Analysis of the chemical network in a volume-production high-current negative hydrogen ion source

The relative contributions of the reactions to the creation and loss of particle species (i.e. H⁻, electrons, ${\mathrm{H}}_{3}^{+}$, ${\mathrm{H}}_{2}^{+}$, H⁺, and H(n = 1–3)) will be studied. For certain NHIS, special attention is usually paid to the production of H⁻. Therefore, only the relative contributions of different reactions to the creation and loss of H⁻ as well as electrons are shown in the main text, while the detailed discussions on the evaluation of physicochemical processes of ${\mathrm{H}}_{3}^{+}$, ${\mathrm{H}}_{2}^{+}$, H⁺, and H(n = 1–3) can be found in the appendix. The reactions with very small contributions in the investigated pressure range are not shown in the following figures.

The relative contributions of the reactions to the creation and loss of H⁻ as well as electrons at different pressures are shown in figures 2(a) and (b). Figure 2(a) shows that the dissociative attachments of the ground state and low vibrational levels dominate the creation of H⁻ at relatively high pressures, for example, dissociative attachments of H₂(υ = 0) and H₂(υ = 1) respectively contribute 58.4% and 26.5% (84.9% in total) at 100 Torr, while the high vibrational levels contribute more at relatively low pressures, for example, dissociative attachments of H₂(υ = 6–9) individually contribute 13.4%, 24.0%, 29.9%, and 17.4% (84.7% in total) at 1 Torr. The dissociative attachments of H₂(υ = 11–14) contribute very little in the investigated pressure range due to very low densities of H₂(υ = 11–14) (see figure 3), and they can therefore be neglected.

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The associative detachment by hydrogen atoms, i.e. H⁻ + H → e + H₂ (no. 32 in table 1), dominates the loss of H⁻ with a contribution of 50.5%–87.8% in the investigated pressure range. It is very different from the case in the usual NHIS operated at low pressures [18], where the mutual neutralization of ${\mathrm{H}}_{3}^{+}$ with H⁻ is the most important channel. This is because there is around a one order of magnitude higher density of hydrogen atoms in the HCNHIS than in the usual NHIS [18]. The contributions of mutual neutralizations of H⁻ with ${\mathrm{H}}_{3}^{+}$, i.e. ${\mathrm{H}}_{3}^{+}+{\mathrm{H}}^{-}\to \mathrm{4}\mathrm{H}$, ${\mathrm{H}}_{3}^{+}+{\mathrm{H}}^{-}\to \mathrm{2}{\mathrm{H}}_{\mathrm{2}}$ and ${\mathrm{H}}^{-}+{\mathrm{H}}_{3}^{+}\to \mathrm{2}\mathrm{H}+{\mathrm{H}}_{2}$ (nos. 30, 31 and 38 in table 1), show a similar trend with increasing pressure, and the maximum contribution (${\mathrm{H}}_{3}^{+}+{\mathrm{H}}^{-}\to \mathrm{4}\mathrm{H}$) is 22.7% at 1 Torr. Electron detachments, i.e. e + H⁻ → H + 2e and ${\mathrm{H}}^{-}+{\mathrm{H}}_{\mathrm{2}}\left(\upsilon =2\right)\to {\mathrm{H}}_{\mathrm{2}}\left({\upsilon }^{\prime }=0\right)+\mathrm{H}+\mathrm{e}$ (no. 22 in table 1 and no. 1094 in table 2), are respectively responsible for the loss of H⁻ at relatively low and high pressures, for example, reaction e + H⁻ → H + 2e contributes 10.4% at 1 Torr. Electron detachments of H₂(υ > 2) (nos. 1095–1098 in table 2) can be neglected due to their small contributions in the investigated pressure range.

