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Passivity-based Robust Control of Aerospace Systems

This report provides a brief summary of the research work performed over the duration of the cooperative research agreement between NASA Langley Research Center and Kansas State University. The cooperative agreement which was originally for the duration the three years was extended by another year through no-cost extension in order to accomplish the goals of the project. The main objective of the research was to develop passivity-based robust control methodology for passive and non-passive aerospace systems. The focus of the first-year's research was limited to the investigation of passivity-based methods for the robust control of Linear Time-Invariant (LTI) single-input single-output (SISO), open-loop stable, minimum-phase non-passive systems. The second year's focus was mainly on extending the passivity-based methodology to a larger class of non-passive LTI systems which includes unstable and nonminimum phase SISO systems. For LTI non-passive systems, five different passification. methods were developed. The primary effort during the years three and four was on the development of passification methodology for MIMO systems, development of methods for checking robustness of passification, and developing synthesis techniques for passifying compensators. For passive LTI systems optimal synthesis procedure was also developed for the design of constant-gain positive real controllers. For nonlinear passive systems, numerical optimization-based technique was developed for the synthesis of constant as well as time-varying gain positive-real controllers. The passivity-based control design methodology developed during the duration of this project was demonstrated by its application to various benchmark examples. These example systems included longitudinal model of an F-18 High Alpha Research Vehicle (HARV) for pitch axis control, NASA's supersonic transport wind tunnel model, ACC benchmark model, 1-D acoustic duct model, piezo-actuated flexible link model, and NASA's Benchmark Active Controls Technology (BACT) Wing model. Some of the stability results for linear passive systems were also extended to nonlinear passive systems. Several publications and conference presentations resulted from this research.

Kelkar, Atul G.↗

Precision Pointing for the Wide-Field Infrared Survey Telescope(WFIRST)

The Wide-Field Infrared Survey Telescope (WFIRST) mission, scheduled for a mid-2020's launch, is currently in its definition phase. The mission is designed to investigate essential questions in the areas of dark energy, exoplanets, and infrared astrophysics. WFIRST will use a 2.4-meter primary telescope (same size as the Hubble Space Telescope's primary mirror) and two instruments: the Wide Field Instrument (WFI) and the Coronagraph Instrument (CGI). In order to address the critical science requirements, the WFIRST mission will conduct large-scale surveys of the infrared sky, requiring both agility and precision pointing (11.6 milli-arcsec stability, 14 milli-arcsec jitter). This paper describes some of the challenges this mission profile presents to the Guidance, Navigation, and Control (GNC) subsystem, and some of the design elements chosen to accommodate those challenges. The high-galactic-latitude survey is characterized by 3-minute observations separated by slews ranging from 0.025 deg to 0.8 deg. The need for observation efficiency drives the slew and settle process to be as rapid as possible. A description of the shaped slew profile chosen to minimize excitation of structural oscillation, and the handoff from star tracker-gyro control to fine guidance sensor control is detailed. Also presented is the fine guidance sensor (FGS), which is integral with the primary instrument (WFI). The FGS is capable of tracking up to 18 guide stars, enabling robust FGS acquisition and precision pointing. To avoid excitation of observatory structural jitter, reaction wheel speeds are operationally maintained within set limits. In addition, the wheel balance law is designed to maintain 1-Hz separation between the wheel speeds to avoid reinforcing jitter excitation at any particular frequency. The wheel balance law and operational implications are described. Finally, the candidate GNC hardware suite needed to meet the requirements of the mission is presented.

Stoneking, Eric T.↗

Proximity Magnetism in Mn(Bi,Sb) 2 Te 4 –(Bi,Sb) 2 Te 3 /MnTe Natural Heterostructures

