Advances in Force Probe Techniques

Advances in Force Probe Techniques IEPC-2025-153 Presented at the 39th International Electric Propulsion Conference, Imperial College London, London, United Kingdom 14-19 September 2025 Thomas Trottenberg∗and Holger Kersten† Institute of Experimental and Applied Physics, University of Kiel, 24098 Kiel, Germany We present a novel type of force probe for measuring momentum transfer from beams of ions and neutral atoms to solid targets. In contrast to earlier cantilever-based designs, which are briefly reviewed, the new probe employs a membrane as the deflecting element, enabling a compact, longitudinal geometry. Like previous probes, the instrument permits spatially resolved diagnostics of momentum flux in thruster plumes from electric spacecraft propulsion systems. This contribution focuses on mechanical design, ex situ characterization, and calibration. Measurements in a beam environment had not yet been performed at the time of the conference. Nomenclature A= amplitude of oscillation b= width of a cantilever with rectangular cross-section (perp. to bending plane) B= magnetic flux density d= deflection E= Young’s modulus dE/dT= temperature coefficient of Young’s modulus F= force acting on the target g= gravitational acceleration h= thickness of a cantilever with rectangular cross-section (in the bending plane) I◦= second moment of area of a circular cross-section Irect = second moment of area of a rectangular cross-section k= calibration constant (spring constant) k◦= spring constant of a single membrane m= mass applied to the target (calibration weight) ro= outer radius of a cantilever tube or rod ri= inner radius of a cantilever tube t= time τ= decay constant of oscillations ∗Senior Researcher and Lecturer, Plasma Technology Group, [email protected]. †Professor, Head of Plasma Technology Group, [email protected]. The 39th International Electric Propulsion Conference, Imperial College London London, United Kingdom 14–19 September 2025 Copyright 2025 by the Electric Rocket Propulsion Society. All rights reserved. Page 1 I. Introduction Diagnostics of the exhaust plumes from electric spacecraft propulsion systems are usually based on electric probes such as Faraday cups1and retarding potential analyzers.2, 3 Optical techniques, including emission spectroscopy,4laser absorption spectroscopy,5and laser-induced fluorescence,6are also applied but are more demanding and therefore less common. Electrostatic probes are limited to measuring only the charged component of the plume, which is a shortcoming because charge-exchange collisions with residual gas generate a substantial population of energetic neutral atoms. These neutralized beam particles remain invisible to electrically sensitive diagnostics,7–10 although they can carry a significant fraction of the total momentum. Force probes11, 12 offer a complementary approach: They respond to the total momentum flux, regardless of the charge state of the incident particles. In contrast to electrostatic diagnostics, which measure current, force probes directly quantify the mechanical impact of the plume. This makes them especially valuable in scenarios where charge-exchange (CEX) collisions alter the charge composition of the beam. Over the past decade, cantilever-based force probes have been developed and applied in our lab for indirect thrust measurements with spatial resolution in the plume, and for studying the angular distribution of momentum flux carried by sputtered particles released from surfaces.13–16 These probes rely on an interferometric measurement of the deflection of a target mounted on an elastic cantilever. The target material is chosen for its low sputter yield, as particle release would cause additional momentum transfer. In recent years, other laboratories have also applied force measuring techniques. For example, Scharmann et al. mounted a target on a hanging pendulum.17 Astruc et al. developed a probe that is derived from a quartz MEMS accelerometer.18, 19 In this contribution, we introduce a new kind of force probe that replaces the cantilever with a compact, cylindrical membrane-based structure. A central rod is suspended by two membranes, each with a diameter of 20 mm. This design reduces the exposure of mechanical components to the beam and improves the thermal robustness of the probe. The elastic deformation of the structure is detected via fiber-coupled interferometry, as in the earlier cantilever force probes. The article is structured as follows. Sec. II reviews the design evolution of interferometric cantilever force probes. Sec. III introduces the membrane-based force probe prototype and its operating principle. Sec. IV presents ex situ characterization results, including vibrations, damping, and calibration. In Sec. V, we discuss results, prospects, and potential applications. II. Design Evolution of Cantilever Force Probes So far, our force probes were based on a sensitive cantilever whose elastic deflection is measured interferometrically. Over the past decade, we have developed two main generations of interferometric force probes: an initial, rather bulky setup with single-axis and two-axis variants (see Fig. 1), and a more compact design (see Fig. 2) that represents the current state of the art. The latter can be deployed