Analytically Optimising Muon Diffusion Experiments with Fisher information

TL;DR

This work addresses how to maximize information gained from muon diffusion experiments under limited beamtime by adopting a Fisher-information framework to bound parameter errors. It models the muon asymmetry with a dynamic Kubo-Toyabe function and derives how the Fisher information projects onto diffusion parameters such as the hopping rate ν and field-disorder Δ, enabling calculation of the minimum required muon decays for a desired precision. Across simulated and copper-diffusion data, the study finds that a two-field strategy (zero field plus one optimally chosen longitudinal field) generally minimizes uncertainties, with the optimal field growing with ν and Δ and about half the counts allocated to the fielded condition. The approach provides a fast, analytical planning tool that can guide experiment design and scheduling, though it relies on accurate a priori asymmetry models and does not account for systematic errors; a publicly available codebase accompanies the method.

Abstract

One of the key challenges in performing muon experiments is knowing which temperatures and applied fields to measure at, and how many muon decays should be measured at each temperature/field combination to get the most useful dataset. We have developed a technique using Fisher information which, for a given muon asymmetry function, can analytically calculate the number of muon decays required to obtain a given error on the parameters of the asymmetry model. Here, we report on the results of our project, in particular applying our methodology to the problem of knowing the best choice of applied longitudinal fields for ionic diffusion experiments.

Figures (3)

• Figure 1: The predicted error in $\Delta$ and $\nu$ (both in µ s$^{-1}$) achieved by measuring a series of longitudinally applied fields, visualized as a series of triangular plots where each vertex corresponds to measuring 100% of the muon statistics at the specified longitudinal field strength. Within each triangle, ellipses are placed at discrete positions (with black circles to represent the centre of these) representing a differing proportion of muon decay statistics measured in the the three field environments. Positions closer to a vertex indicates a larger proportion of measurements taken with the corresponding field. The ellipses within the black circles represent the error covariance matrix as described in the text: the size and colour represent the magnitude of the errors (with red ellipses having the largest error, and green having the smallest) and the shape of the ellipse represents the structure of the matrix as described in the text. The smallest ellipse in each triangle is marked in blue with a yellow star overlaid for clarity.
• Figure 2: (a), the relative error in $\nu$ vs applied field strength ($B_{\rm app}$ in the text) plotted for $\nu=0.1$ µ s$^{-1}$ and $\Delta=0.37$ µ s$^{-1}$) (b), heatmaps over a range of $\nu$ and $\Delta$ (both in µ s$^{-1}$) of optimal longitudinal field strength values (in Gauss) and (c) the percentage of the overall muon decay statistics that should be taken at the optimum non-zero applied field.
• Figure 3: (a), LF data taken at 100 K [the ZF data is omitted for clarity]. (b), scaled plot showing how the relative error in the hopping rate in copper (obtained from the fit to the data), varies with LF strength ($B_{\rm app}$) and temperature.