On the Importance of the Grid Geometry in a Retarding Potential Analyzer

On the Importance of the Grid Geometry in a Retarding Potential Analyzer IEPC-2024-216 Presented at the 38th International Electric Propulsion Conference, Toulouse, France June 23-28, 2024 T. Trottenberg∗ , H. Kersten† , V. Schneider‡ , B. Schuster§ , L. Seimetz¶and R. F. Wimmer-Schweingruber‖ Institute of Experimental and Applied Physics, University of Kiel, 24098 Kiel, Germany F. Bansemer∗∗ and M. Hesse†† von Hoerner & Sulger GmbH, Schwetzingen, Germany J. Laube‡‡ OHB System AG, Bremen, Germany Abstract: In this study, a four-grid retarding potential analyzer (RPA) with drilled grids is investigated, focusing on correlations between grid orientations and resulting characteristics. An ion beam with small divergence and a narrow energy distribution directed perpendicularly to the RPA grid system is used. It is found that for certain grid configurations, particularly when grids are aligned, the characteristics deviate from the expectation of strictly monotonic behavior in plots of the collector current against the discriminator voltage. Specifically, aligning the third and the fourth grid leads to a distinct hump at voltages below the falling flank. If the second grid is also aligned with them, the hump becomes even stronger. Several models are presented to reproduce and understand these observations. Preliminary suggestions for the design of RPAs are derived based on the findings. Nomenclature Udiscr,Urep = voltage applied to the discriminator grid and the two repeller grids j= current density T= geometrical transparency of a single grid r,z= radial and axial coordinates, relative to the RPA or a specific grid hole ∆U= potential difference between adjacent grids Φ(r, z) = potential in the vicinity of a grid hole, coordinates rand zrelative to the hole Φ0(z) = potential without the disturbance of a grid hole, rand zas before Φ1(r, z) = potential disturbance caused by a grid hole, rand zas before E+,E−= field strength of the undisturbed electric field above and below a grid plane  E(r, z) = electric field strength vector at position (r, z) ∆t, ∆s= time step and targeted spatial step width of the numerical integration ∗Research and Teaching Associate, Plasma Technology, [email protected]. †Professor, Plasma Technology, [email protected]. ‡Research Associate, Plasma Technology, [email protected]. §Electrical Engineer, Extraterrestrial Physics, [email protected]. ¶Mechanical Engineer, Extraterrestrial Physics, [email protected]. ‖Professor, Extraterrestrial Physics, [email protected]. ∗∗Development Engineer. ††Development Engineer, [email protected]. ‡‡Propulsion Engineer, Propulsion Department, jens-laub[email protected]. 1 The 38th International Electric Propulsion Conference, P. Baudis Convention Center, Toulouse, France, June 23-28, 2024 Copyright 2024 by the Electric Rocket Propulsion Society. All rights reserved. I. Introduction The Electric Propulsion Diagnostic Package (EPDP) for the Heinrich Hertz communications satellite, launched in July 2023, was designed to measure the dilute plasma backflow from the thruster plume.1, 2 The Heinrich Hertz satellite is the first spacecraft equipped with Highly Efficient Multistage Plasma Thrusters (HEMPT), and it also carries two Hall thrusters for redundancy. One of the three EPDP diagnostics is a Retarding Potential Analyzer (RPA). The grids are not woven wire meshes, which are often used for RPAs, but rather consist of plates with drilled holes. The RPA was tested with a variety of grid arrangements. Some of the arrangements resulted in anomalies in the collector current vs. discriminator voltage characteristics when monoenergetic and directed ions were used, although the impact on the derived distribution functions remained relatively small. Systematic tests and simulations indicate that the observed behavior can be attributed to more or less pronounced ‘channelling’ of the ions through the grid system. The term is intended to draw an analogy with charged particles passing through a thin crystalline solid, where the stopping power is strongly influenced by the direction of the particle relative to the crystalline axes. This effect arises when the finite scale length of the grid structure leads to certain directions with significantly higher or lower transparency for ions within a specific range of energies. When looking at a stack of grids, there are certain directions in which one can see through aligned holes. Changing the configuration of the grids may make this possible only at different angles or not at all. Of course, ions generally do not follow straight paths, so that the situation becomes more complicated. However, there may still be strongly direction-dependent transparencies. Such effects have been studied by Baloniak et al.3and van den Ven et al.4for the case of woven grids. In this conference proceedings paper, we report on the experimental observations of anomalies in RPA characteristics and