Figure 2(b) shows that in addition to the ionization of H₂, i.e. $\mathrm{e}+{\mathrm{H}}_{\mathrm{2}}\to {\mathrm{2}\mathrm{e}+\mathrm{H}}_{2}^{+}$ (no. 4 in table 1), with the minimum contribution of 26.2% at 100 Torr and the maximum contribution of 42.8% at 1 Torr, the collisions between neutral species H(n = 2) and H₂, i.e. $\mathrm{H}\left(n=2\right)+{\mathrm{H}}_{2}\to {\mathrm{H}}_{3}^{+}+\mathrm{e}$ (no. 34 in table 1), significantly affect the creation of electrons. $\mathrm{H}\left(n=2\right)+{\mathrm{H}}_{2}\to {\mathrm{H}}_{3}^{+}+\mathrm{e}$ plays a decreasingly important role as pressure increases, with the contribution being from 42.3% at 1 Torr to 22.0% at 100 Torr, due to the very rapidly decreasing density of H(n = 2) (see figure 8). The associative detachment by hydrogen atoms, i.e. H⁻ + H → e + H₂ (no. 32 in table 1), plays a more important role in the creation of electrons at higher pressure due to the increasing density of hydrogen atoms (see figure 8), and becomes the most important source of electrons with a contribution of 47.6% at 100 Torr. The ionization of hydrogen atoms, i.e. e + H → 2e + H⁺ (no. 9 in table 1), only applies at relatively low pressures with a contribution of 8.1% at 1 Torr. The electron detachment, i.e. ${\mathrm{H}}^{-}+{\mathrm{H}}_{\mathrm{2}}\left(\upsilon =2\right)\to {\mathrm{H}}_{\mathrm{2}}\left({\upsilon }^{\prime }=0\right)+\mathrm{H}+\mathrm{e}$ (no. 1094 in table 2), contributes very little to the creation of electrons.

The electrons escaping the discharge through the walls are the most important loss channel, and the contribution increases with decreasing pressure from 38.6% at 100 Torr to 92.4% at 1 Torr. As mentioned in figure 2(a), the dissociative attachments of the vibrational levels mainly contribute to the creation of H⁻. Even if they contribute less to the loss of electrons, the dissociative attachments of H₂(υ = 0) and H₂(υ = 1) still play important roles at relatively high pressures, with respective maximum contributions of 31.7% and 14.4% at 100 Torr. The recombination of ${\mathrm{H}}_{3}^{+}$ and electrons, i.e. ${\mathrm{e}+\mathrm{H}}_{3}^{+}\to \mathrm{3}\mathrm{H}$ (no. 20 in table 1), contributes very little to the loss of electrons.

Figure 3 shows the VDF at different pressures. For the lowest pressure of 1 Torr, the VDF is relatively well represented by a non-Boltzmann distribution with a plateau for υ = 3–9. As the pressure increases, the densities of higher vibrational levels decrease very rapidly and a single temperature Boltzmann distribution is observed. The density of H₂ in the ground state, i.e. H₂(υ = 0), surpasses by six orders of magnitude the density of H₂(υ = 10) even at relatively low pressures, such as 1 Torr. That is the reason why higher vibrational levels (υ > 10) hardly contribute to the creation of H⁻ ions and the lower vibrational levels contribute more at higher pressures, as shown in figure 2(a). Unlike the usual NHIS operated at low pressures [20], the density difference between lower and higher levels is larger in the HCNHIS due to more frequent collisions between H₂(υ) molecules in the higher pressure range.

Even though the relative contributions of the reactions to the creation and loss of all vibrational levels are evaluated, only the cases of H₂(υ = 1), H₂(υ = 4) and H₂(υ = 7) marked with green circles in figure 3 are shown in figures 4(a)–(c). The electron excitations of H₂(υ = 0–14) molecules mainly include two kinds of process: one is through a resonant mechanism (called the eV process, e + H₂(υ) → e + H₂(υ'), nos. 1–210 in table 2) and the other is through an indirect mechanism followed by radiative decay (called the EV process, e + H₂(υ) → e + H₂(υ') + hν, nos. 211–435 in table 2). The eV process usually only contributes to the creation and loss of relatively low vibrational levels, such as H₂(υ < 4).

