1. INTRODUCTION
Among other applications, intense pulsed electron beams can be used to improve the surface properties of metallic materials. Irradiation of metal surfaces leads to local heating and melting of the surface layer followed by rapid cooling and re-solidification, which induces stresses and a refined microstructure. By this technique, the surface hardness, wear, and corrosion resistance can be improved (Mueller et al., Reference Mueller, Schumacher and Strauss1998, Reference Mueller, Schumacher and Zimmermann2000; Proskurovsky et al., Reference Proskurovsky, Rotshtein, Ozur, Markov, Nazarov, Shulov, Ivanov and Buchheit1998; Archiopoli et al., Reference Archiopoli, Mingolo and Mingolo2008; Zou et al., Reference Zou, Zhang, Hao, Dong and Grosdidier2010). Also alloying of a previously applied coating with the bulk material is feasible (Mueller et al., Reference Mueller, Engelko, Weisenburger and Heinzel2005; Batrakov et al., Reference Batrakov, Markov, Ozur, Proskurovsky and Rotshtein2008; Fetzer et al., Reference Fetzer, Weisenburger, Jianu and Mueller2012; Weisenburger et al., Reference Weisenburger, Jianu, An, Fetzer, Del Giacco, Heinzel, Mueller, Markov and Kasthanov2012). One particular application of pulsed electron beam treatment is the improvement of the corrosion resistance of steels in high temperature heavy liquid metal environment such as GenIV fission reactors or accelerator driven systems. Of special importance are here the cladding tubes, which will have to resist the highest temperatures in the reactors. The cylindrical pulsed electron beam facility GESA IV was specifically designed for treatment of the cladding tube surface (Engelko et al., Reference Engelko, Kuznetsov and Mueller2009). The target, that is, the cladding tube, serves as anode and is surrounded by a cylindrical cathode and controlling grid. The converging radial electron beam is generated at the explosive emission cathode, which consists of individual carbon fiber bundles. Application of a high voltage pulse leads to electron emission at the fiber bundles and the formation of plasma in the cathode region. The grid voltage controls the beam current density emitted from the cathode plasma. The beam is then accelerated towards the target, where it induces heating and melting of the cladding tube surface. Beam–target interaction also results in ion generation and, after a few microseconds, in the formation of plasma in the vicinity of the target surface. No external magnetic field is applied. However, the axial current flow along the target induces an azimuthal magnetic field, which may become relevant.
Operation of the cylindrical GESA IV was found to be a rather delicate task; current instability and beam inhomogeneity are often observed (An et al., Reference An, Engelko, Mueller and Weisenburger2009). In order to improve the beam performance and to extend the region of treatment, a new design with a 1-m long cathode has been developed (Engelko, Reference Engelko2014). In a recent numerical study (Altsybeyev et al., Reference Altsybeyev, Engelko, Ovsyannikov, Ovsyannikov, Ponomarev, Fetzer and Mueller2016), the feasibility of this new design was investigated and a stable operation regime for bipolar flow could be identified. In the present paper, the numerical study is extended to explore by particle-in-cell (PIC) code simulations the different operation regimes of cylindrical triode-type electron accelerators. In addition to bipolar flow, also the case of unipolar electron flow is investigated in detail. Unipolar flow is relevant in the first few microseconds of beam application, when only a very limited number of ions is generated at the target and the flow of ions towards the cathode can be neglected. For both unipolar and bipolar flow, operation regimes beyond laminar flow are identified and the influence of different target materials is investigated.