our ongoing efforts to understand them using computer simulations. The insights gained so far can be helpful for the design of an RPA with drilled grids. II. Description of the Retarding Potential Analyzer The RPA has four titanium grids, each consisting of numerous hexagonally arranged circular apertures with diameters of 0.5 mm. The distance between adjacent holes along one of the six grid axes, the “grid constant”, is 0.7 mm. The spacing between the grids is 4.8 mm, and the grids themselves have a thickness of 0.2 mm, see Fig. 1. The entrance grid has a total of 349 holes, while the inner stack grids have 649 holes. The hexagonal arrangement of apertures provides the grids with a 6-fold rotational symmetry, while the square frames of the grids exhibit 4-fold rotational symmetry. Consequently, each grid can be mounted in two different orientations: twisted by 90 degrees or not twisted. The geometric transparency of a single grid resulting from hole radius, grid constant, and the hexagonal arrangement is T= 0.4627. The circular collector is divided into four segments: one central circular segment and three outer 120degree ring segments, allowing for the detection of oblique incidence, as shown in Fig. 1. Figure 1. (a) The Plasma Sensor of the EPDP with the RPA and a plane Langmuir probe, which is not discussed in this paper. (b) The drawing shows the stack of four grids and the collector plate. We introduce the following 4-digit nomenclature to designate the configurations. The entrance grid is usually not rotated (except for one configuration) and defines the orientation of an ‘untwisted’ grid. 2 The 38th International Electric Propulsion Conference, P. Baudis Convention Center, Toulouse, France, June 23-28, 2024 Copyright 2024 by the Electric Rocket Propulsion Society. All rights reserved. Figure 2. Arrangement of the holes on the grids and the segmented collector. (a) The entrance grid in its usual orientation, (b) an inner grid in ‘0’ (untwisted) orientation, (c) an inner grid in ‘1’ (twisted) orientation. (d) The four segments of the collector. All sketches have the same scale and orientation within the sensor housing. The square black areas have a side length of 25.6 mm. Untwisted grids are coded with the digit ‘0’, whereas grids twisted by 90 degrees are coded with the digit ‘1’. Figure 2 shows sketches of the two grid types (entrance grid and inner grids) with the two possible orientations of an inner grid, and the segmented collector, in their relative orientations. Thus, all configurations can be specified by indicating a four-digit binary number, where the first digit encodes the entrance grid, the second encodes the next deeper grid, and so on. We investigated the configurations 0000 to 0111, and 1111. The grids serve their typical purposes in four-grid RPAs. The entrance grid is integrated into the grounded sensor head. The second grid repels electrons from the plasma that may have passed through the entrance grid. The subsequent discriminator grid only permits ions with kinetic energies exceeding the variable grid potential to overcome the barrier and reach the collector. The fourth grid reflects ion-induced secondary electrons generated at the collector surface by impacting ions. Both repeller grids are maintained at a constant potential of Urep =−25 V throughout the measurements presented in this paper. The collectors are at ground potential by design. Further details, as well as the other two diagnostics of the EPDP (Langmuir probe and erosion sensor), have already been described in a previous IEPC proceedings paper1and in an open access journal.2 III. The Test Environment The experimental part of this study makes use of our previously established test environment.2The setup was designed to mimick the artificial plasma created by the electric thruster that surrounds the spacecraft and in particular the associated ion flow onto the spacecraft surface. On the Heinrich Hertz satellite, the retarding potential analyzer is not directly exposed to the thruster plume. Instead, it measures the dilute backflow from the thruster plume. The unintentionally created secondary plasma consists of ions resulting from charge-exchange (CEX) collisions between accelerated propellant ions and neutral, cold gas atoms leaking out of the thruster. As accelerated ions lose their charge and continue as energetic neutral atoms, the ions produced by CEX collisions are cold. Due to the positive local potential relative to the satellite ground where CEX ions are generated, these ions fall back towards the satellite, gaining several tens of electronvolts. In our test environment (Fig. 3), the instrument can either be operated outside the energetic beam (>1 keV) emitted by an industrial ion beam source, or within the “idling” beam. With idling beam we denote the beam of lower energies up to approximately 100 eV that is emitted when no acceleration voltage is applied to the anode of the ion source. All measurements that we present in this paper were conducted in such an idling beam. The RPA is positioned on the beam axis, pointing directly at the ion source. We conduct repeated tests with eight different grid configurations, and a ninth configuration resembling the one with none twisted grid, where all four grids are twisted by 90 degrees. We would like to remark at this point that there was a screen with an opening of 80 mm located between the ion source and the RPA, see Fig. 3. This served to clearly spatially distinguish the regimes ‘in the beam’ and ‘out of the beam’ from each other, as we discussed in Ref. 2. 