Importance of stepwise ionization from the metastable state in electron cyclotron resonance ion thrusters

Abstract

In electron cyclotron resonance (ECR) thrusters, the plasma mode transition is a critical phenomenon because it determines the maximum thrust performance. In ECR ion thrusters, ionization generally occurs in the magnetic confinement region, where electrons are continuously heated by ECR and confined by magnetic mirrors. However, as the flow rate increases, ionization is also observed outside the magnetic confinement region, and this induces the plasma mode transition. In our previous work, two-photon absorption laser-induced fluorescence (TALIF) analysis revealed that the stepwise ionization from the metastable state plays an important role in the ionization process. However, the distribution of the stepwise ionization has not yet been revealed because of the long lifetime of the metastable state. In this study, this distribution was investigated using one experimental and two numerical approaches. First, TALIF was applied to two types of gas injection with clear differences in thrust performance and ground-state neutral density distribution. In the first simulation, the metastable state particle simulation was used to estimate the excitation rate distribution. In the second study, simulations of the electric field of microwaves were used to estimate the contribution of the stepwise ionization to the plasma density. The experimental and numerical results revealed that the stepwise ionization spreads outside the magnetic confinement region because of the diffusion of metastable particles, and this spread induces the plasma mode transition, explaining the difference between the two types of gas injection.

Introduction

In recent years, electric propulsion thrusters have been used in many space missions [1,2,3,4,5]. The design of thrusters is very important for thrust performance and lifetime; thus, the geometries of the magnetic field and wall have been investigated [6,7,8,9]. To investigate the plasma parameters, plasma diagnostics such as electrical measurements [10,11,12], laser spectroscopy [13,14,15,16,17], and optical emission spectroscopy [18,19,20] have also been developed.

However, measurement of the local excitation and ionization rates is still challenging. Generally, the ionization and excitation rates can be written as k_{i, j} = ⟨σ_{i, j}v_e⟩n_en_j, where k_{i, j} is i-th production rate from the lower state j (e.g., the ground state), ⟨σ_{i, j}v_e⟩ is the rate coefficient from state i to state j, n_e is electron density, and n_j is the number density at the lower state j. To evaluate ⟨σ_{i, j}v_e⟩, the cross-section σ_{i, j} and electron velocity distribution function are necessary. Of course, this function can be theoretically evaluated using the Langmuir probe and LTS; however, it is very difficult to measure non-Maxwellian electrons because of the low signal ratio. Additionally, in low-temperature plasmas, not only direct ionization from the ground state, but also ionization from the lower excited state, i.e., stepwise ionization, can be important because relatively low-energy electrons can ionize the excited neutral particles. For example, plasma simulation models including these excited states have been proposed by Boeuf [21], Hagelaar [22], and Hara [23]. Chaplin suggests that the stepwise ionization from the metastable state is very important in Hall effect thrusters by collisional radiative model (CRM) [24].

Hence, an experimental evaluation of the ionization process that includes direct and stepwise ionization is strongly desirable. In our group, understanding the ionization process of the ECR ion thrusters used in the Hayabusa [25] and Hayabusa2 [26] missions is one of the most important issues. In the thrusters, the plasma mode transition occurs as the input flow rate increases, which restricts the maximum thrust performance. Generally, in ECR ion thrusters, ionization occurs in the magnetic confinement region where electrons are continuously heated by the magnetic mirror confinement. However, we previously reported that ionization is also observed outside the magnetic confinement region as the flow rate increases, which induces plasma mode transition [27]. Therefore, to investigate the ionization process, we proposed a method for investigating excitation process by the ground-state neutral density and spontaneous emission intensity at that point [28]. Then, an analysis of the excitation process was used to estimate the ionization process. The measurement revealed that stepwise ionization from metastable neutral particles 1s₅ (Paschen notation) plays an important role in the ionization process [28].

However, the distribution of the stepwise ionization is not yet clear because the metastable state particles have a long natural lifetime [29]. In this study, to investigate this distribution, one experimental and two numerical approaches were used. In the experiment, the ground-state neutral density and spontaneous emission intensity were measured using two-photon absorption laser-induced fluorescence (TALIF) for two types of gas injections that are substantially different not only in thrust performance [30], but also in metastable neutral density [31], ground-state neutral density [32], and spontaneous emission intensity [33]. These differences can be useful for revealing the distribution of stepwise ionization. In one simulation, a metastable state particle simulation considering de-excitation due to collision and wall diffusion was performed to specify the excitation rate distribution in the metastable state. Then, the simulation results were validated using the density distribution obtained by laser absorption spectroscopy (LAS) [30, 31]. Second, to estimate the plasma density distribution, the electric field of the microwaves was simulated because it strongly depends on the plasma density (e.g., cut-off phenomenon). Here, the plasma density profile was calculated using the plasma diffusion equation [34], thus revealing the contribution of the stepwise ionization. The simulated electric field was also validated using experimental data measured by an electro-optic (EO) probe [35]. These numerical approaches are useful for estimating unobserved parameters (e.g., the excitation rate distribution and stepwise ionization distribution) and the correlation between each pair of parameters.

