Simulation of runaway electron kinetics in magnetized gas diodes with a strongly inhomogeneous electric field Open Access

The “runaway” electron (RE) phenomenon is that, under certain conditions, free electrons in gas or plasma can gain more energy from an electric field than they lose in collisions. As a result, REs are continuously accelerated by an electric field. The RE essence was formulated for the first time in the 1920s by Wilson^1,2 for the β-particle acceleration in thunderclouds. The conditions of the electron's continuous acceleration in high-temperature plasmas were revealed by Dreicer^3,4 at the end of the 1950s. The criterion of the electron “running away” in dense gases was proposed by Gurevich in the early 1960s.⁵ The existence of continuously accelerated electrons in pulsed gas discharges was experimentally confirmed in works.^6,7 The results and history of the RE phenomenon investigation in gases were summarized and generalized in the monography⁸ by Babich. Nowadays, RE participation is believed to be the explanation for the development of lightning and other discharges in the atmosphere.^9,10 Within laboratory experiments, REs are usually generated at the delay stage of nano- and subnanosecond discharges of a high gas medium pressure up to 40 atm.¹¹ In the case of a pulsed discharge of high pressure, the main features of generated REs are (1) extremely short duration of the RE current pulse of ∼10 ps;^12–20 (2) the possibility of generating REs with “anomalous” energies exceeding to that determined by the voltage applied to a discharge gap;^21–23 (3) preionization of a gas medium by REs leading to extremely fast volume switching of the gap.^24–27 These features make REs attractive from the point of view of their possible applications.

At the moment, pioneering research on focusing RE beams utilizing magnetic fields of various spatial configurations is carried out in the Institute of Electrophysics of the Ural Branch of the Russian Academy of Sciences.^28–32 The research is aimed at improving the operational parameters (primarily, the current density) of generated RE beams to make them suitable for applications, in particular, for the excitation of electromagnetic pulses in the range from radiofrequencies to x rays. Employing a guiding homogeneous magnetic field with an induction of up to 4 T allowed the experimental group to achieve a record-high value of the RE beam current density of 0.65 kA/cm² in atmospheric pressure air.²⁸ Earlier, without employing a magnetic field, comparable values of the RE current density were achieved only under low gas pressures,³³ when the critical electric field strength value, being the threshold one for electron running away, is known to be about two orders of magnitude lower than that at an atmospheric pressure air. The experimental setup configuration from Ref. 28 was studied in detail using the Monte Carlo simulation in Ref. 30.

In addition to the experimental setup configuration with a homogeneous guiding magnetic field, the experimental group tested focusing systems with an inhomogeneous magnetic field.²⁹ In Ref. 29, the magnetic system provided a gradual increase in the concentration of magnetic field lines along the path of the RE beam. Accordingly, the magnetic field induction B varied from 0.05 to 0.25 T in the region of the RE beam formation near the cathode to 1–5 T at the collector position. As a result, a three- to fourfold radial compression of a tubular RE beam was achieved with a maximum beam current density of up to 100 A/cm².²⁹ However, from the set of tested tubular cathodes with diameters of 22, 12, 8, and 4 mm, the 8‐mm‐diameter cathode provided the highest RE beam current density and integral electron current, although it was expected that the 22‐mm‐diameter cathode would give a maximal current due to the largest area of the RE emission. Qualitatively similar effects were observed in Ref. 34 in the absence of a magnetic field; however, considering the experimental setup described in Ref. 34, we believe the cause was a non-optimal combination of tubular cathode diameters and an aperture of an electron current detector utilized in Ref. 34. In Ref. 29, it was suggested that the main factor limiting the passage of the RE beam through the gap was the “magnetic mirror” effect, i.e., the reflection of electrons with a sufficiently large transverse velocity component from the region of concentration of magnetic field lines. Therefore, a proper analysis is necessary to find ways of suppressing the “magnetic mirror” effect for the RE transport optimization. In this sense, the problem of the RE beam radial compression by an inhomogeneous magnetic field appeared to be inverse to the issue of the magnetic confinement of high-temperature plasma, where the maximization of the “magnetic mirror” effect is required.

It should be noted that the idea of improving the generated RE beam parameters is not new. Within the works,^33,35 the experimental results on the generation of RE beams with increased current density and integral beam current values are given. In Ref. 33, the authors achieved the RE current density of ∼2 kA/cm² without employing a longitudinal magnetic field. In Ref. 35, the possibility of the RE generation with a beam current of up to 300 A was shown. However, in Refs. 33 and 35, the experiments were carried out under low gas medium pressures of units or tens of Torr. When the gas pressure was decreased below this level (i.e., the transition to the vacuum diode mode was performed), the integral RE beam current and current density inevitably decreased; the current decrease was also observed with the gas pressure growth above some optimal value. Analogous regularities were demonstrated in the later paper.³⁶ Moreover, from Refs. 33, 35, and 36, a decrease in the gas pressure led to an increase in the RE current pulse duration to hundreds of picoseconds. In contrast, the results described in Refs. 28 and 29 were obtained at an air of an atmospheric pressure when the RE current pulse duration was about tens of picoseconds. Therefore, we believe it is important to investigate the generation of powerful RE beams at atmospheric pressure air when there is no necessity for vacuum pumps, and the ultra-short (∼10 ps) duration of the RE current pulse takes place.