Magnetic topological insulators and their heterostructures provide significant opportunities to couple band topology with a nontrivial spin configuration for enhanced spintronic device performance, as well as designing magnetoelectric systems and functionalities. We find that Mn interdiffusion from MnTe when interfaced with (Bi,Sb) 2 Te 3 stabilizes as self-organized Mn(Bi,Sb) 2 Te 4 septuple lamellae among alternating (Bi,Sb) 2 Te 3 quintuple layers, as observed using scanning transmission electron microscopy and depth-sensitive polarized neutron reflectometry. We further demonstrate a valuable combination of magnetic and topological orders in these naturally formed Mn(Bi,Sb) 2 Te 4 –(Bi,Sb) 2 Te 3 heterostructures, which are exchange-coupled with MnTe. Magnetotransport experiments and quantum magnetism simulations reveal that, above its own Néel temperature T N ∼ 20 K, Mn(Bi,Sb) 2 Te 4 mediates the exchange field leading to an anomalous Hall effect at the (Bi,Sb) 2 Te 3 /MnTe interface, with an enhanced interfacial T N exceeding 200 K, approaching that of the bulk MnTe. This magnetic interface, in turn, allows a robust and deterministic spin–orbit torque switching without an external magnetic field at a low critical current density of 3 × 10 5 A cm –2 . The antiferromagnetically coupled architecture of Mn(Bi,Sb) 2 Te 4 –(Bi,Sb) 2 Te 3 /MnTe, featuring magnetic and topological proximity effects across a chalcogenide backbone, is rich in fundamental interface physics and holds the potential for practical applications in spintronics.

magnetic topological insulator↗

Topological order and entanglement dynamics in the measurement-only XZZX quantum code

We examine the dynamics of a ( 1 + 1 ) -dimensional measurement-only circuit defined by the stabilizers of the [[5,1,3]] quantum error correcting code interrupted by single-qubit Pauli measurements. The code corrects arbitrary single-qubit errors and it stabilizes an area law entangled state with a D 2 = Z 2 × Z 2 symmetry protected topological (SPT) order, as well as a symmetry breaking (SB) order from a twofold bulk degeneracy. The Pauli measurements break the topological order and induce a phase transition into a trivial area law phase. Allowing more than one type of Pauli measurement increases the measurement-induced frustration and the SPT and SB order can be broken either simultaneously or separately at nonzero measurement rate. This yields a rich phase diagram and unanticipated critical behavior at the phase transitions. Although the correlation length exponent ν = 4 3 and the dynamical critical exponent z = 1 are consistent with bond percolation, the prefactor of the logarithmic entanglement growth may take noninteger multiples of the percolation value. Remarkably, we identify a robust transient scaling regime for the purification dynamics of L qubits. It reveals a modified dynamical critical exponent z * ≠ z , which is observable up to times t ~ L z * and is reminiscent of the relaxation of critical systems into a prethermal state.

36 MATERIALS SCIENCE↗

Carboxylating Elastomer via Thiol‐Ene Click Reaction to Improve Miscibility with Conjugated Polymers for Mechanically Robust Organic Solar Cells with Efficiency of 19%

Abstract Incorporating flexible insulating polymers is a straightforward strategy to enhance the mechanical properties of rigid conjugated polymers, enabling their use in flexible electronic devices. However, maintaining electronic characteristics simultaneously is challenging due to the poor miscibility between insulating polymers and conjugated polymers. This study introduces the carboxylation of insulating polymers as an effective strategy to enhance miscibility with conjugated polymers via surface energy modulation and hydrogen bonding. The carboxylated elastomer, synthesized via a thiol‐ene click reaction, closely matches the surface energy of the conjugated polymer. This significantly improves the mechanical properties, achieving a high crack‐onset strain of 21.48%, surpassing that (5.93%) of the unmodified elastomer:conjugated polymer blend. Upon incorporating the carboxylated elastomer into PM6:L8‐BO‐based organic solar cells, an impressive power conversion efficiency of 19.04% is attained, which top‐performs among insulating polymer‐incorporated devices and outperforms devices with unmodified elastomer or neat PM6:L8‐BO. The superior efficiency is attributed to the optimized microstructures and enhanced crystallinity for efficient and balanced charge transport, and suppressed charge recombination. Furthermore, flexible devices with 5% carboxylated elastomer exhibit superior mechanical stability, retaining ≈88.9% of the initial efficiency after 40 000 bending cycles at a 1 mm radius, surpassing ≈83.5% for devices with 5% unmodified elastomer.

