Estimation of Ion Density of Helium Plasma using Collisional-Radiative Models in Linear Plasma Device NUMBER

Abstract

In the linear plasma experimental device NUMBER, the densities of ions and atoms in helium plasma were estimated using passive spectroscopic measurements. The collisional-radiative models were employed to derive the densities of atoms and ions from the measured line intensities of each species. The estimated densities of helium species showed that (1) the atom density is one order of magnitude smaller than that estimated from gas pressure before discharge, (2) the density of singly charged ions is one order of magnitude smaller than that of atoms, (3) the doubly charged ions have a density 4 orders of magnitude smaller than the singly charged ions.

Fractional abundance of ions in plasmas is an important parameter to understand transport, instability, and atomic processes in magnetically confined plasmas [1], as well as in cylindrical plasmas and scrape-off layer (SOL) plasmas. While ionization equilibrium between two adjacent charge states is well established in magnetically confined torus plasmas, it is rarely achieved in the open-field plasmas. Direct measurement of specific charge states using spectroscopy is a powerful tool to obtain the charge state distribution. However, it is difficult to measure the fully ionized charge state (bare ions) with spectroscopy. In this paper, a practical method to estimate the charge state distribution, including neutral atoms and bare ions, in helium ionizing plasma is proposed.

The experiment has been performed using a linear plasma experimental device NUMBER [2], which has a stainless steel vacuum chamber with a diameter of 0.2 m and a length of 2 m. The plasma source utilizes electron cyclotron resonance (ECR) driven by 2.45 GHz microwaves. The experimental conditions for helium plasma were set to a gas pressure of 0.04 − 0.06 Pa and a microwave power of 6 kW. At an axial position $z$ = 0.85 m from the microwave injection window, a Langmuir probe was used to measure the electron temperature and electron density, which were in ranges of $T_{\mathrm{e}}$ = 8.3 − 12.0 eV and $n_{\mathrm{e}}$ = 5.7 − 9.8 × 10¹⁶ m⁻³, respectively. The current-voltage characteristic curves exhibited deviations of less than 20% from the Maxwellian behavior at the energy $E$ ≃ 20 eV. The deviations in the energy range $E$ > 20 eV, where the helium excitation cross-section is most pronounced, were difficult to evaluate due to low signal-to-noise ratio. A fiber optic cable was also installed at $z$ = 0.85 m, and the line-integrated emission intensities of He I (728 nm $(2^1\mathrm{P}-3^1\mathrm{S})$ and 504 nm $(2^1\mathrm{P}-4^1\mathrm{S})$) and He II (468 nm $(n=3-4)$) along the radial chord were measured using an ICCD camera.

The excited state population density $n_p^z$ of level $p$ for a charge state $z$ (including neutral atoms, $z$ = 0) in ionizing plasma can be expressed via the population coefficient $R_p^{\mathrm{ion.}}=R_p^{\mathrm{ion.}}(T_{\mathrm{e}},n_{\mathrm{e}})$ and the ground state density $n_g^z$ of charge state $z$, $n_p^z=R_p^{\mathrm{ion.}}{n}_{\mathrm{e}}{n}_g^z$. A collisional-radiative model (CRM) for helium atoms [3, 4] is used to calculate the population coefficients $R_p^{\mathrm{ion.}}$ of helium atoms ($z$ = 0) for given electron temperature $T_{\mathrm{e}}$ and density $n_{\mathrm{e}}$. For singly charged ions ($z$ = 1), a model based on charge-state scalings from a hydrogen atom CRM [5] is similarly used. The line emission intensity $I_{pq}$ resulting from a transition from an excited state $p$ to $q$ is proportional to the upper level population density and the spontaneous emission probability $A_{pq}$. Therefore, the ground state density of charge state $z$, $n_g^z$, can be determined as

  

$$n_g^z=I_{pq}/\left(A_{pq}{R}_p^{\mathrm{ion.}}{n}_{\mathrm{e}}\right).$$(1)

The ground state density $n_g^z$ represents the density of the ion $n^z$. Then, the density ratio of ${\mathrm{He}}^{\mathrm{+}}$ to $\mathrm{He}$ is

  

$$\frac{n^{{\mathrm{He}}^{\mathrm{+}}}}{n^{\mathrm{He}}}=\frac{I_{pq}^{{\mathrm{He}}^{\mathrm{+}}}{A}_{p^{\prime }{q}^{\prime }}^{\mathrm{He}}{R}_{p^{\prime }}^{\mathrm{ion.He}}}{I_{p^{\prime }{q}^{\prime }}^{\mathrm{He}}{A}_{pq}^{{\mathrm{He}}^{\mathrm{+}}}{R}_p^{\mathrm{ion.}{\mathrm{He}}^{\mathrm{+}}}}.$$(2)

Among a variety of transitions in He I, those with upper state of $n^1\mathrm{S}$ are selected for observation. This is because these transitions minimize the effects of radiation trapping and the quasi-steady-state approximation for metastable states.

