Results for “VACUUM TUBE” | Engineering Papers

DOE OSTI2025

NOVEL METHODS FOR IN SITU HIGH DENSITY SURFACE CLEANING SCRUBBING OF ULTRAHIGH VAC LONG NARROW TUBES TO REDUCE SECONDARY ELECTRON YIELD AND OUT GASSING

This project developed new methods for cleaning the inside surfaces of very long, narrow vacuum tubes used in particle accelerators. Traditional cleaning approaches are expensive, slow, or difficult to implement in accelerator tunnels. We designed and tested a portable plasma discharge cleaning system that uses lower-cost microwave and magnetron technologies to reduce outgassing and secondary electron emission from stainless steel and copper surfaces. The system, called the Plasma Discharge Test System (PDTS), allows accelerator components to be scrubbed more efficiently, which can improve performance and reduce maintenance costs for research and industrial applications.

POOLE, JOE HENRY [PRESIDENT]↗

DOE OSTI2025

Designing a User Interface for Real-Time Magnetometer Data Acquisition

The Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100) is a next-generation quantum sensor designed to search for ultralight dark matter and explore new frontiers in quantum mechanics. Due to the experiment s sensitivity to magnetic interference, a magnetometer trolley system was developed to scan magnetic fields along a vacuum tube. Interacting with the system required command-line inputs, creating usability challenges. To improve accessibility and streamline data acquisition, I developed a graphical user interface (GUI) using Python and the customtkinter library. The GUI supports real-time data display, state/mode switching, command execution, and CSV file management. I collaborated with another intern to integrate data visualization features into the GUI, allowing users to generate 3D plots of post-acquisition magnetic field data. In the future, I aim to fix the real-time plotting feature as it results in an unresponsive GUI.

Mendez, Milagros [DuPage Coll.]↗

DOE OSTI2025

Designing a User Interface for Real-Time Magnetometer Data Acquisition

The Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100) is a next-generation quantum sensor designed to search for ultralight dark matter and explore new frontiers in quantum mechanics. Due to the experiment’s sensitivity to magnetic interference, a magnetometer trolley system was developed to scan magnetic fields along a vacuum tube. Interacting with the system required command-line inputs, creating usability challenges. To improve accessibility and streamline data acquisition, I developed a graphical user interface (GUI) using Python and the customtkinter library. The GUI supports real-time data display, state/mode switching, command execution, and CSV file management. I collaborated with another intern to integrate data visualization features into the GUI, allowing users to generate 3D plots of post-acquisition magnetic field data. In the future, I aim to fix the real-time plotting feature as it results in an unresponsive GUI.

Mendez, Milagros [DuPage Coll.]↗

DOE OSTI2026

15.3% AM1.5G Efficiency GaAs Solar Cells Fabricated via an Epitaxy-Free Process

Here, we report simple and potentially low-cost techniques for creating high-quality n-type gallium arsenide (GaAs) and GaAs p/n junctions and fabricate GaAs p/n junction solar cells. Detailed-balance modeling suggests that 20% AM1.5G efficiency p/n homojunction devices may be possible if the surface doping concentration can be limited to values less than ∼ 5 × 10 19 cm −3 . Our process exploits an open-tube, vapor-phase, deposition-free, zinc diffusion technique for forming p-type layers in melt-grown n-GaAs substrates that results in sheet resistances less than 1 kΩ/$\square$. In addition, we have improved the minority carrier diffusion lengths of melt-grown GaAs from less than one micron to over five microns using an open-tube, vacuum-free, annealing process which reduces the density of EL2 midgap defects. Finally, we have combined these advances to fabricate epitaxy-free, GaAs solar cells with a validated AM1.5G efficiency of 15.3%.

14 SOLAR ENERGY↗

DOE OSTI2025

Advances in understanding vacuum break dynamics in liquid helium-cooled tubes for accelerator beamline applications

