Measurement of wave-particle interaction and metastable lifetime using laser-induced fluorescence

Laser-induced fluorescence (LIF) is a nonintrusive plasma diagnostic technique used to probe the dynamics of ion motions in phase space. As long as the effect of photon momentum on ion orbits is negligible,¹ LIF measurements do not disturb the plasma, enabling the accurate study of velocity-space diffusion,² ion acoustic wave reflection,³ ion heating,^4,5 and other related phenomena in plasmas.^6,7 For example, LIF is extensively employed to determine the concentrations of chemical species in industrial etch reactors where nonperturbative measurements are required.^8,9 This laser diagnostic is also frequently adopted to examine the plume plasma properties of Hall effect thrusters to study the physical processes that control their operation.^10,11

In order to induce fluorescence, a laser is tuned to a specific wavelength to resonate with metastable ions produced from direct ionization of neutral atoms as well as ions in other electronic states. Metastables produced from existing ions have a history as “typical” ions. However, the newly produced metastables from neutral atoms are representative of the neutral velocity distribution and only become “typical” over time through ion-ion coulomb collisions. Due to the finite metastable lifetime, metastable ions can have a different velocity distribution from the ground-state ions (the majority ion population in many laboratory plasmas), causing errors in LIF detections of any observable quantities that are derived from the ion velocity distribution function, such as the ion temperature and amplitude of electrostatic waves. Here, rises a fundamental issue in LIF measurements: when can Doppler-resolved LIF on metastable ions be used to infer the velocity distribution of ground-state ions? This important matter is the basis of LIF that concerns anyone who uses LIF as a plasma diagnostic.

Numerical simulations based on our newly developed Lagrangian model for LIF^12–14 show that under circumstances where the metastable ion population is produced from direct ionization of neutral atoms, the velocity distribution measured using LIF will only faithfully represent processes which act on the ion dynamics in a time shorter than the metastable lifetime. For instance, the LIF measured ion temperature on these metastables is only accurate when they live longer than the ion-ion collision mean free time. For the purposes of wave detection,^15,16 the wave period has to be shorter than the metastable lifetime for the perturbed distribution f₁(v, t) measured on this metastable population to be correct. However, the LIF measurements made on the metastables produced from pre-existing ions are not affected by their lifetime.

A good understanding of the behavior associated with each metastable population helps us avoid the systematic errors caused by the finite metastable lifetime. When these errors become inevitable, correction of the LIF measurements requires knowledge on the metastable lifetime and fraction of metastables produced from pre-existing ions as opposed to directly from neutral atoms.

In this paper, we present the first experiment on determining the metastable lifetime as well as the relative fraction of metastables produced from pre-existing ions in an Argon multidipole plasma. The technique relies on measuring the ionic wave response. The first numerical study of the hole-burning effect on the metastable distribution perturbed by an electrostatic wave is also reported. This paper is organized as follows: Sec. II presents a simple theory for the metastable lifetime effects and the comparison between the theory and the simulation results, Sec. III gives a description of the experimental setup, Sec. IV presents the experimental results, and Sec. V provides a summary.

As mentioned earlier, metastable ions are produced from direct ionization of neutral atoms as well as ions in other electronic states. Assuming that the plasma is weakly collisional and spatially uniform, in the presence of an electrostatic wave, the velocity distribution functions of metastable ions produced from neutrals and pre-existing ions, p and q, can be obtained by solving the modified Vlasov equation for each population,

where S_n and S_i are the birth rates of metastables produced from neutrals and pre-existing ions, E is the amplitude of the electrostatic wave, ω is the wave angular frequency, and m_i is the ion mass. The velocity distributions of the neutral atoms and ground-state ions are denoted by f_n and f_i, respectively. The metastable ions produced from pre-existing ions have the same velocity distribution as ground-state ions, i.e., q = f_i. According to Ref. 14, the metastable lifetime τ is limited by metastable quench rate r, electron-collisional excitation rate u, and optical pumping rate W (from the metastable state to the upper state),

where γ = 1 – A₂₁/A_T, A₂₁ is the Einstein coefficient of spontaneous decay from the upper state to the metastable state, and A_T is the total spontaneous decay rate of the upper state.

Laboratory plasmas are often in the regime where the ion sound speed is much larger than the ion thermal speed. If we further assume that the neutrals and ions have the same temperature (both neutrals and ions have the same zeroth-order distribution f₀) and ignore the wave vector k, one can compute the first-order perturbation of p and q as

Since LIF directly measures the distribution of the metastable ions, therefore, the first-order perturbation of the LIF measured ion distribution in the presence of an electrostatic wave can be written as

where the densities of the metastable ions produced from neutrals and pre-existing ions are denoted by n_meta-n and n_meta-i, respectively. By following the same procedure, the second-order perturbation of the LIF measured ion distribution is

In a general case where the neutral atoms and ions are not in thermal equilibrium, it is difficult to find the exact solutions of f_1-LIF and f_2-LIF. However, their numerical solutions can still be calculated using the Lagrangian model for LIF as described in Sec. II B.

