Simulation and optimization of the Active Magnetic Shield of the n2EDM experiment

This paper presents a high-accuracy finite element simulation of the n2EDM experiment's Active Magnetic Shield, demonstrating its utility in optimizing the placement and number of feedback sensors via genetic algorithms to ensure magnetic stability within the magnetically shielded room.

The Gist

The Big Picture: Keeping a Neutron Calm

Imagine you are trying to balance a very delicate, spinning top (the neutron) on a table. If the room shakes, or if a giant fan turns on nearby, the top wobbles and falls. Scientists want to study this top to see if it has a tiny, hidden "tilt" (called an electric dipole moment) that could explain secrets about the universe.

To do this, they need the room to be perfectly still and the magnetic "wind" to be perfectly calm. The n2EDM experiment at the Paul Scherrer Institute is this high-stakes room.

The Problem: A Noisy Neighborhood

The experiment is located in a busy scientific neighborhood. Nearby, there are massive superconducting magnets (like the SULTAN and COMET machines) that act like giant electromagnets. When these machines ramp up or down, they create huge magnetic "storms" that would completely ruin the delicate measurement of the neutron.

The Solution: A Double-Layer Defense

To keep the room calm, the scientists built a two-part defense system:

1. The Passive Shield (The Fortress): They built a special room called a Magnetically Shielded Room (MSR). Think of this as a fortress made of seven layers of a super-magnetic metal called mu-metal. It acts like a thick, heavy blanket that absorbs most of the magnetic noise coming from the outside world.
2. The Active Shield (The Noise-Canceling Headphones): Even the best blanket has tiny leaks. To fix this, they added an Active Magnetic Shield (AMS).
   • How it works: Imagine the MSR is surrounded by eight giant, invisible "magnetic hands" (coils).
   • The Sensors: Small devices called fluxgates (like tiny magnetic ears) are placed around the room. They listen to the magnetic noise.
   • The Feedback Loop: When the "ears" hear a disturbance (like a nearby magnet ramping up), a computer instantly tells the "hands" to push back with an equal and opposite magnetic force. It's exactly like noise-canceling headphones: they hear the outside noise and generate a "anti-noise" to cancel it out perfectly.

The Challenge: The Shield Changes the Sound

The scientists realized that the "fortress" (the mu-metal room) doesn't just block noise; it also distorts it.

• The Analogy: Imagine shouting into a cave. The walls of the cave bounce the sound around, making it echo louder in the corners and quieter in the middle.
• The Reality: The mu-metal walls of the MSR bend the magnetic fields. This means the magnetic "noise" isn't uniform; it gets amplified at the corners of the room. If the scientists just guessed where to put their "ears" (sensors), they might miss the loudest spots or try to cancel a noise that isn't actually there.

The Simulation: A Virtual Twin

To solve this, the team built a digital twin of their entire experiment using computer software (COMSOL).

• They created a virtual version of the fortress and the eight magnetic hands.
• They tested how the "hands" would push back against the "noise" while the "fortress" distorted the waves.
• The Result: The computer simulation matched their real-world experiments almost perfectly. This proved that their math was right and that the system behaves in a predictable, linear way (like a simple volume knob: turn it up, the sound gets louder; turn it down, it gets quieter).

The Optimization: Finding the Perfect Spot

Once they had a working digital twin, they asked: "Where is the absolute best place to put our magnetic ears?"

• The Old Way: They used a standard algorithm to guess the positions.
• The New Way: They used Genetic Algorithms. Think of this as "digital evolution."
  • The computer created thousands of random arrangements of sensors.
  • It tested which arrangements worked best at canceling noise.
  • It kept the "fittest" arrangements (the ones that canceled noise best) and mixed them together to create even better generations.
  • The Goal: They wanted to minimize the "condition number." In plain English, this is a score that tells you how stable and easy-to-control the system is. A lower score means the system is less likely to get confused or unstable.

The Outcome:
The genetic algorithm found a new arrangement of sensors that was mathematically superior. However, the perfect spot was physically impossible to build (there wasn't enough space). So, the scientists picked the best possible spot that fit in the real room.

• They moved the sensors to these new spots.
• The system worked exactly as the computer predicted. The "condition number" improved, meaning the system is now more stable and better at canceling out the magnetic storms.

Summary

The paper describes how scientists built a high-tech "noise-canceling" system for a neutron experiment. They realized the room itself warped the magnetic fields, so they built a super-accurate computer simulation to understand the distortion. Using that simulation and a "digital evolution" algorithm, they figured out the perfect places to put their sensors to ensure the system remains stable and can successfully cancel out massive magnetic disturbances from nearby machines.

Technical Summary

1. Problem Statement

The n2EDM experiment at the Paul Scherrer Institute (PSI) aims to measure the neutron electric dipole moment (nEDM) with extreme precision. This measurement requires a highly stable and uniform magnetic field within the Ultra-Cold Neutron (UCN) storage chambers.

• Requirements: Temporal magnetic field fluctuations must be kept below 10 pT, and the vertical field gradient across the chambers must be within 0.6 pT cm⁻¹ over a measurement cycle of ~180 seconds.
• Challenges: The experiment is located near facilities with strong, fluctuating magnetic fields, specifically the SULTAN superconducting magnet (frequent ramps) and the COMET cyclotron (sporadic ramps).
• Existing Solution: A passive Magnetically Shielded Room (MSR) consisting of seven layers (six mu-metal, one aluminum) provides high attenuation (shielding factor $10^5$ to $10^8$). However, passive shielding alone is insufficient for low-frequency stability and does not prevent the outermost layer of the MSR from changing magnetization, which propagates inward.
• The Gap: An Active Magnetic Shield (AMS) was installed to actively compensate for external disturbances. However, the interaction between the AMS coils and the high-permeability mu-metal of the MSR distorts the magnetic fields, making the control system complex. A precise simulation was needed to understand these interactions and optimize the feedback control system, specifically the placement of fluxgate sensors.

