Runaway electrons during subnanosecond breakdowns in high-pressure gases

The parameters of runaway electrons produced in nanosecond high-voltage discharges in different gases (air, nitrogen, sulphur hexafluoride, krypton, argon, methane, neon, hydrogen, helium) at atmospheric and higher pressure were studied. An optical analysis was also performed to investigate the ionisation dynamics in diffuse discharges in nitrogen and nitrogen-containing mixtures. At breakdown of a point-to-plane gap by nanosecond (≃2 ns) high-voltage (≃200 kV) pulses of negative voltage polarity and gas pressure above 0.1 MPa, a supershort avalanche electron beam (SAEB) was detected by a collector behind the flat anode. For pressure >0.1 MPa of nitrogen and other gases it is shown that the maximum pressure for SAEB registration decreases with increasing the voltage pulse rise time. Therefore, to detect a SAEB at atmospheric and higher gas pressure, one should use voltage pulses with an amplitude of hundred kilovolts and rise time of ∼1 ns and shorter. The experimental research in the dynamics of optical radiation from the discharge plasma shows that the breakdown in which runaway electrons are produced develops as an ionisation wave.


Introduction
Now, much attention is being focused on research and application of nanosecond discharge plasmas produced at atmospheric pressure [1][2][3].For practical application, it is promising to use diffuse discharges at atmospheric pressure which are formed in an inhomogeneous electric field with no preliminary gas ionisation (preionisation) by an external source.Back in the last century, it was shown that this type of discharge develops with the participation of X-rays and runaway electrons which arise early in the discharge and provide gas preionisation [4][5][6].In recent years, interest in the phenomenon of runaway electrons has greatly increased, see collection book [3], reviews [7,8], and papers .There are four main groups of discharges in gases, with exception of those used in plants for controlled thermonuclear fusion, in which runaway electrons and/or X-rays were registered.
The first group is discharges in the lower and upper layers of the Earth atmosphere [9][10][11][12][13].X-rays were detected in lightning, blue jets, and sprites.The possibility of high-energy electrons occurring in different types of atmospheric discharges was confirmed by simulation data.
The second group is repetitive pulsed discharges at comparatively low pressures and small interelectrode distances, including those in a homogeneous electric field [14,15].
The third group is discharges in large (metre) gaps with an inhomogeneous electric field distribution [16][17][18].Under these discharge conditions, X-rays were detected in atmospheric pressure air at applying voltage pulses of negative polarity with an amplitude of ∼1 MV and rise time of hundred nanoseconds to several microseconds to an electrode with a small curvature radius.
The fourth group is runaway electron preionised diffuse discharges [3][4][5][6][7][8] or shortly REP DDs [7].For the formation of diffuse discharges in high-pressure gases, including air, in an inhomogeneous electric field, it is required to apply short voltage pulses, normally of several to tens of nanoseconds, with an amplitude of tens to hundreds of kilovolts.This discharge mode at atmospheric pressure without preionisation was first time realised in [4,5].A diffuse discharge with X-rays was formed in helium [4] and in air [5].In both cases, voltage pulses of negative polarity were applied to an electrode with a small curvature radius.In another study, runaway electrons were detected using a Faraday cup [6].A detailed description of the discharge modes can be found in [3,7,8,[19][20][21][22].
Note that diffuse discharges at atmospheric pressure can also be formed at positive voltage polarity applied to an electrode with a small curvature radius [3,7,21,22,[26][27][28][29].This was demonstrated in [29].Detection of X-rays in a REP DD at positive voltage polarity of an electrode with a small curvature radius is much more difficult than at negative polarity, but in some studies this was done [21,22].Difficulties in the registration of X-rays and, especially, a runaway electron beam, at positive polarity in a gap formed by an electrode with a small curvature radius and a flat electrode are due to escape of runaway electrons to the anode or to the dense plasma the front of which propagates from the anode to the flat cathode.As a result, the electrons in the region of electric field amplification gain a comparatively low energy (several to tens of kiloelectron-volts) and only soft X-rays, which are hard to detect, are generated.
The formation of nanosecond diffuse discharges in an inhomogeneous electric field at atmospheric pressure was studied in many papers, see, for example [31][32][33][34][35][36][37][38], but neither X-rays nor runaway electron beams were detected.This can be explained by low voltages used, complexity of detecting runaway electron beams and X-rays at high gas pressures, and lack of necessary equipment and measuring techniques.Note that until our earliest studies in this field [7,39], only one scientific group managed to reliably detect a runaway electron beam behind the foil anode in atmospheric pressure air using a Faraday cup [6], see also the monograph [40].The number of beam electrons measured by behind the foil was ∼10 9 ; however, detection of any larger amounts of electrons failed during 40 years [41].No report was also made on detection of a runaway electron beam at a gas pressure higher than 0.1 MPa using a collector.