The rest of this paper is organized as follows. The configuration and an explanation of the plasma mode transition are presented in “ECR ion thruster” section. The TALIF measurements of the ground-state neutral density and spontaneous emission are given in “Ground-state neutral density and spontaneous emission intensity measurements” section. The numerical models are described in “Numerical simulation” section. The experimental results, simulation results, and discussion are presented in “Results” section. Finally, the conclusions are summarized in “Conclusion” section.

ECR ion thruster

Configuration

A schematic of the ECR ion thruster is shown in Fig. 1. Microwaves are transmitted into the discharge chamber from the antenna at the bottom of the waveguide. In the discharge chamber, two magnet rings generate a mirror magnetic field. Larmor radius of electrons in the mirror magnetic field is the order of 0.001–0.01 cm. Hence, electrons are strongly magnetized and confined by the mirror magnetic field. At that time, electrons are continuously heated by ECR in the mirror magnetic field, which is called the magnetic confinement region. Then, the electrons heated by ECR ionize neutral particles, and then plasma is generated. Larmor radius of ions is the order of 0.1–1.0 cm, which indicates that ions are weakly magnetized. As a results, ions can be transported to the grid and ions are accelerated by a high voltage between the screen grid and accelerator grid.

Fig. 1

Schematic of a gridded microwave discharge ion thruster. WG and DC injections were evaluated. The ECR region is where the electron cyclotron frequency and microwave frequency match

In this study, two types of gas injection were used: waveguide (WG) injection and discharge chamber (DC) injection. In WG injection, the propellant is injected from the bottom of the waveguide, whereas in DC injection, the propellant is injected between the two magnet rings. The microwave power was fixed at 34 W, and the flow rate was varied from 1 to 4 standard cubic centimeters per minute (sccm). The screen and accelerator voltages were fixed at 1500 and − 350 V, respectively. The background pressure was 2.0 × 10^− 3 Pa and contained 3.0 sccm of xenon. The vacuum facilities are described in detail elsewhere [27].

Plasma mode-transition

There are two main types of magnetic field in ECR plasma sources: one mirror magnetic field (our study [28] and magnetic nozzle [36]), and multipole type in azimuthal direction [37, 38]. Plasma mode transition has been observed in both types of magnetic field [36,37,38]. The mode transition in an ECR ion thruster is shown in Fig. 2 [27]. Here, when the flow rate is optimal, the thrust (ion beam current) is maximized. At a low flow rate (below the optimal flow rate), the plasma is concentrated in the magnetic confinement region. In contrast, as the flow rate increases, plasma is also generated outside the magnetic confinement region. At the optimal flow rate, red luminescence is generated inside the boundary of the waveguide. At rates higher than the optimal flow rate, the red luminescence further increases. This transition indicates that the extraction rate distribution changes drastically as the flow rate increases. Additionally, because the ionization process strongly correlates with the excitation process, the ionization distribution should also change.

Fig. 2

Changes in the outside thruster (a) below, (b) at, and (c) above the optimal flow rate. The red dotted line indicates the boundary of the waveguide. At the optimal flow rate, the thrust (ion beam current) is maximized

Ground-state neutral density and spontaneous emission intensity measurements

Principle

Using the TALIF experimental setup, the ground-state neutral density and spontaneous emission intensity were measured at the same point, as shown in Fig. 3. Our idea is based on the fact that the wavelengths of fluorescence and spontaneous emission by electron-impact excitation are the same. For instance, in normal laser-induced fluorescence on metastable ions, the wavelength of the fluorescence is different from that of spontaneous emission [13, 14]. Thus, in TALIF measurement, the main background noise is the spontaneous emission at the same wavelength, and hence both parameters can be measured using the same setup. Generally, the spontaneous emission intensity I_j can be expressed as follows [39].

$$ {I}_j={I}_0\left({k}_{g,j}{n}_g+{\sum}_{i=1}^N{k}_{i,j}{n}_i\right){n}_e. $$

(1)

Fig. 3

Principle of measuring (left) the ground-state neutral density and (right) the spontaneous emission intensity using electron-impact excitation

Here, I₀ is a constant and n_e is the electron density. The subscripts i, j, and g are the upper, lower, and ground states, respectively. This equation states that the spontaneous emission is the sum of the direct excitation from ground state g and the stepwise excitation from lower excited state i. Because the spontaneous emission intensity is proportional to n_g, the stepwise excitation can be evaluated by the correlation between n_g and spontaneous emission I_j. Noted that Eq. (1) assumes optically thin, that is, the radiation trapping is neglected [24]. Additionally, because the ionization process is strongly related to the excitation process, the ionization process is estimated using an analysis of the excitation process.

Determination of excitation process

Generally, there are many selections of two-excitation processes as summarized in [40]. We propose that the excitation process should be selected following three perspectives.

