Determining Metastable Ion Lifetime and History through Wave-Particle Interaction
Laser-induced fluorescence (LIF) performed on metastable ions is frequently used to probe the dynamics of ground-state ion motions in many laboratory plasmas. However, these measurements place restrictions on the metastable ion lifetime. Metastable states are produced from direct ionization of neutral atoms and ions in other electronic states, of which the former will only faithfully represent processes that act on the ion dynamics in a time shorter than the metastable lifetime. We present here the first experimental study of this “new” type of instrumental errors using wave-particle interaction in an Argon multidipole plasma. The metastable lifetime and relative fraction of metastables produced from pre-existing ions, necessary for correcting the LIF measurement errors, can be determined by fitting the experimental results with the theory we proposed.
Introduction.—Laser-induced fluorescence (LIF) is a nonintrusive, nominally nonperturbative diagnostic technique that has found application in the study of a wide range of fundamental and applied problems in plasma physics. For example, LIF is often adopted to probe the plasma parameters in the plume of a Hall effect thruster to study the physical processes that control its operation W. A. Hargus and Cappelli (2001); Mazouffre et al. (2009). In semiconductor fabrication, LIF is employed to measure the plasma ion velocity distribution in the silicon etching process Jacobs et al. (2010). A reliable phase-space diagnostic is also required in the study of plasma sheath formation Severn et al. (2003), ion heating McChesney et al. (1987), and velocity-space diffusion Bowles et al. (1992).
For practical purposes, LIF is frequently performed on metastable states that are produced directly from neutral gas particles and also from ions in other electronic states Cherrington (1979); Goeckner et al. (1991). Here rises an important question: when can Doppler-resolved LIF on metastable ions be used to infer the velocity distribution of ground-state ions, the majority ion population in many laboratory plasmas?
Previous experimental results Nakano et al. (1992); Sadeghi et al. (1991) suggest that there are limitations of this laser diagnostic technique due to the finite lifetime of metastable ions. Simulations based on our newly developed Lagrangian model for LIF Chu et al. (2017); Chu and Skiff (2018) show that under circumstances where the metastable ion population is produced from direct ionization of neutrals, 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 in wave measurements, the perturbed distribution on these metastables cannot be correct if the wave period is greater than the metastable lifetime. However, the LIF performed on the metastable population produced from pre-existing ions is not affected by metastable lifetime.
Understanding the behavior of each metastable population is crucial in LIF applications as it provides a guideline for avoiding the systematic errors caused by the finite metastable lifetime. In the case where these errors are 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. However, unlike other well-known instrumental errors existing in LIF measurements such as optical pumping broadening Goeckner and Goree (1989); Goeckner et al. (1993), the metastable lifetime effects have never been explored experimentally before. In addition, it is a long-standing problem to trace the production history of metastable ions.
In this Letter, we report the first experimental measurement of metastable ion lifetime in a plasma as well as the relative fraction of metastables produced from pre-existing ions. The technique relies on measuring the ionic wave response. A theory is also presented to demonstrate that the LIF measurement errors can be corrected when the metastable lifetime effects become critical.
Theory and simulation.—Laboratory plasmas are often in the regime where the ion sound speed is much larger than the ion thermal speed. By solving the Vlasov equation and ignoring the wave vector k one can obtain the first- and second-order perturbation of the LIF measured ion distribution in the presence of an electrostatic wave
where is the amplitude of the electrostatic wave, is the wave angular frequency, is the ion mass, is the Maxwellian distribution, and the densities of the metastable ions produced from neutrals and pre-existing ions are denoted by and , respectively. The metastable lifetime is controlled by metastable quench rate , electron-collisional excitation rate , and optical pumping rate
In the limit when the metastable lifetime is long compared to the wave period (), the LIF measured first-order perturbation in Eq. (1) is proportional to the total metastable density . On the contrary, when the metastable lifetime is short (), is only proportional to with no contribution from at all. The difference between these two metastable populations is resulted from their distinctive history. The lifetime of the metastables produced from neutrals sets the typical time scale they experience the wave field. With a history of being neutral particles, this metastable 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 . On the other hand, as metastables produced from pre-existing ions have already fully interacted with the wave field before produced, the perturbed distribution measured using LIF is independent of their lifetime. This analysis can also be applied to in Eq. (2).
A numerical simulation based on a Lagrangian model for LIF is performed to test the theory for and . In the Lagrangian interpretation, one must follow each individual ion orbit as it moves through space and time. This approach achieves a large computational advantage by exploiting the separation of the classical dynamics of the ions from the quantum mechanics of the electronic states, reducing a system of coupled partial differential equations in the traditional Eulerian model to ordinary differential equations. Furthermore, since this model does not impose constraints on the ion orbits, it can be applied to systems with complicated ion dynamics. A detailed description of the model and its application are presented in Chu et al. (2017) and Chu and Skiff (2018). The simulation results for and demonstrate a good agreement with the theoretical predictions, as shown in Fig. 1.
