Evidence of the ionization instability and ion acoustic turbulence correlation in sub-ampere hollow cathodes

Plasma instabilities in the plume of hollow cathodes have been extensively researched in particular for high-current operation. The rise of plume mode ionization-like instability leads to a degradation of cathode’s performance along with the emergence of highly energetic ions that can produce sputtering of various cathode’s surfaces. Numerical simulations using 2D fluid or hybrid codes brought forward an interesting correlation between the evolution of ion acoustic turbulence (IAT) and emergence of plume mode oscillations. Such numerical findings were proven to be true by experimental measurements of wave dispersion and plume mode-IAT correlation in the plume of cathodes emitting currents >10 A. This study brings forward evidence of the correlation between plume mode oscillations and IAT in the plume of low-current cathodes operating with Kr at sub-ampere current levels. It is shown that at <1 A the plume mode instability is highly correlated with the IAT and the anomalous electron collision drives the electron transport in the cathode plume. The fluctuations in IAT wave energy lead to large temperature oscillations which then drive fluctuations in the density via ionization.


Introduction
Thermionic hollow cathode operating at sub-ampere (< 1 A) current levels remain instrumental for the operation of low-power (< 250 W) Hall thrusters. Such devices create highly dense plasmas, n e ∼ O(10 19 − 10 22 ) m −3 , that expand into collisionless plumes in the cathode-anode space with large gradients in the neutral and electron density. Two modes of operation were observed, i.e. spot and plume, with spot-to-plume mode transition taking place at larger discharge currents or insufficient propellant flow rates [1,2]. Both modes and the mode transition were observed to occur also at currents below 1 A [3,4]. The mode transition comes with the distinct change in plume morphology, i.e. bright spot-like structure next to the cathode in the case of spot mode and divergent diffuse plume attached to the anode in the case of plume mode. Moreover, the plume mode is characterized by the emergence of low-frequency oscillations in the range 10-250 kHz as seen in the discharge current spectra. The cathode discharge stability influences the efficiency of the propulsion system and operation in plume mode is detrimental to the lifetime of the cathode due to the emergence of high energy ions that sputter the surface of the cathode and accelerate erosion. Therefore, understanding the mechanism that triggers the spot-to-plume mode transition and the underlying phenomena that sustain and enhance plume mode operation remains an important issue for the electric propulsion community.
Over the years, several studies focused on the origin of plume mode [1,2,5,6]. The widely accepted interpretation suggests an ionization related process with a predator-prey relation between the neutral and electron density [7]. The plume mode waves are believed to be driven unstable by large fluctuations in the electron temperature and plasma potential. Furthermore, the unusually large values for the electron temperature and plasma potential along the cathode-anode gap when the cathode operates in plume mode could be explained by a primarily anomalous elecron transport as suggested by numerical results [8][9][10]. The anomalous electron Ohmic heating was attributed to ion acoustic turbulence (IAT). The coexistence of low-frequency plume mode ionization instability and high-frequency IAT content in the plume of such devices was experimentally proved [11][12][13][14][15][16][17][18][19]. Nevertheless, the mentioned studies, except for the one of Tsikata et al. [19], were conducted with cathodes operating at high-current in excess of 10 A. In their recently published study, Tsikata et al. employed the coherent Thomson scattering method to evaluate the plasma wave content in the plume of a cathode operating with a Hall thruster at 1 A. The cathode was not in self-sustained mode and operated in the magnetic field environment of the thruster. Their study showed the presence of MHz IAT waves propagating towards the thruster and the cathode, along and across the magnetic field lines, as well as kHz fluctuations which may be attributed to drift-driven instabilities in the plane perpendicular to the magnetic field.
The present work tries to answer the following questions: i) does the plume mode oscillation have the same structure at sub-ampere current operation in comparison to high-current operation? ii) is IAT present in the plume of hollow cathodes operating at < 1 A? iii) is there still a correlation between the IAT and plume mode oscillation? and finally iv) quantitatively, are the oscillations amplitudes in the same order of magnitude as compared to high-current operation? Here, Kr is used as propellant due to the increasingly interest for this gas driven by its much lower cost as compared to Xe, making it the ideal fuel for large constellations.
This communication brings novel insights on the ionization instability in a low-current hollow cathode's plume where ion saturation probes are used to construct a better understanding of the main wave phenomena. Theory of the ionization instability and IAT section includes a brief presentation of the ionization instability and IAT theory. The experimental setup and methods used here are described in Experimental setup and methods section and the main results and discussion are included in Results and discussion section which comprises of: a macroscopic depiction of the correlation between the two aforementioned classes of instabilities in Macroscopic correlation of low-and high-frequency oscillations section, a study of the plume plasma wave dispersion in Plume plasma wave dispersion section, evidence of temporal and spatial correlation of low-and high-frequency waves in Correlation of ionization instability and IAT section, and the evolution of plasma parameters over the plume mode oscillation time-scale in Plasma parameters evolution over the ionization instability time-scale section. In the end, Conclusions section outlines the conclusions.