2. METHODS
For the numerical studies, the PIC code software package MAGIC by Orbital ATK, USA, is used (Goplen et al., Reference Goplen, Ludeking, Smithe and Warren1995). MAGIC is an electromagnetic finite-difference time-domain PIC code capable of describing charged particle beams and pulsed power devices. The time-dependent Maxwell equations are solved to obtain electromagnetic fields. The particle trajectories are calculated using the relativistic Lorentz force equation. The continuity equation yields charge and current densities for the Maxwell equations. Azimuthal symmetry of the cylindrical system is assumed as well as mirror symmetry at z = 0. The system geometry is shown in Figure 1. The target (anode) has a radius of 5 mm. The cathode has a total length of 1 m and an inner radius of 15 cm. The controlling grid has a total length of 1.2 m and a radius of 10 cm. It is composed of individual conductor rings, which are connected via line resistors. In the shown case, each ring is 2 mm wide (the chosen cell size in axial direction) and the periodicity is 26 mm. Thus, the transparency of the grid is 92.3%. The line resistors connecting the rings fix the electric potential between the rings, which keeps the electric field distortion around the grid rings to a minimum. Note that smaller cell sizes and thinner rings do not change the presented results noticeably, apart from the obvious change of shaded regions. Also note that in the experimental realization the grid is made from wires with 0.5 mm diameter and no shaded regions are expected.
Fig. 1. System geometry. The accelerator with mirror symmetry at z = 0 is composed of conductors and line resistors. Conductors are the outer chamber walls, cathode and cathode holder, grid rings and grid holder, and the target. Line resistors R cg and R g connect cathode holder, grid holder, and target. Additional line resistors connect the individual grid rings. The system is connected to a generator via a transmission line at the input port.
A high voltage pulse is applied to the cathode via a transmission line at the input port, which is an open boundary of the simulation area. The generator is impedance matched to the connecting transmission line. The input voltage is given by a respective function in time. In the cases considered here, the generator voltage is smoothly ramped to a specified value, at which it remains constant for the rest of the simulation. The potentials of cathode and grid against the target are controlled via the resistors R cg and R g. The outer boundary of the simulation area (z = 1.1 m) is grounded. Beam current reaching the target flows in axial direction along the target to the outer boundary. Primary electrons (beam electrons) are generated at the cathode via the explosive emission routine (space charge limited emission). Their initial velocity distribution at emission is controlled by the temperature of the emission plasma, which is set to 10 eV. The beam electrons then move in accordance with the relativistic Lorentz equation in the free space of the simulation area until they hit a conductor. There is no background gas or ionization/recombination processes included in the simulation. Thus, electrons are deleted if and only if they hit a conductor. For some conductors such as the target, backscattering of electrons may be considered. In this case, when electrons hit the surface, secondary electrons (backscattered electrons) are emitted according to the specifications of the conductor material, that is, the number of backscattered electrons and their kinetic energy distribution obey the corresponding statistics. The third type of charged particles considered in the simulation are ions, here protons in particular. If included, they are emitted from the target surface obeying the space charge limit. Once emitted, ions move according to the Lorentz equation and are deleted from the simulation if and only if they hit a conductor.
In all simulations presented throughout this study, the cathode potential is fixed at U c = −120 kV. The resistors R cg and R g obey R cg + Rg = 64 Ω, while their individual values are varied against each other to tune the grid voltage. The grid current, that is, the part of the beam current striking the grid and conducted along R g to the target, also has some influence on the grid voltage. The simulations are run until a stationary or quasi-stationary regime is obtained.
