🔬 materials science

Ion Jump Motion as the Background for Muon Diffusion in Battery Materials Research Using $μ$ SR

This paper demonstrates that anomalous features in conventional $\mu$ SR analyses of ion diffusion arise from the temperature-dependent ratio between ion and muon jump rates, suggesting that reanalyzing data with an extended Kubo-Toyabe function incorporating Edwards-Anderson autocorrelation is essential for accurately evaluating ion dynamics.

Original authors: Ryosuke Kadono

Published 2026-02-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ryosuke Kadono

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Case of Mistaken Identity

Imagine you are trying to watch a busy dance floor to see how fast the dancers (ions) are moving. You have a special camera (Muon Spin Relaxation, or µSR) that takes snapshots of the room.

For the last 15 years, scientists have been using this camera to study battery materials. They wanted to know: How fast are the Lithium or Sodium ions hopping around inside the battery? This speed is crucial for making better, faster-charging batteries.

However, the author of this paper, Ryosuke Kadono, argues that for many years, scientists have been looking at the wrong dancer. They thought they were watching the ions, but they were actually watching the camera itself (the muon) moving around, or they were misinterpreting the blur.

The Characters in Our Story

The Ions (The Dancers): These are the Lithium or Sodium atoms inside the battery. They are heavy and move slowly, hopping from one spot to another only when they get enough heat energy.
The Muon (The Camera/Observer): This is a tiny particle scientists shoot into the material. It acts like a spy. It stops in a spot and "watches" the magnetic fields of the surrounding ions.
The Old Map (The Kubo-Toyabe Function): This is the mathematical formula scientists used for decades to translate the camera's blurry photos into a speed for the ions. It assumed the camera (muon) stayed perfectly still while the dancers (ions) moved.

The Problem: The Camera is Moving Too!

The paper points out a flaw in the old map. The old formula assumed the Muon was a statue, standing perfectly still. But in reality, the Muon is also a dancer! It hops around, too, just like the ions.

The Old View: "The picture is blurry because the ions are moving fast."
The New Reality: "The picture is blurry because both the ions and the Muon are moving, and they are moving at different speeds."

The "Anomaly" (The Weird Peak)

When scientists analyzed their data using the old map, they saw something strange that didn't make sense:

As they heated up the battery, the ions should move faster and faster, like a car accelerating on a highway.
But the data showed a weird peak: The calculated speed would go up, hit a maximum at a specific temperature, and then suddenly drop.

The Analogy: Imagine you are watching a race.

Expected: The runner gets faster as the race goes on.
Observed: The runner speeds up, hits a peak, and then suddenly seems to slow down, even though the race is getting hotter.

Scientists were confused. They couldn't explain why the ions would suddenly slow down when it got hotter.

The Solution: The "Extended" Map

Kadono created a new, more accurate map (the Extended Kubo-Toyabe function). This new map accounts for the fact that the Muon is also jumping around.

He ran a computer simulation (a virtual experiment) where:

The Ions move slowly at first, then speed up massively as it gets hot.
The Muon moves very slowly at first, but speeds up differently.

The "Aha!" Moment:
When Kadono fed this realistic simulation into the old map (the one everyone was using), the old map produced that exact weird peak and the sudden drop in speed that scientists had been seeing in real life!

The Metaphor:
Think of it like a traffic jam.

Low Speed: Cars (ions) are moving slowly. The traffic looks chaotic.
Medium Speed: Cars speed up. The chaos peaks.
High Speed: Cars are zooming so fast that they actually look like a smooth, steady stream (this is called "motional narrowing").

The old map didn't know how to handle the "smooth stream" phase. It thought the cars had stopped or slowed down. The new map realizes that the "slowing down" was actually just the cars moving too fast to see individually.