The perturbed ion distributions measured using LIF in Eqs. (6) and (7) reveal different behaviors of the two metastable populations. In the limit when the metastable lifetime is much longer than the wave period (τ ≫ 1/ω), the LIF measured first-order perturbation f_1-LIF is proportional to the total metastable density n_meta-i + n_meta-n,

However, in the other limit, when the metastable lifetime is short (τ ≪ 1/ω), f_1-LIF is proportional to n_meta-i with no contribution from n_meta-n at all,

The difference between these two metastable populations results from their distinctive histories. The lifetime of the metastables produced from neutrals sets the time scale over which they experience the wave field. With a history of being neutral particles, this population cannot react to the electric field until they become ions. If the lifetime is shorter than the wave period, these metastables will not live long enough to interact with the wave, resulting in a reduction in the measured f₁. On the other hand, as metastables produced from pre-existing ions have already fully interacted with the wave field before becoming metastables, the perturbed distribution measured using LIF is independent of their lifetime. This analysis can also be applied to f_2-LIF in Eq. (7).

According to Eqs. (6) and (7), the LIF measured amplitude of the electrostatic wave can be calculated as

As expected, E_LIF is affected by the metastable lifetime effects as well. By comparing E_LIF with the measurement of the same wave electric field using a different method not relying on metastable ions, such as an electric field probe, the metastable lifetime effects can therefore be observed experimentally.

To test the theory for f_1-LIF and f_2-LIF in Eqs. (6) and (7), a numerical simulation is performed based on a Lagrangian approach for LIF, developed specifically to model the LIF measurements in laboratory plasmas. A detailed description of the model and its application are presented in Refs. 13 and 14, and so, here, we only give a brief overview.

The Lagrangian approach is an interpretation of LIF where one follows each individual ion orbit as it moves through space and time. Unlike the traditional Eulerian model which extends the rate equations to a system of coupled kinetic equations, the Lagrangian model separates the classical dynamics of the ions from the quantum mechanics of the electronic states. This separation provides a large computational advantage, as it reduces the coupled partial differential equations in the Eulerian model to ordinary differential equations. In addition, since the Lagrangian model does not impose constraints on the ion orbits, it can be applied to systems with complicated ion dynamics. The simulation results for f_1-LIF and f_2-LIF demonstrate a good agreement with the theoretical predictions, as shown in Fig. 1.

In LIF measurements, the fluorescence signal is produced from allowed transitions of plasma ions that are optically pumped to excited states. This optical pumping process depletes the metastable ion population through “burning a hole” in its distribution function.¹⁸ Since the LIF measured perturbed ion distributions f_1-LIF and f_2-LIF are used to construct E_LIF in Eq. (10), it is necessary to understand how the metastable distribution is affected by optical pumping in the presence of an electrostatic wave.

We simulate the hole-burning effect using the Lagrangian model for LIF, and the results are presented in Fig. 2. To our knowledge, this is the first numerical study of the optical pumping process involving an electrostatic wave. The frequency of the laser selected in the simulation is in resonance with ions at v = 0. The time evolution of the metastable distribution perturbed by an electrostatic wave is shown in Fig. 2(a). The metastable ions initially have a Maxwellian distribution, and then, a hole starts to form at v = 0 as optical pumping pumps those ions to the excited state. The size of the hole keeps growing until optical pumping reaches an equilibrium with velocity-space diffusion and the appearance of new metastables in the laser beam. The metastable distribution perturbed by electrostatic waves with different wave amplitudes is shown in Fig. 2(b). This plot demonstrates that the size of the hole is affected by not only optical pumping but also the wave amplitude. Figure 2(c) shows the metastable distribution at different optical pumping rates. When the metastable distribution is not perturbed by the electrostatic wave, optical pumping always burns a hole in the distribution function. However, when affected by wave electric fields, the resulting shape of the distribution function in the optical pumped region can be either concave or convex. The metastable distribution at different times is plotted in Fig. 2(d). It shows that optical pumping burns a hole in the metastable distribution at the beginning; however, the hole is averaged out later as the ions are sloshed back and forth in velocity space by the wave field.

Since the second derivative of a function directly reflects its concavity, the LIF measured second-order perturbation f_2-LIF is more sensitive to the shape of the metastable distribution function than f_1-LIF.