2. Methodology

The authors developed a comprehensive workflow combining finite element simulation, experimental verification, and algorithmic optimization.

A. System Design

• AMS Architecture: Eight feedback-controlled coils arranged on an irregular grid surrounding the MSR.
  • Coil Types: 3 coils for homogeneous fields (X, Y, Z) and 5 coils for first-order gradients.
  • Capabilities: Designed to compensate static/variable fields up to ±50 µT (homogeneous) and ±5 µT/m (gradients).
• Feedback Mechanism: Uses eight 3-axis fluxgate magnetometers (Sensys FGM3D/125). The system employs an integral control function based on a linear model:
  $$\mathbf{B} = \mathbf{B}_0 + \mathbf{M}\mathbf{I}$$
  Where $\mathbf{B}$ is the measured field, $\mathbf{B}_0$ is the background, $\mathbf{I}$ are coil currents, and $\mathbf{M}$ is the proportionality matrix. The control current is calculated using the Moore-Penrose pseudoinverse ($\mathbf{M}^\dagger$).
• Optimization Metric: The stability of the feedback loop depends on the condition number ($\sigma$) of the matrix $\mathbf{M}^\dagger$. A lower $\sigma$ implies a more stable and well-defined control system.

B. Simulation Implementation (COMSOL)

• Modeling: A full 3D finite element model was created in COMSOL Multiphysics 6.1 using the Python-MPh package.
• Geometry: The AMS grid (approx. $10.3 \times 8.6 \times 9.8$ m) and the MSR were modeled.
• Simplification: The six-layer mu-metal MSR was approximated as a single 5 cm thick wall with an effective relative permeability ($\mu_r$) of 2000 cm⁻¹ (yielding $\mu_{eff} = 10,000$). Simulations confirmed that values above this threshold do not significantly alter the external field behavior.
• Linearity Verification: The simulation confirmed that the system behaves linearly, allowing the magnetic field response to be scaled from a reference current rather than re-simulating for every current value.

C. Optimization Strategy

• Algorithm: Genetic Algorithms (GA) were used to solve a high-dimensional multi-objective optimization problem.
• Objectives:
  1. Minimize the condition number ($\sigma$) of the feedback matrix.
  2. Minimize the average distance of sensors to the MSR corners (where field distortion is highest).
• Parameter Space: The GA optimized the number of sensors ($n$), their 3D positions, and their orientation (azimuthal and polar angles), creating a 6$n$-dimensional search space.

3. Key Results

A. Simulation vs. Experiment

• Accuracy: The COMSOL simulation showed excellent agreement with experimental data.
  • Linearity: The relationship between coil current and magnetic field was confirmed to be linear.
  • Condition Number: The simulated condition number ($\sigma_{sim} = 10.41$) matched the experimental value ($\sigma_{exp} = 10.18$) within 1.37 standard deviations, accounting for sensor positioning uncertainties (±2.5 cm).
• Field Distortion: The simulation visualized how the mu-metal MSR distorts the homogeneous fields generated by the AMS, amplifying the field strength at the corners and edges of the room (up to >10 µT residual fields despite compensation).

B. Optimization Outcomes

• Pareto Front: The GA generated a Pareto front showing the trade-off between the condition number and sensor proximity to the MSR corners.
• Optimal Configuration:
  • The theoretical optimum suggested a condition number of $\sigma \approx 6.3$.
  • Due to physical space constraints around the experiment, a practical configuration was selected with $\sigma = 7.85$.
  • This configuration utilized 8 fluxgates placed closer to the MSR corners than the previous setup.
• Experimental Validation: The new configuration was implemented, yielding an experimental condition number of $\sigma_{exp} = 8.15$, validating the simulation and optimization workflow.

C. Performance

• The AMS successfully suppresses external magnetic disturbances (e.g., SULTAN ramps) down to the level of a few µT in the sub-Hertz frequency range.
• While the new sensor configuration resulted in slightly higher residual fields at the sensor locations (due to being closer to the high-distortion corners), it significantly improved the controllability and stability of the feedback loop.

4. Significance and Conclusion

• System Understanding: The paper provides the first full finite element simulation of the AMS-MSR interaction, proving that the mu-metal distorts fields but preserves the linearity required for control algorithms.
• Methodological Advance: It demonstrates a successful workflow for using Genetic Algorithms to optimize sensor placement in complex magnetic environments, moving beyond trial-and-error commissioning.
• Impact on n2EDM: The optimized AMS ensures the magnetic stability required for the n2EDM experiment to reach its sensitivity goals, minimizing systematic uncertainties related to magnetic field fluctuations.
• Broader Applicability: The developed simulation and optimization framework is transferable to other active magnetic shielding systems in high-precision physics experiments and medical instrumentation.

In summary, the authors successfully bridged the gap between theoretical design and experimental reality for the n2EDM Active Magnetic Shield, using advanced simulation to refine the control system and ensure the stability necessary for next-generation fundamental physics measurements.