What This Means for Battery Research

The Peak is a Good Sign: If you see that weird peak and the drop in the data, it actually means ions are moving! It's a signature that the ions are hopping around.
No Peak? No Movement: If a study shows a smooth, boring line with no peak, it likely means the ions aren't moving at all. The scientists were just watching the Muon (the camera) move around, not the battery ions.
Re-evaluating the Past: Many past studies claimed to measure how fast ions move. This paper suggests those numbers were wrong because they used the old map. We need to re-analyze all that old data with the new map to get the real speeds.

The "Hidden Hydrogen" Twist

There is one more interesting side note. The paper suggests that in many of these battery materials, the Muon might actually be acting like a fake Hydrogen atom.

Since Hydrogen is hard to find in materials, scientists use Muons as a stand-in. The paper suggests that even if we can't measure the battery ions perfectly yet, these past studies might have actually told us a lot about how "hidden" Hydrogen behaves in these materials, which is also very important for science.

Summary

The Mistake: Scientists used a formula that assumed the observer (Muon) was frozen in place.
The Reality: The observer was moving too, creating a confusing blur in the data.
The Result: The "weird peaks" in old data weren't errors; they were the specific signature of ions finally starting to move fast.
The Takeaway: We need to redo the math on old battery data to get the true speed of the ions. If the data looks "boring" (no peaks), it probably means the ions are stuck, and we were just watching the Muon dance.

1. Problem Statement

Muon Spin Rotation/Relaxation (µSR) has been widely used over the past 15 years to investigate ion diffusion in battery materials (e.g., Li-ion and Na-ion conductors). The standard analysis relies on the Kubo-Toyabe (KT) function, $G_{KT}^z(t; \Delta_{KT}, \nu_{KT})$ , to interpret the relaxation of muon spins caused by fluctuating local magnetic fields from diffusing ions. In this conventional framework:

$\nu_{KT}$ is interpreted as the ion jump frequency ( $\nu_i$ ).
$\Delta_{KT}$ represents the static linewidth.

The Anomaly: Previous studies on various battery materials frequently reported an anomalous behavior where $\nu_{KT}$ exhibits a sharp peak at a specific temperature $T^*$ , accompanied by a significant decrease (or "kink") in $\Delta_{KT}$ at temperatures above $T^*$ . This contradicts the physical expectation for thermally activated diffusion, where the jump rate should increase monotonically with temperature. Despite these inconsistencies, no satisfactory explanation was previously offered, and the data were often forced into the conventional KT model to extract activation energies.

2. Methodology

The author proposes that the anomaly arises because the conventional KT function assumes the muon is immobile and that the entire local field fluctuates (a "strong collision" model where $Q=1$ ). However, in reality, only a fraction of ions surrounding the muon jump, while the muon itself may also diffuse.

To address this, the study employs the following methodology:

Extended Model: The author utilizes an extended Kubo-Toyabe function, $G_{EA}^z(t)$ $G_{E A}^{z} (t)$ , which incorporates an Edwards-Anderson type autocorrelation function. This model accounts for:
- Ion Diffusion ( $\nu_i$ ): Only a portion of the local field fluctuates, characterized by a parameter $Q$ ( $0 < Q < 1$ ).
- Muon Diffusion ( $\nu_\mu$ ): The muon itself jumps, causing the total amplitude of the field to fluctuate.
- The autocorrelation function is defined as: $C(t) \approx [(1-Q) + Qe^{-\nu_i t}]e^{-\nu_\mu t}$ .
Numerical Simulation:
- Monte Carlo simulations were performed to generate theoretical µSR time spectra ( $G_{EA}^z(t)$ ) over a temperature range of 10–400 K.
- Parameters were set based on physical expectations for battery materials:
  - Ion jump: $\nu_i = \nu_{i0} \exp(-E_i/k_B T)$ with $E_i \approx 0.35$ eV.
  - Muon jump: $\nu_\mu = \nu_{\mu0} \exp(-E_\mu/k_B T)$ with $E_\mu \approx 0.02$ eV (phonon-assisted tunneling).
  - Static linewidth $\Delta = 0.3$ MHz and $Q = 0.8$ .
Re-analysis Procedure: The simulated spectra were analyzed using the conventional KT function (mimicking the standard literature procedure) to extract "apparent" parameters $\Delta_{KT}$ and $\nu_{KT}$ . This allows for a direct comparison between the "true" input physics and the "observed" experimental artifacts.