Theory of the ionization instability and IAT
The cathode discharge stability when operated in diode mode with an external anode or keeper electrode can be assessed by analyzing the temporal evolution of the discharge/anode current, I d , and potential, V d , and their respective power spectral densities (PSDs) [20,21]. Spot mode operation is characterized by quiescent, less energetic spectra, with no distinct frequency peaks or harmonics. Plume mode is expected to be characterized by noisy and energetic spectra, with higher amplitudes and distinct frequency peaks and harmonics in the low-frequency range 10-150 kHz for Xe [20,21]. Moreover, the plume mode instability shows large scale and amplitude (up to 100% of the average) oscillations that can be observed in the discharge current, plasma density, and potential. The oscillations are quasi-coherent and propagate mainly in longitudinal direction. Due to their large amplitude, there is a common acceptance that such oscillations must be related to ionization, hence, their name of ionization-like instabilities [7]. Nevertheless, in the presence of an applied magnetic field that simulates the thruster environment, the amplitude of the ionization instability decreases with higher field strengths [22] and other instabilities arise in the plasma, such as the drift instability propagating azimuthally across the magnetic field lines [19,22,23] and the kink instability at large discharge current [24].
Due to the strongly drifting electrons in the plume of hollow cathodes, the growth of high-frequency turbulence such as IAT can be facilitated through inverse electron Landau damping [25]. Such waves propagate mainly in the direction of the electron current and are known to induce non-classical resistivity acting on electrons [12] and non-classical ion heating [14]. They have low amplitudes and present an incoherent and broadband continuous spectrum from 200 kHz up to 10 MHz, with cut-off at low frequency and decaying power with increasing frequency. Being a current-driven oscillation, IAT is strongly dependent on discharge current, mass flow rate, and facility pressure [26,27]. Increased charge exchange events at higher flow rate or facility pressure tend to damp the IAT.
The plume of a hollow cathode is characterized by a low collisional environment with large electron drifts that challenge the assumption of Maxwellian plasmas. Therefore, numerical fluid approaches fail to predict the experimentally observed trends in plasma potential and electron temperature in this region. Non-classical behaviors, such as the IAT anomalous resistivity, are usually added to fluid codes to match the trend and magnitude of plasma parameters in the cathode plume to the experimentally observed ones. The most striking result of the implementation of IAT contribution to the cathode plume plasmadynamics is far from being the high fidelity prediction of the plasma temperature and potential, but the emergence of low-frequency oscillations in the cathode plume, similar to the experimentally found plume mode instability [9,10,28]. This suggests that the cathode plume plasma is primarily described by non-classical effects of acoustic turbulence, where there may be a correlation between the IAT and the plume mode instability.
Assuming the frequency of the waves in the IAT spectrum are much lower than the ion plasma frequency, ω pi , the following expressions are found for the real and imaginary parts of the dispersion relation of IAT [22]: where k is the wavevector, u i and u e are the ion and electron drift velocities respectively, c s = √ eT e /M is the ion sound speed with M the ion mass and e the elementary charge, v g is the group velocity, v e is the electron thermal velocity, and M e the electron Mach number. Equation 2 can be seen as the electron contribution to the IAT growth rate and shows that the source of energy for the IAT growth is the electron drift velocity.
An established approach in modeling the anomalous collision frequency induced by the IAT follows a proportionality law [29,30]. Assuming that the IAT energy density scales with the thermal energy density of the plasma [31] and that the plasma potential fluctuations,φ p , can be correlated to the fluctuations in electron density,ñ e , the following expression can be obtained [16]: where m is the electron mass and n e is the electron density. However, in simulation context, a priori knowledge of the IAT spectrum is needed to implement an anomalous collision frequency as described thus far. To alleviate this caveat, fluid-like approximations for ν an are used to self-consistently solve plasma parameters in the plume region. Georgin and Jorns [17] give a comprehensive review of the main closure models used for IAT anomalous frequency in fluid codes and subject them to a comparison with experimental data. One of the established models was introduced by Mikellides et al. and gives the following expression for ν an [8]: where the proportionality constant α IAT is typically O(10 −2 ) and ω pe is the electron plasma frequency. The ion temperature, T i , is usually assumed to be 0.1 T e . It was shown for a high-current cathode operating at > 15 A that the IAT amplitude is modulated on the time-scale of the plume mode instability with a correlation between the two classes of instability [22]. The phase difference observed between the IAT and density might imply a causal relation. Moreover, the IAT oscillations are correlated with the plasma parameters where the anomalous collision frequency leads in phase the electron temperature, which then leads the electron density, suggesting an ionization process. In the presence of IAT-driven resistivity, a predator-prey wave oscillates near the ionization frequency and grows when the temperature is above a critical threshold and fluctuating [22], suggesting a link between the IAT-driven non-classical electron transport and the low-frequency ionization instability.
The current hypothesis that the ionization-like waves are driven unstable by nonclassical effects such as enhanced resistivity and electron Ohmic heating induced by the growth of IAT is explored in the following section for hollow cathodes operated with Kr at < 1 A and in the absence of a magnetic field.

Experimental setup and methods
In this section, a brief description of the vacuum facility, cathode, and probe array used for this work is presented.

