Spectroscopic Measurement of Hydrogen Atom Density in a Plasma Produced with 28 GHz ECH in QUEST

Abstract

The spatial distribution of the hydrogen atom density was evaluated in a spherical tokamak (ST) plasma sustained only with 28 GHz electron cyclotron heating (ECH). The radially resolved H_δ emissivity was measured using multiple viewing chord spectroscopy and Abel inversion. A collisional-radiative (CR) model analysis of the emissivity resulted in a ground-state hydrogen atom density of 10¹⁵–10¹⁶ m⁻³ and an ionization degree of 1–0.85 in the plasma.

1. Introduction

Q-shu University Experiment with Steady-State Spherical Tokamak (QUEST) is a medium-sized spherical tokamak (ST) device operated for the understanding of plasma–wall interactions in steady-state discharges [1] and for the development of non-inductive ST start-up schemes [2,3]. This study contributes to the latter by investigating the effect of neutrals (atoms) on an initial plasma produced with electron cyclotron heating (ECH).

In the ST discharge produced with ECH, a cylindrical plasma in an open field equilibrium is initially produced around the resonance layer [2,4,5]. Then, by ramping up the plasma current, an ST plasma sustained only with ECH is formed [2,4,6]. Among the characteristics of the initial ST plasma is a large variation in ion toroidal velocity in the range of 0–20 km/s with the size and position of the last closed flux surface (LCFS) [7]. Since this characteristic could be used for the active control of the flow profile, we intended to further investigate the driving and damping mechanisms of the flow. In this study, we measured the radial distribution of the atom density and the degree of ionization (DOI) in order to evaluate the loss and transport of the ion momentum through the charge-exchange reaction (e.g., [8]), since the bulk electron temperature and density of the initial plasmas are typically less than 1 keV and 1 × 10¹⁹ m⁻³ [2,6,9], respectively, and these relatively small plasma parameters can result in a greater effect of atoms.

2. Experimental Setup

We performed experiments using a discharge sustained only with 28 GHz ECH. This discharge is described in detail elsewhere [2]. Briefly, the plasma was mainly heated by second harmonic resonance ECH at R = 0.32 m with a power of 120 kW, where R is the major radius. We used two discharges performed with an identical setup, i.e., one for Thomson scattering (#41176) and the other for spectroscopy (#41178) measurements. The reproducibility of plasma parameters for these discharges is within their uncertainties. Figure 1a shows the temporal evolutions of the plasma current, electron temperature $T_{\mathrm{e}}$, and density $n_{\mathrm{e}}$ at the core [9], and the chord-integrated H_α intensity measured on a radial viewing chord on the midplane. We conducted all the measurements in the time period t = 2.7–3.0 s denoted by the shaded area in Figure 1a. The period was in the flattop phase, where the magnetic field coil currents were kept constant. The reconstructed equilibrium magnetic flux surfaces are shown in Figure 1b; these surfaces were obtained from the measured magnetic flux and plasma current using a modification of the method described in [10]. The plasma minor radius was reduced to approximately 0.1 m to increase the volume power density of the heating. The magnetic axis was located at R = 0.32 m and z = −0.1 m, and the last closed flux surface (LCFS) was located at R = 0.25 and 0.4 m on the midplane.

Figure 2 is a schematic illustration of the visible spectroscopic system. We measured the H_δ emission line spectrum (p = 2–6), where p is the principal quantum number, using 24 fan-shaped viewing chords on the midplane. The distance between the viewing chord and the torus axis, which we define as $\eta$, ranged between 0.24 and 1.01 m. The diameters of the viewing spots were approximately 30 mm at the positions closest to the torus axis.

The emission was collected using two sets of a collimator (Edmund Optics Japan, Japan, TS achromatic lens; 15 mm focal length, $\varphi$14 mm) and an optical fiber bundle (Mitsubishi Cable Industries, Japan, STU230D; 230 and 250 μm core and cladding diameters, respectively, 0.2 NA). The collected emission was transferred to a Czerny–Turner spectrometer (Acton Research, USA, AM-510; 1 m focal length, F/8.7, 1800 grooves/mm grating) equipped with astigmatism compensation optics. The spectra were recorded with a CCD (Andor Technology, UK, DU440-BU2; 2048 × 512 pixels, 13.5 μm pixel size, 16 bit). We set the entrance slit width of the spectrometer at 50 μm and the central output wavelength at 412 nm. The instrumental width at the full width at half maximum and the reciprocal linear dispersion were 40 pm and 6.5 pm/pixel, respectively. The sensitivity of the system was absolutely calibrated using a standard lamp while separately considering the window transmittance.

