Muon-spin spectroscopy: a primer
Note
This is a short, self-contained introduction, not a substitute for the literature — see the references at the foot of the page. For how Asymmetry relates to the established μSR programs, see Asymmetry in the μSR software landscape.
Muon-spin spectroscopy (μSR) has grown from a specialist method of particle physics into a mainstream probe of condensed matter, where it is used to study magnetism, superconductivity, and molecular and ionic dynamics. A spin-polarised beam of positive muons is implanted into the sample, and the subsequent evolution of the muon spin reports on the local magnetic field where each muon comes to rest. This page summarises how that works and what the measured signal means; it assumes undergraduate physics but no prior knowledge of the technique.
The muon as a local probe
The positive muon (μ⁺) is a spin-½ particle with charge \(+e\), a mass about one-ninth that of the proton, and a mean lifetime of \(\tau_\mu = 2.197\) μs. Implanted into a solid it thermalises within nanoseconds — far faster than its lifetime, and with no measurable loss of spin polarisation — and comes to rest at an interstitial site, where it acts as a sensitive magnetometer for the local field. Two properties make the measurement possible: the muon beam is produced already spin-polarised, and the muon decay is anisotropic, so the direction of the spin at the moment of decay can be read out.
Spin precession
A muon in a local magnetic field \(B\) precesses about that field at the Larmor frequency
where \(\gamma_\mu / 2\pi \approx 135.5\) MHz/T is the muon gyromagnetic ratio. Because \(\gamma_\mu\) is known precisely, a measured precession frequency is a direct measurement of the field at the muon site: a spontaneous frequency in zero applied field, for instance, measures the internal field of a magnetically ordered state, and its temperature dependence follows the magnetic order parameter.
Measuring the polarisation
The muon decays to a positron (and two neutrinos), and the parity-violating weak decay emits the positron preferentially along the direction of the muon spin at that instant. Recording positrons in detectors placed around the sample — typically a forward and a backward group, ahead of and behind the sample relative to the initial spin — and histogramming their arrival times recovers the time evolution of the spin polarisation as the asymmetry between the groups (Key concepts). The asymmetry is proportional to the muon spin polarisation function \(P(t)\), the quantity that carries the physics.
Static and dynamic fields
The shape of \(P(t)\) reflects the distribution of local fields and how it evolves in time. A single well-defined internal field gives a coherent oscillation; a spread of static fields dephases the ensemble and relaxes the polarisation; fluctuating fields relax it differently again, and the distinction between static and dynamic disorder can be settled by applying a longitudinal field. Two limiting cases recur throughout this documentation:
In a transverse field, applied perpendicular to the initial spin, the muon precesses at the total field; the precession amplitude and its damping measure, for example, the field distribution of the vortex lattice in a superconductor.
In zero field, randomly oriented nuclear dipolar fields produce the Kubo–Toyabe relaxation function — the characteristic dip and recovery whose width measures the width of the static field distribution (Kubo–Toyabe).
Asymmetry provides polarisation functions for these and many other static and dynamic field distributions; they are catalogued in Fit functions.
Relation to other probes
As a local probe of magnetism, μSR yields information similar to nuclear magnetic resonance (NMR), electron spin resonance (ESR), or Mössbauer spectroscopy, but with two distinctions: no resonant electromagnetic field is required, since the precessing muon is followed directly in the time domain; and the muon is a sensitive, essentially universal probe — it stops in any material, responds to very small fields, and works across the full temperature range.
A note of caution
The muon’s chief limitation is the mirror image of its strength: because it is an implanted, positively charged interstitial, its stopping site is not known a priori, and it can perturb its local environment. Quantitative interpretation therefore often rests on a calculation of the stopping site and the distortion it induces. This is an active area, and results should be checked against the primary literature and, where possible, an established analysis tool (Asymmetry in the μSR software landscape).
Beyond this primer, this account is deliberately confined to the time-domain picture that underlies every μSR measurement; the specialist topics each have a home in the reference manual — muonium and muoniated radicals in Muonium, avoided level crossings in Integral scan mode (avoided-level-crossing field scans) (with the terms defined in Glossary), the Knight shift in transverse-field work, and the rotating-reference-frame view for high-field precession in Rotating-reference-frame display, while the frequency-domain counterpart to this time-domain account is introduced in Fourier analysis.
References
S. J. Blundell, R. De Renzi, T. Lancaster, and F. L. Pratt, Muon Spectroscopy: An Introduction (Oxford University Press, Oxford, 2022).
A. Yaouanc and P. Dalmas de Réotier, Muon Spin Rotation, Relaxation, and Resonance: Applications to Condensed Matter (Oxford University Press, Oxford, 2011).