3. Key Contributions

Identification of the Origin of Anomalies: The paper demonstrates that the anomalous peak in $\nu_{KT}$ and the subsequent drop in $\Delta_{KT}$ are artifacts caused by the sharp increase in the ion jump rate ( $\nu_i$ ) relative to the muon jump rate ( $\nu_\mu$ ) as temperature rises.
Mechanism of Crossover: The peak occurs at a characteristic temperature $T^*$ where the ion jump rate $\nu_i$ crosses a threshold (approximately $3\sqrt{Q}\Delta$ ). Below $T^*$ , the system exhibits "motional broadening"; above $T^*$ , it transitions to "motional narrowing" for the fluctuating component, while the non-fluctuating component ( $1-Q$ ) dominates, leading to a reduced effective linewidth.
Critique of Previous Data: The study reveals that in many cases where these anomalies are not observed, the µSR signal likely reflects muon self-diffusion rather than ion diffusion, meaning previous claims of observing ion dynamics in those specific datasets may be incorrect.
New Analysis Protocol: The author advocates for re-analyzing historical data using the extended $G_{EA}^z(t)$ function to properly decouple ion and muon diffusion rates.

4. Key Results

Reproduction of Anomalies: The simulation successfully reproduced the characteristic peak in $\nu_{KT}$ and the drop in $\Delta_{KT}$ observed in experimental literature (e.g., in $Li_xCoO_2$ , $LiFePO_4$ , etc.) solely by applying the conventional KT fit to data generated by the extended model.
Temperature Dependence:
- $\nu_{KT}$ : Increases with temperature, peaks at $T^*$ (where $\nu_i \approx 3\sqrt{Q}\Delta$ ), and then decreases monotonically.
- $\Delta_{KT}$ : Remains relatively constant at low $T$ , then drops sharply above $T^*$ , asymptotically approaching $\sqrt{1-Q}\Delta$ .
Validity of Activation Energy:
- Extracting activation energy ( $E_i$ ) from the rising part of $\nu_{KT}$ (below $T^*$ ) is unreliable because this region is dominated by muon diffusion ( $\nu_\mu$ ), not ion diffusion.
- However, the temperature dependence of the linewidth drop ( $\Delta_{KT}$ ) can be fitted to a two-state transition model to provide a rough estimate of the ion activation energy ( $E_i \approx 0.4$ eV in the simulation, close to the input 0.35 eV).
Role of Parameter $Q$ : The magnitude of the anomaly depends heavily on $Q$ . If $Q$ is small (e.g., in materials with few magnetic ions near the muon), the anomaly is suppressed, and µSR becomes insensitive to ion diffusion, potentially leading to false negatives.

5. Significance and Implications

Paradigm Shift in Interpretation: The paper suggests that a significant portion of µSR literature on battery materials, previously interpreted as evidence of ion diffusion, may actually be describing muon self-diffusion.
Corrective Action: It calls for a systematic re-evaluation of "vintage" data using the extended model to correctly extract ion jump rates ( $\nu_i$ ) and distinguish them from muon dynamics ( $\nu_\mu$ ).
Hidden Hydrogen: The author notes that even if ion diffusion was not detected, these studies are not in vain. Since muons act as "pseudo-hydrogen," the observed muon diffusion data provides valuable insights into the behavior of trace hydrogen impurities in battery oxides, which significantly influence material properties.
Future Directions: The work establishes a rigorous framework for analyzing ion dynamics in non-magnetic materials, ensuring that future µSR studies on battery materials accurately reflect ion transport mechanisms rather than muon artifacts.