• (1). Trade-off between fluorescence intensity and spontaneous emission intensity

• (2). Lower state is resonant state

• (3). Difference between energy of excitation and singly-charged ionization

Because the spontaneous emission at the same wavelength of fluorescence is the inevitable noise, (I) is required for measuring the ground-state neutral density with high signal to noise (S/N) ratio as pointed in C. Eichhorn [41]. (II) is required for minimizing the uncertainly of optically depth. Generally, if the optical depth is not sufficiently thin, the spontaneous emission intensity decreases because the excited neutral density at lower state reabsorbs in the spontaneous emission [24]. Hence, the lower state should be resonant state because the density is generally lower than that of metastable state. Last, (III) is required for minimizing the difference of energy between the excitation and singly-charged ionization. In this thruster,

the singly charged ions account for more than 90% of all ions [9], and hence we focus on the ionization process of singly-charged ions. In this experiment, the ionization process is investigated by using the analysis of the excitation process; thus, the discrepancy of electron population that can react excitation and ionization should be small.

Based on the above criteria, we recommend the excitation process of 1s₀ to 2p₃ as shown in Fig. 3. In this excitation process, the fluorescence at 834.68 nm emits from the transition 2p₃ to 1s₂. From the previous result of spectra [32], the wavelength of 834.68 nm is the best choice in view of point (I). Then, because lower excited state 1s₂ is resonant state, (II) is satisfied. Actually, we confirmed that the assumption of optically thin establishes within 1% of the spontaneous emission intensity under the typical electron density of 10¹⁶ − 10¹⁷ m⁻³ [23, 24, 42]. In point (III), the excitation energy is 11.1 eV and the difference in energy between excitation and singly-charged ionization is only 1 eV [43], which indicates that if the stepwise excitation is non-negligible, the stepwise ionization of singly-charged ions is also non-negligible [28].

Experimental setup

Details of the experimental setup can be found in [28, 32]. Hence, only a short explanation is provided here. A Nd:YAG laser (NL310, EKSPLA) at a wavelength of 532 nm that emitted 300 mJ was utilized as the pump laser. Then, the 532 nm laser was first converted to 672.87 nm by a dye (LDS-698) and its third harmonic (672.87/3 = 224.29 nm) was extracted using a dye laser (LiopStar-E-N-D2400, LIOP-TEC). Two quartz windows with diameters of 4 mm were opened for laser injection and detection while the thruster performance was maintained. The laser was focused on the center of the waveguide. To detect the fluorescence and spontaneous emission at 834.68 nm, a band-pass filter and a photomultiplier tube were used. The full width at half maximum of the band-pass filter (Sigma Koki Co., Ltd.) is less than 2 nm to eliminate the emissions at 828.0 nm and 840.9 nm [28]. To reduce the noise due to fluctuations in spontaneous emission, 3000 fluorescence signals were averaged. When measuring the spontaneous emission intensity, the laser was stopped and the signal was recorded. The calibration to obtain the ground-state neutral density is given in [28, 44]. The ground-state neutral density and spontaneous emission were measured at the three distances from the screen grid: − 7, − 11.5, and − 16 cm.

Numerical simulation

Metastable state particle simulation

Metastable neutral particles 1s₅ have a long natural lifetime (40 s) [29], but their lifetime can be reduced by collision and wall diffusion significantly. The de-excitation frequency ν_de is the sum of the collision with other particles ν_coll and wall diffusion ν_wall, that is, ν_de = ν_coll + ν_wall. Considering thruster length L ≈ 10 cm and thermal velocity of metastable neutral particles v_th ≈ 10² m/s, ν_wall can be estimated as ν_wall ≈ v_th/L ≈ 10³ Hz. Two types of collisions are possible: collisions with ground-state neutral particles ν_g and collisions with electrons ν_e. However, ν_g can be negligible compared to ν_e [45] thus, ν_coll ≈ ν_e. ν_e can be expressed as ν_e = σ_totalv_en_e, where σ_total is the total cross-section of de-excitation, v_e is the electron particle velocity, and n_e is the electron density. The total cross-section can be expressed as follows [39, 46].

$$ {\sigma}_{total}\approx \sum \limits_i{\sigma}_{1,i}+\sum \limits_k{\sigma}_{2,k}. $$

(2)

The sum over i occurs for the transition from the upper states (2p_i, 3d_i, 2s_i, 3p_i) to 1s₄, which is optically coupled to the ground state, and intra-transition from 1s₅ and 1s_i (i = 1, 2, 3, 4). The sum over k occurs in stepwise ionization including singly charged and multiply charged ionization. From the probe measurement and full kinetic simulation [42], the electron temperature in the discharge chamber and waveguide was varied from 1 to 20 eV. From the simulated cross-section using the relativistic distorted wave approach [20, 47], we have σ_{1, k} ≈ 10⁻²⁰ − 10⁻¹⁹ m² in the range of T_e= 1–20 eV. Then, from experimental measurements, the stepwise ionizations are σ_{2, k} ≈ 10⁻²⁰ − 10⁻¹⁹ m² in the range of T_e= 1–20 eV [48]. Additional factors must also be considered. First, there is the possibility that the de-excited particles will reproduce, but this reproduction is not dominant [46]. Other reactions such as Penning ionization and three-body collision are negligible because of the low pressure [21]. As a result, σ_total ≈ 10⁻¹⁹ − 10⁻¹⁸ m² is valid in the range of T_e= 1–20 eV, which is consistent with the results of Karabadzhak [39].