Chemistry↗

Autonomous spacecraft attitude control using magnetic torquing only

Magnetic torquing of spacecraft has been an important mechanism for attitude control since the earliest satellites were launched. Typically a magnetic control system has been used for precession/nutation damping for gravity-gradient stabilized satellites, momentum dumping for systems equipped with reaction wheels, or momentum-axis pointing for spinning and momentum-biased spacecraft. Although within the small satellite community there has always been interest in expensive, light-weight, and low-power attitude control systems, completely magnetic control systems have not been used for autonomous three-axis stabilized spacecraft due to the large computational requirements involved. As increasingly more powerful microprocessors have become available, this has become less of an impediment. These facts have motivated consideration of the all-magnetic attitude control system presented here. The problem of controlling spacecraft attitude using only magnetic torquing is cast into the form of the Linear Quadratic Regulator (LQR), resulting in a linear feedback control law. Since the geomagnetic field along a satellite trajectory is not constant, the system equations are time varying. As a result, the optimal feedback gains are time-varying. Orbit geometry is exploited to treat feedback gains as a function of position rather than time, making feasible the onboard solution of the optimal control problem. In simulations performed to date, the control laws have shown themselves to be fairly robust and a good candidate for an onboard attitude control system.

Musser, Keith L.↗

Improved Whole-Blood-Staining Device

Dramatic improvements have been made in NASA s Whole Blood Staining Device (WBSD) since it was last described in "Whole-Blood-Staining Device," NASA Tech Briefs, Vol. 23, No. 10 (October 1999), page 64. The new system has a longer shelf life, a simpler and more effective operational procedure, improved interface with instrumentation, and shorter processing time. More specifically, the improvements have targeted bag and locking clip materials, sampling ports, and air pocket prevention. The WBSD stains whole blood collected during spaceflight for subsequent flow cytometric analysis. In short, the main device stains white blood cells by use of monoclonal antibodies conjugated to various fluorochromes, followed by lysing and fixing of the cells by use of a commercial reagent that has been diluted according to NASA safety standards. This system is compact, robust, and does not require electric power, precise mixing, or precise incubation times. Figure 1 depicts the present improved version for staining applications, which is a poly(tetrafluoroethylene) bag with a Luer-lock port and plastic locking clips. An InterLink (or equivalent) intravenous- injection port screws into the Luer-lock port. The inflatable/collapsible nature of the bag facilitates loading and helps to minimize the amount of air trapped in the fully loaded bag. Some additional uses have been identified for the device beyond whole blood staining. The WBSD has been configured for functional assays that require culture of live cells by housing sterile culture media, mitogens, and fixatives prior to use [Figure 2(a)]. Simple injection of whole blood allows cell-stimulation culture to be performed in reduced gravity conditions, and product stabilization prior to storage, while protecting astronauts from liquid biohazardous materials. Also, the improved WBSD has reconstituted powdered injectable antibiotics by mixing them with diluent liquids [Figure 2(b)]. Although such mixing can readily be performed on Earth by shaking in glass vials, it cannot readily be performed this way in outer space without entraining air bubbles. The present device can be preloaded with the powder and diluent(s) in separate compartments. The powder and diluent( s) can be mixed, without introducing air bubbles, by removing the clip(s), then shaking. This use of the device could also be advantageous in terrestrial applications because it maintains the isolation of the constituents until the time of use.

Sams, Clarence F.↗

Optimal Recursive Digital Filters for Active Bending Stabilization

In the design of flight control systems for large flexible boosters, it is common practice to utilize active feedback control of the first lateral structural bending mode so as to suppress transients and reduce gust loading. Typically, active stabilization or phase stabilization is achieved by carefully shaping the loop transfer function in the frequency domain via the use of compensating filters combined with the frequency response characteristics of the nozzle/actuator system. In this paper we present a new approach for parameterizing and determining optimal low-order recursive linear digital filters so as to satisfy phase shaping constraints for bending and sloshing dynamics while simultaneously maximizing attenuation in other frequency bands of interest, e.g. near higher frequency parasitic structural modes. By parameterizing the filter directly in the z-plane with certain restrictions, the search space of candidate filter designs that satisfy the constraints is restricted to stable, minimum phase recursive low-pass filters with well-conditioned coefficients. Combined with optimal output feedback blending from multiple rate gyros, the present approach enables rapid and robust parametrization of autopilot bending filters to attain flight control performance objectives. Numerical results are presented that illustrate the application of the present technique to the development of rate gyro filters for an exploration-class multi-engined space launch vehicle.