in electric propulsion test facilities as conveniently as a Faraday cup or a retarding potential analyzer (RPA). All designs share the principle that a small target is mounted at the free end of an elastic tube, rod, or metal strip serving as a bending cantilever. The target is exposed to the particle beam, for example in a vacuum chamber for testing electric space propulsion systems, while the cantilever itself remains shielded from direct particle impact. When a force acts on the target, the cantilever bends elastically (typically by only a few micrometers) so that every point along the cantilever is slightly displaced from its initial position. In terms of elasticity theory, or more specifically Euler-Bernoulli beam theory, the cantilever can be regarded as a beam with one fixed end.20 To avoid confusion, in the following the term beam will be used exclusively for the particle beam or thruster plume, while cantilever will denote the solid bending element. Early implementations demonstrated the feasibility of this approach for thruster plume diagnostics, but also revealed limitations in mechanical robustness and in the integration into vacuum test chambers. In recent years, successive design iterations have addressed these shortcomings, resulting in a more compact and mechanically robust probe design suitable for integration in vacuum test chambers. This section summarizes the evolution of cantilever force probes, while the membrane-based probe will be introduced in the next section. Figure 1(a) shows the original interferometric force probe.11, 21 The cantilever consisted of a ceramic (Al2O3) tube, which allowed a wire to be threaded through its interior for biasing and for measuring the The 39th International Electric Propulsion Conference, Imperial College London London, United Kingdom 14–19 September 2025 Copyright 2025 by the Electric Rocket Propulsion Society. All rights reserved. Page 2 Figure 1. (a) Original interferometric force probe with a ceramic tube cantilever and mirror for interferometric readout. (b) Two-axis variant with an eddy-current damping unit. (c) 3D view illustrating the assembly of the essential components. (d) Photograph of the assembled probe with casing. current collected by the target. For practical reasons, the interferometer measured the deflection at an intermediate point between the target and the fixed end rather than at the target itself. To enable this measurement, a small mirror (actually a polished stainless-steel block) was attached to the cantilever. The Fabry–P´erot interferometer employed the path between the cleaved end of a single-mode fiber and the mirror as the cavity, with a collimator lens placed between fiber and mirror. A detailed description of this setup is given in Ref. 11. We also implemented a two-axis variant of the probe, which enabled force measurements as a vectorial quantity [see Fig. 1(b)]. Such a two-axis force probe has proven very useful in our studies of ion beam sputtering.11, 15, 16 Figure 1(c) illustrates how the essential components are assembled. The cantilever was protected by a shielding to prevent any additional force from impinging beam particles, while the optical elements were enclosed in a casing, which can be seen in the photograph in Fig. 1(d). The overall dimensions of the shown setup are 100 mm in width, 70 mm in depth, and 270 mm in height. The targets of these probes have a circular front face with a diameter of 20 mm. Note the damping unit shown in Fig. 1(b). The masses of the mirror, the target with its holder, and the ceramic tube give rise to undesired mechanical vibrations. In particular, the pumps of vacuum chambers excite eigenmodes, and an abrupt change of the applied force can cause a ballistic overshoot beyond the equilibrium position followed by weakly damped oscillations. We will return to this issue in the next section in connection with the new membrane force probe. Vibrations are not a principal problem, since the force can still be derived from the change of the equilibrium position about which the deflections oscillate. However, to keep the required averaging time short, the probe is equipped with a damping mechanism. The figure shows the essential parts of the eddy-current damping unit: a pair of small cylindrical permanent magnets generates a magnetic field that penetrates a thin copper sheet attached to the back side of the target holder. As the target together with the copper sheet oscillates perpendicular to the field lines, eddy currents are induced in the sheet, which produce a counteracting force and dissipate the oscillation energy. Because oscillations around an equilibrium position can be damped but never completely avoided, the equilibrium deflection must be taken as the measurement variable. By averaging over several periods of the fundamental oscillation, reliable results can be obtained even for signals with a low signal-to-noise ratio. A single force measurement therefore consists of at least one reference measurement with the beam off and one measurement with the beam on. The choice of the target material is also crucial. The goal should always be a very low sputtering yield. Sputtering means that energetic particles, either target atoms ejected from the surface or reflected beam particles, leave the target