The present work is devoted to the numerical simulation of the interaction of an RE tubular beam with a focusing inhomogeneous magnetic field. The electrode configurations with 22- and 8-mm-diameter tubular cathodes were considered. Spatial configurations of electric and magnetic fields were analogous to those used in the experimental work.²⁹ The previously developed kinetic 2D‐3V (2 coordinates–3 velocity components) Monte Carlo model^30,37,38 was employed. Within the simulation, the “magnetic mirror” effect features were studied for the electron motion in atmospheric pressure nitrogen. The features were compared to those observed within the simulation for the vacuum case. The conditions of the electron falling into the loss cone of the magnetic focusing system were investigated for both considered configurations of the discharge gap.

For the simulation, we used the developed and implemented Monte Carlo model. The model described the motion of electrons in a gas in electric and magnetic fields of given spatial configurations. The Monte Carlo model is considered in detail in Refs. 30, 37, and 38. The model included two main modules: (1) a module integrating Newton's equations of motion and (2) a module describing collisions of electrons with particles of the gas medium. Within the first module, the relativistic equations of motion were integrated using the “predictor–corrector” scheme of the second order of accuracy. Within the second module, the “zero-collision” algorithm was implemented to describe the collisions of electrons with gas particles.³⁹ Molecular nitrogen at 1 atm pressure was considered as a gas medium. The collision module operated with a set of 19 electron–molecular interaction processes,⁴⁰ including elastic electron scattering, ionization, dissociation of the nitrogen molecule, excitation of various singlet and triplet electron states, as well as excitation of rotational and vibrational states of the nitrogen molecule. The data for the “zero-collision” algorithm operation were the total, differential, and doubly differential cross sections of the corresponding electron scattering processes between electrons and nitrogen molecules.^40–44 

The problem geometry sketch is given in Fig. 1. We considered the acceleration and transport of electrons in a discharge system comprised of an edge of a tubular cathode, which was the termination of the coaxial transmission line inner conductor, and an anode being the outer conductor of the transmission line. The outer conductor had a stepwise changing inner radius from 23.5 to 7.5 mm. Two tubular cathodes were considered: one with an outer diameter of 22 mm (11 mm radius) and the other with a diameter of 8 mm (4 mm radius). The cathode edges were assumed to be rounded; the edge curvature radius was 0.1 mm. The distance from the cathode edge to the line end face at the point of a stepwise decrease in the outer conductor radius was 24 mm for the 22-mm-diameter cathode and 14 mm for the 8-mm-diameter cathode. In Fig. 1, the boundaries of a computational domain are marked with a red dashed line. Outside this domain, the electron motion was not considered. Within the computational domain, a spatial configuration of an electric field was preliminarily calculated in the SAM software package⁴⁵ and then used in the Monte Carlo model.

In the experiments,²⁹ a magnetic field was created by a pulse solenoid placed on the outer conductor of the transmission line in its narrow part (the inner radius of the tube is 7.5 mm). The position of the solenoid edge relative to the other components of the electrode system is marked in Fig. 1 by the point (0,0). This point was also taken as the origin of the coordinate system within the applied numerical model. The model 0z axis coincided with the symmetry axis of the electrode system, i.e., with the axis of the transmission line. The magnetic field spatial configuration within the computational domain was calculated with the SAM software⁴⁵ taking into account actual parameters of an experimental setup.²⁹ 

Thus, the modeled electrode system was as close as possible to the electrode system used in the experimental work.²⁹ Taking into account the system symmetry, the dimension of the problem being solved was 2D-3V (2 coordinates–3 velocity components). The main adjustable model parameters characterizing force fields were an absolute value of a voltage applied to the electrodes U and a maximal induction B of a magnetic field at the collector position inside the narrowest part of the tube is the outer conductor of the transmission line. From Ref. 29, typical values of U and B were 110 kV and 5 T, respectively. Thereinafter, these U and B values are referred to as the “typical experimental conditions.”

First, it should be noted that, when modeling the RE motion through atmospheric pressure nitrogen, the generation of secondary particles through the electron ionizing collisions with gas molecules and the formation of discharge plasma were not considered. Accordingly, the influence of the electron space charge on the electric field distribution was not taken into account. The validity of this simplification is justified by the fact that, as is known,^16,37,46 in subnanosecond discharges, the RE flow moves ahead the ionization wave front propagating from the cathode toward the anode. Corresponding electric field distortion is important for the problem of “anomalous runaway electrons,” however, we believe that these effects may be neglected when considering the processes of RE focusing and reflecting from the “magnetic mirror.” In turn, the RE flow charge density and current are quite small. Therefore, the electron electrostatic repulsion may be neglected, and the magnetic field generated by an RE beam is expected to be much weaker than the external magnetic field. Hence, the field spatial configurations were considered undistorted during the calculation. That is, the single-electron approximation was used.