3 The 38th International Electric Propulsion Conference, P. Baudis Convention Center, Toulouse, France, June 23-28, 2024 Copyright 2024 by the Electric Rocket Propulsion Society. All rights reserved. Figure 3. The top view sketch shows the test environment with ion source, RPA, and a screen in-between. The plasma sensor is mounted together with a Faraday cup (FC) on the translation stage. The cathode is located outside (below) the constricted beam. IV. Experimental Observations The panel in Fig. 4(a) presents RPA characteristics for nine configurations, depicting the total collector current as a function of discriminator voltages ramped from 0 V to 200 V. The collector currents are reported as current densities normalized to those measured by the Faraday cup. The current densities varied slightly as the individual configurations were measured on different days, and amounted to approximately 17.2±1.3 mA/m2. While the reference area of the Faraday cup is clearly defined by the circular aperture, this is not clear in the case of the RFA with its noncircular pattern of 349 apertures. We defined the reference area for calculating the current densities as the area of the circumcircle of the hole pattern, which is 1.56 cm2(radius 7.05 mm). One would expect an RPA characteristic to exhibit a strictly monotonous decrease as the discriminator voltage increases, since the higher the potential barrier at the discriminator grid, the fewer ions can overcome it. However, the data show systematic deviations from the expected behavior in some of the configurations. For example, the configurations 0000, 0111, and 1111 exhibit a distinct hump at discriminator voltages corresponding to energies just below the ion energy. This feature is completely absent for the configurations 0001, 0010, 0101, and 0110. Configurations 0011 and 0100 also show the hump, but much less pronounced. Additionally, increased currents at lower voltages are found together with a steeper negative slope at voltages below the hump. We repeated such measurement series at larger distances (500 mm and 650 mm from the screen), and found essentially similar behaviors. We can categorize the configurations into these three classes based on their ‘grade of monotonicity.’ In this classification, 0001, 0010, 0101, and 0110 would be considered ‘good’; 0000 (as well as its equivalent 1111) and 0111 would be ‘bad’; and 0011 and 0100 would be considered ‘medium’. Interestingly, all the ‘good’ combinations have in common that the fourth (secondary electron repeller) grid is rotated relative to the discriminator (third) grid, whereas the second grid does not play a significant role. All the ‘bad’ combinations have all three inner grids aligned. In the ‘medium’ rated combinations, the alignments of the last two grids are identical, however, the second grid (plasma electron repeller) is rotated relative to them. The first derivatives represent the energy distributions of the ions that reach the RPA – or, at the satellite, the spacecraft surface. The derivatives of the same data discussed above are depicted in Fig. 4(b). The humps in the characteristics cause values below the zero line in the first derivatives, corresponding to negative values in the probability density function, which are unphysical. Despite these artifacts, the detected energies, i.e. the maxima of the first derivatives, remain nearly unchanged. 4 The 38th International Electric Propulsion Conference, P. Baudis Convention Center, Toulouse, France, June 23-28, 2024 Copyright 2024 by the Electric Rocket Propulsion Society. All rights reserved. Figure 4. Measured RPA data for nine different configurations of the RPA grids. (a) Characteristics, (b) first derivatives. 