Generation of runaway electrons in plasma after a breakdown of a gap with a sharply non-uniform electric field strength distribution

The experimental studies were carried out on a setup (figure 1(a)) consisting of a gas-discharge chamber combined with a transmission line (wave impedance Z is 75 Ω), an NPG-18/3500 N high-voltage nanosecond voltage pulse generator, a LeCroy WaveMaster 30Zi oscilloscope (16 GHz, 40 GS s⁻¹) and a HSFC-PRO four-channel ICCD camera. Voltage pulses produced by the high-voltage generator (figure 1(b)) were applied across a discharge gap via a 3 m high-voltage coaxial cable (Z = 75 Ω) and the transmission line.

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Voltage pulses U_CVD were measured with a capacitive voltage divider (CVD) built into the transmission line. The voltage across the gap U_gap was restored from the incident U₀ (figure 1(b)) and reflected U_ref waves which was defined as the difference between U_CVD and U₀. The discharge current I_d can also be defined from the incident and reflected waves: I_d = (U₀ − U_ref)/Z. The short-circuit current I_sc is defined by 2U₀/Z.

A high-voltage electrode was made of a piece of a 5 mm long sewing needle (stainless steel). The diameter of the sewing needle was 1 mm and a radius of curvature of the needle tip was 75 μm. The grounded electrode was flat with a 1 cm in diameter hole in the center. A perforated grid (grid #1 in figure 1(a)) with a transmittance of 0.75 was installed in the hole. The hole edge was rounded and polished to eliminate the non-uniformity of the electric field strength distribution at the edge-grid interface. This made it possible to avoid the closure of the discharge channel to the edge. The distance between the high-voltage and grounded electrodes was d = 8.5 mm.

The current I_REs of REs leaving the discharge gap through the grid #1 was measured with a collector placed downstream of the grounded electrode. A grounded grid #2 with a transmittance of 0.3 was located in front of the receiving part of the collector (see figure 1(a)) to exclude the influence of the electric field penetrating through the grid #1.

Figure 2 shows that the electric field partially penetrates through the grid #1. In the space between the anode and the collector, the electric field strength directly depends on the electric field strength near the anode surface in the gap. When grid #2 is absent, it is possible to measure a displacement current caused by the change in the electric field strength. The appearance and propagation of a streamer causes a redistribution of the electric field in the gap. In the space between the anode and the collector, the electric field begins to change over time and the displacement current flows. We call it the dynamic displacement current I_DDC to separate from a displacement current caused by a change in the voltage and to 'highlight' the role of the streamer [38, 39].

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From the waveforms of I_DDC, it is possible to determine with high accuracy when a streamer appears and when it arrives to the grounded electrode [38, 39]. When there is the waveform of the sum of I_REs and I_DDC, it is possible to determine with high accuracy when REs appear relative to the streamer dynamic [40].

The development of plasma emission in the discharge gap was recorded with the four-channel ICCD camera. This camera takes four consecutive images with a known delay relative to each other per one discharge implementation. The minimum exposure time of the ICCD camera channels is 3 ns. In the experiment, the exposure time for three channels was 3 ns, and for the fourth was 20 ns that corresponds the total voltage pulse duration. Thus, the first three channels (C1–C3) made it possible to study the dynamics of the discharge formation, and the fourth channel (C4) made it possible to obtain an integral image of the discharge.

The oscilloscope recorded signals from the CVD, collector as well as a sync signal from the first channel of the ICCD camera. This made it possible to synchronize the ICCD images and the waveforms of voltage and RE current.

The gas-discharge chamber was pumped out with a fore vacuum pump and then filled with air, helium (He), and sulfurhexafluoride (SF₆). The pressure was varied in the range of 12.5–100 kPa.

A discharge current density j_d is related to an electron concentration n_e by the known equation:

$$\begin{equation}j = e{v_{\text{d}}}{n_{\text{e}}},\end{equation} \tag{ 1 }$$

where e—the elementary charge, v_d—the drift velocity of an electron:

$$\begin{equation}{v_{\text{d}}} = {{\mu}_{\text{e}}}E,\end{equation} \tag{ 2 }$$

$$\begin{equation}E = \frac{{{U_{{\text{gap}}}}}}{d},\end{equation} \tag{ 3 }$$

where µ_e—the electron mobility, E—the electric field strength, U_gap—the gap voltage, d—the distance between electrodes.