Second, the voltage U and the induction B were considered constant throughout the simulation. For the magnetic field, such an assumption is natural since, within the experiment,²⁹ the duration of the solenoid current pulse was orders of magnitude greater than the duration of the subnanosecond voltage pulse. The assumption of a constant voltage applied to the electrodes is also applicable to the system under consideration since the acceleration of electrons is expected to occur, mainly, near the cathode edge, where the electric field strength is maximum. For instance, near the 22-mm-diameter cathode, the electric field strength edge reaches 800 kV/cm at U = 100 kV. It is in a small region of ∼10 mm near the cathode edge where electrons accelerate up to ∼10¹⁰ cm/s and gain the bulk of their energy (from 50 to 80 keV). The simulation result analysis shows that electrons go through this region in <100 ps, which is less than the incident voltage pulse duration (∼300 ps FWHM²⁹) At the periphery of the gap, far from the cathode, the electric field strength decreases very rapidly, especially when approaching the section of the stepwise decrease in the radius of the outer conductor of the transmission line. Inside the anode tube with a radius of 7.5 mm, the electric field is negligibly weak, and electrons move by inertia only. Therefore, we believe the constant voltage assumption cannot introduce significant errors in the simulation results on the high-energy (tens of keV) RE beam passage through the described electrode system. As shown below, the simulation agrees qualitatively well with the experimental results.

In the simulations, an absolute value of the voltage U applied to the electrodes and the maximum induction B of a homogeneous magnetic field at the collector position were set as the input parameters. In Ref. 29, it was assumed that the RE source was a submillimeter semi-toroidal plasma layer near the cathode edge. The reason for the plasma layer formation was gas ionization by field-emitted electrons. Therefore, as in Ref. 29, the rounded edge of the cathode (see Fig. 1) was assumed to be the plasma boundary, which emitted electrons and had the cathode electric potential. When starting the simulation, we posited the presence of some free electrons emitted from the plasma in various directions. The emission point at the cathode edge and the direction of the initial electron velocity were characterized by the electron emission angle θ₀ measured from the direction of the auxiliary axis 0′z′ parallel to the main axis 0z and passing through the center of the cathode edge curvature, point 0′ = (z = −31.1 mm; r = 10.9 mm) for the 22‐mm‐diameter cathode (see Fig. 1) and 0′ = (z = −14.1 mm; r = 3.9 mm) for the 8‐mm‐diameter cathode. The angle θ₀ varied in the range of −75° to +75°. The number of electrons having each considered θ₀ value was identical. The total electron number was ∼200 000. The initial electron energy was specified in the range of 1–50 eV; however, it was found that varying the initial electron energy within this range did not significantly affect the simulation results.

After setting the simulation parameters (U, B, electron initial spatial distribution, etc.), the simulation of electron motion through the discharge gap started. A typical simulation time was 750 ps, and a simulation time step was 10⁻¹⁶ s. During the simulation, the electron kinetics were tracked. In addition, statistics were collected on the electrons reaching the anode-collector with different initial emission angles θ₀ for given values of U and B. It should be noted that, despite the formal anode of the discharge gap being the entire external conductor of the transmission line, only those electrons having reached the anode tube at its end face in the narrowest part of the tube (Z = 20 mm, see Fig. 1) were taken into account when collecting electron statistics. The noted end face was where an electron collector was placed inside the anode tube within the experimental work.²⁹ Thereinafter, the anode-collector denotes exactly this end face of the transmission line external conductor (see Fig. 1). Accordingly, electrons that had reached the outer tube in other areas were excluded from the final statistics and treated as electrons having left the calculation area.

First, it seems important to reveal the regularities of the electron motion in the discharge gap described in Sec. II B without considering electron scattering. Figure 2 demonstrates the electron trajectories in a vacuum for the discharge gaps with 22- and 8-mm-diameter cathodes. Each trajectory corresponds to an electron emitted from the cathode edge at some particular angle θ₀. The trajectories were calculated for typical experimental conditions:²⁹  U = 110 kV, B = 5 T, and the initial electron energy was 1 eV. From the trajectories for the 22-mm-diameter cathode, it is evident that electrons with θ₀ = −64° and θ₀ = −38° successfully reached the anode-collector. On the contrary, for the electrons with θ₀ = −7° and θ₀ = 0°, significant Larmor oscillations led to a decrease in the longitudinal component of the electron velocity V_z and an increase in the radial velocity component V_r. As a consequence, these electrons were reflected from the “magnetic mirror.”

In general, for the vacuum case, a detailed trajectory analysis allowed the conclusion that, for the given U and B value combination, there was a certain maximum angle θ_0max. Electrons emitted from the cathode edge with angles θ₀ < θ_0max reached the anode-collector regardless of the specific value of θ₀, at least within the studied angle range of −75° ≤ θ₀ ≤ +75°. On the contrary, none of the simulated electrons with θ₀ > θ_0max ever reached the anode-collector.

Figure 3 presents the calculated dependences of the θ_0max value on the magnetic field induction B and the voltage U for both cathodes considered. The dependences θ_0max(B) were calculated for U = 110 kV and the dependences θ_0max(U) were calculated for B = 5 T. The error bars given are due to the discrete character of the θ₀ value setting in the simulation. The calculation result analysis showed that, for both the 22- and the 8-mm cathode, the θ_0max value increased with increasing B and decreased with increasing U. Under typical experimental conditions (U = 110 kV, B = 5 T), for the 22-mm-diameter cathode, θ_0max = −13°, and for the 8-mm cathode, θ_0max = +65°. As a result, in a vacuum under typical experimental conditions, for the 22-mm-diameter cathode, only ∼40% of the simulated electrons were focused by the magnetic field without reflection. In turn, for the 8-mm-diameter cathode, this value was found to be greater than 90%.