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For the low pressures in the usual NHIS, only the electron excitations of ground-state molecules H₂(υ = 0) to levels H₂(υ = 1–14) apply, and other electron excitations between different vibrational levels H₂(υ = 1–14) can be neglected, due to the density of H₂(υ = 0) being far higher than that of H₂(υ = 1–14) [20]. For the high pressures in the HCNHIS, it is still available for electron excitations through the indirect mechanism (EV process), but not for those through the resonant mechanism (eV process). This is because the low-energy electrons (<4 eV) facilitate the eV process, while the high-energy electrons (>12 eV) facilitate the EV process. The electron temperatures are lower than 3 eV (see figure 6) in the HCNHIS. Moreover, the densities of low vibrational levels are higher and are closer to the density of H₂(υ = 0) (see figure 3) than that in the usual NHIS operated at low pressures.

As shown in figure 4(a), H₂(υ = 0) is the ground state and the neighboring level of H₂(υ = 1) for the eV process, e + H₂(υ = 0) → e + H₂(υ' = 1). This reaction contributes to the creation of H₂(υ = 1) increasing from 32.4% to 82.5% as pressure decreases from 100 Torr to 1 Torr. H₂(υ = 2) is also the neighboring level of H₂(υ = 1) for the eV process, e + H₂(υ = 2) → e + H₂(υ' = 1), but this reaction contributes to the creation of H₂(υ = 1) with the maximum value of 3.3% at 7.75 Torr, due to the density of H₂(υ = 2) being lower than that of H₂(υ = 0), as shown in figure 3. The transitions through eV processes between neighboring levels have larger reaction cross sections, and therefore the contributions of other levels to the creation of H₂(υ = 1) through the eV processes can be neglected in the investigated pressure range. Figure 4(b) shows that the eV process, e + H₂(υ = 3) → e + H₂(υ' = 4), contributes most to the creation of H₂(υ = 4) in all eV processes with a maximum of 23.7% at 7.75 Torr. Even if H₂(υ = 5) is also the neighboring level of H₂(υ = 4), the density of H₂(υ = 3) is higher than that of H₂(υ = 5). Therefore, the eV process, e + H₂(υ = 5) → e + H₂(υ' = 4), contributes less to the creation of H₂(υ = 4) than e + H₂(υ = 3) → e + H₂(υ' = 4). Note that the contribution of the eV process, i.e. e + H₂(υ = 2) → e + H₂(υ' = 4), is almost larger than the eV process of the neighboring level H₂(υ = 5) in the investigated pressure range, due to the density of H₂(υ = 2) being higher than that of H₂(υ = 5) (see figure 3). Similarly, the contributions of the eV processes of H₂(υ = 0) and H₂(υ = 1) to the creation of H₂(υ = 4) are non-negligible, especially at higher pressures. Figure 4(c) shows that the eV processes are almost not important any more to the creation of H₂(υ = 7) except for the process, e + H₂(υ = 6) → e + H₂(υ' = 7), with the maximum contribution of 8.1% at 1 Torr. Therefore, the eV transitions between levels H₂(υ = 1–7) closer to each other apply in this work, which has been noted in table 2.