As expected, the LIF measured wave amplitude is subject to the metastable lifetime effects. If the same electric field can be measured using a different method which does not rely on metastable ions, such as an electric field probe, then the metastable lifetime effects can be observed experimentally by comparing the results from these two measurements.
Experiment description.—We demonstrate the technique to measure the metastable lifetime and history in an Argon plasma confined in a multidipole chamber of 73 cm length and 49 cm diameter Hood et al. (2016). A hot cathode consisting of lanthanum-hexaboride (LaB6) heated by a resistive graphite bar is biased at V with respect to the chamber walls, emitting primary electrons to produce the plasma through impact ionization. Emission current from the cathode is regulated at 56 mA. The multidipole confinement is provided by 16 rows of magnets with alternating poles covering all inside walls of the chamber. The magnetic field strength is about 1000 G on the surface of the magnets and less than 2 G in the bulk plasma.
Figure 2(a) shows the LIF scheme used in the experiment. It is accomplished by a single mode tunable Rhodamine 6G dye laser (Sirah Matisse-DS). 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 to the state. Fluorescence photons are emitted at 461.086 nm when those electrons decay to the state with a large branching ratio of % Severn et al. (1998); Mattingly et al. (2013). The LIF measurements of and are made at two different places in velocity-space, (ion thermal speed) and 0 respectively, to maximize the fluorescence signal as illustrated in Fig. 2(b).
The experimental setup is depicted in Fig. 2(c). An ion acoustic wave is generated in the plasma by applying a differential sinusoidal signal with V on the double-mesh antenna. This driving signal is also sent to the lock-in amplifier as a reference. The frequency of the wave is scanned from 1 kHz to 45 kHz with an increment of 1 kHz. At each wave frequency, the electric field of the wave is measured using both the double-tip electric field probe and LIF.
The electric field probe is made with a low noise, high speed instrumentation amplifier AD8421 which allows to extract low level differential voltage signals in the presence of high frequency common-mode noise over a wide frequency range. Since plasmas tend to have a large impedance, even a small capacitance from the wires connecting the probe tips and the instrumentation amplifier can significantly reduce the bandwidth of the probe. Therefore, the instrumentation amplifier is placed only 15 mm away from the tips to improve the probe’s performance in the high frequency range. The separation between the probe tips is half a millimeter, providing an excellent spacial resolution for the electric field measurements. The electric field given by the probe is , where is the output voltage of the probe and is the gain of the instrumentation amplifier. The probe measurement needs to be corrected by multiplying a factor to compensate the errors mainly caused by the large plasma impedance comparable to the input impedance of the instrumentation amplifier and difficulties in precisely measuring the tip separation .
Results.—Using LIF, it is found that the ions have a Maxwellian velocity distribution along the direction of the laser beam in the center of the chamber with the temperature eV. This gives an ion thermal speed cm/s. The other plasma parameters are measured using a disc-shaped Langmuir probe. At the neutral pressure mTorr, the typical parameters in the bulk are, electron density , electron temperature eV, and plasma potential V. The ion sound speed is estimated as , which satisfies the assumptions made in the derivation of Eqs. (1)–(2).
The comparison between and , the electric field of the ion acoustic wave measured using electric field probe and LIF respectively, is presented in Fig. 3(a). The probe measurement is multiplied by a correction factor to scale with the LIF measurement. For a small electric field, the probe’s electronic pickup of fast electrons accelerated by the antenna can introduce big errors to the measurement, causing the dips on the two curves to slightly shift from each other around 16 kHz. It is clear that the electric field measured using the two methods make a 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.
To be able to compare with the theory in Eq. (4), the LIF measured electric field needs to be normalized by the probe measurement . The result of this procedure, shown in Fig. 3(b), is the key experimental result of this Letter. The peak at 16 kHz is resulted from the misalignment of the dips in Fig. 3(a). Caused by the metastable lifetime effects, instead of being a constant, the normalized curve starts to roll off as the wave frequency decreases. The metastable lifetime and fraction of metastables produced from pre-existing ions can be determined by fitting the experimental data with the theory. Since there are more ions in the center of the Maxwellian distribution, based on the simulation the optical pumping rate at is about 1.5 times as the rate at . From the best fit, it is found that the inverse metastable lifetime at is , which gives the metastable lifetime . Similarly, at ion thermal velocity and . The sum of the quench rate and collisional excitation rate is estimated as which is at least 3 times smaller than the optical pumping rate , making the latter the dominant factor in controlling the metastable lifetime in the experiment. The relative fraction of the metastables produced from pre-existing ions %, suggesting that the metastable ions are mainly produced by direct ionization of neutrals in this Argon multidipole plasma Goeckner et al. (1991).
