In this study, a four-grid retarding potential analyzer (RPA) with drilled grid holes is investigated, focusing on correlations between grid orientations and resulting characteristics. The individual grids have a hexagonal hole pattern and can be mounted rotated relative to each other in multiples of 90°. An ion beam with a small divergence and a narrow energy distribution directed perpendicularly to the RPA grid system is used. We find 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 two of the inner grids leads to a positive slope and a distinct hump at voltages below the falling edge. When all three inner grids are aligned, the hump becomes significantly more pronounced, with the signal intensity nearly doubling. Several models are presented to reproduce and understand these observations. We find that grid holes can act as scattering centers, and a finite grid thickness mitigates the potential reduction that occurs inside the grid holes. Suggestions for the design of RPAs are derived based on the findings.
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 thruster plumes.1,2 The Heinrich Hertz satellite is equipped with electric propulsion systems, being the first spacecraft to utilize Highly Efficient Multistage Plasma Thrusters (HEMPTs),3,4 along with two Hall thrusters for redundancy.
One of the three EPDP diagnostics is a Retarding Potential Analyzer (RPA). The individual grids are not woven wire meshes, which are often used for RPAs,5–11 but rather individual plates with drilled holes.12,13 The RPA, which consists of four of such grid plates, was tested with a variety of grid arrangements. Some of the arrangements resulted in anomalies in the discriminator voltage vs collector current characteristics when monoenergetic and directed ions were used although the impact on the derived distribution functions remained relatively small. Systematic tests and simulations suggested that the observed behavior can be attributed to more or less pronounced “channeling” 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 crystal axes.14 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, which makes the situation more complex than in the case of this “optical” transparency. However, there may still be strongly direction-dependent transparencies.
To the best of our knowledge, only a few publications address a fundamental understanding of the observations we have made or similar phenomena. The most closely related works are the following. The study by Baloniak et al.8 includes a theoretical analysis of the grid system’s optical transparency. This simplified assumption of straight particle trajectories is justified in the limit of negligible grid potentials. Their research addressed the question of how mechanical tolerances in the lateral positioning of meshes with very fine openings (25 μm) influence the total transparency. Van den Ven et al.10 extended the modeling efforts by simulating ion trajectories through a three-grid system, explicitly accounting for the influence of the grid potentials. The authors demonstrated that at high ion energies, the optical transparency (the limit case of straight trajectories) provides a reasonable approximation, rather than the product of individual grid transparencies. At lower ion energies, however, they observed a strong impact of the applied electric fields. Transparency fluctuations were attributed to alignment and focusing effects, and notably, grid misalignments were found to reduce these fluctuations at low ion energies. Denieffe et al.15 experimentally investigated variations in the total transparency by measuring the loss currents at the individual grids. Based on their measurements, they identified the spreading or focusing of individual beamlets between the grids as a key factor determining the effective grid transparency. The authors emphasized the significance of ion optics, grid geometry, grid alignment, and, at higher gas pressures, collisions as contributing factors. It is important to highlight that while these three studies explore effects somewhat related to those observed in our work, there are significant differences in the experimental conditions. Unlike our RPA, designed for application in space, these studies focus on diagnostics for plasma reactors at non-negligible gas pressures. Such RPAs typically feature, in order to avoid ion-neutral collisions, very thin grids, minimal grid-to-grid distances, and very small, typically square grid openings.
In this article, we report on the experimental observations of anomalies in RPA characteristics and our efforts to understand them using computer simulations. The insights gained so far can be helpful for the design of future RPAs 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 radii of r0 = 0.25 mm. The distance between the bore axes of adjacent holes, referred to as the “grid constant,” is g = 0.7 mm. The distance between the grids is d = 4.8 mm, and the grids themselves have a thickness of s = 0.2 mm, resulting in a total distance of s + d = 5 mm between the top surfaces of the grids (see Fig. 1). The entrance grid has a total of 349 holes, while each of the inner stack grids has 649 holes.
(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 of the RPA.
(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 of the RPA.
The grids serve their typical purposes in four-grid RPAs, as illustrated in the diagram in Fig. 2. The entrance grid (E) is integrated into the grounded sensor head of the EPDP. The second grid (PE) repels electrons from the plasma that pass through the entrance grid. The subsequent discriminator grid (D) only permits ions with kinetic energies exceeding the variable grid potential to overcome the barrier and reach the collector. The fourth grid (SE) reflects ion-induced secondary electrons generated at the collector surface (C) by impacting ions. Both electron repeller grids are maintained at a constant potential of Urep = −25 V throughout the measurements and simulations presented in this paper.
Sketch of the setup and operation of the four-grid RPA: entrance grid (E), plasma electron repeller grid (PE), discriminator grid (D), secondary electron repeller grid (SE), and collector (C). The ions approach the RPA from below. The trajectories illustrate an ion (black) that overcomes the retarding potential and one that does not (red), as well as electrons originating from the plasma (blue) and the collector (green).