From the full-kinetic simulation [42], the electron density is n_e ≈ 10¹⁶ − 10¹⁷ m⁻³ inside the discharge chamber. Thus, considering the electron velocity v_e ≈ 10⁶ m/s, ν_e ≈ 10³ − 10⁵ Hz was used. However, inside the waveguide, the electron density can be less than that in the discharge chamber, especially near the bottom of the waveguide. If electron density n_e ≤ 10¹⁶ m⁻³, ν_e can be smaller than ν_wall, which indicates that wall diffusion is dominant. Some studies have reported that the wall diffusion model is ν_wall = (2.405²D_x/r²)⁻¹ + (ν₀/r)⁻¹ [19, 34]. However, in the thruster, it is difficult to model the diffusion between the waveguide and discharge chamber. Thus, in this study, wall diffusion was directly evaluated by particle simulation. Specifically, the de-excitation due to wall diffusion was simulated by deleting the particles when they reached the wall. The Monte Carlo approach was used to simulate the collision [49]. Simulated particles were deleted using the following probability of collision:P_c = 1 − exp(−ν_collΔt), where Δt is the simulation time step. To keep the calculations within a numerical error of 1%, ν_collΔt = 1/50 was selected [49].

Finally, the excitation rate distribution at the metastable state K_ex was modeled. According to the plasma mode transition, the excitation rate distribution in the metastable state can be expressed using driving parameters η_w and L_w. First, η_w is defined as a linear interpolation of the excitation production rate at waveguide K_{ex, wave} and magnetic confinement region K_{ex, mag} as K_{ex, total} = η_wK_{ex, wave} + (1 − η_w)K_{ex, mag}. Here, K_{ex, total} is the total excitation production rate. Second, to express the axial distribution, the generation length L_w inside the waveguide was introduced, defined as shown in Fig. 4. For simplicity, the metastable neutral particles were generated randomly in each magnetic confinement region and the waveguide. To specify the magnetic field line, the vector potential was calculated using finite element method magnetics [50].

Fig. 4

Model of metastable generation using driving parameters η_w and L_w. Parameter η_w is the ratio of the ionization rates in the waveguide and the discharge chamber, and L_w is the distance of the ionization area from the screen grid

Effect of stepwise ionization on microwave propagation

To estimate the contribution of the stepwise ionization to plasma density, the electric field of the microwaves was simulated. For simplicity, a quasi-1D simulation model was used. By specifying the electromagnetic field and current density as ${E}_y\left(z,t\right)\approx \overset{\sim }{E_y}(z){e}^{i{\omega}_mt}$ and ${J}_y\left(z,t\right)\approx \overset{\sim }{J_y}(z){e}^{i{\omega}_mt}$, the following equation can be obtained [51].

$$ \left({\nabla}_t^2+\frac{\partial^2}{\partial {z}^2}+{\varepsilon}_0{\mu}_0{\omega}_m^2\right)\overset{\sim }{E_y}(z)=i{\omega}_m{\mu}_0\overset{\sim }{J_y}(z) $$

(3)

Here, the electric field is assumed to be only an electromagnetic field, i.e., the electrostatic field is neglected: $\overrightarrow{\nabla}\cdotp \overrightarrow{E}=0$. Moreover, ω_m is the frequency of the microwaves and ${\nabla}_t^2$ is the transverse part of the Laplacian. To simulate the electric field in the central axis, we employed quasi-1D approximation [52]. The detailed derivation and the justification were written in Appendix. Using this approximation, the transverse part can be approximated as ${\nabla}_t^2\overset{\sim }{E_y}(z)\approx -{\left({\rho}_{1,1}/R\right)}^2\overset{\sim }{E_y}(z)$, where ρ_{1, 1} ≈ 1.84 is the first positive root of ∂J_B1/∂r = 0 (J_B1 is first Bessel function) and R is the radius and ρ_{1, 1} corresponds to the eigenvalue of transverse electric (TE) 11 mode [35].

Additionally, the current density was estimated by cold-electron approximation, i.e., $\overset{\sim }{J_y}(z)\approx {\sigma}_p(z)\overset{\sim }{E_y}(z)$, where σ_p(z) is the complex plasma conductivity, expressed as follows [34].

$$ {\sigma}_p=\frac{e^2{n}_p}{m\left[{\nu}_m^2+{\left({\omega}_m-{\omega}_{ce}\right)}^2\right]}\left[{\nu}_m-i\left({\omega}_m-{\omega}_{ce}\right)\right], $$

(4)

where, ν_m is the momentum frequency and ω_ce is electron cyclotron frequency. Noted that the conductivity of ions is neglected because ions plasma frequency are much larger than microwave one. Additionally, because the electron cyclotron frequency is one order smaller than the microwave frequency and the momentum collision frequency is three order small than the microwave frequency in the central axis [28], the plasma conductivity is approximated in the non-magnetized case as follows.

$$ {\sigma}_p\approx -i\frac{e^2{n}_p}{m{\omega}_m}. $$

(5)

Using these approximations, one Helmholtz equation is obtained.