Figure 4 demonstrates electron distributions on the emission angle θ₀ for those having reached the anode-collector under the typical experimental conditions for the 22- and 8-mm-diameter cathodes. Insets show the electron distributions on the radial R coordinate at the anode-collector. Here, the typical experimental conditions were considered (U = 110 kV, B = 5 T), and the electron scattering on molecular nitrogen of atmospheric pressure was taken into account. The distributions are normalized to 1. Obviously, the distributions on the R coordinate appeared to be close to normal ones. The distribution maxima are at R = 2.5 mm for the 22-mm-diameter cathode and at R = 1.3 mm for the 8-mm-diameter cathode. The positions of the distribution maxima agree well with the dimensions of the RE beam imprints on the phosphorous screen having been observed in the experiment,²⁹ where the radii of the luminous rings were ∼2.5 and ∼1.25 mm for the 22- and 8-mm-diameter cathodes, respectively. That is, as in the experimental work,²⁹ the three- to fourfold radial compression of the tubular RE beam was obtained in the simulation. In turn, an analysis of the electron distributions on the angle θ₀ at the anode allowed the conclusion that, for both cathodes considered, the electrons had a non-zero probability of reaching the anode-collector even at θ₀ = +75°, i.e., when, formally, θ₀ > θ_0max in a vacuum case. However, there are maxima at the distributions given in Fig. 4. For the 22-mm-diameter cathode, the electron θ₀ distribution maximum is at θ₀ ∼ −40°, and for the 8-mm-diameter cathode, the maximum is at θ₀ ∼ −10°.

It should be noted that, under typical experimental conditions, for the 22-mm-diameter cathode, the fraction of simulated electrons having reached the anode-collector was ∼20%; for the 8-mm-diameter cathode, the fraction was ∼57%. The simulation result analysis indicates that the main factors limiting the probability for electrons to reach the anode-collectors in nitrogen are (1) an inelastic energy loss of REs, which leads to their thermalization; (2) RE leaving the calculation domain; (3) the “magnetic mirror” effect arising due to the magnetic field inhomogeneity.

Figure 5 depicts the “emission angle θ₀–energy ε” portraits of electrons not having reached the anode-collector by the end of the simulation in atmospheric pressure nitrogen under typical experimental conditions. The portrait vertical axis is the decimal logarithm of the electron energy ε (eV). The portrait is normalized to the maximum concentration over the phase space.

From the portraits given in Fig. 5, for both cathodes, electrons can be separated into two energy-discriminated groups: the first one is with the energy ε ∼ 10 eV, and the second one is with 10⁴ eV ≤ ε ≤ 10⁵ eV limited by the electrode potential difference (∼10⁵ V). Accordingly, we labeled the first group as the “thermalized electrons” and the second group as the “reflected electrons.” We took the energy ε = 10³ eV as the conditional boundary between the groups. For the 22-mm-diameter cathode, both groups contained comparable numbers of electrons. For the 8-mm-diameter cathode, the group of “thermalized electrons” had a significantly larger number of electrons; moreover, the majority of “thermalized electrons” had the emission angle θ₀ < −30°, which correlated with the “fall” at θ₀ < −30° in the electron distribution on θ₀ for the 8-mm-diameter cathode (see Fig. 5). Similarly, for the 22-mm-diameter cathode, the concentration maximum of the “thermalized electron” group was at the region of large negative values of θ₀ < −50°. At the same time, for both cathodes, the maximal concentration of “reflected electrons” was at large positive values of θ₀; however, the concentration maximum was much more pronounced for the 22-mm-diameter cathode at θ₀ ∼ +45°.

Table I illustrates summary data on the electron passing through the discharge gap by the simulation end under various simulation conditions. Varied parameters were the voltage U, the induction B, the cathode unit configuration (the 22- or 8-mm-diameter cathode) and the presence/absence of a gas (atmospheric pressure nitrogen) in a discharge gap. All electrons simulated were separated into four groups. The first group was the electrons that had reached the anode-collector by the end of the simulation (the “anode” electron fraction). The second group was the electrons with ε < 1 keV that had not reached the anode-collector; these electrons were assumed to be thermalized. The third group contained the electrons not having reached the anode‐collector and having the energy ε > 1 keV; these electrons were considered as reflected due to the “magnetic mirror” effect. The fourth group comprised electrons having left the calculation domain out of the anode‐collector, mainly in the radial direction. The percentage given in the table is the fraction of the corresponding electron group within the simulated electron ensemble.