Cathode and vacuum facility
The PSAC-KE cathode is used in the following investigations [4]. The cathode has an 2 mm diameter open-end cylindrical LaB 6 emitter with a 45°knife-edge, as shown in Fig. 1a. The total emission area is 0.66 cm 2 . The emitter is connected to the cathode's stainless steel base via a tantalum gas tube and a graphite holder. The heater comprises of two ceramic parts encapsulating a tungsten-rhenium filament. Two thermal shields made of tantalum foil are placed around the heater and at the base of the cathode respectively to improve the device's thermal design. The keeper electrode is made of stainless steel with a tantalum orifice plate welded at its extremity. The keeper orifice has a diameter of 2r ko = 1 mm. A 6 mm thick complex boron nitride ceramic part (BN-ZrO 2 ) insulates the keeper from the cathode base. Figure 1b shows the general I − V characteristics of the PSAC-KE cathode operated at various Kr mass flow rates in self-sustained mode (i.e. no heating and keeper powers) with the disk anode. The anode potential, V a , varies from 68 to 135 V, increasing with lower mass flow rates and decreasing with higher anode current, I a . The overall power consumption of the cathode ranges between 40 and 86 W. Nevertheless, in order to ensure stable discharges, a small keeper current I k ∼ 0.05 A is applied throughout the following measurements.
The experimental setup is depicted in Fig. 2 and the main testing conditions are summarized in Table 1. The experiments are performed in the CEVAC chamber [3]. The chamber is a stainless steel cylindrical enclosure with a diameter of 0.4 m and a height of 0.3 m equipped with a dry rotary pump and a turbomolecular pump with a capacity of 600 l s −1 for light gases. Inside the CEVAC, the pressure can reach 5 × 10 −6 mbar-N 2 . The chamber is accessed through a 0.4 m diameter hatch. For single point investigations at specific locations in the plume, the cathode is firstly tested with a 30 mm-diameter disk anode placed at L ca ≈ 9 mm downstream of the keeper orifice, whereas a second 51 mmdiameter cylindrical anode at L ca ≈ 22 mm is used for a complete plume swipe along the cathode-anode centerline.

Probe array and estimation of plasma parameters
The probe array consists of two cylindrical Langmuir probes made of 0.15 mm-diameter tungsten wires with a length of 1.5 mm and one emissive probe made of 0.15 mmdiameter tungsten wires with a length of 3.6 mm. All three probes are protruding  Experimental setup for cathode plume wave measurements with the probe array for a the disk anode and b the cylindrical anode. The keeper orifice diameter, 2r ko , plume expansion angle, θ , and direction of measurements, z, are also depicted from a single four-bore alumina tube that provides mechanical support and the electrical insulation between the probes. The distance between the two cylindrical Langmuir probes is x = 1.5 mm. Furthermore, the distance between the emissive probe loop and the two Langmuir probes is also 1.5 mm, leading to a measuring volume of 1.5 mm×1.5 mm×1.5 mm. The probes are inserted orthogonally to the plasma flow as shown in Fig. 2, with one Langmuir probe closer to the cathode and the second one closer to the anode.
The two Langmuir probes can be i) both in ion saturation mode, when they are biased at -36 V by using a pack of batteries to collect ion saturation currents across 1 k resistors or ii) one can be in ion saturation mode while the second one is left floating to measure the local floating potential, φ f , using a high-voltage oscilloscope probe. A 1.8 A current is passed through the emissive probe bringing it to electron emission, while the floating potential of the hot probe, φ em , is collected by a high-voltage oscilloscope probe and used to approximate the local plasma potential, φ p .
Experimentally, electron densities from around 1 × 10 17 to 3 × 10 18 m −3 and electron temperatures < 2 eV are measured in the cathode plume at I d < 1 A. This leads to Debye lengths in the range 6-30 μm, which are at least one order of magnitude smaller than the probes' diameter. By assuming quasi-neutrality, i.e. n i = n e , the thin sheath limited theory can be applied to compute the electron density, n e , in [m −3 ] [16]: where I sat is the ion saturation current collected by one of the Langmuir probes, A p is the probe collection area, and T e is the electron temperature in [eV]. Here, to compute n e , the ion current collected by the Langmuir probe closer to the cathode, I sat,1 , is used. The floating potential of the hot emissive probe, φ em , can be used to estimate the plasma potential, φ p , as follows [16]: Due to sheath effects near the surface of a hot emissive probe, in Eq. 6 a correction parameter, α, is included and is typically between 1 and 1.5. This suggests that the plasma potential is 1T e to 1.5T e above the floating potential of the probe [32]. In this study, α = 1. Finally, for the electron temperature, the following expression stands [33]: where φ f is the floating potential of the Langmuir probe closer to the anode. The oscillations in the anode/discharge current are also recorded using a Pearson wideband current monitor and this signal is used as a trigger reference when computing time-resolved analyses.
The probes' and the discharge current signals are recorded by an oscilloscope for 100 ms at 12.5 MHz, allowing for a cut-off frequency of 6.25 MHz.