In the papers available, one can hardly find any data on direct detection of runaway electron beams by a collector at pressures of air and other gases above 0.1 MPa.The exception is our studies in which a runaway electron beam, which we term a supershort avalanche electron beam (SAEB), was detected by a collector in different gases at pressures of several atmospheres [7,26,42].However, systematic research in SAEB parameters and amplitude-time characteristics of discharges with direct collector measurements has not been conducted so far.
The aim of the present paper is to study the parameters of a runaway electron beam passed through a flat grid anode and foil and the dynamics of ionisation processes in a REP DD in different gases at pressures of 0.1 MPa and higher.These studies were carried out with a time resolution better than 100 ps.
The RADAN-220 pulser produced high-voltage pulses of negative polarity with a FWHM of ≃2 ns and rise time of ≃0.5 ns at a matched load.The voltage pulses were applied via a short transmission line to an electrode with a small curvature radius.This potential electrode (cathode) was 6-mm-diameter cylinder.The lateral surface of the cathode was made of a 100-µm-thickness stainless steel.The grounded flat electrode (anode) in one case was an aluminium disk of diameter 38 mm.In another case the anode was a disk of diameter 43 mm with a 20-mm round hole at its centre.In the hole, either a perforated 200-µm-thickness steel plate with a transparency of 14% or a steel grid with a transparency of 64% was arranged for extraction of runaway electrons from the discharge gap.The gap width d was 8, 12, 13 or 14 mm.
The voltage pulses were registered by a capacitive voltage divider located upstream of the discharge gap.The discharge current pulses were registered by a current view resistor (CVR) composed of low-inductance film chip resistors [23].Behind the grid anode a foil was placed.At d = 8 mm, we used a 10-µm-thickness Al foil, and a 50-µm-thickness AlMg at d = 13 and 14 mm.For SAEB extraction at d = 12 mm, a ketoprofen film (C 16 H 14 O 3 ) plated with a 0.2-µm-thickness Al layer was used; the film thickness was 2 µm.The runaway electron beam (SAEB) passed through the grid anode and foil was registered by a collector located 24 mm away from the grid anode.The diameter of the collector receiving part was 20 mm; the time resolution of the collector was ≃80 ps.
The optical radiation from the discharge plasma was registered as follows.Using a lens, a 2× magnified image of the discharge plasma was obtained in the plane of a screen with a slit of width 1 mm located in front of a Photek PD025 photodiode (LNS20 cathode, rise time ≃80 ps).The Photek PD025 diode and the screen were placed on a movable table, making it possible to register the radiation from zones along the discharge gap axis with a spatial resolution of ∼1 mm.
In studying the amplitude-time characteristics of the voltage, the discharge current, the SAEB current, and the optical radiation from the discharge plasma, the latter was recorded in the vicinity of the tubular cathode.Electrical signals from the capacitive voltage divider, the CVR, the collector and the photodiode were registered simultaneously with the digital real-time oscilloscope Tektronix DSA72504D (25 Hz, 100 GS/s).The oscilloscope was triggered by a signal from the capacitive voltage divider and operated in averaging mode (over 32 pulses).
In studying the dynamics of ionisation processes, we recorded optical radiation from 14 zones along the discharge gap axis.For each zone, 100 radiation pulses and 100 respective pulses of the voltage and discharge current were registered and then averaged.Because the voltage and the discharge current pulses were irrespective of what zone was dealt with, they were averaged over 1400 pulses.In the measurements, a Tektronix DPO70604 oscilloscope (6 GHz, 25 GS/s) was used.As in the previous case, to trigger the oscilloscope a signal from the capacitive voltage divider was used.The resulting waveforms were averaged over 100 pulses.
Time integrated images and spectra of the discharge plasma glow were taken with a SONY A 100 digital camera and a spectrometer EPP-2000C (operating spectral range is 200-850 nm; StellarNet Inc.), respectively.The X-ray exposure dose was determined using an Arrow-Tech dosimeter (Model 138).The electron beam and X-ray patterns were obtained using RF-3 film placed in a 120-µm-thickness black paper envelope.
It should be noted that, at present time, it is known about at least another two scientific groups that succeeded in registering with a collector of the runaway electron beam behind the anode foil in similar conditions with a temporal resolution better than 100 ps [23,24,[44][45][46][47].Before our early studies [7,39,42], as already mentioned, only one scientific group dealt with the registration by a Faraday cup of the runaway electron beam in atmospheric pressure air behind the anode foil [6,40].
3 Experimental results and discussion