Understanding air propagation and condensation following a catastrophic vacuum break in particle accelerator beamlines cooled by liquid helium is essential for ensuring operational safety. This review summarizes experimental and theoretical work conducted in our cryogenics lab to address this issue. Systematic measurements were performed to study nitrogen gas propagation in uniform copper tubes cooled by both normal liquid helium (He I) and superfluid helium (He II). These experiments revealed a nearly exponential deceleration of the gas front, with stronger deceleration observed in He II-cooled tubes. To interpret these results, a one-dimensional (1D) theoretical model was developed, incorporating gas dynamics, heat transfer, and condensation mechanisms. The model successfully reproduced key experimental observations in the uniform tube system. However, recent experiments involving a bulky copper cavity designed to mimic the geometry of a superconducting radiofrequency (SRF) cavity revealed strong anisotropic flow patterns of nitrogen gas within the cavity, highlighting limitations in extrapolating results from simplified tube geometries to real accelerator beamlines. To address these complexities, we outline plans for systematic studies using tubes with multiple bulky cavities and the development of a two-dimensional (2D) model to simulate gas dynamics in these more intricate configurations. As a result, these efforts aim to provide a comprehensive understanding of vacuum breaks in particle accelerators and improve predictive capabilities for their operational safety.

Beamline tube↗

DOE OSTI2025

Deuterium Extraction from Helium with a Vanadium Vacuum Permeator: Commissioning of the Tritium Extraction eXperiment (TEX)

Here, the Tritium Extraction eXperiment (TEX) is a forced-convection lead-lithium (PbLi) loop in the Safety and Tritium Applied Research (STAR) facility at Idaho National Laboratory (INL) with the purpose of providing validation data for the vacuum permeator tritium extraction concept. A vanadium tube of 1000 mm length, 12.7 mm outside diameter, and 0.50 mm wall thickness is installed in the test section of TEX. The installed vanadium tube is characterized to quantify impurity concentrations, surface chemistry, and microstructure to elucidate permeation phenomena observed in experimentation. Herein, the permeation properties of the vanadium tube are characterized by measuring deuterium permeation at 300 °C, 325 °C, and 350 °C at 100 kPa, 125 kPa, and 150 kPa total pressures with 5000 ppm deuterium in helium gas mixture in a once-through flow configuration. The hydrogen isotope permeation through the vanadium tube in the test section is measured with quadrupole mass spectrometers and the hydrogen isotope concentration in the feed and retentate gas stream is measured with gas chromatography. The transient permeation results are modeled with MELCOR-TMAP, a thermal-hydraulic tritium transport code, and compared well with literature data.

70 - PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2025

Extinction Monitoring of Pulsed Proton Beams Using FPGA-Based Peak Detection

The Mu2e experiment at Fermilab imposes stringent requirements on the elimination of out-of-time beam in its pulsed proton beam - a requirement known as "extinction". We present a method to measure the out-of-time particle rates to calculate the level of extinction in the inter-pulse gaps. The proposed method utilizes an array of quartz Cherenkov radiators and photomultiplier tubes to detect particles scattered from a vacuum chamber in the M4 transfer beamline at Fermilab.The measurement will employ a new μTCA-based FPGA system for data acquisition and signal processing, utilizing real-time peak detection algorithms to count scattered beam particles. By integrating data over many transfers, the time profile of the out-of-time beam will be resolved to fractional levels relative to that of the in-time beam. These results are compared with G4beamline simulations to validate models of beam transport, dynamics, and extinction, providing critical input for optimizing beam delivery to Mu2e.

Hensley, Ryan [UC, Davis]↗

DOE OSTI2026

Ultra-High Vacuum Outgassing Characterization of Thermally Processed Low-Carbon Steel for Advanced Particle Accelerator and Gravitational Wave Detector Applications