3. Results

Figure 3a shows a H_δ spectrum measured on the viewing chord at $\eta$ = 0.51 m, which is indicated by the bold line in Figure 2; this position corresponds to the outer scrape-off-layer (SOL). We evaluated the chord-integrated intensity $I_A$ from the area of the spectrum. The evaluated $I_A$ is marked in Figure 3b. The error bars consist of uncertainties in the sensitivity calibration coefficient and the zero level of the spectrum. Note that the pedestal of the OII emission line at 410.32 nm is superposed on that of H_δ, but its effect on $I_A$ is small and we neglected it. Additionally, we omitted $I_A$ obtained on the chord at $\eta$ = 0.84 m owing to the large noise produced by the incidence of X-rays to the CCD.

From the evaluated $I_A$, we calculated the radially resolved emissivity $\epsilon \left(R\right)$ using Abel inversion assuming the toroidal symmetry of the emission. In a tokamak discharge with similar plasma parameters and wall conditions, the variation in H_α (p = 2–3) intensity in the toroidal direction was reported to be small except for an approximately 10 times local increase at the poloidal limiter [11]. Since the present discharge is limited by the center stack, the validity of the assumption of the toroidal symmetry could degrade near the center stack.

For the numerical calculation of the Abel inversion, the discrete data of $I_A$ was interpolated with a smooth spline function in the range $\eta$ = 0.24–1.01 m as shown by the red curve in Figure 3b. $\epsilon \left(R\right)$ was then calculated as follows:

$$\epsilon \left(R\right) = -\frac{1}{\pi }{\displaystyle {\int }_R^{R_{\mathrm{end}}}}\frac{1}{\sqrt{{\eta }^2-R^2}}\frac{d{I}_A\left(\eta \right)}{d\eta }d\eta ,$$

(1)

where $R_{\mathrm{end}}$ is the outermost radius of H_δ emission, which we determined to be 1.01 m because $I_A$ obtained on viewing chords at $\eta \ge$1.01 m was less than the noise level. The calculated $\epsilon \left(R\right)$ is shown in Figure 3c. We evaluated the error bar by the procedure described in [12] using 500 pairs of data. A small peak was found in $\epsilon \left(R\right)$ around R = 0.7 m owing to an increase in $I_A$ at $\eta$ = 0.71 m. We confirmed the reproducibility of this tendency in identical discharges, but we have not identified its origin.

The hydrogen atom density was calculated from $\epsilon \left(R\right)$ by collisional-radiative (CR) model analysis [13]. The atom density in an excited state $p$, $n_{\mathrm{H},p}\left(R\right)$, is expressed from the equation of continuity and the quasi steady-state approximation as follows:

$$\begin{array}{cc}{n}_{\mathrm{H},p}\left(R\right)=& r_{0p}\left(R;T_{\mathrm{e}},n_{\mathrm{e}}\right)n_{{\mathrm{H}}^{+}}\left(R\right)n_{\mathrm{e}}\left(R\right)+r_{1p}\left(R;T_{\mathrm{e}},n_{\mathrm{e}}\right)n_{\mathrm{H}}\left(R\right)n_{\mathrm{e}}\left(R\right)\\ & +r_{2p}\left(R;T_{\mathrm{e}},n_{\mathrm{e}}\right)n_{{\mathrm{H}}_2}\left(R\right)n_{\mathrm{e}}\left(R\right),\end{array}$$

(2)

where $n_{{\mathrm{H}}^{+}}$, $n_{\mathrm{H}}$, and $n_{{\mathrm{H}}_2}$ are the ground-state hydrogen ion, atom, and molecule densities, respectively. $r_{0p}$, $r_{1p}$, and $r_{2p}$ are called population coefficients whose values are calculated with a CR model code [13] as a function of $T_{\mathrm{e}}$ and $n_{\mathrm{e}}$.

Figure 4 shows the calculated population coefficients for $p$ = 6, where the green, red, and blue lines represent $r_{06}$, $r_{16}$, and $r_{26}$, and the solid, dashed, and dash-dotted lines are those at $n_{\mathrm{e}}$ = 10¹⁶, 10¹⁷, and 10¹⁸ m⁻³, respectively. $T_{\mathrm{e}}$ and $n_{\mathrm{e}}$ of the present plasma are in the ranges of 10 $\le T_{\mathrm{e}}\le$ 400 eV and 10¹⁶ $\le n_{\mathrm{e}}\le$ 10¹⁸ m⁻³, respectively (see Figure 5a), and the former is denoted by the shaded area in Figure 4. Considering the relative magnitudes of the population coefficients and the approximated charge neutrality condition $n_{{\mathrm{H}}^{+}}\cong n_{\mathrm{e}}$, we can neglect the first term of Equation (2). We can also neglect the third term by assuming that $n_{{\mathrm{H}}_2}<n_{\mathrm{H}}$, which is reasonable except for near the wall. Equation (2) is then approximated as follows:

$$n_{\mathrm{H},6}\left(R\right) \cong r_{16}\left(R;T_{\mathrm{e}},n_{\mathrm{e}}\right)n_{\mathrm{H}}\left(R\right)n_{\mathrm{e}}\left(R\right).$$

(3)