Uncertainty in plasma parameter measurements
The electrostatic probes used in this study are known to induce plasma perturbations [34] and affect to a certain extent the plasma parameter measurements. The changes in plasma conditions can be linked to changes in the anode voltage when the probe array is inserted in the plasma. For the following results, the anode voltage during probe array insertion in the plasma drifted by 2-3 V, which amounts to a maximum of ±5% change in the operating conditions. This uncertainty is ultimately found in the plasma parameter measurements.
A previous study showed that the amplitude of the IAT waves can be affected by the electrostatic probes [35]. The study suggested that if the cross-sectional area of the probe surpasses 10% of the size of the plasma, the amplitude of the drift waves is affected. In the present setup, assuming a conical expansion of the plasma plume from the keeper orifice to the anode, as shown in Fig. 2, the probe is more likely to influence the IAT spectrum amplitude for locations closer to the cathode where the plume cross-sectional area is smaller. For the current setup, the nearest locations to the cathode probed are z = 2.5 mm, in the case of disk anode, and z = 3.5 mm, in the case of cylindrical anode. At these locations, the plume cross-sectional area can be calculated as follows: where θ is the plume expansion angle. Given the 0.15 mm-diameter of the ion saturation probes and the 1.5 mm-diameter of the entire probe array, their corresponding crosssectional areas remain below 2.5% of the plume cross-sectional area at locations near the cathode. This suggests minimal influence of the probe array on the IAT spectrum amplitude.
In summary, apart from the uncertainty induced by the statistical variations in the measurements, the perturbations induced by the probes to the plasma are the main contributor to the systematic error. Conservatively, in the results herein the following total uncertainty levels are considered: 50% for n e , 30% for T e , and 10% for φ p [21,[36][37][38].

Results and discussion
The following sections present several sets of results that are intended to support the hypothesis of the correlation between the ionization instability, i.e. plume mode instability and IAT at sub-ampere cathode operation. Time-averaged spectral properties of the parameters collected by the probes are shown and a cross-correlation technique is used to investigate wave propagation behavior at a single location in the plume as well as its spatial development in the cathode-anode space. Furthermore, the temporal evolutions of the IAT, ionization instability, as well as of plasma parameters are explored.

Macroscopic correlation of low-and high-frequency oscillations
The Fourier amplitude of the anode current oscillations, Ĩ a /I a,0 2 ω , whereĨ a is the fluctuation in the anode current around its mean value, I a,0 , offers a macroscopic picture of the main oscillatory phenomena in the cathode plume and their response to the change in anode current or flow rate. As shown in Fig. 3 for Kr, both quasi-coherent modes characteristic of the ionization/plume mode instability and turbulent content underlying the existence of IAT are present in the anode current spectra. Moreover, the general observed trends suggest an increase in both low-, < 250 kHz, and high-frequency contents with higher I a or lowerṁ. This simple picture also suggests a correlation between the low-frequency modes and higher frequency oscillations: they both tend to increase or decrease in tandem for a given cathode operational point, with larger amplitudes at higher current or lower mass flow rate as expected.
Next, using the signal collected by the Langmuir probe closer to the cathode and operated in ion saturation mode, the contribution of low-and high-frequency fluctuations to the total spectrum of the saturation current is dissociated and their correlation is investigated. The total spectrum energy of the low-frequency plume mode fluctuations can be computed as follows: and the IAT wave spectrum energy is given by: Fig. 3 Power spectra of the anode current for Kr for various a anode currents and b mass flow rates. The cathode is operated with the disk anode with the cut-off frequency ω c = 250 kHz for Kr. The sum over the low-frequency spectrum is performed from 10 kHz to ω c , while for the high-frequency part of the spectrum from ω c to 2 MHz. Experimentally, it is observed that for Kr, the plume mode fluctuations tend to have a frequency content extending beyond the 150 kHz usually assumed for Xe [25] and in this study the cut-off frequency is set at 250 kHz.