Runaway electrons and X-rays behind the anode foil
The detailed data on the effect of the pressure and the kind of a gas on amplitude-time characteristics and morphology of discharges and on SAEB's amplitude during the subnanosecond breakdown of various gases are presented below.It is shown that the generation of runaway electrons and X-rays at negative voltage polarity is a common phenomenon in gas diodes under pressure above 0.1 MPa.Fig. 2 presents pressure dependences of the beam current for six gases in an interelectrode distance of 14 mm at a voltage rise time of ≃1 ns.
The SAEB was registered behind the 45-µm-thickness AlBe foil and grid with 14% transparency.The voltage pulses produced by the RADAN-220 pulser had the rise time of ≃1 ns.However, even with such rise time value of a voltage pulse, we managed to register a SAEB in different gases, including Kr, at a pressure of 0.15 MPa.As the voltage rise time was decreased to 0.5 ns, a SAEB was registered in N 2 at 0.5 MPa, in xenon and SF 6 at 0.2 MPa, and in He at 1.5 MPa, see also [26].Note that in heavy gases, the ionisation and excitation cross-sections are larger than those in light gases (He, H 2 ), which decreases the number of runaway electrons behind the anode.
Fig. 3 shows the SAEB current amplitude againsts the pressure of air, N 2 , and SF 6 for different gap widths and anode foil thicknesses.
The data of collector measurements demonstrate the presence of runaway electrons behind the anode foil in N 2 , air, and SF 6 at pressures of several atmospheres.The number of electrons in the beam increases when using thin foils with low electron absorption and gaps of optimum width for each gas.All data presented above were obtained on the RADAN-220 pulser.The presence of runaway electrons at pressures above 0.1 MPa was also detected by a collector in experiments on the SINUS pulser at a voltage amplitude of 180 kV and rise time of 0.5 ns [42].In helium, the presence of a SAEB was detected at 0.6 MPa; measurements at higher pressures were not taken due to limited strength of the working chamber.In N 2 it was detected at 0.4 MPa.At higher pressures, the presence of runaway electrons in the discharge gap can be verified using the glow of luminophore placed behind the anode foil.
In most experimental conditions, the SAEB current amplitude decreases with increasing pressure.However, with certain gas diode and cathode designs and certain voltage pulse parameters, a different situation may arise.For example, an increase in SAEB current amplitude with increasing the He pressure from 0.1 to 0.3-0.4MPa was observed [42].In this pressure range, a larger number of cathode sports were formed, as was evidenced by images of the discharge and cathode plasma glow.Likely it was the increase in cathode spots and associated increase in emission current from the cathode which was responsible for the buildup of SAEB currents with increasing the He pressure to 0.3-0.4MPa.Similar result was obtained in air [48].
In experiments with the RADAN-220 pulser at negative voltage polarity, an RF-3 film placed in a black paper envelope behind the AlMg foil was also used for SAEB detection.A single pulse was sufficient to obtain a SAEB imprint on the RF-3 film.The highest degree of film blackening was observed in the region near the discharge gap axis.The X-ray exposure dose at an air pressure of 0.1 MPa and interelectrode distance of 13 mm was 0.6 mR.This value was the average per pulse in a series of 50 pulses.For measuring the X-ray exposure dose, the collector was removed and the AlMg foil was replaced by a 20-µm-thickness Cu foil.The dosimeter in these measurements was located 5 mm away from the foil.The X-rays pattern was also obtained with film while using additional electron absorption filter.Because the film sensitivity to X-rays was much lower than that to electrons, more than 100 pulses were required to obtain an X-ray imprint.
In the experiments described above, the discharge was diffuse but became constricted with increasing the pressure and/or with decreasing the interelectrode distance.A detailed description of spatial discharge forms and their sequence is presented below.From the data reported here and in other studies [3, 7, 8, 19-22, 25, 28, 30, 48, 49] it follows that it is the runaway electrons which provide preionisation of the discharge gap and formation of a REP DD.