The vibrational–translational relaxation of H₂(υ = 0–14) in collisions with hydrogen atoms (mono-quantum and multi-quantum Vt processes, $\mathrm{H}+{\mathrm{H}}_{\mathrm{2}}\left(\upsilon \right)\to \mathrm{H}+{\mathrm{H}}_{\mathrm{2}}\left({\upsilon }^{\prime }\right)$, nos. 464–673 in table 2) and with hydrogen molecules H₂(υ = 0–14) (mono-quantum VT process, ${\mathrm{H}}_{\mathrm{2}}\left(\omega \right)+{\mathrm{H}}_{\mathrm{2}}\left(\upsilon \right)\to {\mathrm{H}}_{\mathrm{2}}\left(\omega \right)+{\mathrm{H}}_{\mathrm{2}}\left({\upsilon }^{\prime }=\upsilon {\pm}1\right)$, nos. 674–1093 in table 2) can be neglected in the usual NHIS operated at low pressures [20]. For the high-pressure range of 1–100 Torr here, it is still available for the multi-quantum Vt process, but not for mono-quantum Vt and VT processes. As shown in figures 4(a)–(c), the mono-quantum Vt processes, $\mathrm{H}+{\mathrm{H}}_{\mathrm{2}}\left(\upsilon =1\right)\to \mathrm{H}+{\mathrm{H}}_{\mathrm{2}}\left({\upsilon }^{\prime }=0\right)$, $\mathrm{H}+{\mathrm{H}}_{\mathrm{2}}\left(\upsilon =4\right)\to \mathrm{H}+{\mathrm{H}}_{\mathrm{2}}\left({\upsilon }^{\prime }=3\right)$, and $\mathrm{H}+{\mathrm{H}}_{\mathrm{2}}\left(\upsilon =7\right)\to \mathrm{H}+{\mathrm{H}}_{\mathrm{2}}\left({\upsilon }^{\prime }=6\right)$ respectively contribute to the loss of H₂(υ = 1), H₂(υ = 4), and H₂(υ = 7) with maximums of 21.6% at 7.75 Torr, 14.3% at 7.75 Torr, and 6.4% at 1 Torr. They play a decreasingly important role in the loss of vibrational levels with increasing level. Similar to the eV transitions, the mono-quantum Vt transitions between neighboring levels of H₂(υ = 1–7) apply in this work, which has been noted in table 2.

Compared with mono-quantum Vt processes, the mono-quantum VT processes play a more important role in the creation and loss of different vibrational levels. Figures 4(a)–(c) show that the mono-quantum VT process, ${\mathrm{H}}_{\mathrm{2}}\left(\omega =0\right)+{\mathrm{H}}_{\mathrm{2}}\left(\upsilon =1\right)\to {\mathrm{H}}_{\mathrm{2}}\left(\omega =0\right)+{\mathrm{H}}_{\mathrm{2}}\left({\upsilon }^{\prime }=0\right)$, ${\mathrm{H}}_{\mathrm{2}}\left(\omega =0\right)+{\mathrm{H}}_{\mathrm{2}}\left(\upsilon =4\right)\to {\mathrm{H}}_{\mathrm{2}}\left(\omega =0\right)+{\mathrm{H}}_{\mathrm{2}}\left({\upsilon }^{\prime }=3\right)$, and ${\mathrm{H}}_{\mathrm{2}}\left(\omega =0\right)+{\mathrm{H}}_{\mathrm{2}}\left(\upsilon =7\right)\to {\mathrm{H}}_{\mathrm{2}}\left(\omega =0\right)+{\mathrm{H}}_{\mathrm{2}}\left({\upsilon }^{\prime }=6\right)$ almost dominate the loss of H₂(υ = 1), H₂(υ = 4), and H₂(υ = 7) at higher pressures and play an increasingly important role with increasing level, with maximum contributions of 74.3%, 83.2%, and 84.1% at 100 Torr. As the vibrational level increases to higher ones, such as H₂(υ = 13) (not shown here), the effect of the mono-quantum VT process on the loss of vibrational level is weakened, due to the lower density of higher vibrational levels (see figure 3). The reversed processes, ${\mathrm{H}}_{\mathrm{2}}\left(\omega =0\right)+{\mathrm{H}}_{\mathrm{2}}\left(\upsilon =0\right)\to {\mathrm{H}}_{\mathrm{2}}\left(\omega =0\right)+{\mathrm{H}}_{\mathrm{2}}\left({\upsilon }^{\prime }=1\right)$, ${\mathrm{H}}_{\mathrm{2}}\left(\omega =0\right)+{\mathrm{H}}_{\mathrm{2}}\left(\upsilon =3\right)\enspace \to {\enspace \mathrm{H}}_{\mathrm{2}}\left(\omega =0\right)+{\mathrm{H}}_{\mathrm{2}}\left({\upsilon }^{\prime }=4\right)$, and ${\mathrm{H}}_{\mathrm{2}}\left(\omega =0\right)+{\mathrm{H}}_{\mathrm{2}}\left(\upsilon =6\right)\enspace \to \enspace {\mathrm{H}}_{\mathrm{2}}\left(\omega =0\right)\enspace +\enspace {\mathrm{H}}_{\mathrm{2}}\left({\upsilon }^{\prime }=7\right)$, play a decreasingly important role in the creation of H₂(υ = 1), H₂(υ = 4), and H₂(υ = 7), with maximums of 48.8%, 30.8%, and 16.8% at 100 Torr as level increases, especially at high pressures. However, the contribution of transitions from higher levels, ${\mathrm{H}}_{\mathrm{2}}\left(\omega =0\right)+{\mathrm{H}}_{\mathrm{2}}\left(\upsilon =2\right)\to {\mathrm{H}}_{\mathrm{2}}\left(\omega =0\right)+{\mathrm{H}}_{\mathrm{2}}\left({\upsilon }^{\prime }=1\right)$, ${\mathrm{H}}_{\mathrm{2}}\left(\omega =0\right)+{\mathrm{H}}_{\mathrm{2}}\left(\upsilon =5\right)\enspace \to \enspace {\mathrm{H}}_{\mathrm{2}}\left(\omega =0\right)\enspace +{\enspace \mathrm{H}}_{\mathrm{2}}\left({\upsilon }^{\prime }=4\right)$, and ${\mathrm{H}}_{\mathrm{2}}\left(\omega =0\right)+{\mathrm{H}}_{\mathrm{2}}\left(\upsilon =8\right)\to {\mathrm{H}}_{\mathrm{2}}\left(\omega =0\right)+{\mathrm{H}}_{\mathrm{2}}\left({\upsilon }^{\prime }=7\right)$, to the creation of H₂(υ = 1), H₂(υ = 4), and H₂(υ = 7), increases, with maximums of 10.2% at 40.6 Torr, 28.5% at 40.6 Torr, and 61.1% at 100 Torr as the level increases. In total, the mono-quantum VT transitions through H₂(υ = 0–14) in collisions with ground-state molecules H₂(υ = 0) and H₂(υ = 5–10) in collisions with H₂(υ = 1) are required to be taken into account in this work, which has also been noted in table 2.