$$ \left\{-{\left(\frac{\rho_{1,1}}{R}\right)}^2+\frac{\partial^2}{\partial {x}^2}+{\varepsilon}_0{\mu}_0{\omega}_m^2-{\varepsilon}_0{\mu}_0{\omega}_{pe}^2\right\}\overset{\sim }{E_y}(x)=0, $$

(6)

where, ω_pe is electron plasma frequency. Finally, to determine the plasma density, the 1D plasma diffusion equation is solved as follows [28].

$$ -\frac{\partial }{\partial x}\left({D}_p\frac{\partial {n}_p}{\partial x}\right)={\nu}_{iz,g}{n}_g+{\nu}_{iz,m}{n}_m, $$

(7)

where ν_{iz, g} and ν_{iz, m} are the collision frequencies of the ground-state and metastable neutral particles, respectively. In this study, to investigate the contribution of stepwise ionization to the plasma density, the ratio of ground-state neutral density to metastable neutral density n_m/n_g was varied.

To solve Eqs. (6) and (7) numerically, a tridiagonal matrix algorithm was utilized. To solve Eq. (6), the perfect electrical conductivity condition $\overset{\sim }{E_y}=0$ was employed as the boundary condition at the grid and the bottom of the waveguide. To solve Eq. (7), the Bohm condition −D∂n_p/∂x = Γ_b was employed at the grid. Here, Γ_b is the Bohm flux, where Γ_b = n_pv_b exp(−1/2)η_sc, v_b is the Bohm velocity, and η_sc ≈ 0.8 is the effective transparency of the screen grid. At the bottom of the waveguide, an artificial boundary condition of n_p = 0 from a certain length z < L_p inside the waveguide was employed. The boundary condition was derived from the radial transport of electrons, i.e., the loss of the side wall at the waveguide. The magnetic field is not parallel to any axial direction except for the central axis (Fig. 1). Because the plasma is strongly restricted along the magnetic field, the plasma is lost at the wall. A length L_p = 9 cm was empirically determined by agreement in the experimental results of the microwaves.

Results

Ground-state neutral density versus spontaneous emission intensity

The ion beam currents with respect to the propellant flow rate are shown in Fig. 5. The red points indicate the results at the optimal flow rate. In the WG injection, the maximum ion beam current is 138–139 mA at the optimal flow rate of 1.8 sccm. In the DC injection, the maximum ion beam current is 168–175 mA at the optimal flow rates of 2.9 sccm or 3.0 sccm. The spontaneous emission intensity with respect to the ground-state neutral density is shown in Fig. 6. In all cases, the spontaneous emission intensity rapidly increases with respect to the ground-state neutral density around the optimal flow rates (red points).

Fig. 5

Ion beam currents with respect to propellant flow rate: (left) WG injection and (right) DC injection. Red point: optimal flow rate

Fig. 6

Spontaneous emission with respect to ground-state neutral density. (Top row) WG injection and (bottom row) DC injection. Each value represents a propellant flow rate in units of sccm. Red point: optimal flow rate

From Eq. (1), the sharp increase can be caused by three components: stepwise excitation, an increase in electron density, and an increase in electron temperature. In a previous study, by eliminating the ECR layer inside the waveguide, the magnitude of the sharp increase was decreased by approximately 50%, but it was still present, and the plasma mode transition was not eliminated [28]. Therefore, the sharp increase cannot be explained only by the increase in electron temperature. Considering the rate coefficient and the density ratio of metastable neutrals in the range of T_e= 3 to 5 eV measured by Langmuir probe [53], the spontaneous emission from metastable neutral particles 1s₅ accounts for 20%–250% of the spontaneous emission from the ground state [28]. In addition, the rate coefficient suggests that stepwise ionization occurs in the same order as that of stepwise excitation, which increases electron density [28].

Comparing WG injection with DC injection, the ground-state neutral density of WG injection is higher than that of DC injection, which is consistent with the direct simulation Monte Carlo (DSMC) results [54]. In WG injection, the neutral density increased towards the bottom of the waveguide. By contrast, in DC injection, the density at 3.5–4.0 sccm decreases towards the bottom of the waveguide. Thus, the absolute value of the density distribution of the ground state differs in WG and DC injection, which leads to a difference in the optimal flow rate. Focusing on the axial dependence of the spontaneous emission intensity, the maximum value of spontaneous emission decreases by only approximately 48% from z = − 7 cm to z = − 16 cm in WG injection, whereas it decreases by approximately 25% in DC injection. In “Spatial structure of plasma mode-transition” section, we discuss how the difference in ground-state neutral density and spontaneous emission generates the difference in thrust performance between WG and DC injection based on the simulation results.

Estimation of excitation rate distribution at metastable state

As explained in “Metastable state particle simulation” section, the de-excitation frequency ν_de can be varied in the range ν_wall ≈ 10³ Hz ≤ ν_de ≤ ν_wall + ν_coll ≈ 10⁵ Hz. Thus, to evaluate the sensitivity of ν_de with respect to the metastable neutral density distribution, four cases, ν_coll = 0, 10³, 10⁴, 10⁵ Hz, were simulated. As an example, Fig. 7 shows the normalized 2D metastable neutral density distribution for two cases: (a) η_w = 0% and (b) η_w = 100%, where L_w = 10 cm. The simulation results indicate that in both cases, ν_coll is very sensitive to the density distribution when ν_coll ≥ 10⁴ Hz because ν_wall is on the order of 10³ Hz.