It was found that in nitrogen, under typical experimental conditions, for the 22-mm-diameter cathode, only ∼20% of electrons reached the anode‐collector. ∼41% of electrons were reflected from the “magnetic mirror,” ∼32% of electrons were thermalized, and ∼7% left the computational domain. Moreover, the fraction of electrons managing to pass through the “magnetic mirror” only slightly increased with increasing B and U (∼23% for U = 110 kV and B = 6 T, and ∼24% for U = 130 kV and B = 5 T). A decrease in B led to a decrease in the “anode” electron fraction to ∼16% and a sharp increase in the fraction of electrons having left the computational domain to ∼18%. For the 8‐mm‐diameter cathode, under typical experimental conditions, the “anode” electron fraction reached ∼57%, and, for this cathode, the main channel of electron losses was not the reflection from the “magnetic mirror” (∼14%), but the electron thermalization (∼29%). The electron losses due to leaving the computational domain were negligible (<1%). Meanwhile, increasing the gap voltage from 110 to 130 kV led to a sharp decrease in the fraction of thermalized electrons to ∼12%, and the “anode” electron fraction increased to ∼73%, however, the fraction of reflected electrons remained almost unchanged (∼14%–15%). An increase in B from 4 to 6 T at U = 110 kV led to a corresponding increase in the “anode” electron fraction from ∼55% to ∼59%, while the fraction of thermalized electrons was almost constant, ∼28%–29%. That is, the simulation gives a three- to fourfold higher probability of the electron passage through the “magnetic mirror” for the 8‐mm‐diameter cathode compared to the 22‐mm‐diameter cathode.

If we ignore the electron thermalization in nitrogen, the “magnetic mirror” effect is the main factor limiting the electron passage through the systems for the runaway electron beam radial compression by an inhomogeneous magnetic field. The effect manifests as arising from the cyclotron oscillations after the initial stage of the electron acceleration near the cathode and the subsequent reflection of relatively high-energy (∼10 keV) electrons from the region of the magnetic field line concentration. The “magnetic mirror” effect is well known, e.g., within the field of high-temperature plasma magnetic confinement. However, the feature of the present investigation is that this effect appears to be negative, and we aim to suppress it maximally. In this sense, the considered problem statement is inverse to the plasma magnetic confinement problem for which the maximization of the “magnetic mirror” effect is required.

For the vacuum case, an analysis of the electron trajectories (see Fig. 2) allows the conclusion that, under particular simulation conditions (cathode configuration, electrode voltage, and maximal magnetic field induction), the possibility of the electron reaching the anode-collector was determined only by the direction θ₀ of the electron emission. This determinism appears as the existence of the maximal electron emission angle θ_0max above which the “magnetic mirror” effect was always observed. It has been shown above (see Sec. III A) that the θ_0max value increased with increasing B and decreased with increasing U (see Fig. 3). These regularities agree well with elementary ideas about “magnetic mirror” systems. Namely, for electrons with relatively large θ₀ values, an increase in U is expected to lead to the growth of the electron velocity component orthogonal to the magnetic field lines and arising cyclotronic oscillations. As a consequence, θ_0max has to decrease with an increase in U. Meanwhile, an increase in B has to lead to a decrease in the electron cyclotronic orbit radius and, hence, the electron velocity component orthogonal to the magnetic field lines. Therefore, θ_0max increases with an increase in B.

On the other hand, it is clear that θ_0max has to connect with the “mirror” loss cone of the system under consideration. Let us put the pitch angle α to be the angle between the direction of the electron velocity near the cathode determined by the emission angle θ₀ and the magnetic field line slope θ_B near the cathode edge, and α_max to be such a value of α that limits the electron “mirror” loss cone. For the vacuum case, we used the following relation between α_max, θ_0max, and θ_B: α_max = θ_0max − θ_B. For the experimental setup,²⁹ in the coordinate system introduced in Sec. II (see Fig. 1), θ_B = −18° for the 22-mm-diameter cathode and θ_B = −8° for the 8-mm-diameter cathode. Therefore, for the typical experimental conditions (U = 110 kV, B = 5 T), θ_0max = −13° and α_max = 5° for the 22-mm-diameter cathode. In turn, θ_0max = +65° and α_max = 73° for the 8-mm-diameter cathode. At the same time, the basic theory of the “magnetic mirror” systems gives the following relation for the α_max estimation: α_max = arcsin [(1/K)^1/2], where K is the “mirror” ratio. For the problem under consideration, K = B/B_min, where B denotes the magnetic field induction at the collector position and B_min is the induction near the cathode edge. For the 22-mm-diameter cathode, K = 20 and for the 8-mm-diameter cathode, K = 10;²⁹ the corresponding α_max values are 13° and 18°. Obviously, these α_max values are far from derived from the simulation (5° and 73°, respectively). In fact, it is not surprising since the considered problem differs significantly from the “magnetic mirror” devices for the high-temperature plasma confinement. The main difference is the strong inhomogeneity of an electric field near the cathode edge. In Figs. 6(a) and 6(b), spatial configurations of electric and magnetic fields near the cathode edges are shown for both considered cathodes. It is obvious that the direction of electric field lines (black dashed lines) continuously changes when moving from the cathode toward the anode. Since the electron trajectories tend to coincide, mostly, with the electric field lines, it leads to the continuous change in the electron pitch angle α as an electron accelerates near the cathode. So, the complexity of the electron interaction with inhomogeneous electric and magnetic fields makes the basic theory inapplicable to the considered problem.