The combined effects of different kinetic processes responsible for forming the VDF were decoupled in the usual NHIS operated at low pressures, where an analytical solution of VDF at low pressures was obtained [20]. Even if the analytical solution of VDF may not be applicable to the high-pressure cases in the HCNHIS, the kinetic processes are still significantly simplified. The reactions with blue rectangles in figure 4 highlight the most important reactions. The sum effect of some reactions mentioned above still applies and cannot be ignored. It is found that with increasing vibrational level, the VT transitions mentioned above dominate over the eV and EV processes and become the most important sources of the higher vibrational levels. This is very different from the case in the usual NHIS operated at low pressures [20], where the EV process is the most important channel for the creation of higher levels. With increasing pressure and level, the vibrational levels lost on the wall play a decreasingly important role, and the VT transitions through collisions with ground-state molecules H₂(υ = 0) become the most important loss channel.

The reactions with very small contributions to the creation and loss of particle species and to electron energy loss will be removed in the simplified model. Therefore, except for the evaluation of the contributions of reactions to the creation and loss of particle species, the energy losses caused by the collisions between electrons and neutral species also need to be evaluated.

The normalized power dissipation is shown in figure 5. Both the sum of electronic excitations of H₂(υ = 0), i.e. $\mathrm{e}+{\mathrm{H}}_{\mathrm{2}}\to \mathrm{e}+{\mathrm{H}}_{\mathrm{2}}\left({X}^{\mathrm{b}}\right)\left. \right)$ (nos. 1099–1105 in table 2), and the sum of EV processes of H₂(υ = 1–14), i.e. e + H₂(υ = 1–14) → e + H₂(υ' = 0–14) + hν, (nos. 226–435 in table 2), contribute less than 4% to the energy loss in the investigated pressure range, and therefore they can be ignored in the simplified model. The dissociation of H₂(υ = 0), e + H₂ → e + 2H (no. 5 in table 1) and EV processes of H₂(υ = 0), i.e. e + H₂(υ = 0) → e + H₂(υ' = 0–14) + hν (nos. 211–225 in table 2), contribute significantly to the energy loss at relatively low pressures where the higher electron temperature facilitates these processes. Their maximums are respectively 40.4% and 23.5% at 1 Torr. However, the eV processes of H₂(υ = 0) to H₂(υ = 1–3), i.e. e + H₂(υ = 0) → e + H₂(υ' = 1–3) (nos. 1–3 in table 2), contribute significantly to the energy loss at relatively high pressures, with a maximum of up to 77.6% at 100 Torr. The elastic collision of H₂(υ = 0) also contributes more at relatively high pressures. The energy loss caused by the surface reactions only applies at relatively low pressures with a maximum of 8.5% at 1 Torr, and the eV processes of H₂(υ = 1–7) to H₂(υ') (υ' refers to table 2) contribute to the energy loss with a maximum of 4.8% at 7.75 Torr. Therefore, the reactions contributing very little to the creation and loss of particle species also play a small role in the electron energy dissipation.

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The chemical reactions are significantly simplified through evaluating their contributions to the creation and loss of different species. In order to obtain the generalized simplification, if only the reactions that apply for certain pressures in the investigated pressure rangeare included in the simplified model noted in tables 1 and 2, the number of chemical reactions is reduced from 1149 to 128. The chemical complexity of negative hydrogen plasmas is therefore reduced by a factor of nine using the simplified model. The accuracy of the simplified model needs to be validated by the full model. The electron temperature as a function of pressure obtained using the full model and the simplified model is shown in figure 6. Excellent agreement in electron temperature is achieved between the models in the investigated pressure range. The electron temperature decreases from 2.75 eV to 1.12 eV as the pressure increases from 1 Torr to 100 Torr.

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The densities of charged particle species and H(n = 1–3) atoms obtained in the simulations with the two models as a function of pressure are respectively shown in figures 7 and 8. The two models are in good qualitative and quantitative agreement for the densities of these particle species in the investigated pressure range, except for the slight difference at relatively low pressures (less than around 8 Torr) where the simplified model predicts a slightly higher value. The small difference in the electron density at relatively low pressures is caused by the collecting effects of ignoring the energy loss of electronic excitation of H₂(υ = 0) (nos. 1099–105 in table 2) and the EV process of H₂(υ = 1–14) (nos. 226–435 in table 2) in the simplified model, which might result in the differences in the other particle densities.

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Figure 9 shows the densities of H₂(υ = 0–14) as a function of pressure obtained with the two models. The two models show good agreement for levels H₂(υ < 3) and H₂(υ > 10) in the investigated pressure range, and slight differences for H₂(3 ⩽ υ ⩽ 10) at different pressures. The small differences for H₂(3 ⩽ υ ⩽ 10) are caused by the collecting effects of ignoring parts of the eV processes and Vt processes. In general, figures 6–9 validate the accuracy of the simplified model. To further determine the scope of application of the simplified model, the robustness of the simplified model is studied by changing the discharge conditions in the following.

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The robustness of the proposed simplified model is evaluated by comparing the simulation results of the simplified model against those of the full model for different diameters and lengths of the driver chamber in the HCNHIS. The error caused by the use of the simplified model is quantified using the root mean squared error (RMSE). The RMSE is expressed as

$$\begin{equation}\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{\frac{1}{N}{\sum }_{i=1}^{N}{\left(\frac{{n}_{\mathrm{s},i}-{n}_{\mathrm{f},i}}{{n}_{\mathrm{f},i}}\right)}^{2}},\end{equation} \tag{ 1 }$$

where n_f,i and n_s,i are the densities of the ith species obtained with the full model and the simplified model, respectively. N is the total number of particles.

Figures 10(a) and (b) show the RMSE under different diameters and lengths of the driver chamber, respectively. The diameters are 0.8725, 1.745, and 3.49 cm for constant length L = 7.39 cm, and the lengths are 5, 7.39, and 10 cm for constant diameter D = 1.745 cm. The RMSE generally decreases with increasing diameter and length. The RMSE is roughly less than 0.2 (20%), which indicates that the simplified model is capable of capturing the main chemical processes even if the chamber size is varied.