Spatial forms of REP DD in different gases
As has been shown in many papers, diffuse discharges in an inhomogeneous electric field can be formed in different gases at atmospheric and higher pressure whatever the polarity of an electrode with a small curvature radius, see books [3,40], reviews [7,8] and papers [19-21, 25-30, 48-54].In this paper, we consider data on diffuse discharges which occur with SAEB generation.Fig. 4 shows images of a REP DD plasma glow in air and N 2 at a pressure of 0.1 and 0.3 MPa, as well as in SF 6 at pressure 0.1 and 0.2 MPa for negative voltage polarity; the images were obtained in a pulse (Figs.4a-e) and in 20 pulses (Fig. 4f ).
It is seen that at these pressures, a diffuse discharge is formed in the gap.At pressures above 0.1 MPa, the REP DD has the form of diffuse channels connected with cathode spots (Figs.4b, d, f ).As the experimental conditions are changed, the discharge can be constricted.This happens with decreasing the interelectrode distance and with increasing the gas pressure, voltage pulse amplitude and duration.The onset of constriction (formation of electrode spots and spark leader) can vary from pulse to pulse [3,27,48].At the same experimental conditions, constriction of the REP DD occurs not in each pulse.Moreover, with increasing interelectrode distance the probability of constriction process in the gap is reduced to near 0%.According to research data on discharges with an electrode of small curvature radius at negative voltage polarity and rise time of 0.5 ns [27,[50][51][52], first come bright spots at the cathode, and then, short anode spark leaders develop from these spots.Next, a bright spot arises at the flat anode and forms a spark leader which bridges the gap.In this paper, similar results are obtained.It is seen from Fig. 4a that at an air pressure of 0.1 MPa, a spark leader approaching the cathode develops on the background of the REP DD; at higher pressures, the leader can be transformed to a spark channel.When the tubular electrode is at positive polarity, the spark leader bridging the gap develops from the tubular anode.Detailed studies of the formation of spark leaders and anode spots in experiments on the RADAN-220 pulser at both voltage polarities with a voltage amplitude higher than 100 kV can be found elsewhere [27,[50][51][52].Also available are data of similar studies with a high-speed CCD camera on a FPG-60 frequency pulser at a voltage of tens of kilovolts [53,54].The appearance of bright spots at the flat anode depends on the polarity of voltage pulses and gas pressure.For the flat electrode at negative polarity, bright spots at the electrode are observed over a wider range of experimental conditions.Thus, the research in spatial discharge forms confirms the formation of a REP DD in different gases at high pressures as nanosecond voltage pulses with a short rise time (<1 ns) are applied to a gap with an inhomogeneous electric field distribution.Decreasing the gap and increasing the pressure causes constriction of the discharge but within several to tens of nanoseconds after SAEB generation.The polarity of the electrode with a small curvature radius only slightly affects the formation of a REP DD in centimeter gaps at a voltage of hundred kilovolts, but at positive polarity the parameter ranges in which it can be observed is  greatly limited (this refers to gas pressure, interelectrode distance, voltage pulse amplitude, rise time, and FWHM).Moreover, with this electrode being positive, detection of runaway electron beams is difficult.However, some studies did allow detection of X-rays for which it was required either to decrease the width of an electrode gap [21] or to increase the sensitivity of a registration system [22].