with momentum. This additional momentum flux enhances the measured force, so that corrections, supported by auxiliary data from experiments or simulations, are required. Such an error-inducing effect would not occur in the case of an ideal absorber with no sputtering.15 The 39th International Electric Propulsion Conference, Imperial College London London, United Kingdom 14–19 September 2025 Copyright 2025 by the Electric Rocket Propulsion Society. All rights reserved. Page 3 “Carbon fiber velvet” (produced by Energy Science Laboratories, Inc.) is one such material. It consists of carbon fibers approximately 2.2 mm in length and 7 µm in diameter, arising perpendicularly from a plane base. This geometry allows ions to penetrate deeply into the structure. As a result, sputtered carbon atoms from one fiber are re-deposited on neighboring fibers with high probability, and reflected ions are likely to undergo multiple interactions with the fibers before leaving the material.11 We have also shown that graphite, even though it does not possess the small filling factor of carbon fiber velvet, is already an excellent absorber.15 Therefore, graphite is an excellent choice for force probes intended to measure momentum fluxes in thruster plumes. Figure 2. (a) Cross-section of the first compact force probe using a tungsten rod as cantilever. (b) Perspective view of the compact probe; the drawing shows a version with a graphite block target. (c) Drawing of the probe with tungsten strip cantilever and a beam dump (highlighted in blue) located behind the target. (d) Photograph of the opened probe with tungsten strip cantilever and cover plate. Now we turn to the compact versions that implement the same principles but achieve a smaller and more robust realization. Figure 2(a) shows a cross-section of the probe in the first attempt.16 The ceramic tube used as a cantilever was replaced by a tungsten rod. The relevant mechanical properties, in particular Young’s modulus Eand its temperature coefficient dE/dT, are similar to those of Al2O3. On the other hand, although tungsten is also a brittle material, it does not fracture as easily as ceramics during assembly and can withstand stronger mechanical shocks, for example when the probe is mounted in a chamber. Figure 2(b) shows the probe in a perspective view. Its outer dimensions are 25 mm in width, 23 mm in depth, and 129 mm in height. In sputtering-related work, carbon fiber velvet was used as the target material,16 whereas the drawing depicts a version equipped with a 3 mm thick graphite block. The targets of these compact probes have a square front face with 20 mm long edges. Cantilevers with a cylindrical cross-section allow independent bending in two orthogonal directions, which is only required for a two-axis force probe. In the case of a single-axis probe, two degrees of freedom are unnecessary, and we expected greater robustness from eliminating the redundant degree of freedom. Therefore, the tungsten rod was replaced by a thin tungsten sheet. The calibration constant (“spring constant”) kremains unchanged as long as the second moment of area is conserved; otherwise it scales proportionally to the moment. The details of how the second moment of area enters into the calculation of the cantilever stiffness and the expected calibration constant can be found in Ref. 11. Here we will only address the influence of the three different cantilever geometries on the second moment of area. I◦=π 4r4 o−r4 i(1) is the second moment of area of the cantilever cross-section in the case of a ceramic tube with outer radius ro and inner radius ri. For a solid tungsten rod there is no inner radius, or ri= 0. Note that I◦is independent of the bending direction because of the rotational symmetry of the cross-section. For a rectangular cross-section The 39th International Electric Propulsion Conference, Imperial College London London, United Kingdom 14–19 September 2025 Copyright 2025 by the Electric Rocket Propulsion Society. All rights reserved. Page 4 Figure 3. Three-dimensional view of the membrane force probe. The target is mounted at the front end of the rod, which is elastically supported by two circular membranes. This arrangement constrains the rod to longitudinal motion against the restoring force of the membranes (highlighted in green). A mirror at the rear of the rod and a collimator are used for the interferometric displacement measurement. of the cantilever, i.e. the tungsten sheet, the moment is Irect =b h3 12 (2) for bending along the thickness dimension h, perpendicular to the width b. In case of the (ceramic) tube with ro= 0.5 mm and ri= 0.25 mm, we obtain I◦= 0.046 mm4. A massive (tungsten) rod with ro= 0.5 mm has an only slightly larger second moment of area, I◦= 0.049 mm4. For the tungsten sheet cantilever, we had b= 6 mm and h= 0.6 mm, resulting in an Irect = 0.108 mm4, approximately twice the value of the cylindrical designs. Figures 2 (c) and (d) show the construction drawing and a photograph of the opened probe, respectively. The bluish area in the drawing represents a little beam dump located behind the target. A thin graphite plate was mounted in this position to suppress sputtering caused by beam particles entering the probe through the gap between target and housing. Without this plate, such