Sketch of the setup and operation of the four-grid RPA: entrance grid (E), plasma electron repeller grid (PE), discriminator grid (D), secondary electron repeller grid (SE), and collector (C). The ions approach the RPA from below. The trajectories illustrate an ion (black) that overcomes the retarding potential and one that does not (red), as well as electrons originating from the plasma (blue) and the collector (green).
The circular collector is divided into four segments: one central circular segment with a radius of 7 mm and three outer 120° ring segments with an outer radius of 14.5 mm, allowing for the detection of oblique incidence, as shown in Fig. 1.2 The segments are separated by a trench with a width of 0.25 mm. The center segment collects the ions that traverse the grid system perpendicularly to the grids. In the case of highly directed ions, only a small portion of the current reaches the outer segments. However, if there is a finite incident angle, the three outer segments will collect different currents in an asymmetric manner. The collector plate consists of a polyimide substrate coated with conductive layers of copper (50 μm), nickel (5 μm), and gold (0.1 μm), arranged from bottom to top. All collector segments are at ground potential by design.
Now, we turn to the features of the grids that are of particular interest in this study. The hexagonal arrangement of apertures (with one central hole) provides the grids with a sixfold rotational symmetry, while the square frames exhibit fourfold rotational symmetry. Consequently, each grid can be mounted in two different orientations: either in a reference orientation or rotated by 90°.
We introduce the following four-digit nomenclature to designate the configurations. The entrance grid is usually not rotated (except in one configuration) and defines the reference orientation for the grids. Grids that are “not rotated” are encoded with digit “0,” while grids rotated by 90° are encoded with digit “1.” Figure 3 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 a four-digit binary number, where the first digit encodes the entrance grid (E), the second encodes the plasma electron repeller grid (PE), the third encodes the discriminator grid (D), and the fourth encodes the secondary electron repeller grid (SE). We investigated the eight configurations from 0000 to 0111, as well as 1111. The 1111 configuration is expected to yield the same result as the 0000 configuration in the case of a cylindrically symmetric ion beam and was measured only for verification purposes.
Arrangement of the holes in the grids and the segmented collector. (a) The entrance grid in its usual orientation, (b) an inner grid in “0” (not rotated) orientation, and (c) an inner grid in “1” (rotated) 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 30 mm.
Arrangement of the holes in the grids and the segmented collector. (a) The entrance grid in its usual orientation, (b) an inner grid in “0” (not rotated) orientation, and (c) an inner grid in “1” (rotated) 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 30 mm.
Further details can be found, together with descriptions of the other two diagnostics of the EPDP (Langmuir probe and erosion sensor), in previous publications.1,2
III. THE TEST ENVIRONMENT
The experimental part of this study makes use of our previously established test environment.2 The setup was designed to mimic the artificial plasma generated by the electric thruster and, in particular, the associated ion backflow onto the spacecraft surface. For this reason, the retarding potential analyzer on the Heinrich Hertz satellite is not directly exposed to the thruster plume but instead measures the dilute backflow from it.
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 toward the satellite, gaining several tens of electronvolts.
In our test environment (Fig. 4), the instrument can either be operated outside the energetic beam ( keV) emitted by an industrial ion beam source16 or within the “idling” beam. With the idling beam, we denote the beam of lower energies up to ∼90–100 eV that is emitted when no acceleration voltage is applied to the anode of the ion source.
The top view of a section of the vacuum chamber shows the test environment with an ion source, RPA, and a screen with an aperture in between. The plasma sensor is mounted together with a Faraday cup (FC) on a translation stage.
The top view of a section of the vacuum chamber shows the test environment with an ion source, RPA, and a screen with an aperture in between. The plasma sensor is mounted together with a Faraday cup (FC) on a translation stage.
All measurements presented in this paper were conducted in such an idling xenon beam at a chamber pressure of (1.6 ± 0.2) × 10−2 Pa. The RPA is positioned on the beam axis, pointing directly at the ion source. We conduct repeated tests with eight different grid configurations (0000–0111) and a ninth configuration (1111), where all four grids are rotated by 90°; this last configuration is similar to the one with no rotated grids (0000). We would like to remark at this point that there was a plate with an aperture of 80 mm located between the ion source and the RPA (see Fig. 4). 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.
IV. EXPERIMENTAL OBSERVATIONS
The panel in Fig. 5 presents RPA characteristics for nine configurations for discriminator voltages ramped from 0 to 200 V. The measurements were conducted on nine different days, with the adjustable operating parameters kept constant. Solid lines represent the total collector current from all segments, while dotted curves indicate the sum of the currents from the three outer collector segments.
Measured RPA characteristics for nine different configurations of the RPA grids, with the same ion beam operating parameters.
Measured RPA characteristics for nine different configurations of the RPA grids, with the same ion beam operating parameters.
The collector currents are reported as a normalized quantity Inorm = jcoll/j0, where the reference density j0 is the one measured by a Faraday cup (see Fig. 4). The latter current densities varied slightly as the individual configurations were measured on different days and amounted to approximately j0 = 17.2 ± 1.3 mA/m2. While the reference area of the Faraday cup is clearly defined by the circular aperture with a diameter of 5.3 mm, this is not clear in the case of the RPA with its noncircular pattern of 349 apertures. We defined the reference area A for calculating the current densities as the area of the circumcircle of the hole pattern, which is A = 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 (see Fig. 5). For example, 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 and 650 mm from the aperture) and found essentially similar behaviors.