The discharge current density can be calculated by

$$\begin{equation}j = \frac{{{I_{\text{d}}}}}{S},\end{equation} \tag{ 4 }$$

where I_d—the discharge current, S—the cross-sectional area of a plasma channel with a diameter D that can be determined in the middle of the gap from the ICCD images taken with the fourth channel (C4).

Thus, the electron concentration can be estimated by a fairly simple equation:

$$\begin{equation}{n_{\text{e}}} = \frac{d}{{e{\mu _{\text{e}}}S}} \cdot \frac{{{I_{\text{d}}}}}{{{U_{{\text{gap}}}}}}.\end{equation} \tag{ 5 }$$

This equation is valid under the assumption that after breakdown the electric field is uniformly distributed in the plasma channel.

As is known, a time-varying electric field E(t) causes a displacement current j_displ(t):

$$\begin{equation}{j_{{\text{displ}}}}\left( t \right) = \varepsilon {\varepsilon _0}\frac{{\partial E\left( t \right)}}{{\partial t}},\end{equation} \tag{ 6 }$$

where —the permittivity of a medium, ₀—the vacuum permittivity.

The displacement current density j_c(t) near the anode surface with area S_a caused by a change in the voltage U_gap(t) across the gap having a capacitance C_gap is determined by the equation:

$$\begin{equation}{j_{\text{c}}}\left( t \right) = \frac{1}{{{S_{\text{a}}}}}{C_{{\text{gap}}}}\frac{{\partial {U_{{\text{gap}}}}\left( t \right)}}{{\partial t}}.\end{equation} \tag{ 7 }$$

The appearance of a streamer in the gap causes a redistribution of the electric field in it and, accordingly, a change in the capacitance C_gap. In this case, a displacement current flows due to the redistribution of the electric field. We call it as the dynamic displacement current I_DDC. The density of I_DDC near the anode surface with area S_a is determined by

$$\begin{equation}{j_{{\text{DDC}}}}\left( t \right) = \frac{{{I_{{\text{DDC}}}}\left( t \right)}}{{{S_{\text{a}}}}} = \frac{1}{{{S_{\text{a}}}}}{U_{{\text{gap}}}}\left( t \right)\frac{{\partial {C_{{\text{gap}}}}\left( t \right)}}{{\partial t}}.\end{equation} \tag{ 8 }$$

The total displacement current density j_Σ near the anode surface is given by

$$\begin{equation} {j_{\Sigma}}\left( t \right) {=} \frac{1}{{{S_{\text{a}}}}}\left( {{C_{{\text{gap}}}}\left( t \right)\frac{{\partial {U_{{\text{gap}}}}\left( t \right)}}{{\partial t}} + {U_{{\text{gap}}}}\left( t \right)\frac{{\partial {U_{{\text{gap}}}}\left( t \right)}}{{\partial t}}} \right) {=} \frac{{{I_{\Sigma} }\left( t \right)}}{{{S_{\text{a}}}}}.\end{equation} \tag{ 9 }$$

From (6) and (9), the E_anode(t) can be calculated as

$$\begin{equation} {E_{{\text{anode}}}}\left( t \right) = \frac{1}{{\varepsilon {\varepsilon _{\text{0}}}{S_{{\text{anode}}}}}}{\int I _\Sigma }\left( t \right){\text{d}}t. \end{equation} \tag{ 10 }$$

The total displacement current I_Σ can be measured experimentally using both a current shunt and a collector placed downstream the grid anode (figures 1 and 2). In this case, it is necessary to take into account the weakening of the electric field by the grid anode, as well as the area of the receiving part of the collector.