5 The 38th International Electric Propulsion Conference, P. Baudis Convention Center, Toulouse, France, June 23-28, 2024 Copyright 2024 by the Electric Rocket Propulsion Society. All rights reserved. V. Hierarchical Models The models aim to reproduce and understand the observed features of the characteristics of the various configurations. For this reason, we did not immediately start with a fully comprehensive model, but instead gradually added details to understand the cause of the different behavior of the individual grid configurations. The simplest model for an RPA consists of partially transparent homogeneous grid surfaces (i.e., without structure, just with a probability of particles crossing the grid plane) and homogeneous fields between the grids. In such a model, the ions move on piece-wise parabolic trajectories. It is evident that such a model represents a perfect RPA and cannot reproduce the observations, as a rotation of a homogeneous grid has no consequences. The first refinement adds discrete grid apertures, the drilled hexagonally arranged holes. This requires that every time a particle reaches a grid plane, it must be checked whether there is a grid opening at that point. Now, Moir´e-like effects are to be expected. Figure 5. RPA section. The equipotential surfaces (depicted as lines) conform to the contours of the grids, thus leading to radial field components. In each gap between two neighboring grids, the potential difference ∆Ubetween adjacent grids was divided into ten equal steps ∆U/10 for the drawn lines; the explicit potentials are derived from the respective ∆Uvalues resulting from the applied grid voltages. The second refinement of the model concerns the physical construction of the grids, which are not infinitely large planes with a circle containing the hexagonally arranged grid holes. Actually, they consist of a rectangular frame (see Fig. 1) with a thickness of 2 mm that tapers conically to a circle in the center, which is only 0.2 mm thick (see the cross-section in Fig. 5). This causes a distorted electric field with deviations from the previously assumed constant field. In order to quantify this effect, we calculated the potential and the electric field using the finite element method (FEM). We made use of the cylindrical geometry, which reduces the problem to two dimensions. Figure 5 shows the calculated potentials in a generic manner: For the four independent regions between the metal surfaces (black) at fixed potentials, the calculation was performed assuming 0 V at the respective lower surface and 1 V at the respective upper surface. This way, the potential and field strengths can simply be scaled by the actually chosen voltages at the grids, without the need to rerun the FEM calculation. The FEM model employs triangular elements with a maximum size of 0.25 mm. The simulated region between each pair of grids extends up to a radius of 12.5 mm. The solution is then mapped onto a grid with steps of 0.1 mm in both directions for utilization in the trajectory integration that will be described below. This approach facilitates the fast computation of local fields through linear interpolation. As shown in the figure, the equipotential lines follow the shape of the grids. Consequently, radial field components arise and affect the ion trajectories. In particular, in the case of perpendicularly incident ions, their trajectories will no longer be straight lines near the edges. A third refinement takes into account the near field at the grid holes. However, we keep the assumption of a thin grid for now and neglect the above mentioned 0.2 mm. In Jackson5(Chapter 3.13), the potential 6 The 38th International Electric Propulsion Conference, P. Baudis Convention Center, Toulouse, France, June 23-28, 2024 Copyright 2024 by the Electric Rocket Propulsion Society. All rights reserved. Figure 6. Potential near a single hole. (a) Potential disturbance Φ1(r, z)due to the hole. (b) Potential Φ(r, z)relative to the potential of the grid. In this case, the discriminator grid is shown, which is at the absolute potential of Udiscr = +100 V, while the adjacent grids have potentials of Urep =−25 V. The dashed square indicates the cylinder in which the electric near field replaces the undisturbed field for the numerical integration of the ion trajectories. on both sides of a conducting plane with a circular hole is discussed as an example of problems with mixed boundary conditions. The solution Φ(r, z) = Φ0(z)+Φ1(r, z) consists of the linear part, which arises without the hole from the constant electric fields perpendicular to the plane above and below the plane with magnitudes E+and E−, and the disturbance Φ1(r, z) caused by the hole. Figure 6 visualizes the functions Φ1(r, z) and Φ(r, z). From the near-field potential Φ(r, z), the electric field  E(r, z) = − ∇Φ(r, z) can be calculated, which, however, results in rather complicated expressions for the rand zdirections. This refinement can be applied either to the first model with homogeneous electric fields between the grid planes, or to the second model with a distorted background field (due to the grid frames). When an ion enters the calculated area, this solution is used for the trajectory integration instead of the coarse background field. The last refinement replaces the previous (third) one. We drop the assumption of thin grids and instead take into account the thickness of 0.2 mm of the material where the grid holes were bored. However, to our knowledge, there is no simple analytical solution for this problem. Instead, we calculated the potential and the electric field in the vicinity of a hole using the finite element method again. This