We can categorize the configurations into 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 “intermediate.”
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 (plasma electron repeller) grid does not play a significant role. All the “bad” combinations have all three inner grids aligned (equally oriented). In the “intermediate” rated combinations, the alignments of the last two grids are identical; however, the second grid (plasma electron repeller) is rotated relative to them.
We show the derivatives I′(Udiscr) of the same data discussed above in Fig. 6. The humps in the characteristics in Fig. 5 cause values below the zero line in the first derivatives. These correspond to unphysical negative values in the distribution functions when interpreted with Eqs. (2) and (3). The reason seems to lie in the assumption of a constant factor Teff as the other factors are either measured quantities or fundamental constants.
First derivatives of the measured RPA characteristics for nine different configurations of the RPA grids. The red vertical lines and annotations specify the voltages of the peak values.
First derivatives of the measured RPA characteristics for nine different configurations of the RPA grids. The red vertical lines and annotations specify the voltages of the peak values.
The maxima of the first derivatives are marked by red vertical lines. All configurations produced sharp peaks though these varied their positions on the voltage axis within a range from ∼90–100 V. One should not prematurely conclude that this variation is due to the grid configurations as the individual configurations were measured on different days under slightly different conditions. Unfortunately, it was not possible to control the ion energy, and ion current density (see above), of the idling beam within narrower limits.
V. HIERARCHICAL MODELS
We developed models to reproduce and understand the observed features of the discriminator voltage vs collector current 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 performances of the individual grid configurations.
A. Model M0
The simplest model for an RPA consists of partially transparent homogeneous grids (i.e., without structure, just with a probability of particles crossing the grid plane) and homogeneous electric fields between the grids. We will refer to such hypothetical grids, which lack any structure and are characterized only by the probability of particles crossing the grid plane, as “gray” grids, in analogy to gray filters in optics. Ions with sufficient energy and a sufficiently steep angle of incidence would pass through all four grids with a probability of T4 = 0.0458. In this simplest model, the ions move on piecewise parabolic trajectories, like in Fig. 2. It is evident that such a model represents an ideal RPA and cannot reproduce the observations as a rotation of a homogeneous grid has no consequences. We refer to this zeroth model as M0 and use the mnemonic gray.
B. Model M1
The first refinement introduces discrete grid apertures: the drilled, hexagonally arranged holes. This requires checking whether a grid opening is present at the location where a particle reaches a grid plane. Moreover, these discrete apertures break the rotational symmetry for arbitrary angles. The fields between the grids are still assumed to be homogeneous. We refer to this model as M1 and use the mnemonic homog.
C. Model M2
The second refinement of the model concerns the physical construction of the grids, which are not infinitely large planes with a circle containing 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 s = 0.2 mm thick (see the cross section in Fig. 7). This causes a distorted electric field with deviations from the previously assumed constant field (see also Ref. 18). 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 7 shows the calculated potentials in a generic manner: For the four independent regions between the metal surfaces (beige-colored) 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.
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 ΔU between adjacent grids was divided into ten equal steps ΔU/10 for the drawn lines; the explicit potentials are derived from the respective ΔU values resulting from the applied grid voltages. This distorted field is essential for model M2, and models M4 and M5 use it as well.
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 ΔU between adjacent grids was divided into ten equal steps ΔU/10 for the drawn lines; the explicit potentials are derived from the respective ΔU values resulting from the applied grid voltages. This distorted field is essential for model M2, and models M4 and M5 use it as well.
The two-dimensional, rotationally symmetric 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 Fig. 7, 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. We refer to this model as M2 and use the mnemonic macro.
D. Models M3 and M4
Potential near a single hole in an infinitesimally thin plane conductor. (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. The near-fields around individual holes in a thin grid are incorporated into models M3 and M4.
Potential near a single hole in an infinitesimally thin plane conductor. (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. The near-fields around individual holes in a thin grid are incorporated into models M3 and M4.
We now return to the description of the model variants. The analytical solution for the microscopic near field around a grid hole can be applied either to model M1 (homog), which assumes homogeneous electric fields between the grid planes, or to model M2 (macro), which includes a distorted macroscopic background field due to the grid frames. We refer to these two models as M3 and M4, and use the mnemonics micro and mima, respectively. When an ion enters the calculated area, this solution is used for the trajectory integration instead of the coarse background field alone (E+ and E− are the local backgrounds in the respective models).
E. Model M5
The final refinement replaces the analytical near-field solution. We drop the assumption of infinitesimally thin grids and instead account for the thickness of s = 0.2 mm of the material where the cylindrical grid holes with a radius of r0 = 0.25 mm were drilled. However, at least 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 10 μm. The edges of the holes were rounded with a fillet radius of 10 μm to avoid numerical artifacts. At the edges, the elements have reduced sizes down to a minimum of 1 μm. 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 1.0 mm above and below the mid plane of a thick (s = 0.2 mm) grid. The solution is then mapped onto a grid with steps of 10 μm in both directions, similar to the method applied for the macroscopic (large-scale) field described above. We applied this refinement only in combination with the large-scale distortions from model M2 (macro), not to the simpler model M1 (homog) with homogeneous fields between the grid planes. Note that in the case of thick grids, the electric large-scale fields are slightly stronger due to the reduced free space (distance) between the grids and, therefore, had to be recalculated. We refer to this model as M5 and use the mnemonic thick.