3. RESULTS AND DISCUSSION
3.1. Unipolar Flow
3.1.1. Laminar flow regime
First, unipolar flow is considered, that is, electrons represent the only relevant particle type. The performance of unipolar flow is of interest for the initial stage of pulsed electron beam application (<3 μs) when ion generation at the target and ion motion towards the cathode is negligible. In Figure 2, the results of one particular simulation with R cg = 10 Ω and R g = 54 Ω resulting in U cg = 18.2 kV are summarized. An aluminum target with respective backscattering of electrons is employed. The distribution of primary beam electrons (blue dots) is axially homogeneous (except the regions shaded by the grid rings), with continuously decreasing density from the cathode towards the target, see Figure 2a. The beam edge, located at the cathode at z = 500 mm, is slightly bent inwards and reaches the target at ~480 mm, that is, the beam is slightly focused. The kinetic energy of the beam electrons (Fig. 2c) monotonically increases on their way from the cathode (r = 150 mm) via the grid (r = 100 mm) to the target (r = 5 mm). From the kinetic energy of the beam electrons, the electric potential distribution as well as the space charge distribution can be easily deduced. From Figure 2d, the radial momentum of all beam electrons is found to be negative, that is, no primary electrons return towards the cathode. This also means that no electrons miss and pass by the target. Note that oscillation of beam electrons around the target can indeed be provoked by increasing the emission plasma temperature to values above ~100 eV, thus generating a large azimuthal momentum due to the angular velocity spread at emission. Backscattered electrons (red dots) are found mainly in the vicinity of the target and have a wider spread in axial position than beam electrons. The kinetic energy of backscattered electrons is below that of the beam electrons, covering a wide range. Backscattered electrons show both positive and negative radial momentum. Starting at the target, they move towards the cathode until they stop and finally return to the target. As shown in Figure 2b, the beam power density measured at the target is homogeneous (except the positions shaded by the grid rings) for axial positions below ~400 mm with a value of 0.74 MW/cm2, and is continuously decreasing to zero in the axial range ~400–500 mm. Note that the quantitative numbers of the energy density given in Figure 2b require normalization to the size of the simulation cells. To summarize the situation shown in Figure 2, laminar flow of beam electrons is obtained. The trajectories of beam electrons do not cross each other.
Fig. 2. Simulation results for unipolar flow with U cg = 18.2 kV: (a) Particle distribution in (z,r) space, (b) accumulated beam energy density at target surface after 95 ns versus axial position, (c) kinetic energy distribution versus radial position, (d) radial momentum versus radial position. Blue dots represent primary beam electrons, red dots indicate electrons backscattered at the target.
As the cathode-grid voltage and the beam current are increased, the beam characteristics shown in Figure 2 change. Due to the increased axial target current flow and the stronger self-induced azimuthal magnetic field, the beam shows a stronger focusing and smaller footprint on the target. Further, higher beam current increases the space charge. As a consequence, a potential well is formed between grid and target, which affects the kinetic energy and radial momentum distributions. Both distributions show a local minimum between grid and target for cathode-grid voltages above ~25 kV. Nevertheless, the electron flow is still laminar.
In Figure 3 various currents of the full accelerator (1 m long cathode) are summarized. Two different operation regimes are found, with the transition occurring at a cathode-grid voltage of about 32 kV (emission current threshold ~4.5 kA). The above described situation of laminar flow is obtained below this threshold. In the following, the currents of the laminar flow regime are analyzed in more detail. Due to the grid transparency of only 92%, about 8% of the current emitted at the cathode (emission current) strikes the grid rings and is conducted along the grid and resistor R
g to ground (grid current). The rest of the beam current reaches the target directly (target current), where it flows along the target to ground (z = 1.1 m). A very small amount of electrons (mainly backscattered electrons) escapes to the sides before reaching target, grid, or cathode. The respective current of escaped electrons is below 40 A and is not shown in the figure. All the three currents of the cylindrical triode shown in Figure 3, increase with increasing cathode-grid voltage. The simulation results are compared with the theoretical expectation of space-charge limited flow in infinite-length cylindrical diodes with cathode radius 150 mm and anode radius 100 mm, that is, the position of the controlling grid in the triode system. The earliest numerical solution of space-charge limited flow in cylindrical diodes with infinite lengths was found by Langmuir and Blodgett (Reference Langmuir and Blodgett1923), while an approximate analytical solution can be found in Ref. (Chen et al., Reference Chen, Dickens, Hatfield, Choi and Kristiansen2004):
$J = (4{\rm \varepsilon} _0 /9)\sqrt {(2e/m)} (U^{3/2} /d^{1/2} r_{\rm c}^{3/2} )(1/\ln (r_{\rm c} /r_{\rm a} ))^{3/2} $
with J the emission current density, U the cathode-anode voltage, d the distance between cathode and anode, ε0 the vacuum permittivity, e the electron charge, m the electron mass, r
c the cathode radius, and r
a the anode radius. In the limit of gap distances much smaller than the cathode radius, both solutions converge to the well-known Child-Langmuir law for planar diodes,
$J = (4{\rm \varepsilon} _0 /9)\sqrt {(2e/m)} (U^{3/2} /d^2 )$
(Child, Reference Child1911; Langmuir, Reference Langmuir1913). In Figure 3 the approximate analytical solution from Chen et al. (Reference Chen, Dickens, Hatfield, Choi and Kristiansen2004) is shown. The slightly higher emission currents obtained in the simulations might be due to edge effects (finite length of cathode), which are not considered in the approximate analytical solution. Apart from this small offset, the simulation results for the cylindrical triode in the laminar flow regime agree very well with the predicted space-charge limit of corresponding diodes. Thus, processes between grid and target do not influence the beam characteristics in the cathode-grid gap and the space charge limited emission current of the triode can be predicted by the corresponding diode configuration.