The excited hydrogen atom density $n_{\mathrm{H},6}\left(R\right)$ is obtained from $\epsilon \left(R\right)$ using the following:

$$\epsilon \left(R\right) = \frac{h\nu }{4\pi }{n}_{\mathrm{H},6}\left(R\right)A,$$

(4)

where $h$ is the Planck constant and $\nu$ and $A$ are the frequency and spontaneous emission coefficient of the H_δ transition, respectively. Then, $n_{\mathrm{H}}\left(R\right)$ is obtained from Equations (3) and (4) as follows:

$$n_{\mathrm{H}}\left(R\right) = \frac{4\pi }{h\nu A{r}_{16}\left(R;T_{\mathrm{e}},n_{\mathrm{e}}\right)n_{\mathrm{e}}\left(R\right)}\epsilon \left(R\right).$$

(5)

We interpolated $T_{\mathrm{e}}$ and $n_{\mathrm{e}}$ measured by Thomson scattering [9] in the same way as $I_A$ as shown in Figure 5a. In addition, the data was linearly extrapolated on the inboard side (R = 0.28−0.34 m) to evaluate $n_{\mathrm{H}}$ around the resonance layer. The extrapolated curves are shown by dashed lines.

$n_{\mathrm{H}}\left(R\right)$ evaluated with the above procedures is shown in Figure 5b. The density is in the range of 10¹⁵–10¹⁶ m⁻³ and monotonically decreases toward the ECH resonance layer. The increase around R = 0.65 m is physically implausible and may have been caused by an error in $\epsilon \left(R\right)$. The evaluated density is less than the hydrogen molecule density of approximately 10¹⁷ m⁻³ at the vacuum chamber wall (R = 1.3 m) obtained from the pressure measured by a fast ionization gauge assuming a temperature of 300 K.

The accuracy of the CR model analysis can be affected by the opacity of Lyman series emission lines (p = 1–q), where q is the upper-state principal quantum number [14]. When the opacity is significant, the reabsorption of Lyman series emission lines increases the excited hydrogen atom density, and ignoring the effect results in the overestimation of $n_{\mathrm{H}}\left(R\right)$. Since the present CR model analysis is mainly affected by the L_ε line (p = 1–6) and the secondary effect, namely, the increase in $n_{\mathrm{H},6}\left(R\right)$ owing to the absorption of the other Lyman series emission lines, is small [14,15], we evaluated the effect of the opacity by calculating the optical depth of the L_ε line at its line center [16] under the assumption of a uniform plasma as follows:

$$\tau =\frac{e^2}{4m_{\mathrm{e}}{ϵ}_0}\sqrt{\frac{m_{\mathrm{H}}}{2\pi k_{\mathrm{B}}{T}_{\mathrm{H}}}}\frac{f_{16}}{{\nu }_{16}}{n}_{\mathrm{H}}L,$$

(6)

where e is the elementary charge, ε₀ is the vacuum permittivity, k_B is the Boltzmann constant, m_e is the electron mass, m_H is the hydrogen atom mass, f₁₆ (=7.8035 × 10⁻³) and ν₁₆(=3.1967 × 10¹⁵ Hz) are the absorption oscillator strength and transition frequency, respectively, n_H is the density of ground-state atoms, and L is the absorption length. We assumed an identical temperature T_H for the excited and ground-state atoms. Assuming T_H = 300 K, n_H = 1 × 10¹⁶ m⁻³, and L = 2 m gives the upper limit of the optical depth τ as 1.5 × 10⁻². The reabsorption with an optical depth of this magnitude is negligible in the CR model analysis.

Finally, as a measure of the effect of atoms on the plasma, we evaluated the degree of ionization (DOI). The DOI is defined as $n_{{\mathrm{H}}^{+}}/\left(n_{{\mathrm{H}}^{+}}+n_{\mathrm{H}}+n_{{\mathrm{H}}_2}\right)$ and we approximated it to $n_{\mathrm{e}}/\left(n_{\mathrm{e}}+n_{\mathrm{H}}\right)$ on the basis of the above-mentioned assumptions, where we used the interpolated curve of $n_{\mathrm{e}}$ in Figure 5a. The evaluated DOI shown in Figure 5b reached nearly unity over the entire core region, suggesting that the effect of atoms on the core plasma is small. The DOI, however, decreased in the SOL region and this could induce the momentum loss of the ions trapped in orbits that intersect the LCFS.

4. Conclusions

In this study, in order to evaluate the ion momentum loss and transport through the charge-exchange reactions in a QUEST plasma sustained only with 28 GHz ECH, we measured the radial distribution of the hydrogen atom density. The obtained density was of the order of 10¹⁵–10¹⁶ m⁻³, and considering the fact that it is less than the hydrogen molecule density evaluated at the vacuum chamber wall and the observed tendency of a monotonic decrease toward the ECH resonance layer, the result is plausible. However, the effect of the violation of the assumed toroidally symmetric emission should be addressed to further improve the reliability of the result. The DOI evaluated using the atom density and $n_{\mathrm{e}}$ measured by Thomson scattering suggested that the effect of atoms is small in the core region.