A note on the numerical implementation: To avoid solving the potentials and electric fields for the various combinations of field strengths E+ and E− above and below the grid multiple times, we utilized the symmetry and linearity of the problem. First, the potential Φ(r, z) is decomposed into two parts, Φ(r, z) = Φ+(r, z) + Φ−(r, z), corresponding to the cases E+ ≠ 0 with E− = 0 and E− ≠ 0 with E+ = 0, respectively. We solved Φ+(r, z) only for the special case E+ = 1 V m−1, which is antisymmetric to Φ−(r, z) (with the z components of the electric fields and z coordinates changing signs), and derived Φ−(r, z) for E− = 1 V m−1 from it. The same approach applies to the electric field components E1r(r, z) and E1z(r, z).
Models M0 to M5 and their distinguishing features are summarized in Table I for easy reference.
Comparison of the model variants.
Model and mnemonic . | Localized apertures . | Large-scale distortion . | Near-field distortion . | Finite grid thickness . |
---|---|---|---|---|
M0 (gray) | No | No | No | No |
M1 (homog) | Yes | No | No | No |
M2 (macro) | Yes | Yes | No | No |
M3 (micro) | Yes | No | Yes | No |
M4 (mima) | Yes | Yes | Yes | No |
M5 (thick) | Yes | Yes | Yes | Yes |
Model and mnemonic . | Localized apertures . | Large-scale distortion . | Near-field distortion . | Finite grid thickness . |
---|---|---|---|---|
M0 (gray) | No | No | No | No |
M1 (homog) | Yes | No | No | No |
M2 (macro) | Yes | Yes | No | No |
M3 (micro) | Yes | No | Yes | No |
M4 (mima) | Yes | Yes | Yes | No |
M5 (thick) | Yes | Yes | Yes | Yes |
The trajectories were integrated for these five model variations using the fourth-order Runge–Kutta method in three dimensions. The initial time step was Δt = 2.5 ns but was adjusted to yield spatial step widths of approximately Δs = 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. The numerical integration and data processing were implemented in Matlab, while the generic FEM calculations were performed with the COMSOL software.
VI. NUMERICAL RESULTS
In this section, we show numerical results for the five model variations introduced above.
A. Individual trajectories
First, examples will be used to illustrate how the trajectories differ between the models. Figure 9 shows trajectories according to the five model variants for two ions starting at different positions with the same velocities. The grid configuration is the same for both cases: 0101.
Trajectories for two ions with kinetic energies of 105 eV, according to the five model variants. The grid configuration is 0101, and the retarding potential is Udiscr = +100 V. The ions approach the RPA from below. (a) The first ion starts at x = +1.275 mm, y = +0.1 mm, and z = −1 mm with a velocity of m s−1, corresponding to an angle ϑ = 0.1° relative to the RPA axis. (b) The stretched xy plane shows where direction-changing forces act on the first ion. (c) The second ion starts at x = +6.45 mm, y = +0.1 mm, and z = −1 mm with the same velocity vector as the first ion. (d) Stretched view for the second ion.
Trajectories for two ions with kinetic energies of 105 eV, according to the five model variants. The grid configuration is 0101, and the retarding potential is Udiscr = +100 V. The ions approach the RPA from below. (a) The first ion starts at x = +1.275 mm, y = +0.1 mm, and z = −1 mm with a velocity of m s−1, corresponding to an angle ϑ = 0.1° relative to the RPA axis. (b) The stretched xy plane shows where direction-changing forces act on the first ion. (c) The second ion starts at x = +6.45 mm, y = +0.1 mm, and z = −1 mm with the same velocity vector as the first ion. (d) Stretched view for the second ion.
We begin with an ion entering the RPA near its axis. In the overview Fig. 9(a), the differences are already noticeable, but it is not clear how they arise. Therefore, Fig. 9(b) stretches the x and y axes while maintaining the z axis. Model M1 (homog) is represented by two curves, denoted as “parabolas” and “homogeneous,” because it was implemented in two ways: semi-analytically using piecewise combined parabolas and alternatively through numerical integration. Both curves overlap completely, as expected. The passage through the grid holes is without any noticeable disruption. The same is true for model M2 (macro). The similarity of the M1 and M2 trajectories indicates that the effect of the radial electric field components is weak, at least near the axis of the RPA. The outcome is very different for models M3, M4, and M5, which take the near-field effects of the grid holes into account. The ions experience a significant “kick” when passing through the discriminator grid. The radial force push is even stronger in the case of M5, the model with finite grid thickness.