In Ref. 29, an analytical estimation for α_max taking into account the employed experimental setup features was proposed: m_eUK⁵sin⁴α_max = 8ed²B². Here, m_e is the electron mass, e is the absolute value of an elementary charge, and d is the conditional length of the gap. The d value was set to be the distance between the cathode edge and the transmission line end face at Z = −7 mm; for the 22-mm-diameter cathode, d = 24 mm, and for the 8-mm-diameter cathode, d = 14 mm (see Fig. 1). The expression was derived at the boundary of the region with the length h ≪ d where the electric force exceeds the magnetic one. In addition, the following approximation for the electric field strength E distribution along the straight line parallel to the 0Z axis and connecting the cathode edge and the transmission line end face was used: E(Z) ≈ U/(4Zd)^1/2. (A detailed expression deriving process is given in Ref. 29.) When using the described estimation, for the typical experimental conditions, the α_max values for the 22- and 8-mm-diameter cathodes are 29° and 63°, respectively. The estimation for the 8-mm-diameter cathode (63°) appears to be close enough for the α_max value obtained in the simulation: 73°. However, for the 22-mm-diameter cathode, the discrepancy between the estimation and the simulation (29° and 5°) is even more than that between the basic theory and the simulation (13° and 5°). Highly likely, it is because of the strong spatial inhomogeneity of an electric field near the 22-mm-diameter cathode, due to which the E(Z) approximation used for the estimations provides poor agreement with the real E spatial distribution. However, it should be noted that there is an adequate qualitative agreement between the α_max (B) dependence obtained in the simulation (see Fig. 2) and that provided by the estimation (see Fig. 3 in Ref. 29).

Unlike the vacuum case, in atmospheric pressure nitrogen (see Fig. 4), electrons reached the anode-collector even if θ₀ > θ_0max for given values of U and B (see Fig. 3). Obviously, electron scattering leads to a chaotic change in the electron velocity direction, and the velocity vector may occasionally appear to be collinear with the magnetic line direction. In this case, the probability for this “lucky” electron to reach the anode-collector has to increase, although the probability still depended strongly on θ₀. As a result, the electron distributions on the angle θ₀ at the anode-collector had maxima. For the 22-mm-diameter cathode, the distribution maximum is at θ₀ ∼ −40° (see Fig. 4), which exceeds the magnetic line slope θ_B near this cathode: θ_B = −18°. In turn, for the 8-mm-diameter cathode, the distribution maximum is at θ₀ ∼ −10°, which is close to θ_B = −8°. Apparently, for the 8-mm-diameter cathode, the maximum probability of reaching the anode-collector was observed for electrons with a near-zero pitch angle α. With an increase in α, the probability of reaching the anode-collector decreased only slightly because, for the 8-mm cathode, the majority of θ₀ values considered met the condition θ₀ < θ_0max. However, for the 22-mm-diameter cathode, the distribution maximum is shifted away from the magnetic line slope. At the same time, for both cathodes, the electron θ₀ distribution maxima match well with the emission angle value corresponding to the minimal electron flight time in a vacuum [see Fig. 6(d)]. One may conclude that these θ₀ values refer to optimal electron trajectories both in a vacuum and in nitrogen. The nature of the optimum is such a balance between electric and magnetic forces acting on an electron that it provides the minimal radius of cyclotron oscillations and the least value of the electron reflection probability. The identity of the optimal angle θ₀ in a vacuum and nitrogen suggests that neglecting the electron scattering is the approved assumption when performing analytical estimations of the runaway electron motion within the considered systems since the main features of the electron motion are similar in a vacuum and gas.

Meanwhile, under typical experimental conditions, for the 22‐mm‐diameter cathode, only ∼20% of simulated electrons reached the anode-collector in nitrogen by the end of the simulation (see Table I). For the 8‐mm‐diameter cathode, this value was ∼57%. The respective fractions of the reflected electrons were ∼41% and ∼14%. That is, the threefold lower probability for a runaway electron to reach the anode-collector and the threefold higher reflecting probability were for the 22‐mm‐diameter cathode compared to those for the 8‐mm‐diameter cathode. We believe that for the 22‐mm‐diameter cathode, the higher reflecting probability relative to the anode-reaching probability was due to the predominance of the electric force acting on electrons over the magnetic force near this cathode. On the contrary, for the 8‐mm‐diameter cathode, the predominance of the magnetic force over the electric force was observed, and the reflecting probability was much smaller than the anode-reaching probability. From the experimental work,²⁹ for the 22‐mm‐diameter cathode, the “magnetic mirror” ratio K was 20; for the 8‐mm‐diameter cathode, K was 10. Hence, with the B value being the same, the magnetic field induction near the 8‐mm‐diameter cathode was twice the induction near the 22‐mm‐diameter cathode. At the same time, an average electric field strength near the 8‐mm‐diameter cathode was ∼20% smaller than the strength near the 22‐mm‐diameter cathode [see Fig. 6(c)]. Therefore, it is natural to expect that the force balance shifts more toward the magnetic force near the 8‐mm‐diameter cathode than near the 22‐mm‐diameter cathode. It explains a higher reflection probability for REs started from the 22‐mm‐diameter cathode.