Fig. 3. Emission, target, and grid current of the full length accelerator as a function of cathode-grid voltage for constant cathode potential −120 kV, Al target, and unipolar flow.
3.1.2. Virtual cathode formation
For cathode-grid voltages above the laminar flow threshold of ~32 kV, the emission current is found to stay below the space charge limit predicted for the cathode-grid gap, see Figure 3. As will be demonstrated in the following, the reason for this behavior is the formation of a virtual cathode between grid and target.
As mentioned in the previous section, a potential well is formed between grid and target for cathode-grid voltages above ~25 kV (beam current above ~3 kA). By further increasing the cathode-grid voltage and thus the emission current, the potential well grows until the kinetic energy of the beam electrons between grid and target drops to zero (i.e., the potential of the well reaches the cathode potential) and a virtual cathode is formed. Note that the cathode potential was fixed at −120 kV for all simulations. As a consequence, for increasing cathode-grid voltage not only the higher beam current and increased space charge but also the lower grid voltage facilitates the formation of the virtual cathode. The transition from the laminar flow regime to a regime where a virtual cathode is formed is shown in Figure 4. Here, the beam electrons are represented in different colors (blue, green, pink, orange, and grey) depending on the position along the cathode where they were emitted, see the particle distribution in Figure 4a. From the radial momentum distribution (Fig. 4b) it becomes clear that mainly “pink” and a few “green” electrons see a virtual cathode between grid and target at around r = 50 mm. Thus, a virtual cathode is formed only in the range of z between ~150 and ~270 mm. The beam electrons that experience the virtual cathode return towards the cathode. Before reaching the cathode, however, they turn back and are again accelerated towards grid and target. As can be observed in the particle distribution, some “pink” electrons move towards the beam center (z = 0) while oscillating around the grid. The beam energy density at the target (Fig. 4c) shows a depression for z values of ~100–250 mm due to the local virtual cathode. Further, beam focusing due to the self-induced magnetic field becomes obvious, which also increases the electron space charge and facilitates the formation of the virtual cathode. Note, however, that the magnetic field at the beam edge is not strong enough to cause magnetic insulation, that is, to prevent “grey” electrons from reaching the target. If that was the case, grey dots would appear firstly in the radial momentum distribution with positive (or at least zero) radial momentum at small radii and secondly in the particle distribution at positions closer to the beam center. Because of the still laminar border between “grey” and “orange” electrons, that is, no grey dots enter the region occupied by orange dots, magnetic insulation can be safely excluded for the present case.
Fig. 4. Virtual cathode formation, observed for unipolar flow with U cg = 33.5 kV. (a) Particle distribution in (z,r) space; (b) radial momentum distribution versus radial position; (c) accumulated beam energy density distribution at target surface after 501 ns versus axial position. In (a) and (b), beam electrons are represented by, respectively, blue, green, pink, orange, and grey dots. Electrons backscattered at the target are represented by red dots.
