Many scientific groups have been and continue to be engaged in development of pulsed electron accelerators with energies from tens to hundreds of keV. High-current electron beams (HCEBs) of nano- and subnanosecond duration with a current of ~1 kA and a power density of 106–108 W/cm2 are widely used for various applications—in particular, in radiation physics and solid state chemistry [1, 2] and for diagnostics of natural and artificial crystals [3, 4] and initiation of explosive decomposition of highly sensitive energetic materials [5, 6].

Recently, new problems that require the implementation of higher power densities of electron beams of ~1010–1012 W/cm2 have appeared. These include the development of methods for atomic spectroscopy with evaporation of an HCEB sample, initiation of explosive decomposition of low-sensitive brisant explosives, and generation of powerful shock waves with an intensity sufficient for spalling of metal targets. Some of these problems were solved using the self-focusing effect of the HCEB generated in a vacuum diode of an electron accelerator by a GIN-600 generator [2]. It should be noted that studies of the self-focusing phenomenon earlier were usually carried out using relativistic electron beams (REBs) with a current exceeding the Alfvén current: IA = 17βγ, where β = \({v}\)/c, γ = 1/(1 – β2)1/2 is the relativistic factor, \({v}\) is the velocity of the electron beam, and c is the speed of light. In REB experiments, the current was 100–200 kA and significantly exceeded the Alfvén current (~ 20 kA) [7–9]. The beam was shown to be focused by its own magnetic field, with the primary role being played by the plasma formed in the cathode-anode gap. However, the mechanisms responsible for filamentation and self-focusing of an electron beam in a vacuum diode of an electron accelerator with the current much less than the Alfvén current have not yet been clarified and require special studies.

It should be noted that heterogeneities of the current density during vacuum breakdown and anode erosion were also observed at currents less than 500 A [10]. At discharging air at atmospheric pressure, microchannels from the tip of negative and positive polarity, as well as the damage to the flat electrode surface, were recorded [11]. An increase in the current density of the runaway electron beam was observed at a decrease in pressure in diodes filled with various gases [12]. Thus, it is known from the literature that there is focusing of the beam current in vacuum as well as in gas diodes. However, experimental results of studying the effects on the anode at accelerator currents of ~1 kA are very scarce.

This work is aimed at studying the conditions for obtaining the filamentation of electron beams with a large local current density in vacuum and gas diodes and evaluating the current characteristics based on the research results: spatial distribution, energy of electrons, and energy density.

Experiments were conducted using two setups. In setup no. 1 (the maximal electron energy is T ∼ 400 keV, the current pulse at half-height is τ0.5 = 12 ns, and the maximal beam current behind Al foil with a thickness of 20 μm is Imax ≈ 2 kA), a GIN-600 generator [2, 13] with a vacuum diode connected was used. The vacuum diode was formed by a tubular cathode and a flat anode. Aspect ratio g = R/d (where R is the cathode radius and d is the interelectrode gap) varied in the range of 0.7–1. The focusing process was studied by traces of erosion on anode plates of various metals (“signatures” of the electron beam) using optical microscopy with a spatial resolution of 10 μm.

In setup no. 2 (T ~ 250 keV, τ0.5 ≈ 0.1–0.5 ns, Imax ≈ 100–500 A), an SLEP-150 generator [12] with a gas diode connected was used. The current pulse duration of an ultrashort avalanche electron beam (UAEB) at half-height depended on nitrogen pressure. The cathode was also made in the form of a tube with a diameter of 6 mm. The anode of the gas diode was made of a grid with a 64% light transparence, on which Al or AlBe foil was placed. The aspect ratio ranged from 0.25 to 1.5. In a number of experiments, the foil was removed, and the discharge shape could be recorded through the grid. When photographing the “signatures” of the electron beam, a luminophore was placed beam behind the grid and Al foil while a quartz window was placed behind it. The gas diode was pumped out with a forepump and filled with nitrogen. The maximum amplitudes of the beam current were achieved at nitrogen pressures of units of kPa.

The most interesting results on the effect of the electron beam on the anode were obtained during filamentation and self-focusing of the HCEB in a vacuum diode. Typical erosion traces (“signatures” of the electron beam) formed on the copper anode surface after a single irradiation pulse at cathode–anode gap d  = 3 mm and radial cylindrical cathode radius R = 3 mm are shown in Fig. 1a.

Fig. 1.
figure 1

Images of erosion traces of individual current channel “signatures” formed on the anode surface during HCEB irradiation in the filamentation and self-focusing mode in the vacuum diode of an accelerator with a GIN-600 generator: (a, b) images of the irradiated and back surfaces of a 180-μm-thick copper foil per irradiation pulse and (c) an erosion mark on the surface of a 2-mm-thick copper plate after multipulse irradiation.

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As can be seen in the figure, a single pulse impact results in the formation of a geometrical figure resembling a star with the average “rays” number of ~20 and a size of ~7–8 mm on the anode surface. A more detailed study of the “signature” obtained on 180-μm-thick copper foil allowed determining the geometrical parameters of individual “rays”: the erosion trace width varied within ∼70–150 μm, the depth varied within ∼10–15 μm, and the diameter of the central crater varied within ∼1.0–1.5 mm. In this case, spalling was observed on the rear of the copper target (Fig. 1b). After single pulse exposure, a hole with a diameter of ~1.3 mm was formed in a brass anode with a thickness of 60 μm. Multipulse irradiation of a copper anode with a thickness of 2 mm led to the appearance of “rays” of the erosion ring on the periphery (Fig. 1c). The merging of two closely spaced “rays” into one was observed in some images of “signatures.” Simultaneously with the “rays,” local erosion traces (incomplete “rays”) extending from the periphery to the center of the self-focusing spot are often formed on the irradiated target surface.