The results for various locations within the cathode-anode space and on the cathode centerline are shown in Fig. 4. The trends observed previously in the anode current spectra are seen for the ion saturation current as well, both at low-and high-frequency. In general, the waves amplitudes, A iz and A IAT , tend to increase with higher I a and loweṙ m. Moreover, the waves amplitudes increase further downstream the cathode orifice. The increasing IAT wave amplitude with the distance from the cathode could be explained by higher electron temperatures in a less collisional environment which leads to higher IAT growth rates as dictated by Eq. 2. At z = 2.5 mm, due to the proximity of the probe to the keeper orifice the plasma is perturbed to a greater extent, particularly at low anode current operation, as suggested by the reversed trends at 0.5 A and 0.7 A. Now, to show that the two classes of waves are indeed correlated, the bivariate coefficient and p-value are computed for data sets corresponding to individual I a values Fig. 4 Evolution of the plume mode wave amplitude, A iz , and IAT amplitude, A IAT , as functions of anode current, I a , mass flow rate,ṁ, and location in the plume, z. The cathode is operated with Kr and the disk anode and across the mass flow rate range and z of 2.5, 4.5, and 6.5 mm. This technique was employed by Georgin [22] for high-current cathode operation. The data are shown in Fig. 5 as a response plot between A iz and A IAT for three values of I a . The dotted lines are linear fits to data at 0.5 A, 0.7 A, and 1 A. The bivariate coefficient for two data sets is given by where cov is the covariance between the parameters of the two data sets and σ is the standard deviation for each data set. Values of ρ A iz ,A IAT closer to 1 suggest a positive, linear correlation between A iz and A IAT , while values closer to -1 suggest negative, linear correlations. If ρ A iz ,A IAT is close to 0, the two quantities are most probably uncorrelated. To add more certainty to this statistical analysis, the p-value is also computed for each data set under the statistical test hypothesis that the two quantities are not correlated. Under such an assumption, a small p-value (∼ 0.05 [22]) suggests that it is highly improbable to compute such results if the quantities in the data sets are not correlated, whereas a high p-value signifies that the results can be produced even if the two quantities are uncorrelated. Table 2 summarizes the results of the statistical analysis for the data sets shown in Fig. 5. At the lowest anode current, I a = 0.5 A, the weakest correlation is found, ρ A iz ,A IAT = 0.27, with a large p-value, 0.52. At higher anode currents, I a = 0.7 A and I a = 1 A, relatively high bivariate coefficients are found, above 0.75, with small p-values, below 0.05. This confirms that the growth in the plume mode oscillations amplitude is correlated with the growth in the IAT waves spectrum amplitude. The low correlation observed at 0.5 A could be explained by weaker manifestation of the plume mode oscillations and IAT and confirms the current-driven nature of the plume mode-IAT phenomena. Moreover, a higher level of plasma perturbation induced by the probe is observed at low anode current which could result in spurious results. Nevertheless, the results confirm a high level of correlation between low-and high-frequency oscillations in the plume of a Kr-fed hollow cathode operated at sub-ampere levels and suggests a mainly current-driven mechanism for the emergence of predator-prey ionization instability.

Plume plasma wave dispersion
To enter deeper into the fabric of cathode plume instabilities, the ion saturation currents collected by the two probes are used to estimate the wave dispersion, S(ω, k), with ω the frequency and k the wavenumber of a plane wave. To achieve this, the cross-correlation technique introduced by Ilić [39] and Beall et al. [40] is used, where the Fourier decompositions of the two ion saturation currents, F I sat,1 and F I sat,2 , lead to the calculation Furthermore, the wavenumber can be found from the cross-phase as follows: with x = 1.5 mm being the inter-probe separation. The physical dimensions of the probe array induce limitations to the current technique. For instance, the highest frequencies that this technique can measure cannot overpass the ion plasma frequency. Moreover, the smallest wavelength that this technique can unambiguously detect is equal to the inter-probe separation, which in this case is λ min = 1.5 mm, leading to a maximum wavenumber of k max = 2090 m −1 .
The aforementioned technique allows a simple visualization of the wave dispersion in the ω − k space in two-dimensional histograms. The entire domain is first discretized in 5 kHz frequency bins and 10 m −1 wavenumber bins, while to each bin the average Fourier amplitude computed using Eq. 11 is assigned. To reduce the noise, the analysis is averaged over 500 data sets. In the following analysis, the data are collected at z = 2.5 mm at I a = 0.5 A andṁ = 0.12 mg s −1 -Kr. Under these operational conditions, the measured plasma density is n e = 2.83 × 10 18 m −3 and the electron temperature is T e = 1.44 eV. The Debye length is then λ D = 5.3 × 10 −6 m and the ion plasma frequency is ω pi ≈ 31 MHz. Therefore, the measurements shown herein are restricted to frequencies well below ω pi .
As shown in Fig. 6a, a linear dispersion relation, ∝ ω/k, is observed for frequencies below 1 MHz. Given that the relation characterizes fairly high frequencies, the observed features are attributed to IAT waves with their dispersion relation given by Eq. 1. The group velocity of those high-frequency waves is approximated to v IAT = 3.7 ± 0.5 km s −1 , which is commensurate with previous measurements for high-current cathodes [15]. The dispersion relation for IAT states that the group velocity of such waves estimates the sum of ion sound speed and ion drift velocity, v IAT = c s + u i . Here, the ion sound speed is c s ≈ 1.1 km s −1 which leads to an ion drift velocity of u i ≈ 2.6 km/s, similar to measurements of ion velocity using laser-induced fluorescence spectroscopy in the plume of low- [41] and high-current cathodes [42,43].
For this particular example, distinctive frequency peaks were observed in the discharge current and ion saturation current spectra at 35 kHz, 70 kHz, and 140 kHz, as shown in Fig. 7. Figure 6b shows a close-up saturated view on the wave dispersion histogram at frequencies below 300 kHz and makes more evident the higher amplitude around 35, 70, and 140 kHz. The results show that these lower frequency modes, most probably related to the plume mode ionization instability, have a phase velocity v iz = 5.5 ± 0.5 km/s.
