A Bayesian approach to magnetic moment determination using muSR

A significant challenge in zero-field muSR experiments arises from the uncertainty in the muon site. It is possible to calculate the dipole field (and hence precession frequency nu) at any particular site given the magnetic moment mu and magnetic structure. One can also evaluate f(nu), the probability distribution function of nu assuming that the muon site can be anywhere within the unit cell with equal probability, excluding physically forbidden sites. Since nu is obtained from experiment, what we would like to know is g(mu|nu), the probability density function of mu given the observed nu. This can be obtained from our calculated f(nu/mu) using Bayes’ theorem. We describe an approach to this problem which we have used to extract information about real systems including a low-moment osmate compound, a family of molecular magnets, and an iron-arsenide compound.

💡 Research Summary

The paper addresses a central difficulty in zero‑field muon‑spin rotation (μSR) experiments: the lack of precise knowledge of the muon stopping site. Since the observed muon precession frequency ν is directly proportional to the local magnetic field B_loc at the muon site, and B_loc is dominated by the dipolar field from the magnetic ions, the authors propose a statistical method that combines dipolar‑field calculations with Bayesian inference to extract the magnetic moment μ of the ordered ions even when the muon site is unknown.

First, the authors review the composition of the local field (Eq. 1). In antiferromagnets the Lorentz and demagnetizing fields vanish, and the hyperfine contact term is neglected for the systems considered. Consequently the dipolar field B_dip, expressed through the dipolar tensor Dαβ ( Eq. 2‑3 ), is the only term that needs to be evaluated. The dipolar sum over an infinite lattice converges rapidly; in practice it is computed by summing over a sphere of sufficiently large radius or by using an Ewald‑summation scheme.

Next, the Bayesian framework is introduced. The conditional probability P(ν|μ) can be obtained from a forward calculation: for a given trial moment μ and a chosen magnetic structure, a large number of random points are generated inside the crystallographic unit cell, subject to physical constraints (e.g., the muon cannot stop too close to positively charged ions, and in oxides it typically resides ~0.1 nm from O²⁻). At each admissible point the dipolar field is calculated, converted to a precession frequency via ν = (γμ/2π)|B_dip| (γμ/2π = 135.5 MHz T⁻¹), and the resulting frequencies are histogrammed. After normalisation this histogram is the probability density function f(ν/μ). Because ν scales linearly with μ, the conditional probability is simply

 P(ν|μ) = (1/μ) f(ν/μ) .

The prior for the magnetic moment is taken to be uniform between 0 and a large upper bound μ_max, which cancels in the numerator and denominator of Bayes’ theorem. The posterior probability for the moment given an observed frequency is therefore

 g(μ|ν) =