The second ion enters the RPA near the edge of the entrance grid. Figure 9(c) provides an overview, while Fig. 9(d) shows a corresponding close-up, with the three spatial axes scaled differently. As expected, no differences between the two implementations of model M1 (homog) are observed. Both trajectories end at the repeller grid, where they miss the next hole. According to model M2 (macro), the ion experiences a force between the entrance grid and the repeller grid that bends its trajectory toward the RPA axis. This can be understood from the equipotential lines plotted in Fig. 7, which show a negative potential gradient from the edge toward the center in the region between the first two grids. The ion acquires enough inward momentum in this area that it misses the hole in the discriminator grid and is lost at its surface. The trajectory for model M3 (micro) is very similar to those of model M1 (homog) though there is a slight difference. Model M3 takes the near field into account; for this short trajectory, only the near field of the hole in the entrance grid can be responsible for the deviations in the trajectories. Models M4 (mima) and M5 (thick) allow the ion to pass through all four grids. In the first gap, between grids 1 and 2, it is primarily the distorted coarse field that bends the trajectories toward the RPA axis. So far, the trajectories are similar to the one in model M2 (macro). However, at the repeller grid, the near field plays a crucial role: this negatively biased grid steepens the trajectory, enabling the ion to reach the hole in the discriminator grid that the ion in model M2 missed. The ion passes through the hole very close to its edge. This positively biased grid pushes the ion away from the edge of the hole. Both trajectories pass through the last grid.
B. Discriminator voltage vs collector current characteristics
The precise energy distributions of the ions and the beam divergence in the experiment are unknown. Therefore, we had to make assumptions for the simulations. We assumed a normally distributed energy distribution of (100 ± 5) eV (see Ref. 2).
The angular distribution of the incident ions is limited by the screen with the aperture (see Fig. 4). 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 N = 10 000 ions was selected according to the described procedure. Their starting points at the RPA entrance were randomly chosen using a uniform distribution over a circle, i.e., a homogeneous ion flux density is assumed at the RPA position. The circle had a distance of 1 mm from the entrance grid and a radius slightly (0.12 mm) larger than the circumcircle of the hole pattern on the entrance grid. The distance was chosen to include the electric near fields of the holes, which, according to Eq. (4), also extend outside the RPA. The slightly larger radius is necessary to allow the edge rays of the beam to reach the outermost grid holes.
The normalized collector current Inorm can therefore be calculated as the ratio of the number ncoll of ions that reach the collector (or one of its segments) to the number n0 of ions that reach the entrance grid within the radius of the circular reference area (see Sec. IV), Inorm = ncoll/n0. In the following, we will simply refer to these ratios as normalized currents. (Note that n0 < N because some trajectories that start from the aforementioned circle of starting points at the RPA entrance do not enter the circular reference area. In our case, n0 ≈ 0.97N.)
The statistical errors ΔInorm can be estimated from ncoll. For sufficiently large ncoll, the Poisson-distributed impact events have a standard deviation of . We use this standard deviation to calculate the statistical errors ΔInorm = ±σ(ncoll)/n0.
In the following, we will show and discuss the resulting RPA characteristics after some helpful remarks: Instead of plotting all eight characteristics for one model in a single plot, we group the configurations into three categories and plot them separately. The first category consists of configurations where the three inner grids have alternating orientations: 0101 and 0010. The second category includes all configurations with exactly two neighboring inner grids that are aligned: 0001, 0011, 0100, and 0110. In the third category, all three inner grids are aligned: 0000 and 0111. The plots show the total collector current, i.e., the sum of all four segments, with color-coded lines. Additionally, the sum of the three outer segment currents is shown as dotted curves of the same color, respectively. The total current for the structureless (“gray”) grids model M0 is depicted differently as a black dashed curve for comparison with the “ideal” RPA.
C. Model M1 (homog)
The characteristics shown in Fig. 10 are based on trajectories composed of analytically calculated parabolic sections between neighboring grids, according to model M1 (homog). The same trajectories were also computed numerically for validation and as a test of the integration method, with no discrepancies found.
Simulated RPA characteristics obtained from model M1 (homog) for different configurations of the RPA grids. This model neglects both large-scale distortions from the grid frames and near-field distortions from the grid holes. The ion trajectories are modeled as piecewise parabolas. The solid lines indicate the total collector current from all segments. The dotted curves represent the sum of the three outer collector segment currents, with their color indicating the model in the same way as for the total currents. The error bars indicate the statistical error for the respective normalized current. For clarity, the error bars are not shown on the curves, but three representative error bars are displayed at different ordinate values to illustrate the variation.
Simulated RPA characteristics obtained from model M1 (homog) for different configurations of the RPA grids. This model neglects both large-scale distortions from the grid frames and near-field distortions from the grid holes. The ion trajectories are modeled as piecewise parabolas. The solid lines indicate the total collector current from all segments. The dotted curves represent the sum of the three outer collector segment currents, with their color indicating the model in the same way as for the total currents. The error bars indicate the statistical error for the respective normalized current. For clarity, the error bars are not shown on the curves, but three representative error bars are displayed at different ordinate values to illustrate the variation.