To clarify the “magnetic mirror” effect within the considered systems, let us investigate electron phase portraits. Figure 7 shows the electron Z–V_z portraits for both cathodes at various instances. For all portraits given: (1) V_z is the Z component of the electron velocity, (2) only electrons with ε > 10³ eV were considered, and (3) black dots indicate an electron portrait in a vacuum. The point to note is that, for the electron phase portraits in a vacuum, the electron spread is due to variation in their emission angle values θ₀ only; in atmospheric pressure nitrogen, the spread is due to electron–neutral collisions and the θ₀ variation. The study of the electron phase portraits for the 22-mm-diameter cathode indicated that after ∼150 ps of the simulation start, both in a vacuum and in nitrogen, the portraits had noticeable oscillations. The oscillations most clearly manifested in the gas by ∼220 ps. In nitrogen, the oscillations arose as distinct electron groups that repeated the shape of the electron phase portrait in a vacuum at 220 and 300 ps. In these instances, the maximal electron concentration corresponded to the advanced group of electrons, which had the largest energy and Z coordinate from the electron ensemble simulated. However, at ∼300 ps, forming a group of electrons with V_z < 0 began. By ∼450 ps, it led to the appearance of a significant fraction of electrons moving in the opposite direction relative to the anode-collector. Moreover, the electron concentration maximum moved backward from the anode-collector to the region with Z ∼ −15 mm. Obviously, this retrogradely moving electron fraction comprised electrons reflected due to the “magnetic mirror” effect. In addition, by 450 ps, even though the electron phase portrait oscillations were still observed in a vacuum, they were smoothed in nitrogen due to electron scattering. At the same time, a general pattern of electron motion in nitrogen coincided with that in a vacuum.

For the 8-mm-diameter cathode, by ∼200 ps, the phase portrait oscillations were not as pronounced as for the 22-mm-diameter cathode by ∼220 ps. In turn, by ∼350 ps, a group of backward-moving electrons was formed in nitrogen, although in a vacuum, it formed only by ∼550 ps. Obviously, the electron scattering may facilitate and obstruct electron focusing by a magnetic field. Moreover, in nitrogen, by ∼550 ps, near the cathode, a group of electrons changed the sign of their V_z velocity component, and, as a result, a group of forward-moving electrons started forming. One can say that the oscillation of initially reflected electrons along the Z axis arose. At the same time, there were two electron concentration maxima corresponding to the instance of 550 ps. The first maximum was at the head of the beam near the anode-collector; this maximum was also observed at the instances of 200 and 350 ps. The second maximum was at Z ∼ −15 mm due to electrons slowing down near the cathode and starting newel reflected electrons accelerating toward the anode-collector.

The cause of the phase portrait oscillations is arising electron cyclotronic oscillations. The oscillation phase and amplitude depend generally on the electron emission angle θ₀. The phase portrait spread in nitrogen around the phase portrait in a vacuum is associated with the electron scattering on gas molecules. Nevertheless, a general tendency of REs in nitrogen to repeat the nature of the electron motion in a vacuum is evident. This motion is characterized by cyclotronic oscillations and the reflection of electrons with large pitch angles. In addition, it is clear that the “magnetic mirror” effect was much more pronounced for the 22‐mm‐diameter cathode than for the 8‐mm‐diameter cathode. We believe the causes were the electric field configuration and the balance between the magnetic and electric forces near the 22‐mm‐diameter cathode. Both factors facilitated the RE cyclotronic oscillations and subsequent RE reflection from the “mirror.” It explains qualitatively the experimental fact of higher RE current density registered for the 8‐mm‐diameter cathode.

In addition to reflection, RE thermalization appeared to be another factor limiting the anode-reaching probability for REs. Under typical experimental conditions, the fraction of thermalized electrons was ∼32% for the 22-mm-diameter cathode and ∼29% for the 8-mm‐diameter cathode. The thermalization was most pronounced for electrons emitting with large negative θ₀ angles from the cathode edge areas with relatively small electric field strength values. Namely, for both cathodes, the absolute electric field strength at θ₀ ∼ −75° was found to be approximately twice as small as the one at θ₀ ∼ 0° [see Fig. 6(c)]. This effect was clearly observed for the 8-mm-diameter cathode (see Figs. 4 and 5), especially for electrons with θ₀ < −40°. Although in a vacuum, these electrons always reached the anode-collector; in atmospheric pressure nitrogen, the anode-reaching probability for electrons with θ₀ < −40° was more than twice lower compared to that for electrons with θ₀ ∼ 0°. However, for the 8‐mm‐diameter cathode, it may be compensated by increasing the electrode voltage U (see Table I). An increase in U from 110 to 130 kV led to a decrease in the fraction of the thermalized electrons from ∼29% to ∼12% and an increase in the anode fraction from ∼57% to ∼73%. In contrast, the fraction of reflected electrons changed insignificantly from ∼14% to ∼15%. At the same time, for the 22‐mm‐diameter cathode, an increase in U to 130 kV gave a decrease in the thermalized electron fraction from ∼32% to 15% and an increase in the anode electron fraction from ∼20% to ∼24%; however, the reflected electron fraction increased from ∼41% to ∼46%. Moreover, an increase in U led to an increase in the fraction of electrons having left the computational domain in a radial direction from ∼7% to ∼15%. The last note additionally confirms that for the 22‐mm‐diameter cathode, the force balance appears to shift toward the electric force, not the magnetic one.