This dissertation investigated AISI 1020 low-carbon steel as an alternative vacuum chamber material to conventional stainless steel for ultra-high vacuum (UHV) and extreme-high vacuum (XHV) applications. After a 400 °C/48 h bake, AISI 1020 tube chambers achieved a hydrogen outgassing rate of 2.4 × 10¿¹6 Torr·L·s¿¹·cm¿², approximately 2,300 times lower than the prebaked 316L stainless-steel comparator, among the lowest hydrogen outgassing rates ever reported for an uncoated metallic vacuum chamber. Bare and magnetite-coated AISI 1020 chambers were then compared using throughput and rate-of-rise methods. The magnetite coating yielded 5× lower water outgassing at room temperature, but this advantage disappeared after 80 °C baking. After full thermal conditioning (400 °C/48 h prebake followed by 150 °C/96 h and 200 °C/110 h), bare steel achieved 25× lower hydrogen outgassing than the magnetite-coated chamber (9.6 × 10¿¹6 Torr·L·s¿¹·cm¿²) and >99% H2 purity with carbon species below RGA detection. Monte Carlo molecular flow simulations of a CEBAF photogun beamline (96 scenarios) showed that replacing 304L stainless steel with AISI 1020 reduces equilibrium H2 pressure by a factor of 833; a single 304L electrode contributes 98.8% of the gas load despite occupying only 9.1% of the internal surface area. A 500-m Einstein Telescope beampipe screening showed that corrugated bellows contribute 18% of the gas load from only 0.7% of the surface area. A five-model adsorption isotherm framework applied to 22 pumpdown datasets (164 fits with AR(1)-GLS correction) established that the experimental protocol, not the material, controls isotherm identifiability: Dubinin–Radushkevich wins isothermal pipe pumpdowns; Langmuir wins thermally dominated chamber bakes. Cross-dataset joint fitting of the AISI 1020 pipe pumpdowns yielded an H2 diffusion activation energy Ed = 7.24 ± 1.28 kcal·mol¿¹, consistent with trap dominated diffusion in commercial low-carbon steels. Two companion innovations were developed: a Variable Conductance Device (VCD, patent pending IDF-00723) for XHV outgassing measurement, and VacuumDesignerPro (VDP), a MATLAB-based design tool validated against LIGO benchmarks.

Al-Allaq, Aiman H [Old Dominion University]↗

DOE OSTI2026

Performance and Stability Characterization of the ICARUS Light Detection System

The ICARUS detector, a key component of the Short Baseline Neutrino (SBN) Program at Fermilab, consists of two identical T300 modules filled with liquid argon. It is equipped with a Light Detection System (LDS) based on 360 8-inch Hamamatsu R5912-MOD photomultiplier tubes (PMTs) arranged behind the wire planes to collect Vacuum Ultraviolet (VUV, $\sim$ 128 nm) scintillation light. Operating under cryogenic conditions ( $\sim$ 87 K), the LDS is essential for determining the event start time (t0) with nanosecond precision for beam spill synchronization, improving longitudinal spatial resolution, and contributing to the event trigger and cosmic-ray mitigation. The performance of the LDS was investigated addressing both hardware and data analysis aspects. Following a progressive degradation in PMT gain observed during operations at FNAL, systematic gain measurements were first carried out from room temperature down to low temperatures. The results show stable performance at room temperature but a significant, irreversible reduction in gain at low temperatures. Based on these findings, a series of mitigation strategies were implemented in the ICARUS detector to preserve PMT performance and ensure reliable cryogenic operation. Currently, ongoing waveform analysis of the PMT signals is being performed to characterize signal shape, charge integration, and timing properties, aiming to refine and improve the agreement between experimental data and Monte Carlo simulations.

Saia, Clara [U. Catania (main); INAF, Catania; Cat↗

DOE OSTI2025

Side-by-side evaluation of the outgassing rate and ultimate pressure achieved inside tubes made of low-carbon and stainless steel

The hydrogen outgassing rate of a vacuum chamber made of AISI 1020 low-carbon steel was found to be notably smaller than a similar chamber made of AISI 316L stainless steel following heat treatment at 150 °C. When the chambers were baked at 400 °C, the outgassing rate of the 1020 low-carbon steel chamber was approximately 2000 times smaller than that of the 316L stainless-steel chamber. After quantifying hydrogen outgassing rates, vacuum chambers made of 1020 low-carbon steel and 316L stainless steel were heated for a prescribed duration and then pumped with an ion pump and nonevaporable getter pump to reach “ultimate pressure.” The 1020 low-carbon steel chamber reached lower pressures than the 316L stainless-steel chamber. Monte Carlo simulation predictions using the measured hydrogen outgassing rates are in good agreement with the ultimate pressure measurements. Finally, our measurements indicate that low-carbon steel can provide pressures within the extreme high vacuum range (<7.5 × 10 -13 Torr) if care is taken to minimize the surface area contribution from stainless steel.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Entanglement entropy of a color flux tube in (2+1)D Yang-Mills theory

