Iext absolute values using a scaling factor s. The figures 9and 10 show the scaling process for different conditions. The variation of the three major parameters influences the outcome of GPM-2. The first two parameters are sand Er. The former quantifies the discrepancy in the extracted ion current, whereas the latter influences the amount of absorbed power Pabs per equation (11)—see section Power deposition 2.2.2. Lastly, a constant value of ξ=375 was set for 35, 52 and 90 mPa despite being pressure dependent as we consider the pressure to vary minimally in this interval. In the figures displayed above, νeff represents the half-width of the resonance curve. As was also stated previously, the exact pressure dependency remains unknown. Nevertheless, we expect ξto reach unity for sufficiently high pressures. This is further reinforced by the following figure 11. In figure 11 we observe the resonance curve at 550 mPa in a narrower interval of Bnorm than in figures 9and 10 pertaining to lower pressures. The resonance curve is therefore wider at lower pressures, implying larger values of ξas the pressure decreases and vice versa. Accordingly, the model has converged at 550 mPa for a value of ξ=80. On the other hand, a discrepancy is noticeable for larger values of Bnorm suggesting ξis also dependent on the strength of the external magnetic field. However, figures 9and 10 imply that this magnetic field dependency is more prevalent at higher pressures. The following table 1features a summary of the main factors influencing the simulated results to paint a clearer picture for the reader. Despite GPMs inherently including a variety of tuning parameters, the authors are convinced that these parameters form a nearly orthogonal system and each of them has a physics-based justification behind it. The following section extrapolates the validated GPM-2 to paint a picture of plasma composition and showcases additional outputs of the GPM which might be leveraged in future research. 4.3. Insight into plasma composition The experimentally validated model is used to gain additional insight into the underlying plasma processes and plasma composition. The results featured in this section are extrapolated to include combinations of input power and pressure which 8 J. Phys. D: Appl. Phys. 57 (2024) 495203 M ˇ St′ astn´ yet al Figure 5. Comparison between 2D graphs of Iext (Bnorm)and Iext (Pin)(left: experimental results, right: GPM-1, p=52 mPa). Figure 6. Comparison between 2D graphs of Iext (Bnorm)and Iext (Pin)(left: experimental results, right: GPM-2, p=52 mPa). Figure 7. Comparison between 2D graphs of Iext (Bnorm)and Iext (Pin)(left: experimental results, right: GPM-1, p=550 mPa). have not yet been achieved experimentally. The GPM is capable of directly outputting information regarding ion composition allowing to investigate the relative abundances of the most prominent ions. It should be noted that the plasma source and the global model are presently run in a mixture of N2and O2as the operating gas, so the ion composition is not representative of the ion composition at VLEO, where most of the oxygen is pre-dissociated. The population of N+ 2and O+ 2in the following figure 12 mirrors the composition of the operating gas at 70% and 22%, respectively and it stays practically constant over the entire power range. This suggests that direct ionization is the 9