Fig. 7

Sensitivity of de-excitation time against metastable neutral density distribution for (from top to bottom) ν_coll = 0 Hz, ν_coll = 10³ Hz, ν_coll = 10⁴ Hz, and ν_coll = 10⁵ Hz. Results for two values of the metastable neutral particles source term are shown. (a) η_w= 0% and (b) η_w= 100% (L_w = 10 cm)

The simulated metastable neutral density was compared with the experimental one in the central axis obtained by LAS [31, 33]. In numerical simulations, it is difficult to accurately match the absolute value of excitation rate with the experiment. Thus, in this study, each absolute value of metastable neutral density is matched with the experimental result. Five cases for L_w, that is, L_w = 6, 7, 8, 9, 10 cm, and four cases for η_w, that is, η_w = 0, 0.25, 0.5, 0.75, 1.0, were simulated. Finally, to obtain the variation of de-excitation frequency due to collision, ν_coll = 0, 10³, 10⁴, 10⁵ Hz were simulated for each case. In total, 5 × 5 × 4 = 100 cases were simulated, and the simulation results that minimized the discrepancies in the experiment al results were then selected. Figure 8 shows the axial distribution of metastable state 1s₅ density. Each simulation result includes the minimum and maximum values due to the variation in ν_coll = 0 – 10⁵ Hz.

Fig. 8

Comparison of metastable neutral density distributions. Black line: LAS experimental results from [31, 33], red and blue lines: simulation results. Simulation results include the minimum and maximum values due to variations in ν_coll = 0~10⁵ Hz

The most notable result is that for at Fig. 8(a), where WG 1 sccm, and Fig. 8(d), where DC 2 sccm, the results for η_w = 0% are in the best agreement with the experimental results. This indicates that the excitation rate distribution in the metastable state is concentrated in the magnetic confinement region. In Fig. 8(b) and (c), the results for η_w = 75% (L_w = 10 cm) are in the best agreement with the experiment. This indicates that 75% of the metastable neutrals were generated in the waveguide. At z=− 10 to − 15 cm, the density gradient is shown, which indicates that the particles diffuse from the production area at z >  − 10 cm. Because the maximum value of the simulation agrees with that of the experiment, it is suspected that the electron density is very low in this region. In Fig. 8(e), η_w = 25 % (L_w = 7 cm) cm is in the best agreement, and in Fig. 8(f), η_w = 25 % (L_w = 10 cm) cm is in the best agreement. Compared to WG injection, η_w inside the waveguide was reduced. Comparing Fig. 8(e) with Fig. 8(f), L_w is larger, which suggests that the production area is greater at the bottom of the waveguide.

Using the comparison between simulation and experiment, the excitation rate distribution in the metastable state can be estimated. At a low flow rate, the excitation is concentrated in the magnetic confinement region, and then the excitation transitions to the waveguide as the flow rate increases. Additionally, the particle simulation firstly revealed that the transition of the excitation distribution corresponds to the diffusion of metastable neutral particles, which suggests that the stepwise ionization can broad by the diffusion of metastable state neutral particles. The ratio of stepwise ionization will be investigated by numerical simulation of electromagnetic field of microwaves.

Contribution of the stepwise ionization to plasma density

Here, an electron temperature of 3 eV was assumed, and hence ν_{iz, m}/ν_{iz, g} = 20 was used [23]. The ground-state neutral density was estimated as ν_{iz, g} ≈ 10³ Hz from the order estimation of n_e ≈ 10¹⁷ m⁻³, v_e ≈ 10⁶ m/s, and σ_iz ≈ 10⁻²⁰ m². Finally, the ground-state neutral density was fixed at n_g ≈ 10¹⁹ m⁻³ from the experimental results near the optimal flow rate (Fig. 6). In these situations, to investigate the contribution of stepwise ionization, n_m/n_g were varied from n_m/n_g= 1%–10%.

Fig. 9 shows the plasma density and electric field of microwaves in four cases: n_m/n_g = 0 % , 1 % , 5 % , 10%. The experimental data is the electric field of the microwave near the optimal flow rate (2 sccm) in WG injection [35]. First, we note that the simulated electric field can reproduce the experiment. Thus, because the simulation case n_m/n_g = 5% is in the best agreement with the experiment, this suggests that the experimental plasma density can be estimated to be on the order of n_e ≈ 10¹⁷ m⁻³. The microwaves cannot be transported to the discharge chamber when n_m/n_g > 5%. Of course, even if stepwise ionization is not considered, the electric field can be reproduced when there is a large ground-state neutral density. In this case, if ${n}_g^{\ast }={n}_g+{\nu}_{iz,m}/{\nu}_{iz,g}\times {n}_m=2\times {10}^{19}\ {\mathrm{m}}^{-3}$ is used, the source term (right-hand side of Eq. (7)) has the same value; thus, the same electric field can be obtained. However, we emphasize that near the optimal flow rate, such a case did not occur in the experiment. As shown in Figs. 6 and 8, the metastable neutral density increases with respect to the flow rate. By contrast, in WG injection, the ground-state neutral density decreases near the optimal flow rate (WG: 1.5–1.8 sccm), and in the DC injection, the density is approximately constant around n_g ≈ 1 × 10¹⁹ m⁻³ (DC: 2.7–3.1 sccm). Thus, the change in the metastable and ground state neutral density indicates that n_m/n_g increases near the optimal flow rate. Under such conditions, the increase in plasma density near the optimal flow rate cannot be explained without the stepwise ionization.