We construct a novel flux tube entanglement entropy (FTE 2 ), defined as the excess entanglement entropy relative to the vacuum of a region of color flux stretching between a heavy quark-anti-quark pair in pure gauge Yang-Mills theory. We show that FTE 2 can be expressed in terms of correlators of Polyakov loops, is manifestly gauge-invariant, and therefore free of the ambiguities in computations of the entanglement entropy in gauge theories related to the choice of the center algebra. Employing the replica trick, we compute FTE 2 for SU(2) Yang-Mills theory in (2+1)D and demonstrate that it is finite in the continuum limit. We explore the properties of FTE 2 for a half-slab geometry, which allows us to vary the width and location of the slab, and the extent to which the slab cross-cuts the color flux tube. Following the intuition provided by computations of FTE 2 in (1+1)D, and in a thin string model, we examine the extent to which our FTE 2 results can be interpreted as the sum of an internal color entropy and a vibrational entropy corresponding to the transverse excitations of the string.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2026

FY26 Progress Report on the Operation of the Activated Materials Laboratory at the Advanced Photon Source as a Nuclear Science User Facilities Partner Facility

The Activated Materials Laboratory (AML), located in the Long Beamline Building of the Advanced Photon Source (APS) at Argonne National Laboratory (ANL), provides a centralized radiological capability for preparing, handling, and supporting synchrotron experiments on activated materials. As a Nuclear Science User Facilities (NSUF) partner facility, the AML enables the receipt of radioactive shipments, open-form sample handling, encapsulation, transport of specimens to and from APS beamlines, and experimentation with dedicated equipment. The partnership includes the APS 1-ID and 20-ID beamlines, which provide high-energy x-ray scattering, tomography, and diffraction microscopy techniques for ex-situ, in-situ, and grainresolved three-dimensional (3D) characterization. All samples supported through the AML and partner beamlines must meet radiological limits of less than 100 mrem/h at 30 cm. This FY26 progress report summarizes the first full year of AML operations as an NSUF partner facility and highlights progress in both user support and capability development. By the end of FY26, the AML had received a total of 10 NSUF-awarded projects, including 2 Consolidated Innovative Nuclear Research (CINR), 7 Rapid Turnaround Experiment (RTE), and 1 Super RTE projects. Beamtime was fully delivered for 3 RTE projects and partially delivered for 1 CINR project, demonstrating successful workflows for receipt, encapsulation, beamline transfer, and radiological experiment execution. These efforts demonstrated safe radiological experiments at APS beamlines for samples with dose rates above the historical 5 mrem/h threshold and now up to 100 mrem/h at 30 cm, marking an important milestone for neutron-irradiated materials research. During FY26, the AML also expanded its experimental capabilities. A Psylotech xTS load frame with in-grip rotation was deployed for room-temperature mechanical testing with threedimensional x-ray characterization during interrupted loading. A Linkam TS1500V vacuum heater was commissioned for thermally driven studies, and temperature calibration experiments were performed to establish specimen-relevant thermal profiles. In parallel, a customized split-tube furnace for high-temperature mechanical testing entered commissioning, and initial work began on robotic sample handling to reduce worker dose and improve operational efficiency. Data workflow and beamline operations continued to mature. Together, these developments demonstrate that the AML is becoming a unique national resource for safe, efficient, and scientifically advanced characterization of activated materials at the APS.

Zhang, Xuan↗

DOE OSTI2024

Realization of a gas puff imaging system on the Wendelstein 7-X stellarator

A system for studying the spatiotemporal dynamics of fluctuations in the boundary of the W7-X plasma using the “Gas-Puff Imaging” (GPI) technique has been designed, constructed, installed, and operated. This GPI system addresses a number of challenges specific to long-pulse superconducting devices, such as W7-X, including the long distance between the plasma and the vacuum vessel wall, the long distance between the plasma and diagnostic ports, the range of last closed flux surface (LCFS) locations for different magnetic configurations in W7-X, and management of heat loads on the system’s plasma-facing components. The system features a pair of “converging–diverging” nozzles for partially collimating the gas puffed locally ≈135 mm radially outboard of the plasma boundary, a pop-up turning mirror for viewing the gas puff emission from the side (which also acts as a shutter for the re-entrant vacuum window), and a high-throughput optical system that collects visible emission resulting from the interaction between the puffed gas and the plasma and directs it along a water-cooled re-entrant tube directly onto the 8 × 16 pixel detector array of the fast camera. The DEGAS 2 neutral code was used to simulate the Hα (656 nm) and HeI (587 nm) line emission expected from well-characterized gas-puffs of H2 and He and excited within typical edge plasma profiles in W7-X, thereby predicting line brightnesses used to reduce the risks associated with system sensitivity and placement of the field of view. Operation of GPI on W7-X shows excellent signal-to-noise ratios (>100 at 2 Mframes/s) over the field of view for minimally perturbing gas puffs. The GPI system provides detailed measurements of the two-dimensional (radial and poloidal) dynamics of plasma fluctuations in the W7-X edge and scrape-off layer and in and around the magnetic islands outside the LCFS that make up the island divertor configuration employed on W7-X.

