Correcting the Energy-Dependent Asymmetry in Low-Energy $\mu$SR

This paper presents updated energy-dependent asymmetry calibrations and a simulation-based correction factor for beam-sample overlap to improve the accuracy of quantitative depth-resolved analysis in low-energy $\mu$SR experiments following recent beamline upgrades at PSI.

The Gist

Imagine you are trying to take a high-resolution photograph of a tiny, delicate painting (a material's surface) using a very specific kind of camera: a muon camera.

In this experiment, scientists shoot tiny particles called muons (which are like heavy, unstable cousins of electrons) at a material. These muons act like microscopic spies. When they land, they spin like tiny tops. By watching how they spin and when they stop spinning, scientists can learn about the magnetic and electronic secrets of the material's surface, layer by layer, from the very top down to a few hundred nanometers deep.

However, there's a problem. The "camera" isn't perfect, and the "light" (the muon beam) changes depending on how hard you shoot it. If you don't calibrate the camera correctly, your photos will be blurry, too bright, or too dark, and you might think you see a monster in the shadows when it's just a trick of the light.

This paper is essentially a user manual update for this muon camera. The team at the Paul Scherrer Institute in Switzerland fixed the camera, changed some lenses, and realized their old instructions for taking perfect photos were now outdated. They wrote a new guide to ensure everyone gets accurate results.

Here is the breakdown of their new "camera manual" using everyday analogies:

1. The Problem: The "Shooting Star" Effect

In the past, scientists thought the brightness of their muon signal (called asymmetry) was a fixed number, like the brightness of a lightbulb. But they discovered it's actually more like a dimmer switch that changes based on how fast the muons are traveling.

• The Issue: If you shoot muons slowly (low energy), many of them bounce off the surface or get lost before they even hit the target. If you shoot them fast, they penetrate deeper. The "brightness" of the signal changes with the speed, making it hard to compare deep layers with shallow layers.

2. The Solution: Two New "Test Targets"

To fix the dimmer switch, the scientists used two special reference targets, like calibrating a scale with known weights.

• The Silver Target (The "Perfect Mirror"): They shot muons at a piece of silver. Silver is non-magnetic, so it shouldn't change the muons' spin. If the signal gets weaker here, they know it's not the silver's fault—it's the machine losing muons along the way (like muons bouncing off the walls or getting neutralized). This helps them map out exactly how much signal is lost at different speeds.
• The Nickel Target (The "Noise Filter"): They also used a piece of nickel. Nickel is magnetic, but so strongly that it instantly stops the muons from spinning (making the signal disappear). However, some muons bounce off the nickel plate and land in the surrounding equipment, creating a "ghost signal" (noise). By measuring this ghost signal, they can subtract it from their real data to clean up the picture.

3. The "Missing Pieces" Problem: The Beam vs. The Sample

Imagine trying to spray paint a small sticker on a wall using a giant, wide spray nozzle.

• The Issue: If your sticker (the sample) is small (like a 10x10 mm square) and the spray nozzle (the muon beam) is wide, a lot of paint misses the sticker and hits the wall behind it. In the experiment, muons that miss the sample land on the metal plate behind it and don't contribute to the data. This makes the signal look weaker than it really is.
• The Fix: The team used a super-computer simulation (like a video game engine) to map exactly where the muons land for different sample sizes. They created a "Correction Factor" (a mathematical recipe) that tells you: "If you are using a small sample, multiply your result by 1.2 to account for the muons that missed."

4. The "Static Electricity" Glitch

There was one more weird glitch. When the muons hit the sample, they sometimes knock off electrons (like static shock). If the sample is charged just right, these electrons fly backward and trick the detector into thinking a muon arrived when it didn't.

• The Fix: They identified exactly when this happens (at very high speeds/energies) and marked those data points as "unreliable" for now, so scientists know to be careful when interpreting results in that specific range.

The Bottom Line: Why This Matters

Before this paper, if you tried to measure the magnetic properties of a thin film using muons, you might have gotten the wrong answer because your "camera" wasn't calibrated for the new lens they installed.

With these new updates:

1. Accuracy: Scientists can now trust their numbers, whether they are looking at the very surface or 200 nanometers deep.
2. Small Samples: They can finally measure tiny samples (like microchips) accurately by applying the "missing pieces" correction.
3. Future Proof: This new guide ensures that as the machine gets upgraded again, the data remains reliable.

In short: The scientists took a complex, finicky scientific instrument, figured out why it was giving inconsistent readings, and wrote a new set of instructions (calibrations and corrections) so that everyone can take clear, sharp "photos" of the microscopic world.

Technical Summary

1. Problem Statement

Low-energy muon spin rotation (LE-µSR) is a powerful technique for depth-resolved studies of magnetic and electronic properties in materials, offering nanometer-scale resolution from the surface to several hundred nanometers. However, a critical limitation exists: the measured transverse-field asymmetry ($A_0$) in LE-µSR is not an intrinsic constant of the spectrometer, unlike in bulk µSR experiments.

Instead, the observable asymmetry is highly sensitive to:

• Implantation energy and beam transport settings.
• Beamline conditions, specifically changes following the 2023 upgrade of the single-muon tagging system at the LEM beamline (Paul Scherrer Institute), which included a new 2 nm carbon foil.
• Geometric factors, such as the overlap between the muon beam spot and the sample size.