particles could sputter the inner rear wall, and the resulting secondary particles might impinge on the back of the target holder, slightly reducing the measured force. Not only because of the larger second moment of area, but also due to the shorter cantilever, the calibration constants of the compact force probe (Fig. 2) are significantly larger than those of its predecessor (Fig. 1). Typical values are k= 965 N m−1and k= 23.4 N m−1, respectively, for the calibration constants.11, 16 III. Membrane Force Probe Prototype The membrane force probe differs fundamentally from the cantilever-type probes. Figure 3 shows the essential components of the probe. The bending element is a circular thin metal sheet, such as the one shown in Fig. 4(a). We refer to this part as the membrane. In analogy to the cantilevers, it also has fixed and free regions, marked in gray and red in Fig. 4(a). The deformation due to the applied force occurs in the beige-colored area. As can be seen in Fig. 3, two such Figure 4. Membrane. (a) Geometry of the circular membrane used as the elastic element of the probe, with fixed (gray), free (red), and deformable (beige) regions. (b) FEM simulation of the membrane deflection under a force of 1 µN applied as a homogeneous pressure on the red area in (a). The calculated deflection of d= 0.22 nm corresponds to a spring constant of k◦= 4575 N m−1for a single membrane. The 39th International Electric Propulsion Conference, Imperial College London London, United Kingdom 14–19 September 2025 Copyright 2025 by the Electric Rocket Propulsion Society. All rights reserved. Page 5 membranes are used. This arrangement provides mechanical stability and prevents tilting of the membrane, in which the red and gray surfaces in Fig. 4(a) would no longer remain parallel. The rod, at the front end of which the target is mounted, can therefore practically only move in the longitudinal direction against the restoring force of the membranes. At the opposite end of the rod, a mirror is mounted, followed by a collimator with an optical fiber positioned at a distance of about 10 mm; the optical setup is therefore very similar to that of the cantilever force probes. While the cantilevers could be treated analytically by Euler-Bernoulli beam theory, the situation is more complex for a membrane of the shape shown in Fig. 4(a). However, a numerical calculation using a static FEM model can readily be performed, even with standard CAD software. Figure 4(b) shows such a result for a force of 1 µN applied as a homogeneous pressure on the red area highlighted in Fig. 4(a). The calculated deflection of d= 0.22 nm corresponds to a spring constant of k◦= 4575 N m−1for a single membrane. Since two membranes act in parallel, the expected calibration constant is approximately k≈9000 N m−1. IV. Ex Situ Characterization of the Membrane Force Probe A prototype of the membrane force probe described above was built and tested in a table-top experiment. Fig. 5(a) shows a photograph of the setup. This is not yet a probe designed for operation in a vacuum chamber and exposure to a particle beam, but rather a device for preliminary ex situ characterizations. A complete probe will eventually include a cylindrical housing, probably with an outer diameter of about 30 mm. The first test addressed the oscillations arising from a sudden change of the load (see Sec. II). In the original cantilever force probe, the intrinsic damping without additional measures was weak. This can be quantified by the time constant τof the exponentially decaying amplitudes, i.e. ∝exp(−t/τ) with time t. The original probe with ceramic cantilever exhibited a long decay time of τ≈6 s, which motivated the development of an eddy current damping system. With this additional braking, the decay time was reduced to only τ≈0.3 s.11 Figure 5. Membrane force probe. (a) Photograph of the table-top setup used for a preliminary ex situ characterization. (b) Time series of the deflection after dropping a weight of 18 mg from a height of about 1–2 cm, yielding a decay time of τ≈0.3 s even without intentional damping. (c) Calibration plot obtained by applying certified weights of 1–10 mg to the target surface. The linear fit yields a calibration constant of k= (8640 ±235) N m−1. Figure 5(b) shows the time series of the measured deflections after a weight of 18 mg was dropped onto the target from a height of approximately 1–2 cm. To our initial surprise, the membrane force probe exhibited a decay time of τ≈0.3 s even without intentional damping. We return to this result in the discussion in Sec. V. This probe was also calibrated, using the same procedure as described earlier.11 Certified weights of 1, 2, 5, and 10 mg (±0.006 mg) were placed on the target surface. Combinations of up to four weights with a total mass mexerted the force F=mg on the target, where g= 9.81 m s−2is the local gravitational acceleration. Figure 5(c) shows the displacements as a function of the applied force. From this we obtain a calibration The 39th International Electric Propulsion Conference, Imperial College London London, United Kingdom 14–19 September 2025 Copyright 2025 by the Electric Rocket Propulsion Society. All rights reserved. Page 6 Figure 6. Time series with corresponding amplitude spectra. (a), (b) Horizontal orientation