Instruments & Instrumentation↗

DOE OSTI2025

Experimental studies of high-temperature thermal dissociation of iso-propanol

iso-Propanol is an important biofuel for transportation and industrial applications, as well as a canonical compound for understanding alcohol reaction chemistry. There are few studies of the thermal decomposition of iso-propanol, hence the objective of this work was to investigate the mechanism of iso-propanol pyrolysis at high-temperature conditions. Shock tube studies were used to investigate the thermal decomposition of iso-propanol and identify the dominant reaction pathways and the intermediate species present in the temperature range of 1395–2053 K and at pressures of 0.3–4 bar. Time-resolved mass spectra were obtained from pyrolysis experiments in shock tube/mass spectrometer apparatuses using electron impact ionization or synchrotron vacuum ultraviolet photoionization. The relative concentrations of the intermediate species were used to identify the dominant reaction channels. iso-Propanol dissociated by water loss to propene and by elimination of a methyl radical yielding a 1-hydroxyethyl radical at the conditions studied. The time-dependent mass spectra allowed secondary products to be identified along with the orders of appearance. Notably, peaks at m/z = 50 and 52 were observed providing insight into secondary reactions. These features have not been previously reported in propanol-pyrolysis literature.

Kim, John H. [Univ. of Michigan, Ann Arbor, MI (Un↗

DOE OSTI2025

Design, Prototyping, Fabrication and Test of the Mu2e AC-Dipole Magnet

Fermilab Magnet Systems is building three High-frequency AC-Dipole Magnets for the Mu2e experiment at Fermilab. These magnets are composed of three single-loop one-meter-long ferrite loaded segments. The excitation consists of a copper tube, which is also used as a means for its cooling with the inherited challenges of Voltage and Frequency uncoupling. These magnets are designed to operate either at 300 kHz or 4.4 MHz via resonance tuning. Following the completion of the design, prototyping, and analysis phases, all ferrites and other components were procured, and the magnets were fabricated to meet vacuum compatibility requirements using strict procedures. Each magnet undergoes thorough baking and testing before being installed in the experiment. This paper discusses the magnets' role in proton background suppression as it represents a unique and essential device for the Mu2e experiment at Fermilab. We present the magnet design choices, modeling approach, and the challenges encountered during fabrication. Additionally, we outline the power supply driving mechanism, the testing that was performed, and our results.

Elementi, Luciano [Fermilab] (ORCID:00000002766372↗

DOE OSTI2026

Determining Optimal Magnetometer Configuration on MAGIS-100

Long-baseline atom interferometers such as the Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100) require stringent control and continuous characterization of background magnetic fields and spatial gradients to prevent systemic phase shifts that mimic ultralight dark matter or gravitational wave signatures. Because direct sensor placement within the ultra-high vacuum beam pipe is infeasible, in-situ magnetic field monitoring relies on external sensor arrays situated in the surrounding annular region. This work demonstrates a field reconstruction framework for a 5.3-meter MAGIS-100 modular section using finite-element Opera simulations. Transverse magnetic fields are expanded using a cylindrical multipole framework as informed by Fermilab’s Muon g-2 experiment, with magnetometer array configurations optimized via Fisher information matrix D-optimality. Inverting external sensor readings through a Gauss-Newton scheme recovers interior tube fields across distinct axial positions. In the discontinuity-averse uniform region (slice pair P4), the model achieves sub-noise-floor performance with a cross-validated root-mean-square error (RMSE) of $6.7227 \times 10^{-4}\text{ A/m}$ ($0.845\times$ sensor noise floor) and an interior field coefficient of variation of $1.71\%$. An elbow criterion in the Fisher bounds establishes $n_{\text{max}} = 2$ as the optimal multipole truncation order to prevent noise amplification from over-parameterization, with $n_{\text{max}} = 3$ (sextupole) order chosen for analysis to demonstrate further complexity and cross-pair comparison. Furthermore, analytical differentiation of the fitted multipole coefficients yields dense spatial maps of the transverse Jacobian gradient matrix $\nabla \mathbf{H}$ along with propagated $1\sigma$ uncertainty bounds across the beam region ($r \le 2.75\text{ in}$). This operational framework confirms that external magnetometer arrays can reliably monitor magnetic field uniformity and spatial gradients along the 100-meter flight path given appropriate sampling for any complexity order.