The experiments of measuring the wave-particle interaction and metastable lifetime are performed in an Argon multidipole plasma confined in a cylindrical chamber of length 73 cm and diameter 49 cm.¹⁹ The plasma is produced through impact ionization by primary electrons emitted from a hot cathode biased at –70 V with respect to the chamber walls, resulting in an emission current of 56 mA. The cathode is made of a sintered lanthanum-hexaboride (LaB6) ring segment which is heated resistively. The multidipole magnetic field is provided by an electrically grounded magnet cage consisting of 16 rows of magnets with alternating poles covering all inside walls of the chamber. The magnetic field strength is about 1000 G close to the magnets and quickly diminishes to less than 2 G in the measurement region. Neutral pressure is regulated by a mass flow controller and measured using an ionization gauge.

The LIF scheme used in the experiment is accomplished by a single mode tunable Rhodamine 6G dye laser (Sirah Matisse-DS). The fluorescence signal is observed through a 25 cm diameter window, filtered using a narrow bandwidth interference filter, and imaged onto a 16-channel photomultiplier tube (PMT). To induce fluorescence, in the rest frame of an ion, the laser is tuned at 611.662 nm to excite electrons in the metastable state $3d2G9/2$ to the upper state $4p2F7/2o$⁠. Fluorescence photons are emitted at 461.086 nm when these electrons decay to the $4s2D5/2$ state with a large branching ratio of 66.5%.^20,21 In principle, the LIF measured electric field E_LIF in Eq. (10) can be obtained by sampling f₁ and f₂ at almost any point in velocity space. However, to maximize the fluorescence signal, the LIF measurements of f₁ and f₂ are made at v_t (ion thermal speed) and the peak of f₀, respectively, as illustrated in Figs. 3(a) and 3(b).

The experimental setup is depicted in Fig. 3(c). A differential sinusoidal signal with V_p = ±2 V is applied on the double-mesh antenna to excite ion acoustic waves in the plasma. The same signal is also sent to the lock-in amplifier as a reference. To measure the response of metastable ions to the wave electric field at various frequencies, the wave is scanned from 1 kHz to 45 kHz with an increment of 1 kHz. At each wave frequency ω, f₁ and f₂ are resolved using the lock-in amplifier by locking the frequency at ω and 2ω, respectively. The LIF measured electric field E_LIF can therefore be calculated using Eq. (10). The same electric field is also evaluated using a double-tip electric field probe to compare with the LIF measurements.

The electric field probe shown in Figs. 4(a) and 4(b) is made with a low noise, high speed instrumentation amplifier AD8421 ideally suited for a broad spectrum of signal conditioning and data acquisition applications. The amplifier features an extremely high common-mode rejection ratio, allowing it to extract low level differential voltage signals in the presence of high frequency common-mode noise over a wide frequency range. The probe tips are designed with an S shape to prevent from shadowing each other in the direction of the wave propagation. As plasmas tend to have a large impedance, even small capacitance from the wires connecting the probe tips and the instrumentation amplifier can significantly reduce the bandwidth of the probe. Therefore, to improve the probe's performance in the high frequency range, the instrumentation amplifier is placed only 15 mm away from the tips. The electric field probe measures the differential voltage between two points in the direction of the wave propagation and gives the electric field E_probe = V_out/dG, where V_out is the output voltage of the probe and G = 100 is the gain of the instrumentation amplifier. The separation d between the probe tips is about half a millimeter, providing an excellent spatial resolution in the electric field measurements. Because of the large plasma impedance comparable to the input impedance of the instrumentation amplifier and difficulties in precisely measuring the tip separation, the probe measured electric field E_probe also needs to be multiplied by a correction factor α to compensate for those errors.

By scanning the laser wavelength from 611.65785 nm to 611.66585 nm, it is found that the ions have a Maxwellian velocity distribution along the direction of the laser beam in the bulk plasma. Based on the halfwidth of the distribution function, the ion temperature is given by T_i = 0.03 ± 0.01 eV, suggesting that both ions and neutrals are close to the room temperature of 0.025 eV. A 0.65 cm diameter disc-shaped Langmuir probe is used to measure electron density n_e, electron temperature T_e, and plasma potential V _p in the experiment. When the neutral pressure is regulated at P = 0.058± 0.006 mTorr, the typical plasma parameters in the bulk are n_e = 2.1 × 10⁹ cm⁻³, T_e = 2.9 eV, and V_p = –4.3 V. The ion sound speed is estimated as $Cs≈Te/mi=2.6×105 cm/s$⁠, which is much larger than the ion thermal speed $vt=Ti/mi≈2.7×104 cm/s$⁠. Therefore, both assumptions made in the derivation of Eqs. (6) and (7) are satisfied.