By further increasing the cathode-grid voltage and beam current, the local virtual cathode axially expands until almost the full length of the beam is subject to a virtual cathode, see Figure 5. Again, electrons oscillating around the grid due to the virtual cathode move towards the beam center. Only electrons around the beam edge (“grey” electrons) do not experience a virtual cathode. This is also confirmed by the beam energy density at the target surface, which shows a peak at z ≈ 400 mm.
Fig. 5. Fully developed virtual cathode formation, observed for unipolar flow with U cg = 44.8 kV. Presentation details see Figure 4.
As soon as a virtual cathode is formed (even locally), the emission current stays below the space charge limit predicted for the corresponding diode, see Figure 3. This is easily explained by the increased space charge in the cathode-grid gap due to electrons returning towards the cathode. Another observation is the increase of the grid current well beyond the geometrical ratio of 8%, which is caused again by the returning electrons that oscillate around the grid. The target current reaches its maximum for a cathode-grid voltage of ~33 kV and saturates at about 4 kA for larger cathode-grid voltages.
3.1.3. Electron backscattering
For all simulations so far, an Al target with backscattering of electrons at the target was considered. In this section, different target materials are introduced and their effect on the observed currents is studied. Note that in general materials with higher atomic number backscatter more electrons and at higher energy. Figure 6 summarizes the emission, target, and grid currents for two different target materials, Al and tungsten (W). Almost no influence of backscattering on the emission current is observed in the laminar flow regime. This may surprise, as backscattering at the target is known to decrease space charge limited emission in the linear GESA triode-type accelerator (Engelko et al., Reference Engelko, Kuznetsov, Viazmenova, Mueller and Bluhm2000). The reason for the negligible influence of backscattering in the present case is the coaxial geometry of the cylindrical triode and the absence of a guiding magnetic field. Due to their lower kinetic energy compared with beam electrons and their initial angular velocity spread, the backscattered electrons mainly remain in the target region. Even backscattered electrons with high enough radial momentum are spatially diluted due to the radial geometry. Therefore, only a very small portion of backscattered electrons enters the cathode-grid gap, which is relevant for the space-charge limited emission process, and their influence on the emission current in the laminar flow regime is negligible. In the regime of high beam current where a virtual cathode is formed, however, an increase of the backscattering of electrons at the target due to a change of target material from Al to W is observed to result in lower emission and target currents. Further, the threshold for virtual cathode formation is shifted from ~32 kV for Al targets to ~28 kV for W targets. This is explained by the increased electron space charge in the region between grid and target due to backscattering, which facilitates the formation of the virtual cathode. The grid current is almost unaffected by backscattering at the target. Not shown in the figure are the electrons escaping to the side. Their current does not exceed 2% of the emission current, even for W targets in the regime of high cathode-grid voltages.
Fig. 6. Emission, target, and grid current of unipolar flow for different backscattering conditions at the target.
3.2. Bipolar Flow
3.2.1. Laminar flow regime
In order to obtain melting of the target surface layer, pulse durations in the microsecond range are usually chosen. On such time scales, ion flow from the target cannot be neglected (Engelko & Mueller, Reference Engelko and Mueller2005). Therefore, space charge limited ion emission from the target is introduced in the simulations and the performance of bipolar flow in the cylindrical triode is studied. In Figure 7, the kinetic energy and radial momentum distributions of beam electrons (blue), backscattered electrons (red) and counter-flowing ions (protons, green) are shown for the case of cathode-grid voltage 17.7 kV (R cg = 10 Ω and R g = 54 Ω). Compared with the situation of unipolar flow shown in Figure 2 the space charge effect of the ion flow becomes evident.
Fig. 7. Simulation results for bipolar flow with U cg = 17.7 kV: (a) Kinetic energy distribution, (b) radial momentum distribution.