Appleby, Darwin [William Rainey Harper Coll.] (ORC↗

DOE OSTI2026

Determining Optimal Magnetometer Configuration on MAGIS-100

Long-baseline atom interferometers such as the Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100) require stringent control and continuous characterization of background magnetic fields and spatial gradients to prevent systemic phase shifts that mimic ultralight dark matter or gravitational wave signatures. Because direct sensor placement within the ultra-high vacuum beam pipe is infeasible, in-situ magnetic field monitoring relies on external sensor arrays situated in the surrounding annular region. This work demonstrates a field reconstruction framework for a 5.3-meter MAGIS-100 modular section using finite-element Opera simulations. Transverse magnetic fields are expanded using a cylindrical multipole framework as informed by Fermilab’s Muon g-2 experiment, with magnetometer array configurations optimized via Fisher information matrix D-optimality. Inverting external sensor readings through a Gauss-Newton scheme recovers interior tube fields across distinct axial positions. In the discontinuity-averse uniform region (slice pair P4), the model achieves sub-noise-floor performance with a cross-validated root-mean-square error (RMSE) of $6.7227 \times 10^{-4}\text{ A/m}$ ($0.845\times$ sensor noise floor) and an interior field coefficient of variation of $1.71\%$. An elbow criterion in the Fisher bounds establishes $n_{\text{max}} = 2$ as the optimal multipole truncation order to prevent noise amplification from over-parameterization, with $n_{\text{max}} = 3$ (sextupole) order chosen for analysis to demonstrate further complexity and cross-pair comparison. Furthermore, analytical differentiation of the fitted multipole coefficients yields dense spatial maps of the transverse Jacobian gradient matrix $\nabla \mathbf{H}$ along with propagated $1\sigma$ uncertainty bounds across the beam region ($r \le 2.75\text{ in}$). This operational framework confirms that external magnetometer arrays can reliably monitor magnetic field uniformity and spatial gradients along the 100-meter flight path given appropriate sampling for any complexity order.

Appleby, Darwin [William Rainey Harper Coll.] (ORC↗

DOE OSTI2026

In-Situ Magnetic Field Reconstruction in the MAGIS-100 Experiment

Long-baseline atom interferometers such as the Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100) require stringent control and continuous characterization of background magnetic fields and spatial gradients to prevent systemic phase shifts that mimic ultralight dark matter or gravitational wave signatures. Because direct sensor placement within the ultra-high vacuum beam pipe is infeasible, in-situ magnetic field monitoring relies on external sensor arrays situated in the surrounding annular region. This work demonstrates a field reconstruction framework for a 5.3-meter MAGIS-100 modular section using finite-element Opera simulations. Transverse magnetic fields are expanded using a cylindrical multipole framework as informed by Fermilab’s Muon g-2 experiment, with magnetometer array configurations optimized via Fisher information matrix D-optimality. Inverting external sensor readings through a Gauss-Newton scheme recovers interior tube fields across distinct axial positions. In the discontinuity-averse uniform region (slice pair P4), the model achieves sub-noise-floor performance with a cross-validated root-mean-square error (RMSE) of $6.7227 \times 10^{-4}\text{ A/m}$ ($0.845\times$ sensor noise floor) and an interior field coefficient of variation of $1.71\%$. An elbow criterion in the Fisher bounds establishes $n_{\text{max}} = 2$ as the optimal multipole truncation order to prevent noise amplification from over-parameterization, with $n_{\text{max}} = 3$ (sextupole) order chosen for analysis to demonstrate further complexity and cross-pair comparison. Furthermore, analytical differentiation of the fitted multipole coefficients yields dense spatial maps of the transverse Jacobian gradient matrix $\nabla \mathbf{H}$ along with propagated $1\sigma$ uncertainty bounds across the beam region ($r \le 2.75\text{ in}$). This operational framework confirms that external magnetometer arrays can reliably monitor magnetic field uniformity and spatial gradients along the 100-meter flight path given appropriate sampling for any complexity order.

Appleby, Darwin [William Rainey Harper Coll.; Ferm↗