Figure 5 shows the electric field of the ion acoustic wave measured using LIF, E_LIF, along with the probe measured electric field, E_probe, which is multiplied by a correction factor α = 11.8 to scale. The fast electrons accelerated by the antenna can be picked up by the probe as well, introducing errors to the electric field measurement especially when the field is small. For this reason, the dips on the two curves are slightly displaced from each other around 16 kHz. The electric fields measured using the two methods are in good agreement above 10 kHz; however, the LIF measurement is systematically smaller than the probe measurement below 10 kHz due to the metastable lifetime effects discussed in Sec. II A.

In the experiment, the efficiency of the antenna in launching the ion acoustic wave varies with the frequency. To compare the experiment with the theory in Eq. (10), the LIF measured electric field E_LIF needs to be normalized by the probe measurement E_probe. The result of this procedure, shown in Fig. 6, is the key experimental result of this paper. The peak at 16 kHz is caused by the misalignment of the dips in Fig. 5. The normalized LIF measurement E_LIF/E_probe, instead of remaining as a constant, rolls off below 10 kHz due to the finite metastable lifetime.

By fitting the ratio E_LIF/E_probe with the theoretical prediction, the metastable lifetime as well as the fraction of metastables produced from pre-existing ions can be determined. Because of the coulomb collisional drag effect (not present for v at the peak of f₀), metastable ions with v ∼ v_t systematically spend less time in resonance with the laser, resulting in a smaller optical pumping rate and a longer lifetime. From the best fit, the inverse metastable lifetime for ions at the peak of f₀ is determined as ξ = (5.63 ± 0.35) × 10⁴ s⁻¹, which gives the metastable lifetime of τ = 17.8 ± 1.1 μs. For metastable ions at thermal velocity, these two parameters are ξ = (4.09 ± 0.23) × 10⁴ s⁻¹ and τ = 24.4± 1.4 μs. The sum of the quench rate and the collisional excitation rate r + γu is estimated to be 1 × 10⁴ s⁻¹, which is at least three times smaller than the optical pumping rate W, making the latter the dominant factor in controlling the metastable lifetime in the experiment (γ ≈ 0.83 for 611.662 nm line). Finally, the relative fraction of the metastables produced from pre-existing ions is estimated as n_meta-i/(n_meta-i + n_meta-n) = 4 ± 2%. This result suggests that the metastable ions are mainly produced by direct ionization of neutrals in this Argon multidipole plasma.²² 

To validate the above results, the theoretical quench rate and the collisional excitation rate are computed to compare with the experimental values. Based on collision theory, the quench rate of metastable ions is given by

where n_n is the neutral density, σ is the quench cross-section, T_n is the neutral temperature in energy units, and m_n is the mass of the neutral particles. According to Ref. 23, σ  = 2.7 × 10⁻¹⁴ cm² when the ions are near room temperature, resulting in r = 1.9 × 10³ s⁻¹. The electron-collisional excitation rate from the metastable state to the upper state can be estimated by considering electron-ion inelastic collisions²⁴ 

where f is the oscillator strength,²⁵ ΔE is the change in energy of the transition (in eV), T_e is the electron temperature (in eV), and n_e is the electron density (in cm⁻³). Since this formula only provides an order of magnitude estimate of the collisional excitation rate, Ref. 24 also provides a more appropriate rate u ≈ 1 × 10⁴ s⁻¹ based on the experiment. The sum of the theoretical quench rate and the collisional excitation rate r + γu can then be estimated to be 1 × 10⁴ s⁻¹, which is consistent with our experimental value within errors.

In this paper, we present the first experimental study of the metastable lifetime effects using wave-particle interaction and LIF in a multidipole plasma. The first numerical study of the hole-burning effect on the metastable distribution in the presence of an electrostatic wave is also reported. We demonstrate that the metastable lifetime and the relative fraction of metastables produced from pre-existing ions can be determined through fitting the experimental results with the theory. The experimental finding verifies the previous simulation results that LIF performed on metastables produced directly from neutral atoms can only be used to infer the velocity distribution of ground-state ions if the ion dynamics is in a time shorter than the metastable lifetime. In the case where the metastable lifetime effects are inevitable, e.g., using LIF to measure the ion acoustic wave amplitude under 10 kHz in our experiment, we show that these systematic errors can be corrected with the theory addressed in this paper. Finally, LIF measurements of f₁ and f₂ provide a new method to determine the absolute electric field without disturbing the plasma. This technique can be employed to calibrate other electric field measurement tools, such as the double-tip electric field probe used in our experiment.

This work was supported by the U.S. Department of Energy under Grant No. DE-SC0016473.