Amplitude-time characteristics of voltage, discharge current, SAEB current, and radiation from REP DD plasma
As already mentioned, we simultaneously registered four parameters: the voltage, the discharge current, the SAEB current, and the radiation intensity of the REP DD plasma.The oscilloscope was operated in a mode with averaging over 32 pulses.The radiation from the gap at the diffuse discharge stage was contributed mostly by the second positive system of N 2 .Figs. [5][6][7][8][9][10][11][12] show the waveforms of the voltage, the discharge current, the SAEB current pulses, and the radiation intensity of the REP DD plasma pulses obtained with high temporal resolution in various gases.The effect of gas pressure on the amplitude and duration of the SAEB is shown.Synchronisation method of SAEB current pulses with voltage and discharge current pulses is described in [55].
Fig. 5 shows waveforms of the voltage, discharge current, and SAEB current pulses at negative polarity of the tubular electrode and interelectrode distance of 8 mm for air at a pressure of 0.1 and 0.3 MPa.Fig. 6 shows respective data for SF 6 at a pressure of 0.1 and 0.18 MPa.
It is seen that in air at 0.1 and 0.3 MPa, the discharge is oscillatory (Fig. 5).In SF 6 , increasing the pressure increases the discharge plasma resistance in the gap, and at >0.15 MPa, the discharge current ceases to oscillate (Fig. 6).Note that the peaking spark gap switch used in the RADAN-220 pulser did not allow matching of its coaxial high-voltage line, charged by a pulse transformer, with the coaxial transmission line (Fig. 1).In the experiments reported, the wave impedance of the transmission line was higher than that of the high-voltage line, and this increased the voltage across the gas diode compared to the breakdown voltage of the peaking switch (about 220 kV).However, due to the mismatch, reflected pulses are generated from the transmission line and from the gas diode, thus lengthening the voltage and discharge current pulses.In a diffuse discharge in SF 6 , the duration and amplitude of the discharge current decreases with increasing the gas pressure and the voltage amplitude across the gap thus increases.The SAEB current amplitude, as a rule, decreases with an increase in pressure (Figs. 2 and 3), and the time till the onset of SAEB generation increases (Figs. 5 and 6).
Figs. 7 and 8 show waveforms of the voltage, discharge current, and SAEB current pulses within the first two nanoseconds for different air and SF 6 pressures; the voltage pulses were reconstructed from incident and reflected waves.It is seen that in atmospheric pressure air, the peaks of the voltage and SAEB current roughly coincide, while the rate of current rise through the gap slows down; at larger gaps and/or higher pressures, the current through the gap even decreases somewhat.We think that this is due to escape of fast electrons to the flat anode with the formation of a reversed electric field between the positive ions left near the anode and the anode itself.On the waveform of the discharge current (Fig. 7b), three characteristic portions are denoted as A, B, and C. Portion A corresponds to capacitive current of a 'cold' diode.Portions B and C, on which the discharge current increases, correspond respectively to the rise of dynamic capacitive current and to an increase in conduction current as the dense plasma produced by the first ionisation wave bridges the gap.A detailed consideration of these discharge phases is given in Section 4.
Fig. 9 shows waveforms of the voltage, discharge current, SAEB current, and radiation intensity from the near-cathode region for different air pressures and gap equal to 13 cm, and initial portions of these waveforms are presented in Fig. 10.
Under these conditions, the discharge is diffuse (Figs.4a and b).In a number of pulses, a spark leader can arise in the gap (Fig. 4a), propagating normally from the flat anode [50].Because the interelectrode distance is increased to 13 cm, all other things being equal, the voltage amplitude across the gap increases and the amplitude of the discharge current and SAEB current decreases; the SAEB current amplitude is also influenced by the anode foil thickness and grid transparency.For both gaps, 8 and 13 cm, increasing the gas pressure lengthens the time till the collector detects a SAEB.The radiation of the second positive system of N 2 from the near-cathode region is delayed by no more than 0.2 ns and its intensity increases with increasing discharge current.Waveforms of the voltage, discharge current, SAEB current and radiation intensity for N 2 are presented in Figs.11 and 12.
The SAEB current amplitudes in N 2 and air differ little, though air contains an electronegative gas (oxygen).
The obtained experimental results confirm the generation of a SAEB in air, N 2 , and SF 6 at atmospheric and higher pressures.Under these conditions, the discharge formed in the gap with an inhomogeneous electric field distribution is diffuse and a SAEB is registered behind the anode foil.At equal pressures, the SAEB amplitudes are highest in light gases He and H 2 (Fig. 2) and are lowest in heavy gases SF 6 , Xe, and Kr.Remind that the SAEB current in Figs.5-12 is limited by the diaphragm of diameter 20 mm and attenuated by the grid either of transparency 14% at d = 13 mm or of 64% at d = 8 mm and also by the 50-µm-thickness AlMg foil at d = 13 mm or 10-µm-thickness Al foil at d = 8 mm.The actual SAEB amplitude behind the entire anode surface is much higher.When using a 10-µm-thickness Al foil, the SAEB amplitude in air and N 2 at atmospheric pressure reaches tens of amperes [7,8,42,56].

UV radiation and discharge current dynamics
The propagation of ionisation waves in a discharge in N 2 and N 2 -containing mixtures at high pressures can be traced by analysing the time behaviour of radiation intensities of the second positive system of N 2 along the discharge gap axis [28].For this purpose, the following equation was derived [28]