The most notable observation is that the grid configurations significantly affect the obtained characteristics, already in this simple model. In general, the normalized currents (grid transparencies) Inorm for all configurations are considerably smaller than those observed in the measurements. While the measured characteristics start at Udiscr = 0 with normalized currents above Inorm = 0.06, the simulated characteristics begin below Inorm = 0.05, in case of the “gray” grid model at approximately T4 = 0.0458.
The normalized currents range from significantly below to slightly above the corresponding “gray” values (only for 0000). This provides strong evidence for the aforementioned “channeling”: There are not only cases where the grid material between the holes obstructs the ion paths to such an extent that the overall transparency falls below T4, but in some cases, the holes are favorably arranged such that the overall transparency exceeds T4.
However, the significantly increased normalized currents observed in the humps of the measured characteristics, exceeding Inorm = 0.08, cannot be reproduced with this model variant. The concentration of the positive slopes near the peak of the characteristic is not reproduced either. The simulated humps are much less pronounced; while they do exhibit a positive slope before the falling edge, this slope extends almost across the entire voltage range up to the falling edge.
In the case of the configurations from the first category [Fig. 10(a)], the dashed curve represents the currents quite well, particularly at the falling edge. There appears to be a slight positive slope of the 0101 curve at the lower voltages, but the corresponding deviations remain within the error bounds.
D. Model M2 (macro)
Model M2 (macro) includes large-scale distortions from the grid frames but neglects near-field distortions from the grid holes.
In comparison with the previous model M1 (homog), the eight characteristics in Fig. 11 show less deviation from each other and never exceed the “gray” values (dashed line) as they occasionally did in the simpler model. This can be attributed to the radius-dependent radial displacements experienced by ions along their individual trajectories: alignments of grid holes along trajectories occurring near the center of the RPA, where radial fields vanish, may not occur at greater radii where radial fields are present, and vice versa. The variable radial field components act as an effective randomization, preventing interferences across the entire grid surface simultaneously. Furthermore, the weak indications of humps observed in the previous model for configurations 0000 and 0111 largely disappear, leaving only a slight increase in the curve before the falling edge.
Simulated RPA characteristics obtained from model M2 (macro) for different configurations of the RPA grids. This model includes large-scale distortions from the grid frames but neglects near-field distortions from the grid holes. The solid lines indicate the total collector current from all segments. The dotted curves represent the sum of the three outer collector segment currents, with their color indicating the model in the same way as for the total currents.
Simulated RPA characteristics obtained from model M2 (macro) for different configurations of the RPA grids. This model includes large-scale distortions from the grid frames but neglects near-field distortions from the grid holes. The solid lines indicate the total collector current from all segments. The dotted curves represent the sum of the three outer collector segment currents, with their color indicating the model in the same way as for the total currents.
The effect of the radial electric field components is also observed in the current to the outer collector segments: compared to the previous model, these currents are much smaller. Evidently, the radial fields redirect ions inward toward the RPA axis (see Fig. 7).
E. Model M3 (micro)
Model M3 (micro) neglects macroscopic distortions from the grid frames but includes microscopic distortions from the grid holes.
In Fig. 12(c), it is immediately noticeable that there is a pronounced hump in the normalized currents in case of the “bad” configurations 0000 and 0111. The maxima are of the same order of magnitude as in the measurements, exceeding Inorm = 0.08. In addition, in the configurations with only two aligned inner grids, Fig. 12(b), the increase is more pronounced compared to the two model variations previously shown. Considering that in this model variant, the large-scale 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.
Simulated RPA characteristics obtained from model M3 (micro) for different configurations of the RPA grids. Note the slightly shifted falling edges. This model neglects large-scale distortions from the grid frames but includes near-field distortions from the grid holes. The solid lines indicate the total collector current from all segments. The dotted curves represent the sum of the three outer collector segment currents, with their color indicating the model in the same way as for the total currents.
Simulated RPA characteristics obtained from model M3 (micro) for different configurations of the RPA grids. Note the slightly shifted falling edges. This model neglects large-scale distortions from the grid frames but includes near-field distortions from the grid holes. The solid lines indicate the total collector current from all segments. The dotted curves represent the sum of the three outer collector segment currents, with their color indicating the model in the same way as for the total currents.
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” configurations 0101 and 0010 [Fig. 12(a)]. On the other hand, they can also act as focusing lenses in case of aligned grids, causing the humps, particularly when all three inner grids are aligned.
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 edge. In the case of the simulation, the “bad” characteristics begin at Udiscr = 0 V with currents even below the T4 value and increase progressively until they reach a sharp peak value. This is still somewhat unrealistic.
There is one more important detail that might easily be overlooked: the falling edges of all eight characteristics are shifted by a few volts to higher values compared to the “gray” reference characteristic. The shift for the two characteristics in Fig. 12(a) is ∼3 V. In model M1 (homog), no such shift was observed, leading to the conclusion that the near fields of the holes are the cause. This can be understood from Eq. (7), which describes the deviation of the potential at the center of a hole, Φ1(0, 0), compared to the potential applied to the grid. Assuming a discriminator grid potential of Udiscr = +100 V and Urep = −25 V for the neighboring repeller grids and with grid spacings of d + s = 5.0 mm for the thin grids, we obtain electric fields of E+ = +25 kV m−1 and E− = −25 kV m−1. This results in a reduced potential up to Φ1(0, 0) = −3.98 V at the center, in good agreement with the observed shifts: The lowered potentials allow even ions with slightly “too low” kinetic energies to overcome the retarding potential barrier. The corresponding effect for grids with woven meshes has been addressed in Ref. 19.