In Figure 8 the relevant currents of unipolar and bipolar flow in the laminar flow regime are compared. The current lost to the side stays below 40 A for both unipolar and bipolar flow and is not shown. The presence of counter-streaming ions results in an increase of the electron emission current by roughly 50% compared with unipolar flow. For planar diodes an enhancement factor due to bipolar flow of 1.86 is theoretically expected, that is, an increase of the electron current by 86% (Miller, Reference Miller1982). For cylindrical diodes with converging beam, the larger the ratio of cathode radius to anode radius is, the larger is the enhancement factor of the electron current in bipolar flow normalized to the current of unipolar flow in the same geometry (Oliver et al., Reference Oliver, Genoni, Rose and Welch2001). Compared with the diode case where the ions start with zero initial velocity at the anode, the ions in the triode configuration considered here enter the cathode-grid gap with a non-zero velocity (kinetic energy ~90–110 keV), which substantially reduces their density. Consequently, a lower enhancement factor is expected and the observed 50% increase is reasonable.
Fig. 8. Emission, target, and grid current as a function of cathode-grid voltage for unipolar (open symbols) and bipolar flow (filled symbols) in the laminar flow regime.
3.2.2. Magnetic insulation
Figure 9 shows the emission, target, and grid electron currents of bipolar flow over the full range of cathode-grid voltages investigated. The current lost to the side stays below 100 A and is not shown. Although virtual cathode formation is absent due to space-charge compensation by the ions, the slope of the emission current data versus the cathode-grid voltage is reduced beyond ~23 kV.
Fig. 9. Emission, target, and grid current as a function of cathode-grid voltage for constant cathode potential −120 kV, Al target, and bipolar flow.
As will be revealed in the following, the reason is the high beam current and axial target current, which induces a large azimuthal magnetic field. The further away from the central axial position, the higher is the accumulated target current and induced magnetic field, and thus the bending of the electron trajectories on their way towards the target. Beyond a certain magnetic field strength, electrons are unable to reach the target and return towards the cathode (magnetic insulation). In the particular case shown in Figure 10, this concerns all electrons emitted at axial positions larger ~350 mm, that is, all “grey” electrons and some of the “orange” electrons. The increased electron space charge in the cathode region at axial positions around 210–250 mm due to returning electrons leads to a locally reduced emission current (center of “pink” electrons), which is also observed in the local minimum of the beam energy density at the target at z ≈ 200 mm (see Fig. 10c). Additionally, returning electrons result in an increase of the grid current beyond the geometrical 8% of the emission current, which further reduces the percentage of emission current reaching the target.
Fig. 10. Magnetic insulation, observed for bipolar flow with U cg = 35.8 kV. Presentation details see Figure 4, with light blue dots added for ions.
For increasing cathode-grid voltage, the z-range of the described magnetic insulation extends until the electron beam reaches the target only in a focused spot at the target center (pinch, Fig. 11). Because not only few electrons from the beam edge but a substantial part of the beam electrons return towards the cathode and oscillate around the grid, the increased electron space charge finally leads to the formation of a virtual cathode between grid and target. In the presented system, electron beam pinching is observed for cathode-grid voltages beyond about 37 kV.
Fig. 11. Electron beam pinching, observed for bipolar flow with U cg = 38.0 kV. Presentation details see Figure 4, with light blue dots added for ions.