: (see (1))
where A is a constant; N 0 is the concentration of N 2 molecules; I(t) is the radiation intensity of the second positive system of N 2 (С 3 Π u → B 3 Π g transition); t eff is the effective lifetime of the С 3 Π u state of a N 2 molecule; k i (E(t)/N 0 ) and k exc (E(t)/N 0 ) are ionisation and excitation rate constants; E(t)/N 0 is the reduced electric field strength; N e (t) is the electron concentration.
The time dependence for the left-hand side of (1) can be calculated from experimental data on radiation intensities of the second positive system of N 2 .Knowing the behaviour of the left-hand side of (1), we can analyse its right-hand side.The latter contains two factors: the term in square brackets and the electron concentration N e (t).The term in square brackets contains two rate constants k i (E(t)/N 0 ) and k exc (E(t)/N 0 ) for ionisation and excitation, respectively, which are directly dependent on the electric field strength E(t)/N 0 : the higher the strength E(t)/N 0 , the higher the constants k i (E(t)/N 0 ) and k exc (E (t)/N 0 ), at least in the E(t)/N 0 range that takes place under experimental condition.In analysing (1), it is assumed that the electron concentration N e (t) at the breakdown stage is a non-decreasing function.Thus, any decrease in the right-hand side of (1) can be due to a decrease of the term in square brackets, i.e. of E(t)/N 0 .
Fig. 13 shows waveforms of the voltage, discharge current, and radiation intensity of the second positive system of N 2 at different distances from the tubular cathode with corresponding diagrams of d(I/t eff + dI/dt)/dt in a discharge in N 2 at 0.4 MPa.Each waveform in Fig. 13c is the average over 100 pulses.
Similar time dependences of the radiation intensity at different distances from the potential electrode are found at other pressures of N 2 and in SF 6 with 2.5% of N 2 .A characteristic feature of the discharge in an inhomogeneous electric field is that the region near the grounded flat electrode starts glowing later than that near the  potential electrode with a small curvature radius (Fig. 13c).For the electrode with a small curvature radius, the time delay of the glow with respect to the voltage pulse did not depend on the pressure and kind of gas, whereas that for the flat electrode with respect to the potential electrode did depend on.The delay of the glow near the flat electrode was shortest in N 2 at the minimum pressures used and increased with pressure in all gases.The curves in Fig. 13d were obtained from the waveforms of radiation intensity in Fig. 13c.It is seen from Fig. 13 that about 1 ns is passed from the instant a voltage is applied to the gap till the onset of conduction current and voltage drop.During this time, the gap is involved in ionisation processes with attendant increase in excitation and ionisation degrees and is broken down.At the breakdown stage, the quantity d(I/t eff + dI/dt)/dt reaches a local maximum due to a local increase in the electric field strength E(t)/ N 0 to its maximum and its further decrease.Such local increase and decrease in the field occurs successively along the gap and is most pronounced at a distance no greater than 7-8 mm from the electrode with a small curvature radius (Fig. 14).
Propagation of the electric field maximum from zone to zone on the way to the grounded electrode is just the one representing an ionisation wave [57].This dynamic confirms that about three forth of the gap is broken down in the form of an ionisation wave, which is denoted as IW in Fig. 14.The rest of the gap is broken down by a somewhat different mechanism.First, the electric field in this part of the gap is enhanced due to potential shifting into the zone near the ionisation wave front.Second, the electron concentration in this part of the gap is increased due to runaway electrons and X-rays.As a result, an almost instant breakdown occurs in this part of the gap or the ionisation wave velocity there increases steeply.Once the first ionisation wave reaches the grounded electrode, a second ionisation wave propagates backward to the electrode with a small curvature radius, and then, a third wave is likely.On arrival of the backward ionisation wave at the electrode with a small curvature radius, the breakdown stage is completed and there begins switching with attendant sharp increase in conduction current, voltage drop across the gap, and radiation enhancement (Figs. 13 and 14).
Fig. 15 shows relative ionisation wave velocities obtained in two series of experiments for N 2 and for SF 6 with 2.5% of N 2 .
At pressures of 0.1-0.3MPa, the ionisation wave velocity was determined from the time delay between the onset of radiation near the flat electrode and that near the tubular electrode.For the onset of radiation, we took the point in time at which a signal from each region reached 5% of its maximum.The signal to noise ratio was greater than 2. For SF 6 with 2.5% of N 2 , the delays of radiation near the flat electrode were much longer, likely because SF 6 is a heavy electronegative gas with large ionisation and excitation cross-sections.The highest ionisation waves velocity was ∼1.5 × 10 8 m/s and was found in atmospheric pressure N 2 and smaller at negative polarity.In the pressure range 0.4-0.7 MPa, the ionisation wave velocities were estimated using (1).The estimates show that the velocities of the forward (first) and backward  (second) ionisation waves in N 2 at 0.4 MPa were ∼3.9 × 10 7 m/s and ∼1 × 10 8 m/s, respectively, and those at 0.7 MPa were ∼3.1 × 10 7 m/ s and 5.8 × 10 7 m/s, respectively.This study of ionisation waves velocity confirms the results obtained earlier [20,28,48,49,58].