Previous calibration methods were insufficient for these new conditions. Without accurate correction, the extraction of quantitative depth-resolved parameters—such as magnetic volume fractions ($V_M$) or diamagnetic fractions ($f_{dia}$)—is compromised, leading to systematic errors in determining material properties.

2. Methodology

The authors developed a comprehensive calibration and correction framework consisting of three main components:

A. Experimental Reference Measurements

To isolate specific instrumental effects, the team performed reference measurements using two specific materials:

1. Silver (Ag) Reference: Used to determine the maximum achievable asymmetry ($A_{Ag}$). Since Ag is non-magnetic with negligible depolarization, deviations in $A_{Ag}$ from a constant value directly reflect instrumental losses (e.g., muon backscattering and neutralization in the carbon foil). Measurements were taken at transport settings of 10, 12, 13.5, and 15 keV.
2. Nickel (Ni) Reference: Used to quantify spurious signals from reflected muons ($A_{Ni}$). Muons stopping in the ferromagnetic Ni layer depolarize too quickly to be detected. Therefore, any observed asymmetry in the Ni measurement originates from muons reflected by the sample plate that stop in the surrounding radiation shield.

B. Simulation-Based Overlap Correction

To address the systematic reduction in asymmetry caused by incomplete beam–sample overlap (where muons miss the sample and hit the non-magnetic sample plate), the authors utilized musrSim/Geant4 simulations.

• An updated electrostatic field map of the sample environment was generated using SIMION and implemented in the simulation.
• The simulations modeled the evolution of the muon beam spot as a function of transport voltage and sample bias.
• An overlap factor, $O(E)$, was calculated for various sample sizes (from $5\times5$ mm$^2$ to $30\times30$ mm$^2$) and implantation energies.

C. Benchmarking

The simulation results were benchmarked against experimental data using Strontium Titanate (SrTiO$_3$ or STO) samples of varying lateral dimensions ($10\times10$, $15\times15$, and $30\times30$ mm$^2$). At 200 K, STO is fully diamagnetic ($f_{dia} = 100\%$), meaning any deviation in measured asymmetry from the theoretical maximum is purely instrumental.

3. Key Contributions

The paper introduces a robust, updated framework for LE-µSR data analysis under current beam conditions:

1. Updated Energy-Dependent Asymmetry Calibration:

   • Established a phenomenological model for the maximum asymmetry using Silver: $A_{Ag}(E) = a_{Ag} + b_{Ag} \ln(E/1\text{keV})$.
   • Quantified the spurious contribution from reflected muons using Nickel: $A_{Ni}(E) = a_{Ni} + b_{Ni} \exp(-E/E_0)$.

2. Sample-Size-Dependent Overlap Correction:

   • Introduced a correction factor $O(E)$ derived from Geant4 simulations to account for muons missing the sample.
   • Provided a parameterized empirical function for $O(E)$ that allows for interpolation across different implantation energies and sample geometries.

3. Comprehensive Correction Formula:
   The authors derived a unified equation to calculate the true diamagnetic fraction ($f_{dia}$) at a given energy $E$:
   $$f_{dia}(E) = \frac{A_{STO}(E) - A_{Ni}(E)}{O(E) \cdot [A_{Ag}(E) - A_{Ni}(E)]}$$
   This formula corrects for background, reflection, and geometric overlap simultaneously.

4. Key Results

• Transport Dependence: The maximum observable asymmetry is strongly dependent on the transport setting (voltage). For example, at 15 keV transport, the asymmetry saturates near 0.24, whereas at 10 keV, it saturates near 0.20. This is primarily due to muon neutralization in the carbon foil, which is more prevalent at lower energies.
• Reflection Effects: Reflected muons contribute a spurious signal that significantly increases the measured asymmetry at very low implantation energies (typically below ~2 keV).
• Overlap Impact: For small samples (e.g., $10\times10$ mm$^2$), a significant fraction of muons misses the sample, leading to a substantial reduction in measured asymmetry. The simulation accurately reproduces the experimental overlap factors for samples larger than $10\times10$ mm$^2$.
• Limitations:
  • Secondary Electron Emission: At high sample biases (below -3.6 kV), secondary electrons can generate spurious triggers, artificially reducing asymmetry. This effect is sample-specific and not included in the universal correction framework.
  • Small Samples: While the correction works for small samples, the associated uncertainty increases markedly as the overlap decreases.

5. Significance and Recommendations

• Quantitative Reliability: This work enables reliable, quantitative extraction of depth-resolved magnetic volume fractions and muonium formation fractions, which was previously hindered by uncorrected energy-dependent asymmetry variations.
• Sample Size Guidelines: The study strongly recommends using sample sizes of $20\times20$ mm$^2$ or larger for standard LE-µSR studies. While smaller samples can be analyzed using the new correction, they suffer from higher statistical uncertainty and require extremely precise sample mounting (lateral positioning accuracy < 1 mm).
• Future-Proofing: The framework is modular and can be adapted for future beamline upgrades, changes in sample environments, or the inclusion of new correction factors (e.g., for secondary electron emission).

In conclusion, this paper provides the essential "standard model" for correcting LE-µSR data at the PSI LEM beamline, transforming the technique from a qualitative tool into a rigorously quantitative method for surface and near-surface physics.