The density of doubly charged helium ions, which cannot be directly measured from line intensities, is determined by the following procedure. The ionization-recombination balance for helium atoms is expressed by the following equation:

  

$$\frac{{\mathit{dn}}^{\mathrm{He}}}{dt}=-S_{\mathrm{CR}}^{\mathrm{HeI}}{n}_{\mathrm{e}}{n}^{\mathrm{He}}+{\alpha }_{\mathrm{CR}}^{\mathrm{HeI}}{n}_{\mathrm{e}}{n}^{{\mathrm{He}}^{\mathrm{+}}}+\frac{n^{{\mathrm{He}}^{\mathrm{+}}}}{{\tau }^{{\mathrm{He}}^{\mathrm{+}}}}+\frac{n^{{\mathrm{He}}^{2\mathrm{+}}}}{{\tau }^{{\mathrm{He}}^{2\mathrm{+}}}},$$(3)

where $S_{\mathrm{CR}}^{\mathrm{HeI}}$ and ${\alpha }_{\mathrm{CR}}^{\mathrm{HeI}}$ represent the effective ionization and recombination rate coefficients calculated by the CRM for helium atoms, respectively. The third and the last terms on the right-hand side correspond to recycling fluxes under assumption of a recycling rate $R$ = 1, where ${\tau }^{{\mathrm{He}}^{\mathrm{+}}}$ and ${\tau }^{{\mathrm{He}}^{2\mathrm{+}}}$ denote particle loss timescales for singly and doubly charged helium ions. The last term on the right-hand side $n^{{\mathrm{He}}^{2\mathrm{+}}}/{\tau }^{{\mathrm{He}}^{2\mathrm{+}}}$ is negligibly small compared to other terms in Eq. (3). Then, in steady state (${\mathit{dn}}^{\mathrm{He}}/dt$ = 0), the timescale for singly charged ions is expressed as:

  

$$\frac{1}{{\tau }^{{\mathrm{He}}^{\mathrm{+}}}}=n_{\mathrm{e}}(S_{\mathrm{CR}}^{\mathrm{HeI}}\frac{n^{\mathrm{He}}}{n^{{\mathrm{He}}^{\mathrm{+}}}}-{\alpha }_{\mathrm{CR}}^{\mathrm{HeI}}).$$(4)

The right-hand side in Eq. (4) can be obtained using probe measurements, spectroscopic measurements, and CR model calculations. Then, ${\tau }^{{\mathrm{He}}^{\mathrm{+}}}$ ≃ 0.5 ms, which is consistent with the timescale for thermal ion escaping along field line, $\tau$ ∼ $L/v$ ∼ 0.1 ms. Similarly, the ionization-recombination balance for doubly charged helium ions is expressed as:

  

$$\frac{{\mathit{dn}}^{{\mathrm{He}}^{2\mathrm{+}}}}{dt}=S_{\mathrm{CR}}^{\mathrm{HeII}}{n}_{\mathrm{e}}{n}^{{\mathrm{He}}^{\mathrm{+}}}-{\alpha }_{\mathrm{CR}}^{\mathrm{HeII}}{n}_{\mathrm{e}}{n}^{{\mathrm{He}}^{2\mathrm{+}}}-\frac{n^{{\mathrm{He}}^{2\mathrm{+}}}}{{\tau }^{{\mathrm{He}}^{2\mathrm{+}}}},$$(5)

where $S_{\mathrm{CR}}^{\mathrm{HeII}}$ and ${\alpha }_{\mathrm{CR}}^{\mathrm{HeII}}$ are the effective ionization and recombination rate coefficients calculated by the CRM for singly charged helium ions, respectively. Assuming the particle loss timescales for singly and doubly charged ions are comparable, the density ratio of ${\mathrm{He}}^{2\mathrm{+}}$ to ${\mathrm{He}}^{\mathrm{+}}$ can be derived from Eq. (5) as:

  

$$\frac{n^{{\mathrm{He}}^{2\mathrm{+}}}}{n^{{\mathrm{He}}^{\mathrm{+}}}}=\frac{S_{\mathrm{CR}}^{\mathrm{HeII}}{n}_{\mathrm{e}}}{{\alpha }_{\mathrm{CR}}^{\mathrm{HeII}}{n}_{\mathrm{e}}+1/{\tau }^{{\mathrm{He}}^{\mathrm{+}}}}.$$(6)

Using the density ratios obtained from Eqs. (2) and (6), and the electron density measured by the probe, the absolute densities of the atoms and ions are calculated based on the charge-neutrality condition, $2n^{{\mathrm{He}}^{2\mathrm{+}}}+n^{{\mathrm{He}}^{\mathrm{+}}}=n_{\mathrm{e}}$.