On the other hand, E_anode(t) can be calculated in relative units. Then, knowing the distribution of the electric field strength in the gap in absolute values (figure 2) and E_anode(t) in relative units in idle mode (I_DDC = 0), the coefficient for converting the E_anode(t) from relative values to absolute ones can be calculated.

When applying voltage pulse across the gap, a large-diameter streamer develops over the entire pressure range. This is due to the extremely inhomogeneous distribution of the electric field strength in the gap (figure 2) and its extremely high strength. As a result, the ionization front moves not only in the direction of the opposite electrode, but also in the radial direction due to the high value of E_r at the initial stage.

Due to the high electric field strength on the cathode surface (figure 2), the initial electrons are provided by field emission from it. The emission current density depends on the electric field strength and is described by the Fowler–Nordheim equation [41].

The cathode surface is not perfectly smooth. There are microprotrusions of various sizes (∼0.1 µm and smaller) on it. The electric field strength increases near the micro-protrusions by a factor β of ∼10–100 [41]. When the emission current density reaches ∼10⁹A cm⁻², micro-protrusions explode in a few nanoseconds as a result of heating, and the explosive electron emission occurs. With β of 100, the emission current density reaches 1 × 10⁹A cm⁻² at a voltage across the gap of 19 kV. In this case, the emission current through an emitter with an area of ≈1 × 10⁻¹¹cm⁻² is ≈10 mA.

Figure 3 shows the waveforms of the voltage and current of an electron beam at a residual pressure in the gas discharge chamber of ≈10 Pa.

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When measuring the electron beam current, a collector 3 mm in diameter was used. It was estimated that the collector captures no more than 10% of the total flow of electrons moving from the pointed cathode. Before the experiment, the tip of the cathode was exposed to light sanding paper. This made it possible to increase the surface roughness and the number of microprotrusions. However, with each pulse, the number of microprotrusions decreased due to their explosion. At the same time, microdroplets scattering during the explosion are capable of forming new microprotrusions.

Figure 3 shows that at the first implementation, the emission current arises at a voltage across the gap of about 20 kV. In subsequent implementations, the emission current pulses were recorded randomly at different times. This corresponds to the random nature of the explosion of microprotrusions on the cathode surface.

A characteristic feature of the dynamics of streamer formation in a sharply inhomogeneous electric field is a high propagation velocity of the streamer when it starts and when it approaches the opposite electrode [42]. Herewith the propagation of the streamer in the gap is accompanied by the flow of current, the magnitude of which directly depends on the streamer velocity, and a decrease in the voltage across the gap, since the current is already flowing in an external circuit. This means that the microprotrusions can explode, in particular, when the streamer propagates in the gap.

Figure 4 shows the waveforms the sum of the dynamic displacement current I_DDC and С_gapdU_gap/dt, measured using the collector (figures 1 and 2) under conditions close to the threshold for breakdown (U_max = 26 kV), as well as shows the corresponding E_anode(t) calculated by the equation (10) for various discharge implementations.

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In implementations without a discharge (idle mode) only a capacitive current is measured with the collector (figure 4, С_gapdU_gap/dt). In these cases, the dependence of E_anode on time (figure 4, $E_{{\text{anode}}}^{{\text{idle}}}$) strictly corresponds to the temporal shape of the voltage pulse (figure 1). When a streamer appears in the vicinity of the pointed electrode, the electric field is redistributed and the corresponding displacement current I_DDC, which we call the dynamic displacement current, flows. From the waveforms of I_DDC it is possible to determine the moment when the streamer appears ('s' in figure 4(a)) and the moment when it arrives at the opposite electrode ('f' in figure 4(a)). In implementations #1–4 (figure 4), the streamer crosses the gap in a few nanoseconds. At the same time, in implementation #4, the electric field strength at the anode barely exceeds the threshold value for breakdown and is barely enough for gas ionization in this region. In implementations #5–8 (figure 4), the streamer stops somewhere in the gap due to the falling edge of the voltage pulse. The electric field strength at the anode has not reached the threshold value.

Thus, the calculated E_anode values are adequate to the observed dynamics of the discharge formation.