second FEM model employs triangular elements with a maximum size of 0.13 mm. The simulated region extends up to a radius of 0.35 mm, which is half the distance between two holes, and to a distance of 0.75 mm above and below the mid plane of a thick (0.2 mm) grid. The solution is then mapped onto a grid with steps of 0.01 mm in both directions, similar to the method applied for the large-scale field described above. The trajectories were integrated for these five model variations using a fourth-order Runge-Kutta method in three dimensions. The initial time step was ∆t= 2.5 ns, but was adjusted to yield step widths of approximately ∆s= 10 µm. In the case of the simplest model, where analytically computable parabolic segments are obtained, the analytical solution was used to verify the numerical integration. VI. Numerical Results In this section, we show numerical results for the five model variations introduced above: 1) Homogeneous electric fields between grids assumed as planar and thin, 2) distorted large-scale fields (caused by the grid frames), 3) like the first case, but supplemented with near-field disturbances in the vicinity of holes, 4) like the second case, but supplemented with near-field disturbances, 5) thick grids with large-scale and near-field distortions. The precise energy distribution of the ions and the beam divergence in the experiment are unknown. Therefore, we had to make assumptions for the simulations. Based on the measured energy distributions, as 7 The 38th International Electric Propulsion Conference, P. Baudis Convention Center, Toulouse, France, June 23-28, 2024 Copyright 2024 by the Electric Rocket Propulsion Society. All rights reserved. shown in Fig. 4(b), we assumed a normally distributed energy distribution of (100 ±5) eV. The angular distribution of the incident ions is mainly limited by the screen (see Fig. 3). At a distance of 350 mm from the screen, the most oblique ions passed near the edge of the aperture, i.e. at a radius of 40 mm. This defines an angle of 6.5◦. The procedure for randomly choosing an ion is as follows: A random starting point at the ion source grid system (diameter 125 mm) is picked, and the hypothetical angle of incidence at the RPA position, considered as a point, is calculated. If the angle exceeds the limit of 6.5◦, the choice is rejected and the procedure is repeated. The random uniform selection of a point on the cross-sectional area of the ion source means that all emission angles toward the RPA are considered equally probable. This seems reasonable since an ion source with a grid system optimized for keV ions emits at a high divergence angle when no acceleration voltage is applied. For each of the 12 discriminator voltages Udiscr = 0,30,60,80,90,95,98,102,105,110,115,120 V, a number of 10 000 ions was selected according to the described procedure. Their impact points at the RPA entrance were randomly chosen with a uniform distribution over the RPA entrance surface (circumcircle of the hole pattern). In the simulation, the normalized collector current can therefore be calculated as the ratio of the number of ions that reach one of the collector segments to the number of ions that reach the entrance grid within the radius of the circular reference area (see Sec. IV). From this, the statistical errors can be estimated (we do not show error bars in order to keep the figures clear). For example, in the case of a normalized current density of 0.04, approximately 400 ions reach the collector. Since the events follow a Poisson distribution, the error is the square root of the number of events, i.e. 20 ions, or 0.002 for the normalized current density. 1) Figure 7 shows characteristics obtained from the homogeneous fields model. The trajectories underlying the displayed characteristics were generated by analytically calculating the parabolic sections between neighboring grids. For validation and as a test of the integration method, the same trajectories were also determined by integration, and no discrepancies were found. It is noticeable that the measured humps of the ‘bad’ combinations 0000 and 0111 are not strongly pronounced in the simulated characteristics. Nevertheless, the characteristics exhibit a positive slope before the falling edge. Furthermore, it can be observed that the normalized currents (grid transparencies) are significantly smaller than those in the measurements. In the case of the ‘good’ configurations, on the left in Fig. 7, the dashed horizontal line seems to represent the currents when the retarding potential can be overcome by all ions (lower voltages). The dashed line represents the normalized current of T4= 0.0458, which is the fourth power of the grid transparency. One would expect from ideal grids, meaning grids that do not exhibit effects from the specific hole pattern, a total transparency of T4for the voltages significantly below the ion energy. The significantly increased normalized currents observed in the humps of the measured characteristics up to 0.08 cannot be reproduced with this model variant. 