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The absorption power affects the electron density and thus the densities of other particle species. Figure 10(c) shows the effect of absorption power on the RMSE as a function of pressure. The absorption powers are 390, 490, 590, 690, and 790 W. The results indicate that lower power decreases the RMSE, and the maximum of RMSE is less than 0.2. For the investigated pressure range, the neutral gas temperature is not kept at room temperature, which has been obtained by the simulation and experiment [16, 17, 44]. The effect of neutral gas temperature on the RMSE is therefore evaluated as a function of pressure as shown in figure 10(d), due to its effect on densities of particle species. The neutral gas temperatures are 1200, 1300, 1400, 1500, and 1600 K. The RMSE increases with decreasing gas temperature and the maximum RMSE is also less than 0.2. For all cases, the RMSE caused by using the simplified model is roughly kept lower than 0.2.

In this section, the electron density, electron temperature, and hydrogen atom density predicted by the model are compared with prior experiments [45, 46] to validate our simulations using the GMNHIS and the established H₂ reaction set. The experimental data are obtained in RF inductively coupled H₂ plasmas [45, 46]. The inductively coupled discharge is ignited in a cylindrical stainless-steel vacuum chamber with four ports used for optical emission spectroscopy (OES) and other diagnostics. The top port contains a dielectric quartz window separating the plasma and a five-turn antenna coil for inductive coupling. An RF power with a frequency of 13.56 MHz is applied and connected to the coil via a P-type inductive matching network. The plasma is confined below the quartz window by a grounded electrode made of stainless steel. The electrode with a diameter of 10 cm is cooled by water. The axial distance between the electrode and the dielectric quartz window is 3 cm. In the model, the ion and neutral temperatures are assumed to be 300 K. The power transfer efficiency, which is defined as the ratio of power absorbed by the plasma over the input power, is assumed to be 70%. The recombination coefficient of hydrogen atoms on the wall is assumed to be 0.05 for the mix of quartz and metal walls [47].

Figures 11(a) and (b) show the electron temperature and electron density as a function of input power. The gas pressure is 15 Pa. The simulations are in very reasonable agreement with Langmuir probe experiments [45]. As shown in figure 11(a), the electron temperature predicted by the model is almost independent of input power, while the experimental data slightly increase with increasing input power. This could be attributed to the fixed neutral temperature assumed in the model. The neutral temperature increases with increasing power in H₂ discharges revealed by previous simulations [48] and experiments [44]. The increased neutral temperature with power can lead to the increase in the electron temperature in the experiments. The model-predicted electron density follows its dependence on the power in the experiments [45], as shown in figure 11(b). The simulations and experiments achieve very good qualitative agreement, and they show that the electron density increases with increasing input power. The simulation results are generally larger than the experimental results. This could be due to the power transfer efficiency assumed in the simulations being higher than that in the experiments [49], which leads to overestimation of the electron density.

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Figures 12(a) and (b) show the hydrogen atom densities as functions of input power and pressure, respectively. The hydrogen atom densities predicted by our model are in qualitative agreement with the OES experiments [46]. Both simulations and experiments show that hydrogen atom densities increase with increasing input power at 10 Pa, and with increasing pressure at 300 W. The increasing rates tend to be moderated with increasing power and pressure in the experiments, which could be due to the neutral heating effects. As stated earlier, the neutral temperature is assumed to be constant in the simulations. Note that the simulation results are generally smaller than in the experiments. The recombination coefficient of hydrogen atoms on the walls is not accurately known, and it can significantly affect the main properties of the hydrogen plasmas [48]. When increasing the coefficient, hydrogen atom density decreases clearly [48]. Therefore, the lower density of hydrogen atoms in the simulations than in the experiments could be due to a larger recombination coefficient of hydrogen atoms on the walls assumed in the model. In addition, the assumption of volume averaged plasma properties of the model itself can cause the difference of magnitude in the plasma parameters between simulations and experiments.

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