Another feature of the dynamics of streamer formation in a sharply inhomogeneous electric field is the generation of REs ( > 10³ eV). An electric field is an accelerating force acting on an electron, while the gas has a braking effect, which is due to the loss of electron energy for ionization and excitation. The braking force F is described by the following equation [12]:

$$\begin{equation}F\left( \varepsilon \right) = 2\pi {e^4}nZ\frac{{\ln \left( {2\varepsilon /I} \right)}}{\varepsilon }\end{equation} \tag{ 11 }$$

where e—the elementary charge, n—the gas concentration, Z—the number of electrons in a molecule (atom), —the electron energy, I—the average energy of inelastic loss.

Figure 5 shows F()/e (V cm⁻¹) for nitrogen at atmospheric pressure. Air contains electronegative oxygen which slightly increases the value of F.

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It is seen that the braking force has a maximum F_max() at an electron energy of ≈10² eV, then it decreases. This is explained by the fact that all interaction cross sections have a maximum and decrease at high electron energies. This means that with an increase in the electron energy, the probability of its collision with gas molecules decreases, and it can go into the runaway mode. The value of F_max()/e determines the critical electric field strength E_cr, at which any electron can become a runaway. For air at atmospheric pressure, E_cr ≈ 5⋅10⁵ V cm⁻¹. In the vicinity of the pointed electrode (cathode), within a radius of ≈0.1 mm, the electric field strength exceeds the critical value for generation of REs (figure 2). As a result, some electrons emitted from the cathode are able to go into the runaway mode, simultaneously ionizing the gas in the vicinity of the cathode. Secondary electrons are also in a strong electric field, and some of them also become runaway. It should be noted that, due to the highly inhomogeneous potential distribution, some electrons emitted from the cathode are able to gain energy more than 10³ eV after passing a distance of ≈0.01 mm from the cathode (figure 6).

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Thus, in the prebreakdown stage of a discharge, a RE beam is formed, which preliminarily ionizes the gas in the gap [43].

However, it was found that REs can be generated not only in the prebreakdown stage [36]. The generation of REs can occur after bridging the gap with plasma. This will be discussed below.

Figure 7 shows ICCD images of a discharge formed in air at atmospheric pressure, as well as the gap voltage U_gap and RE current I_REs obtained in the experiment with the ICCD camera (the experiment #1). In a separate series of experiments (the experiment #2), waveforms of the sum of the dynamic displacement currents I_DDC and RE current I_REs were recorded. This made it possible to synchronize the RE current, gap voltage and ICCD images with each other. Additionally, the electron concentration n_e in the plasma channel after breakdown as well as the electric field strength at the grounded electrode (anode) E_anode were calculated.

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In each experiment, ∼10² waveforms were recorded. Then, the implementations with the same breakdown delay were selected. It is seen from figure 7 that if in two different experiments the ionization processes start under the same voltage, then the dynamics of the discharge formation is reproduced. Even the smallest voltage fluctuations are reproduced (figures 7, 9, 10). This indicates that they are not accidental, but caused by certain ionization processes.

The ionization processes begin, as expected, near the cathode tip and the formation of a large-diameter streamer is observed. However, after breakdown, the diameter of the discharge channel becomes smaller. This is due to the fact that over time, the discharge current is concentrated mainly in the axial zone. In [44], the stage of formation of large-diameter negative and positive streamers was not observed due to its short duration (<100 ps), and the emission intensity of the discharge channel after breakdown was higher. The later stage of the discharge formation was probably captured (figure 4 in [44]). The formation of such streamers is inevitable due to a sharply inhomogeneous distribution of the electric field strength and high overvoltage. For example, in simulation [45], the diameter of a negative streamer is comparable to the length of a point-to-point gap with d = 2.5 mm.