In experiments with an aluminum anode with a thickness of 100 μm, spalling was observed on both the rear and the irradiated surface with the formation of a through hole in a self-focusing spot after a single pulse. In this case, the “rosette” of the breakage located on the back target surface has opened in the direction of electron beam propagation while the “rosette” located on the irradiated surface has opened to meet the electron beam. Consequently, the maximal energy release of a self-focused electron beam was located close to the center of the irradiated target (i.e., at a depth of ~40–50 μm), which leads to simultaneous spalling of both the irradiated and the back surface of the target with the formation of a through hole. Based on the experimentally determined values ​​(the maximum energy release in the aluminum target and the spall strength of copper), the average electron energy in the filaments and the bulk energy density in the self-focusing spot were estimated to be 50–100 keV and 109 J/m3, respectively.

The total energy of the electron beam with the SLEP-150 generator was less than two orders of magnitude at small pressures in the gas diode and 104-fold at a nitrogen pressure of 100 kPa than those in the case of an electron beam with a GIN-600 generator. However, even under these conditions, damage to the anode foil was observed as both the interelectrode gap and pressure in the diode decreased. At the same time, the foil damage was observed in a wider pressure range as the interelectrode gap decreased. Figure 2a shows holes in the Al foil with a thickness of 10 μm at an interelectrode gap of 4 mm and a nitrogen pressure of 100 kPa. Figure 2b shows the signatures of the electron beam at an interelectrode gap of 8 mm and a nitrogen pressure of 100 kPa. The foil thickness and the number of pulses were increased.

Fig. 2.
figure 2

Images of (a, b) damage to the anode foil and (c) luminescence of the luminophore behind the foil under the effect of an electron beam. (a) Nitrogen, 760 Torr, 4-mm gap, Al foil (10 μm), and 20 pulses; (b) nitrogen, 760 Torr, 8-mm gap, AlBe foil (60 μm), and 200 pulses; and (c) nitrogen, 9 Torr, 12-mm gap, Al foil (10 μm), and a single pulse.

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At an increase in the interelectrode gap of up to 12 mm and a nitrogen pressure of up to 11.8–100 kPa, the discharge consisted of several diffuse conelike jets with a base diameter of several millimeters, which originated from the cathode. In images of the integral glow of the discharge in nitrogen and other gases, bright spots were visible only on the cathode edge. The damage to the Al-foil anode was not observed under these conditions. The “signature” of the beam current obtained from the luminescence of the luminophore behind the Al foil is a luminescence uniformly distributed over the anode surface. However, a decrease in the pressure in the gas diode led to the appearance of diffuse channels and the focusing of the beam current. Figure 2c shows the luminophore luminescence at a pressure of 1.2 kPa and a gap size of 12 mm. A bright spot can be seen in the center of the foil. This was also observed for thick Al foils, which confirms the generation of high-energy runaway electrons.

The performed studies show that the conditions when the beam current density increases significantly are developed in vacuum and gas diodes at a total current from hundreds of A to units of kA (primarily at decreasing interelectrode gaps and the formation of an anode plasma). This continues the generation of high-energy electrons and damages the anode. In a vacuum diode with a GIN-600 generator, the generation of high-density electron beams led to the anode surface erosion and the formation of holes in thin foils, as well as spalling from the forward and back sides, even after a single current beam pulse. Consequently, spalls from both sides of the irradiated target were simultaneously produced by varying the Al-foil thickness.

Based on the results obtained in this study and the known data on self-focusing of dense high-current REBs [7–9, 14], a sequence of processes leading to filamentation and self-focusing of an electron beam in a vacuum diode of an electron accelerator with a GIN-600 generator can be imagined. At the beginning, a laminar electron flux produced by a cathode plasma is observed in a diode with a hollow cylindrical cathode before the appearance of an anodic plasma. An anode plasma, which is formed as a result of the desorption of gases from the anode surface and the evaporation of dielectric and semiconductor inclusions with their subsequent ionization, is produced by electron bombardment. Filamentary instability leading to the filamentation of the electron beam (splitting it into 18–20 filaments of current channels) develops at the interaction of an electron beam with the anode plasma. Apparently, as in the case of high-current REBs, each channel is tied to the explosive electron emission center. As the current increases in the filaments due to the avalanche-like multiplication of fast electrons, the magnitude of the magnetic field increases, leading to the collapse of the filamentous microbeams into the central focus spot. Thus, two electron beams with different space–time and energy characteristics are formed in a vacuum diode: a uniform high-energy beam and a filamentous beam with less electron energy.

Similar processes, the onset of which is facilitated by the presence of a gas in the diode, occur in gas diodes. We believe that the most probable mechanism for generating superdense filamentous electron beams in vacuum and gas diodes is runaway-electron breakdown developing in the cathode–anode gap plasma.

In conclusion, we note that damage to the inner surface of the chamber, which is filled with deuterium and tritium in the TOKAMAK setup [15], can also occur due to the filamentation and self-focusing of runaway electron beams.

Acknowledgments. This work was supported by the Russian Science Foundation (project no. 18-19-00184) and partly by the Competitiveness Enhancement Program of Tomsk Polytechnic University (project no. VIU_IFVT_73/2017).