Fig. 9

Results for (a) plasma density and (b) electric field of microwaves for various ratios n_m/n_g. The experimental electric field was measured by EO probe near the optimal flow rate (2 sccm) in WG injection [55]

Spatial structure of plasma mode-transition

Based on the simulation and experimental results, we propose the spatial structure for the plasma mode transition shown in Fig. 10. Specifically, in this spatial structure, plasma mode transition consists of the following three steps:

• (i). At low flow rates (WG: 1sccm, DC: 2 sccm), direct ionization and excitation occur in the magnetic confinement region. Then, the generated metastable neutral particles are transported to the inside of the waveguide and its exit as shown in Fig. 7.

• (ii). Stepwise ionization and excitation occur in and at the exit of the waveguide by the transportation of the metastable neutral particles to the waveguide, and more particles are transported into the waveguide.

• (iii). When the plasma density inside the waveguide becomes high enough to prevent the propagation of microwaves, plasma mode transition occurs.

Fig. 10

Spatial structure of plasma mode transition. (i) At below optimal flow rates, direct ionization and excitation occurs in the magnetic confinement region. (ii) The generated metastable neutral particles are transported to the inside and exit of the waveguide. From the transported metastable neutral particles, stepwise excitation and ionization occurs in and at the exit of the waveguide, and more particles are transported inside the waveguide. (iii) When the plasma density inside the waveguide is high enough to prevent the propagation of microwaves, the plasma mode transition occurs

Step (i) is justified by the good agreement in the LAS measurement results at low flow rates (Fig. 8 (a) and 9(d)). Step (ii) is also consistent with the comparison between simulation and experiment (Fig. 8 (b) and 9(e)), and can be explained by the sharp increase in the spontaneous emission intensity around the optimal flow rate (Fig. 6). Step (iii) is consistent with the comparison between simulation and experiment (Fig. 8 (c) and 9(f)), and is consistent with the electric field of the microwaves measured by the EO probe [35].

Additionally, using steps (i)–(iii), we discuss the difference between WG and DC injection. WG injection enhances steps (ii) and (iii) because of the larger ground-state neutral density (Fig. 6), which causes plasma mode transition at lower flow rates. Then, focusing on the axial dependence of the spontaneous emission intensity, we find that the ground-state neutral density distribution in WG injection induces a large ratio in excitation rate η_w as shown in Fig. 8. Because of the higher metastable neutral density, the maximum value of spontaneous emission decreases by only approximately 48% from z = − 7 cm to − 16 cm in the WG injection, whereas it decreases by approximately 25% in the DC injection. Therefore, steps (i)–(iii) can explain the difference between WG and DC injection in terms of ground state neutral density, spontaneous emission intensity, and metastable neutral density.

Finally, we discuss whether the proposed mechanism of plasma mode transition can be applied to different types of magnetic field or other applications. Because the diffusion of metastable state neutral particles from magnetic confinement region occurs in any magnetic field, it is considered that proposed mechanism of plasma mode-transition can be applied to other different types of magnetic field, such as multipole ECR plasma sources [37, 38]. Another thing that some ECR plasma sources are used to generate multiply charged ions [55, 56]. Then, some paper reported that stepwise (step by step) ionization from lower charged ions is important to generate multiply charged ions [55, 56]. In this situation, it is considered that the diffusion of lower charged ions, i.e., ion leakage, is more important than that of metastable state neutral particles. In summary, our proposed mechanism should be effective under any magnetic field in ECR plasma sources where single-charged ions are dominant.

Conclusion

In this study, to investigate the stepwise ionization distribution from the metastable state that can induce plasma mode transition, one experimental and two numerical approaches were performed. First, to investigate the ionization process in two types of injection, the ground-state neutral density and spontaneous emission intensity were measured. Then, a metastable neutral particle simulation including de-excitation was used to estimate the excitation rate distribution in the metastable state. From this estimation, it was found that at a low flow rate, the excitation is concentrated in the magnetic confinement region, and the excitation then transits to the waveguide as the flow rate increases. Second, the electric field of the microwaves was simulated to estimate the contribution of stepwise ionization to plasma density. It was revealed that the simulation results can reproduce the measured data. The experimental results for n_m/n_g demonstrate that the increase in plasma density near the optimal flow rate cannot be explained without stepwise ionization.