The results presented thus far depict a clearer picture of the coexistence of both lowfrequency ionization-like waves and high-frequency IAT content in the plume of a Kr-fed Fig. 6 Wave dispersion at z = 2.5 mm a full dispersion plot and b low-frequency content. The cathode is operated with the disk anode at I a = 0.5 A andṁ = 0.12 mg s −1 -Kr cathode operated at sub-ampere current levels. Both classes of instabilities are propagating towards the anode with the ionization instability waves having a larger velocity. The IAT group velocity suggests an ion drift velocity commensurate with the one experimentally measured for higher current cathodes. From a quantitative perspective, albeit the anode current here being 1 to 1.5 orders of magnitude lower than in similar studies, the high operating mass flow rate leads to comparable electron densities and temperatures in the cathode plume and thus, similar ion and electron behaviors to the ones observed for higher current cathodes. The following sections further explore the relationship between the two classes of instabilities along with their influence on the temporal evolution of plasma parameters.

Correlation of ionization instability and IAT
The last two sections shown that both the ionization instability and IAT coexist in the plume of hollow cathodes operated at currents < 1 A, while showing similar features to the high-current cathodes operation. Moreover, a first attempt at proving their correlation showed that the average amplitude of the coherent plume mode oscillations is Herein, the technique described by Georgin et al. [15] is used, where the average amplitude of the IAT spectrum, A IAT defined in Eq. 10, is monitored on the time-scale scale of the low-frequency ionization instability. To capture the low-frequency coherent part of the spectrum, the time-averaged ion saturation current is computed, I sat τ . The measurements take place over one temporal cycle of the plume mode oscillation, 1/f 0 , defined by the frequency peak, f 0 , observed in the anode current spectra. The ion current signal collected by one probe together with the anode current signal are used to perform peaktriggered averages, since both signals show the same characteristic oscillations. First, the anode current signal is filtered using a bandpass filter to keep only the low-frequency oscillations below ω c which is set to 250 kHz. Then, the ion saturation current signal is high-pass filtered for frequencies above ω c . Next, the ion saturation current signal for one period of the ionization instability is divided in smaller intervals each with the time width τ = 1/(5f 0 ). For each interval, the average ion saturation current, I sat τ , is computed and a fast Fourier transform is applied to compute A IAT . To increase the signal-to-noise ratio of the results, around 5000 averages are performed.
In Fig. 8, the time variations of the IAT,Ã IAT /A IAT,0 , and ionization, Ĩ sat τ /I sat,0 , contents in the ion saturation probe signal are shown, with A IAT,0 and I sat,0 being the The analysis led to highly correlated fluctuations in IAT and ion saturation current over the time-scale of the ionization instability. The fluctuation in IAT tends to be higher than the one in the ion current and both quantities are up to two orders of magnitude lower than the ones reported for high-current cathodes using the same analysis technique [15]. This can be explained by the much lower discharge current, 0.5 A, as compared to 22.5 A, and ten times lower mass flow rates, ∼ 0.1 mg s −1 against 1 mg s −1 . The growth in IAT leads the coherent component in phase and since Ĩ sat /I sat,0 ∝ñ e /n e,0 it can be inferred that the fluctuations in IAT lead to fluctuations in plasma density. For this example, the IAT leads the density in phase by 118.6°.
The results presented in Fig. 8 are representative of a single location in the cathodeanode space. It is important to prove that this behavior remains consistent throughout the plume or in other words the IAT and ion saturation current fluctuations are correlated spatially along the cathode's discharge plume. Figure 9 depicts the evolutions of A IAT /A IAT,0 and Ĩ sat τ /I sat,0 along the cathode discharge axis when the cathode operates at I a = 0.5 A andṁ = 0.088 mg s −1 -Kr together with the disk anode. The color maps show that the IAT and the coherent oscillations are correlated in time and space along the cathode discharge axis, with the IAT leading in phase the fluctuation in ion saturation current. Furthermore, the velocity of the IAT is measured to be around v = 3.4 ± 1.12 km s −1 and shows a displacement towards the anode as depicted in Fig. 9a, while the structures observed for the density fluctuation in Fig. 9b seem to move towards the anode with a similar velocity, v = 3 ± 1.12 km s −1 . The results are commensurate with the IAT velocity revealed by the dispersion plot in Fig. 6a, where the cathode is operated at the same anode current but a slightly higher mass flow rate. On top of that, the results show that the level of fluctuations for both the IAT and coherent structure seem to increase in the far-field plume, towards the anode, if one ignores the high fluctuation in IAT measured at z = 2.5 mm which may be a spurious result due to higher plume perturbation by the probe array. This features are different from the ones reported in literature [15] for cathode operation with a cylindrical anode, where both the IAT and the coherent fluctuations decrease in intensity towards the anode. It is possible that the disk anode used in this Fig. 9 Oscillatory behavior of a IAT and b ion saturation current over the ionization instability time-scale for Kr at I a = 0.5 A andṁ = 0.088 mg s −1 along the cathode-anode space on the discharge centerline. The cathode is operated with the disk anode placed at L ca ≈ 9 mm from the cathode orifice. The maps show three concatenated waveforms study induces higher neutral and plasma densities in the far-field plume which may sustain a higher turbulent content at the anode proximity. In the next section, results with a cylindrical anode are presented.