The formation of a dense plasma causes a redistribution of the electric field in the gap. As a result, a current begins to flow in the circuit and the gap voltage decreases. An increase in the dynamic displacement current I_DDC caused by the appearance of a streamer is observed. At the same time, the electric field strength at the anode E_anode begins to increase. The RE current I_REs appears ≈80 ps later than I_DDC appears. However, it is important to take into account the fact that the flow of electrons formed near the cathode moves slower than the electromagnetic wave caused by the appearance of a dense plasma also near the cathode. Taking this factor into account, the RE current pulse should be shifted to the left by ≈75 ps. This means that the generation of REs occurs in the vicinity of the cathode tip at the start of gas ionization processes initiated by field emission. The generation of REs probably continues until the electric field in the vicinity of the cathode weakens due to the formation of a dense plasma (streamer). At the same time, the electric field strength at the front of the forming streamer also decreases due to an increase in its diameter.

As the streamer moves in the gap, the current flowing through the gap decreases (I_d in figure 7). This correlates with measurements of the instantaneous streamer velocity [42]. The lower the streamer velocity, the slower the redistribution of the electric field in the gap occurs.

When the streamer approaches the anode, a change in the polarity of I_DDC is observed (figure 7). At this moment, the electric field strength near the anode E_anode reaches a maximum. As the gas ionizes near the anode, E_anode decreases. The electric field is redistributed along the plasma channel—a backward ionization wave is formed. A characteristic propagation of the emission front in the opposite direction was observed at applying a streak-camera [42, 46] and in simulation [47]. It precedes the formation of a secondary streamer. The propagation of the backward ionization wave was observed during the development of a discharge at a voltage across the gap of 20 kV (figure 8). Under these conditions, the propagation velocity of the backward ionization wave was rather low and it was observed using the four-channel ICCD camera. From a comparison of frames 2 and 3 in figure 8, it can be seen that the emission front moves towards the pointed electrode (cathode). At the same time, the emission intensity is increased in the vicinity of the cathode tip. This may indicate an increase in the electric field strength near the cathode because nitrogen molecules are efficiently excited at a high reduced electric field strength (≈200 V cm⁻¹ Torr⁻¹).

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Under experimental conditions corresponding to figure 7, the electric field is redistributed very quickly. This follows from the duration of the positive part of signal of I_DDC (≈0.2 ns) in figure 7. A similar dynamics of the electric field redistribution in the gap was observed in a recent simulation [48]. The authors of [48] also note the high rate of this process (tens of picoseconds).

The analysis of hundreds of discharge implementations shows that the backward ionization wave is always formed, and its propagation velocity is an order of magnitude higher than that of the primary wave (streamer propagation). The backward ionization wave begins to form at the stage of gas ionization in the vicinity of the grounded electrode and is caused by the redistribution of the electric field strength along the plasma channel having an inhomogeneous distribution of an electron concentration. In the near-electrode regions, the electron concentration can be an order of magnitude higher than that in the middle of the gap [48]. As a result, after bridging the gap with plasma the highest electric field strength is observed in the middle of the gap as well as in the cathode region.

After the breakdown of the gap, when the distribution of the electric field strength is close to uniform, it is possible to calculate the electron concentration n_e(t) in the discharge channel using equation (5). Figure 7 shows the dynamics of n_e(t). It can be seen that n_e is ≈1 × 10¹⁴ cm⁻³ after the breakdown and increases at least twofold during discharge. The increase in the electron concentration after the breakdown of the gap is probably due to the fact that the average electric field strength U/d exceeds the threshold value (32 kV cm⁻¹). From the dynamics of U(t)/d and n_e(t), it is seen that n_e grows until U/d becomes less than the threshold value (figure 7).

Figure 9 shows ICCD images of a discharge formed in air at a pressure of 50 kPa, as well as the gap voltage U_gap and RE current I_REs (the experiment #1). With a twofold decrease in the air pressure, the discharge formation time, measured by DDC taking into account the forward and backward ionization waves, decreased ≈1.7 times (from ≈850 to ≈500 ps). The generation of REs still occurs at the very beginning of the streamer formation, but the amplitude of the RE current pulse has doubled.

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It should be noted that a twofold decrease in air pressure did not significantly affect either the electron concentration n_e in the discharge channel after breakdown, or the maximum value of the electric field strength near the anode E_anode. Thus, a decrease in pressure only affected the rate of ionization processes.