These experimental and simulation results indicate that the location of the stepwise ionization transits to outside of the magnetic confinement region because of the diffusion of metastable particles. Additionally, this transition enhances the plasma density inside the waveguide, resulting in plasma mode transition. This finding can explain the difference between the two types of gas injection from the ground-state neutral density distribution and the axial dependence of spontaneous emission intensity. We believe that this finding will lead to the development of future performance improvements, and our proposed mechanism should be effective under any magnetic field in ECR plasma sources where single-charged ions are dominant.

Appendix

Derivation and validity of quasi-1D approximation of electric field

In this study, the electromagnetic field is expressed as follows.

$$ \overset{\sim }{E_y}(z)=\overset{\sim }{E_{y0}}(z){e}^{-i{k}_zz} $$

(8)

Eq. (8) means the electromagnetic field is expressed as the wavenumber k_z and the amplitude $\overset{\sim }{E_{y0}}(z)$. Of course, this expression is also imposed in other direction of electromagnetic field, e.g., $\overset{\sim }{H_z}(z)$. Then, we impose following two approximations.

$$ \left|\frac{\partial \overset{\sim }{E_{\mathrm{y}0}}(z)}{\partial z}\right|\ll \left|{k}_z\overset{\sim }{E_{y0}}(z)\right|,\left|\frac{\partial^2\overset{\sim }{E_{y0}}(z)}{\partial {z}^2}\right|\ll \left|{k}_z\frac{\partial \overset{\sim }{E_{y0}}(z)}{\partial z}\right|. $$

(9)

The approximation means that the change of amplitude $\overset{\sim }{E_{y0}}(z)$ is very smaller than that of k_z. In such a condition, the second derivative in z direction ${\partial}^2\overset{\sim }{E_y}(z)/\partial {z}^2$ can be approximated as follows.

$$ \frac{\partial^2}{\partial {z}^2}\overset{\sim }{E_y}(z)=\left(-{k}_z^2\overset{\sim }{E_{y0}}(z)-2i{k}_z\frac{\partial \overset{\sim }{E_{y0}}(z)}{\partial z}+\frac{\partial^2\overset{\sim }{E_{y0}}(z)}{\partial {z}^2}\right){e}^{-i{k}_zz}\approx -{k}_z^2\overset{\sim }{E_y}(z). $$

(10)

The TE mode is determined by the boundary condition of $\overset{\sim }{H_z}(z)$. Considering the approximation (10), the Helmholtz equation on $\overset{\sim }{H_z}(z)$ under cylindrical coordinate.

$$ \left(\frac{\partial^2}{\partial {r}^2}+\frac{1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial {\theta}^2}+{k}_c^2\right)\overset{\sim }{H_{z0}}(z)=0 $$

(11)

Here, ${k}_c^2={k}_z^2-{k}_m^2$, where ${k}_m^2={\mu}_0{\varepsilon}_0{\omega}_m^2$ and then boundary condition; $\partial \overset{\sim }{H_{z0}}(z)/\partial r=0$ must be satisfied at r = R. The analytical solution of Eq. (11) is Bessel function and then it is satisfied with following condition.

$$ {k}_c^2={\left(\frac{\rho_{m,n}}{R}\right)}^2 $$

(12)

where ρ_{m, n} is the n-th positive root of ∂B_{s, m}(ρ_n)/∂r = 0, where B_{s, m} is m-th the Bessel function. By using this approximation, the transverse part of Laplacian can be expressed as follows.

$$ \left(\frac{\partial^2}{\partial {r}^2}+\frac{1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial {\theta}^2}\right)={\nabla}_t^2=-{\left(\frac{\rho_{1,1}}{R}\right)}^2 $$

(13)

Eventually, by substituting Eq. (13) into Eq. (3), following equation is established.

$$ \left\{-{\left(\frac{\rho_{1,1}}{R}\right)}^2+\frac{\partial^2}{\partial {z}^2}+{\varepsilon}_0{\mu}_0{\omega}_m^2\right\}\overset{\sim }{E_y}(z)=i{\omega}_m{\mu}_0\overset{\sim }{J_y}(z) $$

(14)

Thus, proposed approximation means that the transverse part of Laplacian is estimated under approximation of Eq. (9) and then $\overset{\sim }{E_y}(z)$ is calculated by the Eq. (14). Thus, if the change of amplitude on the electric field is large, the approximation generates discrepancy. In the absence of plasma ($\overset{\sim }{J_y}=0\Big)$, the electromagnetic field of microwaves in central axis is shown in Fig. 11. The electric field of the microwaves in the central axis without plasma ($\overset{\sim }{J_y}(z)=0$) is shown in Fig. 11. The simulation results indicate that the quasi-1D approximation is in good agreement with the 3D FDTD simulation and EO probe measurement inside the waveguide [35]. In contrast, a 10% discrepancy exists inside the discharge chamber, which is caused by the difference in the approximation (Eq. (9)). However, this approximation is very effective because simulation cost is less than 1 s and simulation code is simple.

Fig. 11

Parallel component E_y of the electric field of the microwaves in the central axis in the absence of plasma. The quasi-1D approximation are compared with the results of the 3D FDTD simulation and EO probe from [35]