Plasma parameters evolution over the ionization instability time-scale
Plume mode ionization-like instability is related to ionization processes in the cathode plume. Thus far, it was shown that a periodic fluctuation in the IAT content modulated over the ionization instability time-scale induces fluctuations in the coherent content of the signal collected by an ion saturation probe. It was also shown that the two fluctuations are correlated in time and space, with the IAT leading in phase the coherent oscillations. Therefore, it is important to investigate how other plasma parameters such as density and temperature respond to the temporal variation in the IAT and coherent oscillations. The collisional environment is explored here in terms of total electron collision frequency and its spatial and temporal evolution over the ionization instability time-scale.
For this study, the cathode is operated with the cylindrical anode as shown in Fig. 2b. Based on the collected data, the plasma density, n e , and electron temperature, T e , are calculated using Eqs. 5 and 7. Next, the electron behavior in the cathode plume can be modeled by the generalized Ohm's law for electron momentum: where ν e is the total electron collision frequency, E = −∇φ p is the electric field with φ p the local measured plasma potential computed using Eq. 6, and u e is the electron drift velocity. The total electron collision frequency can be written as the sum of the Coulomb collision frequency, ν ei , electron-neutral collision frequency, ν en , which are both considered as classical collisions, while the ionization frequency, ν iz , tends to have a much lower contribution to the electron dynamics [16].
The classical collision frequencies are found using the following equations for ν ei : where the Coulomb logarithm is: and for ν en : where σ en is the electron-neutral collision cross section. The electron-neutral scattering cross section as a function of electron temperature for Kr, σ en (T e ) [m 2 ], is given by (for T e > 0.4 eV) [44]: Next, the ionization frequency is estimated by ν iz = n n U iz , where the ionization reaction rate for Kr, U iz , given by (for T e < 5 eV) [44]: (18) and the neutral density is taken as (assuming a conically expanding plume from the keeper orifice to the anode): where A z is the plume cross-sectional area at the location z computed using Eq. 8 and v n = 8k B T g /(πM) is the neutral velocity with T g ≈ T i ≈ 0.1 T e . Finally, the contribution of the IAT to the total electron collision frequency, known as the anomalous collision frequency, ν IAT an , can be computed using Eq. 3. However, in an experimental context the next approximation can be madeñ e / n e ≈Ĩ sat /I sat,0 [16], where I sat is the ion saturation current collected by the probe with its mean value I sat,0 . With this, Eq. 3 becomes: where 0.25 ≤ ω r ≤ 2 MHz. Figure 10 shows the time-averaged results. When the cathode operates with Kr at I a = 0.8 A andṁ = 0.145 mg s −1 , a distinct frequency peak is observed in the anode current PSD at f 0 = 85 kHz. The fluctuations in the IAT and coherent content as measured by the ion saturation probe show a high degree of correlation, with the IAT leading in phase the density as depicted by Fig. 10a-b. As opposed to the disk anode, for the cylindrical anode case the two parameters tend to decrease with the distance from the cathode's keeper orifice. The density fluctuations in Fig. 10b, seem to emerge from two distinct locations in the plume: around z = 6 mm and moving towards the cathode and the anode with a velocity around v = 3.57±1.12 km s −1 and at z = 14 mm and moving towards the anode at a slightly lower velocity of v = 2.92 ± 1.12 km −1 . The time-averaged plasma parameters along the cathode discharge axis are shown in Fig. 10c. The electron density drops almost one order of magnitude from slightly over 1.2 × 10 18 m −3 to 1.8 × 10 17 m −3 and depicts two distinct changes in gradient at z = 6 mm and z = 14 mm, in correspondence to the higher fluctuations in saturation current shown in Fig. 10b. The electron temperature follows a slightly increasing trend from 1.3 eV at z = 3.5 mm to 1.76 eV at the anode. The two parameters are used to approximate the gradient ∇(n e eT e ) as well as the other quantities used to compute the various collision frequencies. As presented in Fig. 10d, the sum of classical collision frequencies is orders of magnitude lower than the total electron collision frequency computed using Ohm's law of electron momentum. Moreover, ν en + ν ei shows a monotonically decreasing trend further away from the cathode, while ν e seems to slightly increase before reaching a plateau. The ionization frequency, ν iz , is the lowest contributor, decreasing with the distance from the cathode due to lower collisionality and steep drop in neutral density. This parameter shows a slight increase after z = 14 mm, in correlation with the high fluctuations in ion saturation current at this location. The results prove that the main contributor to the electron dynamics in the plume of a sub-ampere hollow cathode operated with Kr remains the anomalous collision frequency, ν IAT an , which estimates both the magnitude and spatial evolution trend in the total electron collision frequency. Figure 10d depicts also the estimated anomalous collision frequency at the Sagdeev-Galeev limit, ν SG an , computed using Eq. 4 where the electron drift velocity at a given location z can be estimated with u e ≈ I a /(en e A z ) and the electron thermal velocity is given by v e = √ 8eT e /(πm). This parameter follows closely the classical components, showing a decreasing trend most probably due to the rapid decay in plasma electron frequency, ω pe , and it is apparent that a coefficient α IAT = 0.01 and the assumption of T e /T i ≈ 10 does not capture the magnitude of the total electron collision frequency especially for locations beyond z = 6 mm towards the anode.