Orr, Jeb S.↗

Stabilizing dynamic subsea power cables using Bi-stable nonlinear energy sinks

This study investigates vibration mitigation of dynamic subsea cables through passive bi-stable nonlinear energy sinks (B-NESs). These devices suppress vibration energy in a broadband way, and can be regarded as extensions of classical linear tuned mass dampers (TMDs) which are narrowband devices. Through the open-source MoorDyn library, we simulated the vibrations of a vertical subsea cable equipped with a set of B-NESs. Multi-objective optimization was performed to detect the B-NES parameters for optimal mitigation of the cable vibrations. Advanced signal processing verified the efficacy of the optimized B-NESs not only to rapidly absorb and locally dissipate vibration energy, but also to nonlinearly scatter vibration energy from low to high frequencies within the cable itself. This last feature is especially beneficial for vibration mitigation of the undersea cable, as at higher frequencies the cable vibrations exhibit drastically reduced amplitudes and are more effectively dissipated by inherent structural damping and hydrodynamic radiation damping. This contrasts with traditional TMDs which can mitigate vibration only at a single frequency. Furthermore, our robustness study confirms the B-NES's effectiveness under even varying environmental conditions. Overall, the B-NES's capacity for broadband vibration mitigation renders it a promising retrofit solution for improving the performance and operational safety of dynamic power cables in offshore wind farms and other marine applications.

17 WIND ENERGY↗

Toward Long-Life, Ultrahigh-Nickel Layered Oxide Cathodes for Lithium-Ion Batteries: Optimizing the Interphase Chemistry with a Dual-Functional Polymer

Boosting the Ni content in LiMO 2 (M = Ni, Co, Mn, etc.) layered oxides is a promising way to establish high-energydensity, low-cost cathodes, but the poor cathode surface stability is a daunting challenge for their practical viability. Herein, by constructing a dual-functional binder framework with a conductive polymer - polyaniline (PANI), the ultrahigh-Ni layered oxide cathode (LiNi 0.94 Co 0.06 O 2 ) exhibits significantly improved cyclability, with a capacity retention greatly increased from 47% to 81% over 1,000 cycles in full cells. It is demonstrated that the acidic species (e.g. HF) in the electrolyte can be efficiently scavenged through a protonation process of PANI, hence the cathode surface reactivity is greatly suppressed and the rock-salt phase propagation into bulk structure is considerably alleviated. Furthermore, the PANI binder system effectively prevents both the cathode-electrolyte interphase (CEI) and anode-electrolyte interphase (AEI) from degrading to a thick “triple-layer” architecture upon extensive cycling, resulting in more robust, thinner CEI and AEI with regulated interphasial chemistry. Moreover, the delocalized π-conjugated electrons along the backbone of PANI facilitate fast electron transfer and promote rate capability even at low temperatures (-20 °C). Finally, this work sheds light on rational binder engineering for developing high-energy-density lithium-ion batteries with acceptable cycle life.

25 ENERGY STORAGE↗

CD206 + Trem2 + macrophage accumulation in the murine knee joint after injury is associated with protection against post-traumatic osteoarthritis in MRL/MpJ mice

Post-traumatic osteoarthritis (PTOA) is a painful joint disease characterized by the degradation of bone, cartilage, and other connective tissues in the joint. PTOA is initiated by trauma to joint-stabilizing tissues, such as the anterior cruciate ligament, medial meniscus, or by intra-articular fractures. In humans, ~50% of joint injuries progress to PTOA, while the rest spontaneously resolve. To better understand molecular programs contributing to PTOA development or resolution, we examined injury-induced fluctuations in immune cell populations and transcriptional shifts by single-cell RNA sequencing of synovial joints in PTOA-susceptible C57BL/6J (B6) and PTOA-resistant MRL/MpJ (MRL) mice. We identified significant differences in monocyte and macrophage subpopulations between MRL and B6 joints. A potent myeloid-driven anti-inflammatory response was observed in MRL injured joints that significantly contrasted the pro-inflammatory signaling seen in B6 joints. Multiple CD206 + macrophage populations classically described as M2 were found enriched in MRL injured joints. These CD206 + macrophages also robustly expressed Trem2 , a receptor involved in inflammation and myeloid cell activation. These data suggest that the PTOA resistant MRL mouse strain displays an enhanced capacity of clearing debris and apoptotic cells induced by inflammation after injury due to an increase in activated M2 macrophages within the synovial tissue and joint space.