The theoretical quench rate and collisional excitation rate are also computed to compare with the experimental values. According to Moore (1963), the quench rate , where is the neutral density, Skiff et al. (2001) is the quench cross section, is the the neutral temperature in energy units, and is the mass of the neutral particles. The collisional excitation rate is estimated as Curry et al. (1995). The sum of the two rates can then be calculated as , which is consistent with our experimental value within errors. This crosscheck demonstrates that we have successfully identified the metastable lifetime in the experiment.
Conclusions.—We have presented the first experimental study of the metastable lifetime effects using wave-particle interaction and LIF in a multidipole plasma. The experimental finding verifies that LIF performed on metastable ions 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. By fitting the theory with the experiment results, the metastable lifetime and relative fraction of metastables produced from pre-existing ions can be determined. Under circumstances where the metastable lifetime effects are inevitable, in our case probing the wave electric field under 10 kHz with LIF, the measurement errors can be corrected using the theory addressed in this Letter. Last but not least, LIF measurements of and provide a new method to determine the electric field in a plasma. Since this technique does not perturb the local field, it can be used to calibrate other electric field measurement tools, such as the double-tip electric field probe.
We acknowledge R. Hood for designing and constructing the electric field probe and the double-mesh antenna. This work was supported by the U.S. Department of Energy under Grant No. DE-SC0016473.
- W. A. Hargus and Cappelli (2001) J. W. A. Hargus and M. A. Cappelli, Appl Phys B 72, 961 (2001).
- Mazouffre et al. (2009) S. Mazouffre, D. Gawron, and N. Sadeghi, Physics of Plasmas 16, 043504 (2009).
- Jacobs et al. (2010) B. Jacobs, W. Gekelman, P. Pribyl, and M. Barnes, Phys. Rev. Lett. 105, 075001 (2010).
- Severn et al. (2003) G. D. Severn, X. Wang, E. Ko, and N. Hershkowitz, Phys. Rev. Lett. 90, 145001 (2003).
- McChesney et al. (1987) J. M. McChesney, R. A. Stern, and P. M. Bellan, Phys. Rev. Lett. 59, 1436 (1987).
- Bowles et al. (1992) J. Bowles, R. McWilliams, and N. Rynn, Phys. Rev. Lett. 68, 1144 (1992).
- Cherrington (1979) B. E. Cherrington, Gaseous Electronics and Gas Lasers (Pergamon Press, 1979).
- Goeckner et al. (1991) M. J. Goeckner, J. Goree, and T. E. Sheridan, Physics of Fluids B: Plasma Physics (1989-1993) 3, 2913 (1991).
- Nakano et al. (1992) T. Nakano, N. Sadeghi, D. J. Trevor, R. A. Gottscho, and R. W. Boswell, Journal of Applied Physics 72, 3384 (1992).
- Sadeghi et al. (1991) N. Sadeghi, T. Nakano, D. J. Trevor, and R. A. Gottscho, Journal of Applied Physics 70, 2552 (1991).
- Chu et al. (2017) F. Chu, S. W. Mattingly, J. Berumen, R. Hood, and F. Skiff, J. Inst. 12, C11005 (2017).
- Chu and Skiff (2018) F. Chu and F. Skiff, Physics of Plasmas 25, 013506 (2018).
- Goeckner and Goree (1989) M. J. Goeckner and J. Goree, Journal of Vacuum Science & Technology A 7, 977 (1989).
- Goeckner et al. (1993) M. J. Goeckner, J. Goree, and T. E. Sheridan, Review of Scientific Instruments 64, 996 (1993).
- Hood et al. (2016) R. Hood, B. Scheiner, S. D. Baalrud, M. M. Hopkins, E. V. Barnat, B. T. Yee, R. L. Merlino, and F. Skiff, Physics of Plasmas 23, 113503 (2016).
- Severn et al. (1998) G. D. Severn, D. A. Edrich, and R. McWilliams, Review of Scientific Instruments 69, 10 (1998).
- Mattingly et al. (2013) S. W. Mattingly, J. Berumen, F. Chu, R. Hood, and F. Skiff, J. Inst. 8, C11015 (2013).
- Moore (1963) W. J. Moore, Physical Chemistry (1963).
- Skiff et al. (2001) F. Skiff, G. Bachet, and F. Doveil, Physics of Plasmas 8, 3139 (2001).
- Curry et al. (1995) J. J. Curry, F. Skiff, M. Sarfaty, and T. N. Good, Phys. Rev. Lett. 74, 1767 (1995).