Data analysis
The research data suggest that using a collector, it is rather easy to detect a SAEB behind the anode foil in different gases at atmospheric and higher pressure.However, the maximum pressure at which a SAEB is detectable decreases with increasing the voltage pulse rise time and decreasing its amplitude.For reliable SAEB detection at pressures above 0.1 MPa, voltage pulses with an amplitude of hundred kilovolts and rise time shorter than 1 ns are required.
Let us consider the dynamics of the main processes occurring in a discharge gap during the generation of a SAEB and formation of a REP DD.The breakdown of a high-pressure gas in the gap between an electrode with a small curvature radius and a flat electrode at subnanosecond and nanosecond voltage rise times begins with field emission from the cathode having a small curvature radius [59].Because the critical electric field strength for electron emission is higher than that for electron runaway [60], part of the electrons near the cathode switches to a runaway mode.These fast runaway electrons provide initial gas preionisation near the cathode tip.The primary electrons resulting from gas preionisation by runaway electrons form electron avalanches with overlapped heads.Thus, a dense diffuse plasma is formed near the electrode with a small curvature radius.The concentration of electrons and ions in the plasma is sufficient to greatly decrease the electric field strength in the plasma, such as in the case of a streamer [57].The front of the dense diffuse plasma propagates to the flat electrode due to electric field amplification ahead of the plasma front and to primary electrons which result from gas preionisation by runaway electrons, and also X-rays and VUV radiation.In other words, early in the breakdown, a dense plasma cloud much larger in cross-section than a streamer propagates from the electrode with a small curvature radius to the flat electrode.The ions are left almost immobile, and the ionisation wave front to the flat electrode propagates due to electron multiplication in a narrow layer with amplified electric field.See also the papers [19,20,25].
At negative polarity of the electrode with a small curvature radius, a region with an excess negative charge is formed at the boundary of the dense plasma, as in the head of an electron avalanche and streamer.However, the plasma in an ionisation wave during the formation of a REP DD and the plasma in an electron avalanche differ greatly in cross-sectional dimensions and electric field strength on their fronts.Part of the electrons at the boundary of the region with an excess negative charge is accelerated by an external electric field and by a field formed in the region.As the ionisation wave front propagates in the gap, conditions for a polarisation  self-acceleration are created.It promotes the generation of runaway electrons in the region of electric field amplification.This mechanism of particle acceleration at the front of a streamer was proposed by Askar'yan [61].During the propagation of the ionisation wave front, the number of runaway electrons moving from the dense plasma boundary (from the ionisation wave front) increases, thereby increasing the SAEB amplitude behind the foil anode.The electron energy increases also.Thus, the propagation of the first ionisation wave is provided by runaway electrons, responsible for gas preionisation, and by high electric field strength ahead of its front.Increasing the gas pressure decreases the parameter E/p, where E is the electric field strength and p is the gas pressure, ahead of the ionisation wave front with the result that the number of runaway electrons as well as the SAEB current amplitude behind the anode decreases.On the other hand, the formation and motion of the first ionisation wave provides the generation of runaway electrons.
On the waveforms of the discharge current in Fig. 14b and also in Figs.7b, 8b, 10b, 12b, and 13b one can see characteristic portions denoted as A, B, and C. Portion A corresponds to capacitive current of a 'cold' diode.The term cold diode means that the gap is empty of dense plasma with an electron concentration sufficient for a notable decrease in electric field strength.The capacitive current decreases with decreasing the time derivative of voltage dU/dt.At subnanosecond voltage rise times, this current can reach tens to hundreds of amperes, depending on the cathode design and voltage amplitude [62,63].The increase in discharge current on portion B owes to the rise of dynamic capacitive current after the formation of dense diffuse plasma near the pointed electrode.The dynamic capacitive current occurs due to ionisation wave propagation to the flat electrode and reaches hundred amperes within tens to hundreds of picoseconds [25], and with large cathode dimensions, despite the decrease in dU/dt, its value reaches to several kiloamperes [63].The electron concentration in zone behind of ionisation wave front reaches 10 13 -10 15 cm −3 [25,49,[64][65][66], and the electric field strength in the dense plasma region decreases.As this takes place, explosive electron emission, which is described in detail elsewhere [67], becomes the main mechanism of emission from the cathode.The participation of explosive electron emission in the discharge processes is evidenced by bright cathode spots arising in a time of hundred picoseconds [19,62] and by cathode erosion.Further, the plasma of the first ionisation wave bridges the gap.As the electron beam reaches the anode, the total current through the gap decreases, and after escape of runaway electrons to the anode, it increases again.
Once the forward ionisation wave bridges the gap, a backward wave is initiated.The front of the forward wave reaches the flat electrode at the end of portion B (Fig. 14b) and the current in the gap is determined mostly by conduction current.On portion C, a  backward ionisation wave propagates and the conduction current increases, thus increasing the discharge current in the gap.The total current in the gap, in this case, is also contributed by dynamic capacitive current between the front of the backward wave and the electrode with a small curvature radius (cathode).However, no preionisation ahead of the backward wave front is required because by this time the electron concentration in the gap reaches ∼10 14 cm −3 .From Figs. 7b, 8b, 10b, and 12b it is also seen that increasing the gas pressure delays the rise of dynamic capacitive current on portion B and conduction current on portion C.