60 APPLIED LIFE SCIENCES↗

Seismic Assessment of a Long-Term Lunar Habitat

The establishment of secure earth-independent long-term lunar habitats has been envisioned by numerous government agencies and private companies. Recent advancements in assessing seismic hazards caused by shallow moonquakes have highlighted the importance of incorporating this phenomenon into the design of robust and resilient lunar structures. However, further research is required to explore lunar habitat design that considers seismic loads. This paper proposes assessing the structural response of a lunar habitat made of sulfur concrete covered with a regolith layer. The numerical model of the structure is subjected to gravitational, internal pressure and seismic loads. The seismic analysis of the structure is carried out using spectral and nonlinear time history methods. Conditional mean spectra for shallow moonquakes with return periods of 75, 475, 970 and 2475 years are used in the seismic analysis. The records used for the temporal analyses were ground motions that agree with a preliminary seismic hazard on the Moon. The results of temporal analyses reveal that shallow moonquakes with return periods greater than 475 years can lead to the loss of the global stability of the structure. Consequently, the findings imply that seismic loads have the potential to impose unacceptable demands on lunar structures constructed from in-situ materials like sulfur concrete. Hence, it is imperative to incorporate seismic considerations in the design process for developing resilient and long-term lunar habitats.

Lunar habitat↗

Criticality in the Shallow-to-Deep Transition of Simulated Tropical Marine Convection

Idealized simulations of tropical, marine convection depict shallow, nonprecipitating cumuli located beneath the 0°C level transitioning into cumulonimbi that reach up to 12 km and higher. The timing of the transition was only weakly related to environmental stability, and 13 of the 15 simulations run with 5 different lapse-rate profiles had rain develop at nearly the same time after model start. The key quantity that apparently controlled deep convective formation was vertical acceleration inside cloudy updrafts between cloud base and the 0°C level. Below a critical value of updraft vertical acceleration, little rainfall occurred. Just as the domain-mean updraft acceleration reached the critical value, the first convection quickly grew to past 12 km altitude. Then, as acceleration increased above the critical value, rain rate averaged in the model domain increased quickly over about a 3-h-long period. The specific value of the critical updraft acceleration depended on how updrafts were defined and in what layer the acceleration was averaged; however, regardless of how criticality was defined, a robust relationship between domain-mean updraft vertical acceleration and rain rate occurred. Positive acceleration of updrafts below the 0°C level was present below 2.75 km and was largest in the 500 m above cloud base. However, the maximum difference between updraft and environmental temperatures occurred between 2 and 3 km. The domain-mean Archimedean buoyancy of updrafts relative to some reference state was a poor predictor for domain-mean rain rate. In conclusion, the exact value of the critical updraft acceleration likely depends on numerous other factors that were not investigated.

54 ENVIRONMENTAL SCIENCES↗

Impact of Cross-Axis Structural Dynamics on Validation of Linear Models for Space Launch System

A feasibility study was performed to examine the advisability of incorporating a set of Programmed Test Inputs (PTIs) during the Space Launch System (SLS) vehicle flight. The intent of these inputs is to provide validation to the preflight models for control system stability margins, aerodynamics, and structural dynamics. During October 2009, Ares I-X program was successful in carrying out a series of PTI maneuvers which provided a significant amount of valuable data for post-flight analysis. The resulting data comparisons showed excellent agreement with the preflight linear models across the frequency spectrum of interest. However unlike Ares I-X, the structural dynamics associated with the SLS boost phase configuration are far more complex and highly coupled in all three axes. This presents a challenge when implementing this similar system identification technique to SLS. Preliminary simulation results show noticeable mismatches between PTI validation and analytical linear models in the frequency range of the structural dynamics. An alternate approach was examined which demonstrates the potential for better overall characterization of the system frequency response as well as robustness of the control design.