F. Model M4 (mima)
Model M4 (mima) includes both microscopic near-field distortions from the grid holes and macroscopic distortions from the grid frames.
In comparison with the previous model variant, not much has changed in Fig. 13 despite the inclusion of the large-scale fields. Surprisingly, the characteristics of configurations 0100 and 0110 now exhibit a negative slope at the lower voltages and developed a small hump just before the falling edge [Fig. 13(b)]. The other two configurations in this category, 0001 and 0011, remain with the positive slope at the lower voltages.
Simulated RPA characteristics obtained from model M4 (mima) for different configurations of the RPA grids. Note the slightly shifted falling edges. This model includes both large-scale distortions from the grid frames and near-field distortions from the grid holes. The solid lines indicate the total collector current from all segments. The dotted curves represent the sum of the three outer collector segment currents, with their color indicating the model in the same way as for the total currents.
Simulated RPA characteristics obtained from model M4 (mima) for different configurations of the RPA grids. Note the slightly shifted falling edges. This model includes both large-scale distortions from the grid frames and near-field distortions from the grid holes. The solid lines indicate the total collector current from all segments. The dotted curves represent the sum of the three outer collector segment currents, with their color indicating the model in the same way as for the total currents.
This supports the previously stated hypothesis, derived from the comparison of models M2 (macro) and M3 (micro), that the near fields play a dominant role compared to the large-scale fields. The potential reduction Φ1(0, 0) = −3.98 V at the hole centers, previously observed in model M3, can also be noted here.
G. Model M5 (thick)
In model M5 (thick), the grids have a finite thickness of s = 0.2 mm and include large-scale distortions from the grid frames as well as near-field distortions from the grid holes with finite length.
From Fig. 14, one notices only one qualitative difference to the previous model, which fortunately is an improvement. The shift between the eight characteristics and the gray reference curve disappeared. This can also be understood by considering the extreme case of a very long borehole: the potential inside a long hollow conductive cylinder equals the potential of the cylinder itself. Figure 15 illustrates this by a surface plot of the two-dimensional potentials in the rz plane for the thin and the thick discriminator grids. The applied voltages are as in the example above, Udiscr = +100 V and Urep = −25 V. The saddle point for the thin grid has already been discussed; it is Φ1(0, 0) = −3.98 V relative to Udiscr. In case of the potentials from the FEM calculation for the thick discriminator grid, the saddle point has a potential of only Φ1(0, 0) = −1.85 V relative to Udiscr.
Simulated RPA characteristics obtained from model M5 (thick) for different configurations of the RPA grids. In this model, the grids have a finite thickness of s = 0.2 mm and include large-scale distortions from the grid frames as well as near-field distortions from the grid holes with finite length. The solid lines indicate the total collector current from all segments. The dotted curves represent the sum of the three outer collector segment currents, with their color indicating the model in the same way as for the total currents.
Simulated RPA characteristics obtained from model M5 (thick) for different configurations of the RPA grids. In this model, the grids have a finite thickness of s = 0.2 mm and include large-scale distortions from the grid frames as well as near-field distortions from the grid holes with finite length. The solid lines indicate the total collector current from all segments. The dotted curves represent the sum of the three outer collector segment currents, with their color indicating the model in the same way as for the total currents.
Comparison of potential profiles near a grid hole of the discriminator grid. (a) Infinitesimally thin grid and (b) grid with a thickness of 0.2 mm. The red arrows indicate the potential at the saddle point in the center of the hole, which in this case is lower than the potential of the grid (−3.98 and −1.85 V). (Assumed grid potentials Udiscr = +100 V and Urep = −25 V.)
Comparison of potential profiles near a grid hole of the discriminator grid. (a) Infinitesimally thin grid and (b) grid with a thickness of 0.2 mm. The red arrows indicate the potential at the saddle point in the center of the hole, which in this case is lower than the potential of the grid (−3.98 and −1.85 V). (Assumed grid potentials Udiscr = +100 V and Urep = −25 V.)
VII. DISCUSSION AND CONCLUSION
We studied a four-grid retarding potential analyzer using drilled grids with respect to correlations between the grid orientations and the resulting discriminator voltage vs collector current characteristics. The hole diameter to grid distance ratio was ∼0.1. The individual holes were arranged in hexagonal patterns, with all grids having the same hole size and grid constant. The RPA was exposed to a perpendicularly incident ion beam (90–100 eV) with narrow energy distribution (∼5 eV) and small (limited) divergence. We have shown that the characteristics can exhibit pronounced deviations from the expected strictly monotonic behavior and that these are determined by the orientations of the grids to each other. Grids aligned in the same direction are disadvantageous, which may be surprising but was already observed in a previous study.10 In particular, arrangements where the fourth grid (secondary electron repeller) is aligned with the third (discriminator) grid exhibit a peak just before the falling edge. When the second grid is also aligned with the two latter, the peak is even more pronounced. The “good” grid combinations are those with the third and fourth grids rotated relative to each other.