Figure. 1(a) shows the estimated densities of helium atom and ions. For neutral atoms, both estimated densities using He I 728 and 504 nm are consistent. For comparison, Fig. 1(a) also includes the electron density measured by the probe and the neutral particle density. The latter is evaluated from the gas pressure before plasma discharge using the ideal gas law, assuming an ambient temperature of 300 K. Error bars in Fig. 1(a) indicate uncertainty for a 10% variation in the electron temperature. Up to 40% uncertainty in the atom density is estimated. The results indicate that the density of helium atoms is approximately ten times higher than that of singly charged ions, while the density of doubly charged ions is almost four orders of magnitude lower than that of singly charged ions. CRM calculations assuming ionization equilibrium without particle losses predict that doubly charged helium ions are dominant, $n^{{\mathrm{He}}^{2\mathrm{+}}}/n^{{\mathrm{He}}^{\mathrm{+}}}$ ∼ 10¹ − 10², as shown in Fig. 1(b). The result suggests the importance of considering the particle loss timescales for ion abundance measurement.

Fig. 1.  (a) Helium atom and ion densities estimated in this study. Filled circle and filled diamond plots represent the atom density $n^{\mathrm{He}}$ using emissions of $\lambda$ = 504 and 728 nm, respectively; open square, singly charged ion density $n^{{\mathrm{He}}^{\mathrm{+}}}$; filled triangle, doubly charged ion density $n^{{\mathrm{He}}^{2\mathrm{+}}}$. Atom density estimated from gas pressure before discharge (open circle) and electron density measured with a Langmuir probe (plus) are also plotted. (b) Straightforward result of the ionization equilibrium for densities of helium atoms (filled circle), singly charged ions (open square) and doubly charged ions (filled triangle).

Additionally, the density of neutral helium atoms in the plasma was found to be about one-tenth of the neutral particle density before the discharge. One potential reason is the effective increase in the temperature of neutral atoms due to charge-exchange (CX) interactions between atoms and high-temperature ions at several eV. We can estimate a power balance between the CX heating and the cooling by the wall. Assuming a uniform cylindrical volume that contains plasma and neutral atoms and is surrounded by the wall of vacuum chamber, the power balance of neutral atoms in the steady state is given as $2\pi aL{n}^{\mathrm{He}}{v}_{\mathrm{n}}(k_{\mathrm{B}}{T}_{\mathrm{wall}}-k_{\mathrm{B}}{T}_{\mathrm{n}})+\pi a^{2}L{n}^{\mathrm{He}}{n}^{{\mathrm{He}}^{\mathrm{+}}}{C}_{\mathrm{CX}}(k_{\mathrm{B}}{T}_{\mathrm{i}}-k_{\mathrm{B}}{T}_{\mathrm{n}})=0$, where $k_{\mathrm{B}}{T}_{\mathrm{wall}}$ ≃ 300 K, $k_{\mathrm{B}}{T}_{\mathrm{i}}$ ≃ 4 eV, and $k_{\mathrm{B}}{T}_{\mathrm{n}}$ are the temperatures of wall, ions, and neutral atoms, respectively. The radius and length of the cylindrical region are $a$ ≃ 0.1 m and $L$ ≃ 1 m. Thermal velocity of neutral atoms $v_{\mathrm{n}}$ ≃ 1 km/s. The rate coefficient for CX process is $C_{\mathrm{CX}}$ ≃ 2.3 × 10¹⁵ m³/s. Then, $k_{\mathrm{B}}{T}_{\mathrm{n}}\simeq k_{\mathrm{B}}{T}_{\mathrm{wall}}+{10}^{-2}{k}_{\mathrm{B}}{T}_{\mathrm{i}}$ is obtained, which is a few times higher than the wall temperature. We also note that the thermal accommodation coefficient for helium atoms on metal surfaces $\alpha \equiv (T_{\mathrm{n}}^{\prime }-T_{\mathrm{n}})/(T_{\mathrm{wall}}-T_{\mathrm{n}})$ is less than unity [6], while conservatively $\alpha$ = 1 is applied in the estimation above. Therefore, the estimated value is the lower bound of $k_{\mathrm{B}}{T}_{\mathrm{n}}$. Density reduction in the plasma by ionization does not explain, since the mean free path of the atoms for the ionization collisions, ∼ 10 m, is larger than the plasma radius.

In summary, a spectroscopic measurement method for helium ion densities, including that of bare ions ${\mathrm{He}}^{2\mathrm{+}}$, is proposed. The essence of the method is considering the loss timescale in particle balance; the timescale is estimated from observables. The experiment shows consistency of the method. Doubly charged helium ions exist, but the density is four orders of magnitude lower than that of singly charged ions. A one order of magnitude decrease of neutral density in the plasma compared with that before the discharge is observed.

This work was supported by JSPS KAKENHI under Grant Nos. 23H01148 and 23K25845.

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