Conclusions
Thus, we studied the parameters of a nanosecond high-voltage discharge (voltage, discharge current, runaway electron beam current, radiation intensity) in a point-to-plane gap with different gases at atmospheric and higher pressures.It is shown that at negative polarity of an electrode with a small curvature radius, it is easy to detect a SAEB by a collector behind the grid and foil anode.The maximum pressure at which a SAEB is detectable decreases with increasing the voltage rise time, and for reliable SAEB detection at pressures above 0.1 MPa, one should use voltage pulses with an amplitude of hundred kilovolts and rise time shorter than 1 ns.
The amplitude-time characteristics of the voltage, discharge current, SAEB current, and radiation intensity were studied with subnanosecond time resolution.The ionisation waves velocity increases with decreasing the gas pressure and in going from heavy (SF 6 ) to lighter (air, N 2 ) gases.In N 2 at 0.1 MPa, the ionisation waves velocity reaches ∼1.5 × 10 8 m/s.It is also found that in N 2 at 0.4 and 0.7 MPa, a second (backward) ionisation wave with higher velocity starts propagating from the flat electrode as it is reached by the first (forward) wave.The velocities of the forward and backward waves at a N 2 pressure of 0.4 MPa are ∼3.9 × 10 7 m/s and ∼1 × 10 8 m/s, respectively, and their velocities at a pressure of 0.7 MPa are ∼3.1 × 10 7 m/s and 5.8 × 10 7 m/s, respectively.It should be noted that, such values of velocity of the first ionisation wave was obtained for helium and air in [20,48].Most of the SAEB electrons are generated during the propagation of the first ionisation wave due to electric field amplification at its front.This study confirms the results obtained earlier in [7,28,58].

Acknowledgment
The work was supported by the Russian Science Foundation (the project #14-29-00052).

Fig. 1
Fig. 1 Scheme of experimental setup

Fig. 3
Fig. 3 SAEB current amplitude against pressure for different gases.The grid anode transparency is 64%.Values for SF 6 is multiplied by 100 a d = 8 mm, thickness of an Al foil is 10 µm b d = 12 mm, a 2-μm-thickness ketoprofen (C 16 H 14 O 3 ) film coated by 0.2-μm-thickness aluminium layer was located behind the grid anode

Fig. 9
Fig. 9 Waveforms of recorded signals.Discharge in air at 0.1 MPa (1) and 0.3 MPa (2) at negative polarity.d = 13 mm.Grid anode transparency is 14%.AlMg foil thickness is 50 µm а Voltage pulses b Discharge current pulses c SAEB current pulses d Intensity of a discharge plasma radiation

Fig. 11
Fig. 11 Waveforms of recorded signals.Discharge in N 2 at 0.1 MPa (1) and 0,3 MPa (2) at negative polarity.d = 13 mm.Grid anode transparency is 64%.AlMg foil thickness is 50 µm а Voltage pulses b Discharge current pulses c SAEB current pulses d Intensity of a discharge plasma radiation

Fig. 13
Fig. 13 Waveforms of recorded signals.Discharge in N 2 at 0.4 MPa at negative polarity.d = 13 mm а Voltage pulses b Discharge current pulses c Radiation intensity of second positive system of N 2 at different distances from tubular cathode d Left part of (1) at different distances from tubular cathode

Fig. 15
Fig. 15 Ionisation wave to light velocity ratio vs gas pressure.Negative polarity