3.2.3. Electron backscattering
Figure 12 summarizes the emission, target, and grid currents of bipolar flow for two different target materials, Al and W. Higher Z materials (i.e., stronger backscattering) are found to results in an overall current increase. This applies both to the laminar flow regime and to the regime of magnetic insulation. Because of the higher beam current, the onset of magnetic insulations shifts to a smaller cathode-grid voltage for stronger backscattering. A threshold of ~23 kV was found for Al targets, whereas W targets yield a threshold of ~20 kV. On the first glance the current increase may surprise, as backscattering should increase the electron space charge and thus should lower the space charge limited electron emission at the cathode. However, the observations are easily explained by the reduced kinetic energy and large initial angular velocity spread of backscattered electrons and the radial dilution due to the coaxial geometry. Backscattered electrons remain close to the target, while the gap between cathode and grid is almost unaffected by backscattering at the target. Thus, backscattered electrons do not significantly reduce electron emission at the cathode, see also the section on backscattering in case of unipolar flow. However, the presence of backscattered electrons in the vicinity of the target considerably increases the space charge limited emission of ions at the target. The increased ion flow to the cathode then results in a higher electron emission. Similar results were obtained in a theoretical and numerical study by Oliver et al. (Reference Oliver, Genoni, Rose and Welch2001), who found a substantial increase of the emission current for bipolar flow in coaxial diodes due to backscattering.
Fig. 12. Emission, target, and grid current of bipolar flow for different backscattering conditions at the target.
4. FINAL REMARKS
Simulations of the beam performance of a cylindrical triode-type electron beam accelerator showed different operation regimes depending on the cathode-grid voltage applied. For low voltages and low beam current, laminar flow conditions are found with homogeneous energy density at the target. The laminar flow regime exists for both unipolar electron beams and bipolar beams with counter-streaming ions generated at the target. It is the only feasible operation regime for homogeneous treatment of metal surfaces. In order to achieve melting of the target surface and alloying of a previously applied coating, the deposited energy density needs to be ~20–40 J/cm2 for the considered case of 120 kV accelerating voltage (Weisenburger et al., Reference Weisenburger, Jianu, An, Fetzer, Del Giacco, Heinzel, Mueller, Markov and Kasthanov2012; Fetzer et al., Reference Fetzer, An, Weisenburger and Mueller2013). In the laminar flow regime, a beam power density of up to 1.4 MW/cm2 can be obtained for unipolar flow (with U cg = 29 kV) and up to 1.2 MW/cm2 for bipolar flow (with U cg = 20 kV). Thus, a pulse duration of at least ~30 µs is necessary to achieve the required energy density for homogeneous melting and alloying, which is feasible.
The most severe issue expected in the experiments is an inhomogeneity in the ion and plasma generation at the target. Slight variations of the beam current density in the initial (unipolar flow) stage of target treatment or inhomogeneity of the target surface could lead to local differences in ion generation. Because the beam current density is further increased by the generated ions, the positive feedback results in an enhancement of the initial inhomogeneity. On the timescale of microseconds, ion generation at the target is then followed by plasma formation and expansion, which lowers the accelerator's impedance and may cause unstable operation.
5. CONCLUSION
The performance of a cylindrical pulsed electron beam accelerator of triode type with converging radial beam was investigated by means of PIC code simulations. Depending on the cathode-grid voltage applied, different operation regimes could be identified. For low cathode-grid voltages, laminar flow with homogeneous power density at the target was found for both unipolar and bipolar beams. Ion generation and emission at the target results in an increase of the electron emission current by about 50% compared with the case of unipolar electron flow. For bipolar flow, backscattering at the target further increases the beam current significantly; an increase by about 20% was found for tungsten targets compared with Al targets.
For the cathode potential fixed at −120 kV, in case of unipolar flow a virtual cathode is formed between grid and target for cathode-grid voltages above ~32 kV (Al target), which results in an inhomogeneous beam energy density at the target. By increasing backscattering at the target, the cathode-grid voltage threshold is shifted to ~28 kV for tungsten targets. The virtual cathode forms only locally for voltages slightly above the threshold and has a larger extent for higher voltages. For bipolar flow, the laminar flow regime already ends at cathode-grid voltages of ~23 kV (Al target) and ~20 kV (W target), respectively. Due to the larger target current compared with unipolar flow, magnetic insulation at the beam edge is found; beam electrons are not able to reach the target but return towards the cathode. By further increasing the cathode-grid voltage, the magnetically insulated region extends until beam pinching occurs.