Pei, Jing↗

Environmentally Robust Quantum Dot Down Converters for Highly Efficient Solid State Lighting

This project contributes fundamental understanding of the chemical synthesis of cadmium free quantum dots (QDs) designed for solid state lighting (SSL). Methods to incorporate gallium into quantum dot architectures were explored, which led to more stable and brighter solid state lighting devices. These findings will help advance the incorporation of quantum dots in highly energy efficient lighting devices with a warm color spectrum. Such Cd-free lighting devices can greatly accelerate the adoption of high efficiency, warm spectrum lighting in buildings. The project targeted InP/GaP/ZnS colloidal quantum dots that more effectively absorb the blue light produced by light emitting diode packages and emit monochromatic red light with near perfect efficiency. While significant improvements in the QD absorptivity were demonstrated, the absorbance increase achieved was 3x rather than the targeted 10x increase. Similarly, the luminescence from these QDs achieved about half of the target photoluminescence quantum yield (PLQY) and twice the desired spectral width, a measure of the color purity of their luminescence. While the target characteristics and some of the year 2/2 milestones were not met, the new QD architectures did lead to greater stability on SSL packages, a sign of the long term promise of the technology.

36 MATERIALS SCIENCE↗

The Value of Reversible Carbon Storage in a Zero-Emissions World

Atmospheric carbon dioxide removal (CDR) is required to stabilize global temperature. CDR can be achieved via ecosystem-based approaches that are cost-effective but reversible (e.g., soil and forest management) or by more durable but expensive approaches (e.g., direct air capture coupled with geologic storage). Here, we examine trade-offs between these approaches, focusing on timing, climate impacts, and cost. We simulated reversible carbon accrual for a range of CDR contract structures using a general minimalist model of ecosystem carbon cycling, and parameterized it to simulate US agricultural soil management─specifically cover cropping─as a case study. We then quantified the resulting impact on atmospheric carbon and global temperature using a climate model emulator. We find that maintaining a patchwork of reversible CDR projects by replacing lapsed projects with new projects can reduce warming by 22–195 μ°C in 2100 and that the magnitude of this cooling effect depends on how effectively the patchwork is maintained. Long-term maintenance of reversible CDR projects requires institutional stability that cannot be guaranteed over multiple decades. Consequently, effective CDR ultimately requires replacing reversible projects with durable projects. To address this problem, we modeled the cost of replacing reversible agricultural soil CDR with geologic CDR. We found that using reversible CDR as a bridge to durable CDR is potentially more cost-effective as a global cooling strategy (0.20–0.81 billion USD per μ°C avoided) than perpetual maintenance of reversible CDR (0.32–1.31 billion USD per μ°C avoided) or an immediate transition to durable CDR (1.37–2.19 billion USD per μ°C avoided). However, we emphasize that institutional commitments to maintain reversible CDR projects cannot be guaranteed. Reliance on reversible CDR as a bridge to durable CDR therefore carries an unknown amount of risk and will only function if efforts to maintain reversible CDR are robust.

carbon capture and storage↗

Roles of Dynamic State Estimation in Power System Modeling, Monitoring and Operation

—Power system dynamic state estimation (DSE) remains an active research area. This is driven by the absence of accurate models, the increasing availability of fast-sampled, time synchronized measurements, and the advances in the capability, scalability, and affordability of computing and communications. This paper discusses the advantages of DSE as compared to static state estimation, and the implementation differences between the two, including the measurement configuration, modeling framework and support software features. The important roles of DSE are discussed from modeling, monitoring and operation aspects for today’s synchronous machine dominated systems and the future power electronics-interfaced generation systems. Several examples are presented to demonstrate the benefits of DSE on enhancing the operational robustness and resilience of 21st century power system through time critical applications. Future research directions are identified and discussed, paving the way for developing the next generation of energy management systems.

Zhao, Junbo↗

Versatile tape-drive target for high-repetition-rate laser-driven proton acceleration

We present the development and characterization of a high-stability, multi-material, multi-thickness tape-drive target for laser-driven acceleration at repetition rates of up to 100 Hz. The tape surface position was measured to be stable on the sub-micrometre scale, compatible with the high-numerical aperture focusing geometries required to achieve relativistic intensity interactions with the pulse energy available in current multi-Hz and near-future higher repetition-rate lasers ( > kHz). Long-term drift was characterized at 100 Hz demonstrating suitability for operation over extended periods. The target was continuously operated at up to 5 Hz in a recent experiment for 70,000 shots without intervention by the experimental team, with the exception of tape replacement, producing the largest data-set of relativistically intense laser–solid foil measurements to date. This tape drive provides robust targetry for the generation and study of high-repetition-rate ion beams using next-generation high-power laser systems, also enabling wider applications of laser-driven proton sources.

47 OTHER INSTRUMENTATION↗

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