The aim of the design was to make the grid spacing to hole diameter ratio large enough so that the structuring of the grids no longer becomes apparent; in other words, the ions emerging from one grid hole should diverge sufficiently to spread over several holes in the subsequent grid (see Ref. 15). As revealed by the “bad” alignments, this goal was not fully achieved: The ions retain a “memory” of which hole they originated from as they travel from one grid to the next, allowing for correlation effects (or interferences) to occur. In this sense, the uniform hexagonal arrangement of the apertures, which allows for the highest possible hole density, turned out to be disadvantageous: it results in stronger spatial correlations or anticorrelations between the beamlets emerging from one grid and the apertures of the subsequent grid; these correlations can either enhance or reduce the transparency. This suggests that different hole sizes and hole distances in adjacent grids, different patterns (such as orthogonal and hexagonal) in adjacent grids, or irregular patterns could help mitigate such correlations and, consequently, fluctuations in transparency.
The most pronounced deviation of the characteristics from an ideal characteristic, namely, the hump at voltages just below the falling edge, occurred when the third and fourth grids had matching alignments. Hence, this observation suggests that the holes of the discriminator grid become particularly transparent and focusing in the case of highly decelerated, i.e., slow, ions. This hypothesis is the subject of our ongoing numerical investigations, which examine the “scattering” of ions at the holes in more detail (see Ref. 10). The goal is to determine the optimal grid-to-grid distances that allow for sufficient ion spreading as they pass from one grid to the next, effectively making the grid behave like a “gray” grid, ensuring monotonically decreasing characteristics as in model M0 without anomalies such as humps.
We have presented models that incorporate various aspects of real RPAs, namely, large-scale radial field influences arising from the construction of the grids and the near fields from the grid holes. In particular, the latter, as could already be suspected from experimental observations, seem to have a significant impact. The grid hole near fields alone were already able to reproduce the hump-like shapes similar to the measurements.
It is important to note that the humps indicate a significant enhancement of the transparency by the near fields compared to the values observed at low discriminator voltages. This suggests that the near fields exert a steering effect on the ions, which, in the case of the fully aligned “bad” configurations 0000 and 1111, enables a greater number of particles to pass through all four grids (see Ref. 15).
Moreover, in models considering the near fields of the holes in the thin grids, a shift of the falling edge by ∼3 V toward higher voltages (energies) occurred. This shift can be explained by a reduced retarding potential inside the holes, as indicated by the theoretical solution.
In the most detailed model, we also took the thickness of the grid material into account, i.e., the length of the bores, with a thickness-to-radius ratio of 0.8. While the finite bore length had no significant effect on the overall shape of the characteristic, the shift of the falling edge, which occurred in case of thin grids, disappeared. Whether the grid thickness also systematically affects the ion scattering in the holes, either enhancing or attenuating it, remains subject to further investigation.
Another interesting question concerns multiply charged ions, which are produced, for example, in Hall thrusters.22 A Z-fold charged ion, accelerated to Z times the kinetic energy of a singly charged ion, would follow exactly the same trajectory as the singly charged ion but at a higher speed. (This result stems from the concept of mechanical similarity.23) In particular, the effects of the near fields do not alter the well-known feature of RPAs that multiply charged ions are indistinguishable from singly charged ions. Of course, the situation is different for initially multiply charged ions that maintain their kinetic energy but lose part of their charge in CEX collisions after being accelerated in the thruster (see Ref. 24). This case requires a more detailed analysis, which is beyond the scope of this work.
From our study, several conclusions can be drawn regarding the design of RPAs:
Alignment of the holes in adjacent grids should be avoided to prevent alignment effects (“channeling”), which can lead to significant deformations of the characteristics.
The hole diameter to grid distance ratio should be as small as possible to mitigate the graininess of the grids.
The grid material should not be too thin in comparison to the hole radii to prevent a significant reduction of the retarding potential.
We plan to conduct further investigations into the RPA performance under oblique ion incidence for various grid configurations. Additionally, we aim to examine the scattering effects of ions at the grid holes in greater detail and to explore alternative hole patterns to assess their impact on performance.
ACKNOWLEDGMENTS
The experimental part of this work was financially supported by European Space Agency (ESA) under GSTP Contract No. 4000126205/19/NL/RA and the German Aerospace Center (DLR), Project No. 50 RS 2003.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Thomas Trottenberg: Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Florian Bansemer: Data curation (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal). Marcel Hesse: Methodology (equal). Holger Kersten: Funding acquisition (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). Jens Laube: Funding acquisition (equal); Supervision (equal). Viktor Schneider: Methodology (equal). Björn Schuster: Methodology (equal). Lars Seimetz: Methodology (equal). Robert F. Wimmer-Schweingruber: Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.