The next set of results presented in Fig. 11 offer information on the temporal variation of plasma parameters such as density, electron temperature, IAT collision frequency, and electron drift velocity with respect to the anode current phase, ϕ. A peak-triggered average technique similar to the one described in Correlation of ionization instability and IAT section is used here, where the anode current is bandpass-filtered below Fig. 11 Normalized fluctuations in plasma parameters as functions of the phase angle with respect to the anode current. The measurements are for a z = 9 mm and b z = 22 mm, while the cathode is operated with Kr at I a = 0.8 A andṁ = 0.145 mg s −1 with the cylindrical anode ω c = 250 kHz, and its peaks are used to define averaging intervals. Each interval is further divided into 20 points. In this way, the fluctuation in each parameter is reconstructed over the ionization instability time-scale, which in this case is 11.76 μs (f 0 = 85 kHz). Three cycles are concatenated to produce the plots in Fig. 11 for two locations in the cathode plume at z = 9 mm in Fig. 11a and at z = 22 mm in Fig. 11b. At both locations, the results show that the IAT wave energy, represented as ν IAT an , leads the electron temperature in phase by 36°at z = 9 mm and by 126°at the anode location. The electron temperature leads then the density in phase by around 55°at both locations. This behavior was observed at high-current cathode operation [15] and suggests that the plume mode is indeed an ionization related process where the IAT growth in the plume heats the electrons which then induce increased ionization. The density and the electron drift velocity are out of phase, with the velocity leading the density by 270°at 9 mm and 200°a t the anode location. This is expected since this lag allows current continuity, whereas the synchronized parts of the two waveforms are responsible for the fluctuations in the anode current. The anomalous collision frequency is correlated with the total electron collision frequency showing oscillations of similar amplitudes. The large oscillations in the total electron collision frequency, led by the anomalous component, induce large temperature oscillations, up to 20% nearer to the cathode and 10% at the anode location. Moreover, at the anode, the density and electron drift velocity show much larger fluctuations as compared to the location closer to the cathode. Nevertheless, here the out-of-phase relationship between the two parameters leads to lower fluctuations in I a , of 10% as compared to 20% at z = 9 mm. The complex structures of the waveforms at the location nearer to the cathode and the more cnoidal evolution of the parameters at the anode are proof of the nonlinear inter-dependance between the plasma oscillations, plasma parameters, and macroscopic discharge parameters such as the anode current.

Summary
The results presented in the previous sections bring more insights on the various oscillatory content found in the plume of a low-current hollow cathode operated with Kr. The correlation between the high-frequency IAT content and low-frequency coherent oscillations is visible macroscopically in both anode and ion saturation current spectra respectively and its sensitivity to the increase in anode current or flow rate suggests an underlying ionization-related process. The dispersion plot proved the existence of both classes of waves in the cathode plume, with the low-frequency component traveling towards the anode at velocities higher than the IAT. Interestingly, the ion drift velocities for sub-ampere operation are comparable to high-current cathode operation. As the anomalous collision frequency dictates the electron dynamics as shown by the results in Fig. 10d, the large oscillations in the IAT wave energy over the ionization instability time-scale plotted in Fig. 11 are most probably responsible for the anomalous electron heating given the lag in the electron temperature. The density lags behind the electron temperature, supporting the hypothesis of a predator-prey mechanism for the ionization instability at very low current too. Low anode currents induce low electron drift velocities, while relatively large plasma potentials lead to collision frequencies comparable to the ones obtained in the plume of high-current cathode, O(10 8 − 10 10 ) Hz.

Conclusions
This work provided an investigation on the main instabilities in the plume generated by low-current hollow cathodes operated with Kr. It was shown that the nature of the plume mode for low-current operation remains similar to the one of high-current cathodes: predator-prey relation between the temperature and density. The onset of such instability is still intrinsically related to the growth of IAT in the cathode plume. Both IAT and coherent fluctuations reside in the plume of a cathode operating in plume mode as observed in the dispersion plot and are highly correlated especially at larger currents, suggesting a current-driven process. The dispersion plot suggested that the ionization instability traveled towards the anode at velocities in the order of v iz ∼ 5 km s −1 . The same dispersion plot showed that the IAT waves traveled also towards the anode at velocities in the order of v IAT ∼ 4 km s −1 , implying that the ion drift velocities were u i ∼ 2.5 km s −1 , commensurate to ion velocities measured in the plume of higher-current cathodes. The growth and damping of the IAT is still modulated in time over the time-scale of the coherent lowfrequency oscillation. The correlation remains valid throughout the plume where the IAT leads the fluctuation in temperature in phase, which then leads the density, suggesting a possible ionization-related process. Nevertheless, quantitatively at sub-ampere levels, the coherent wave energy and the IAT energy are 1.5-2 orders of magnitude lower than in the case of high-current cathode operation and proportional to the discharge current ratio between the two classes of cathodes. The results presented here are meant to expand the understanding of plasma instabilities and the link between the plume mode oscillations and IAT from high-current to low-current cathode operation. They can help the effort for higher fidelity cathode numerical modeling at sub-ampere operation.