0 25 50 75 100 125 Discriminator voltage (V) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Normalized collector current 0001 0010 0101 0110 0 25 50 75 100 125 Discriminator voltage (V) 0011 0100 0 25 50 75 100 125 Discriminator voltage (V) 0000 0111 Figure 7. Simulated RPA characteristics obtained from the homogeneous-field model (piecewise parabolas) for different configurations of the RPA grids. 2) Figure 8 shows characteristics obtained from the model with fields distorted due to the grid frames, but not from the near fields of the grid holes. In comparison with the previous model, the eight characteristics deviate less from each other. This can be 8 The 38th International Electric Propulsion Conference, P. Baudis Convention Center, Toulouse, France, June 23-28, 2024 Copyright 2024 by the Electric Rocket Propulsion Society. All rights reserved. attributed to the radius-dependent radial displacements experienced by ions on their individual trajectories: Alignments of grid holes along trajectories that occur in the center of the RPA might not occur at greater radii and vice versa. The variable radial field components act as an effective randomization, which destroys interferences. Furthermore, the calculated normalized currents do not or not significantly exceed the T4value (dashed line) as they do in the measurements (humps). These are shortcomings of this model variant. 0 25 50 75 100 125 Discriminator voltage (V) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Normalized collector current 0001 0010 0101 0110 0 25 50 75 100 125 Discriminator voltage (V) 0011 0100 0 25 50 75 100 125 Discriminator voltage (V) 0000 0111 Figure 8. Simulated RPA characteristics obtained from the distorted-field model for different configurations of the RPA grids. 3) Figure 9 shows characteristics obtained from the homogeneous fields model extended by the near fields caused by the holes. It is immediately noticeable that there is a pronounced hump in the normalized current densities in case of the ‘bad’ configurations. The maxima are of the same order of magnitude as in the measurements, even slightly higher. Also, in the ‘medium’ configurations, the increase is more pronounced compared to the two model variations previously shown. Considering that in this model variant the largescale field distortion caused by the grid frames is no longer present, but the near fields of the holes are included, one concludes that the latter have a quantitatively and qualitatively stronger influence. The observation may lead to the following hypothesis: The grid hole fields act, on the one hand, as scattering centers, randomizing and destroying interferences in case of the ‘good’ and ‘medium’ configurations. On the other hand, they can also act as focusing lenses, causing the humps in the ‘bad’ configurations. The introduction of the near fields has already brought significant progress in explaining the observed humps. However, the shape of the characteristics with humps at lower voltages still deviates significantly from the measurements, where the hump was limited to approximately the last third of the voltage range from zero to the voltage of the falling flank. In the case of the simulation, the ‘bad’ characteristics begin at Udiscr = 0 V with currents even below the T4value and increase progressively until they reach a sharp peak value. This is still somewhat unrealistic. 4) Figure 10 shows characteristics obtained from the model with large-scale fields distorted by the frames and the near fields caused by the holes. In comparison with the previous model variant, not much has changed. The characteristics of the ‘medium’ rated configurations still have a slight bump, but less pronounced than in the measurements. This shows again that the near fields play a dominant role compared to the large-scale fields. 5) Figure 11 shows characteristics obtained from the model with thick grids that includes both large-scale and near-field inhomogeneities. One notices that the ‘good’ and ‘bad’ configurations reproduce characteristics very similar to the previous model. When ions pass rapidly through the holes, there seems to be little difference between a thin and a thick grid. However, in case of the ‘bad’ configurations, there is a noticeable difference: The hump is less sharp. Assuming that the significant increases in normalized current density beyond T4is the result of a focusing effect of the holes in the discriminator grid, it becomes understandable why thick and thin grids differ: The heavily decelerated ions at Udiscr ≈100 V spend comparatively more time near and in the holes, so differences in the field profiles have much stronger effects than in the case of fast ions at Udiscr ≪100 V. 9 The 38th International Electric Propulsion Conference, P. Baudis Convention Center, Toulouse, France, June 23-28, 2024 Copyright 2024 by the Electric Rocket Propulsion Society. All rights reserved.