Meanwhile, for both cathodes, it was found that, due to the “magnetic mirror” effect, in nitrogen, REs may pass through the gap at least three times during the simulation. Initially, electrons emitted from a cathode are accelerated by an electric field and focused by a magnetic field. At the region of the magnetic field line concentration, some electrons are reflected and start moving toward the cathode. Approaching the cathode, electrons decelerate and may, again, change the sign of their Z velocity component because of the interaction with gas and magnetic and electric fields. Thus, electrons start newel moving toward the anode. During their motion, electrons inevitably ionize a gas medium. Therefore, multiple passages of high-energy (∼10 keV) electrons through the gap may lead to a rapid formation of a discharge plasma channel and the subnanosecond gap switch. However, the last points require additional investigations.

A numerical 2D-3V Monte Carlo simulation of runaway electron (RE) focusing by an inhomogeneous magnetic field in atmospheric pressure nitrogen was carried out. From the point of the “magnetic mirror” effect manifestation, the problem statement was inverse to the problem of plasma magnetic confinement. The Monte Carlo model used to study electron kinetics included a module for integrating the Newton motion equations and a collision module employing the null-collision technique.²⁹ We considered two electrode gap configurations. The first configuration had a 22-mm-diameter tubular cathode, and the second one had an 8-mm-diameter cathode. For both configurations, spatial distributions of electric and magnetic fields were preliminarily calculated using the SAM software package.⁴⁵ Simulated electrons were assumed to be emitted from a semi-toroidal cathode edge associated with a near-cathode submillimeter plasma layer. The simulation results were data on electron kinetics and statistics on anode-collector reaching by electrons with various emission angles θ₀ relative to the gap axis for given values of an absolute voltage U applied to the cathode and a maximal magnetic field induction B near the collector position.²⁹ The results obtained for the RE motion in nitrogen were compared to those for the case of the electron motion in a vacuum.

It was found that, as in a vacuum, for the 22-mm-diameter cathode, the main reason preventing electrons from reaching the anode-collector in a gas was the “magnetic mirror” effect. Namely, in nitrogen, under typical experimental conditions (U = 110 kV, B = 5 T), no more than 20% of the simulated electrons reached the anode-collector, and ∼41% of electrons were reflected from the “magnetic mirror.” Moreover, the “anode” fraction increased only by a few percent with an increase in the voltage U to 130 kV or the magnetic field induction B to 6 T. The electron phase portrait analysis showed that, in general, runaway electrons in nitrogen repeated the pattern of electron motion in a vacuum. In particular, significant phase portrait oscillations caused by electron cyclotron oscillations were observed. Arising oscillations led to a decrease in the electron velocity component longitudinal to the electrode system axis and, eventually, to the reflection of a significant electron fraction from the “magnetic mirror.” In contrast, for the 8-mm-diameter cathode, no apparent phase portrait oscillations were observed in a vacuum and nitrogen. Under typical experimental conditions, in nitrogen, the fraction of electrons reaching the anode-collector was ∼57%, and the fraction of reflected electrons was no more than ∼15%. We believe that such a large difference in the probability of reaching the anode-collector was due to the features of the electric and magnetic field configurations near the considered cathodes. For the 22-mm-diameter cathode, the higher reflecting probability relative to the anode-reaching probability was due to the predominance of the electric force acting on electrons over the magnetic force near this cathode. In turn, for the 8-mm-diameter cathode, the predominance of the magnetic force over the electric force was observed, and the reflecting probability was much smaller than the anode-reaching probability. Moreover, for both cathodes, the electrons emitted along the trajectory corresponding to the minimum time of flight of electrons from the cathode to the anode in a vacuum had the highest probability of reaching the anode-collector in nitrogen. This trajectory also provided the minimal amplitude of electron cyclotron oscillations. On the other hand, for both cathodes, the RE thermalization probability is relatively high, being ∼30% under typical experimental conditions. For the 8-mm-diameter cathode, an increase in U from 110 to 130 kV allowed the thermalized electron fraction to decrease to ∼12%. It caused the “anode-collector” fraction of electrons from ∼57% to ∼73%, while the fraction of reflected electrons changed insignificantly. For the 22-mm-diameter cathode, an analogous increase in U gave only a weak growth in the “anode” electron fraction since, besides a twofold decrease in the fraction of thermalized electrons (from ∼32% to ∼15%), it led additionally to an increase in the fraction of reflected electrons from ∼41% to ∼46% and a sharp increase in the fraction of electrons leaving the computational domain from ∼7% to ∼15%.

The simulation result analysis explained qualitatively the experimental fact of higher runaway electron current density registered for the 8-mm-diameter cathode when compared to the one registered for the 22-mm-diameter cathode. Moreover, the numerical model predicted the same three- to fourfold radial compression of the runaway electron beam as was observed in the experimental work.²⁹ Therefore, we believe that the developed numerical model is suitable for analyzing currently developed systems for the runaway electron beam focusing and compression by inhomogeneous magnetic fields, and the simulation results will be used to optimize the experimental setup.²⁹ 

The research was carried out with the financial support of the Russian Science Foundation under Grant No. 23-19-00053.

The authors have no conflicts to disclose.

Yu. I. Mamontov: Formal analysis (equal); Investigation (equal); Software (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). M. I. Yalandin: Data curation (equal); Investigation (equal); Methodology (equal); Resources (lead); Software (equal). N. M. Zubarev: Conceptualization (lead); Data curation (equal); Investigation (equal); Project administration (lead); Supervision (lead); Validation (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.