of the probe. (c), (d) Vertical orientation of the probe. constant of k= (8640 ±235) N m−1. Finally, we examined the oscillations caused by ambient mechanical noise. Measurements were performed for two different orientations, including the horizontal case, since this will be the typical orientation of the probe in test chambers. Figure 6 shows excerpts from 10-s-long time series together with the corresponding amplitude spectra. Figures 6 (a) and (b) represent the horizontal orientation, while Figs. 6 (c) and (d) show the data for the vertical orientation, as in Fig. 5(a). V. Discussion and Outlook Force probes have proven to be a useful tool for measuring momentum flux in thruster plumes. The two major benefits can be summarized as follows: (1) The method is charge independent and accounts for both ions and neutral particles produced by charge-exchange collisions in the thruster or in the plume, which is an advantage over Faraday cup measurements. (2) The method provides information on how the thrust (or more precisely, momentum) is spatially distributed in the plume, which represents a valuable complement to global thrust measurements with thrust balances. After a short summary of cantilever-based interferometric force probes developed in our laboratory at Kiel University, we introduced a new concept for a membrane-based interferometric force probe. This design employs a pair of circular thin stainless steel sheets with rotationally symmetric cutouts serving as the bending or “spring” element. A rod transfers the force applied to the target to the central areas of the membranes, while their outer rims are clamped and connected to the housing. Table 1 summarizes selected material properties that are relevant for the mechanical response and dampThe 39th International Electric Propulsion Conference, Imperial College London London, United Kingdom 14–19 September 2025 Copyright 2025 by the Electric Rocket Propulsion Society. All rights reserved. Page 7 ing behavior of the different probe designs discussed in this article. Table 1. Selected material properties relevant to mechanical response and damping. Elasticity data for Al2O3 from Wachtman and Lam;22 temperature coefficients for tungsten and stainless steels calculated from Keulegan and Houseman;23 other values from material suppliers and the CRC Handbook of Chemistry and Physics.24 Material E(GPa) dE/dT (MPa/K) ρ(g/cm3) Internal friction Al2O3(polycrystalline) 309 ≈ −40 3.71 very low Tungsten (W) 411 ≈ −39 19.3 low Stainless steel 193 ≈ −(52 −83) 7.9 high A surprising outcome of the ex situ characterization was the short decay time of the membrane force probe oscillations, τ≈0.3 s, despite the absence of intentional damping [see Fig. 5(b)]. This value is comparable to the decay time of the cantilever probe only after the eddy-current damping unit had been installed.11 A straightforward explanation for this markedly different behavior could be sought in the use of stainless steel as the bending element: there seems to be a striking difference in damping behavior arising from the underlying material properties. Alumina (Al2O3), with its ionic–covalent bonding and very rigid crystal lattice, offers hardly any microscopic mechanisms for energy dissipation during elastic deformation. Tungsten, with its very high melting point and similarly high elastic modulus E, likewise exhibits low internal friction. Stainless steel, by contrast, is polycrystalline and contains many grain boundaries and alloying elements (e.g. Cr, Ni, Mo), which provide multiple pathways for microplastic deformation processes and thus enhanced energy dissipation.25 However, other mechanisms may be more decisive for the decay of the very low-frequency oscillations observed here. In practice, the clamping of the fixed parts of the bending elements is never perfectly fixed, so that vibrational energy is transferred into the housing of the force probe. The situation is therefore closer to a piano or guitar string, which dissipates most of its vibrational energy not through internal friction of the string material, but via its termination at the bridge and resonance body. Therefore, the choice of material may be of only subordinate importance for the damping. A direct comparison of geometrically identical membrane pairs, one made of alumina and the other of stainless steel, could provide further clarification. In any case, this intrinsic damping property of membrane-based probes may offer a practical advantage, as it reduces the need for additional damping units. The longitudinal and slender design is also advantageous, as it exposes only the target and a small rim to the thruster plume. This reduces the thermal load and allows for long-term operation even at closer distances to the thruster. The next steps will be the design and construction of a housing for the probe and tests in a beam environment. In addition, several membrane variants will be explored experimentally and through simulation. This will primarily involve thinner membranes to achieve smaller calibration constants, i.e. higher sensitivity. We also plan to investigate different membrane materials, including ceramics and tungsten, in order to address the open question of the dominant damping mechanisms. 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