After the breakdown of the gap, the electron concentration in the plasma channel continues to grow slowly. It increases by ≈4 times until U/d becomes less than the threshold value.

Qualitative changes in the generation of REs began to occur at an air pressure of 25 kPa and below. Implementations of a discharge with a repeated generation of REs (after breakdown of the gap) were observed. In this case, the amplitude and duration of the RE current pulse could be several times greater than that of the RE current pulse generated at the start of ionization processes near the cathode tip. An example of such a discharge implementation is shown in figure 10.

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The waveform of I_DDC + I_REs (figure 10) proves that the second RE beam is generated after the gap breakdown; the second RE current pulse is observed after the formation of the streamer and the propagation of the backward ionization wave.

The data obtained indicate that conditions for electron runaway can arise in a gas discharge even after breakdown. The peculiarity of this phenomenon is that after breakdown, as a rule, the electric field in a plasma channel is almost uniform and cannot exceed the critical value. The fact that REs are generated after the breakdown of the gap (figure 10) indicates that the electric field strength is still high in some region of the discharge. The vicinity of the cathode can be such a region. As the plasma forms and electrons drift towards the anode, a positive space charge of the ions is formed. If this charge is not compensated by electrons emitted from the cathode, then a cathode potential drop zone with a high electric field strength appears. An additional increase in the electric field strength in this zone is provided by the backward ionization wave moving towards the cathode. This leads to the fact that the electrons in the cathode layer are able to gain energy sufficient to run away in a weak electric field in the plasma channel.

The decrease in pressure still did not significantly affect either E_anode or n_e. At the active stage of the discharge, the electron concentration slowly increases by a factor of 3–4 until U/d becomes less than the threshold value (figure 10).

At an air pressure of 12.5 kPa, the generation of the second RE beam lasts longer than at 25 kPa (figure 11).

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The question arises: what limits the duration of the generation of the second RE beam? On the one hand, it should be determined by the voltage pulse duration. However, the cathode layer can disappear as a result of the explosion of the microprotrusions and the formation of an explosive-emission plasma with a high concentration of electrons. An example of such a discharge implementation is shown in figure 12.

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It can be seen that after the breakdown, the generation of the second RE beam begins. However, the appearance of a cathode spot abruptly interrupts it (frame C3 in figure 12). The formation time of the cathode spot t_f is determined by the equation [41]:

$$\begin{equation}{j^2}{t_{\text{f}}} = {\text{const}}\end{equation} \tag{ 12 }$$

where j—a current density. It follows from (12) that t_f strongly depends on the density of the current flowing through the microprotrusion.

At a low gas pressure, the plasma is adjacent not only to the tip of the cathode, but also to its lateral surface. As a result, a current is distributed over a larger area; the current density becomes less. For this reason, the duration of the current pulse of the second RE beam at a pressure of 12.5 kPa is longer than at a pressure of 25 kPa.

However, the microrelief of the cathode surface and the size of microprotrusions changes randomly from implementation to implementation due to the explosion of the ones. For this reason, microprotrusions of a small area can be formed. The density of a current flowing through such a microprotrusion is greater than a certain average value characteristic of most other microprotrusions and it can explode quickly even during streamer propagation. Therefore, the generation of REs after the streamer propagation (after breakdown) does not occur.

The experiments have shown that the generation of REs both during breakdown and after it occurs not only in air, but also in helium at a pressure of 100 kPa and sulfur hexafluoride (SF₆) at 1.5 kPa (figure 13).

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From the point of view of generation of REs, these gases are polar. At atmospheric pressure, the critical electric field strength for helium is ≈100 kV cm⁻¹, and for SF₆ more than 5000 kV cm⁻¹ [13].

The result obtained allows us to make the assumption that REs and accompanying x-ray radiation can be generated in helium plasma jets devices with a pointed potential electrode when a negative streamer (a plasma bullet) arrives at the surface of an object to be treated with plasma.

Thus, the generation of REs after the gap breakdown is an ordinary phenomenon. It strongly depends on the emission characteristics of the cathode, its microrelief, and the electric field strength in its vicinity.