Muoniated-radical correlation spectrum
The correlation spectrum reads the muon hyperfine coupling of a muoniated radical straight off a transverse-field FFT. A muoniated radical’s transverse- field spectrum shows a pair of precession lines, not one; the correlation spectrum collapses that pair onto a single peak at the coupling \(A_\mu\) that produced it. It is the frequency-domain tool for identifying a muoniated radical and pinning \(A_\mu\).
It lives in the Fourier panel as the Correlation (radical) display mode, alongside the other spectrum views — a specialist tool, off the common path.
What a muoniated radical is
When a positive muon stops in a molecule with an unsaturated bond, it does not always end up diamagnetic. If it adds to a C=C double bond, an aromatic ring, or a C=O group, it sits at a β-position next to a carbon that now carries an unpaired electron — a muoniated radical. The muon spin then couples, through that electron, to the molecule, and the strength of the coupling is the isotropic muon hyperfine coupling constant \(A_\mu\) (in frequency units, MHz). For organic radicals \(A_\mu\) runs from a few MHz up to about 700 MHz, and it is a fingerprint of where the muon sits and how the spin density is distributed.
The muon acts as a polarised spin label reporting on a single, prompt radical, so muoniated-radical μSR sees clean first-order kinetics that conventional EPR, which detects a mixture of products, cannot.
Why a radical gives a line pair
The muon and the radical’s unpaired electron form a coupled two-spin system — the same Breit–Rabi physics as muonium. In a high transverse field the muon precession is not one line at the diamagnetic Larmor frequency \(\nu_\mathrm{d} = (\gamma_\mu/2\pi)B\) but a pair of lines, \(\nu_{12}\) and \(\nu_{34}\), straddling it. The two frequencies are fixed by the field and the coupling, and their sum is the coupling itself:
Equivalently, the splitting between the two lines is \(A_\mu\) (one of the pair is often a negative frequency, so the measured spacing equals the sum). At the kilogauss fields where radicals are measured the system is deep in the high-field (Paschen–Back) regime, where this simple relation is exact and the two lines carry equal weight.
So a radical’s transverse-field FFT shows the diamagnetic line plus a symmetric pair about it. Read the two line positions, add them, and you have \(A_\mu\) — that is the whole method.
How the correlation spectrum works
Reading two lines by eye is easy for one clean radical and hard for a noisy spectrum or several overlapping radicals. The correlation spectrum automates it. For every candidate coupling \(A\) it computes the exact Breit–Rabi pair \((\nu_{12}, \nu_{34})\) that would arise at the measurement field, looks up the spectral amplitude at both frequencies, and multiplies them together (with a ratio penalty that rewards pairs of comparable height). The product is large only when there really is a line at each of the two frequencies — a genuine pair — and small otherwise. Plotted against \(A\), the result peaks at the true coupling of each radical present.
The horizontal axis is therefore a hyperfine-coupling axis in MHz, not a precession-frequency or field axis: a peak at 514 MHz means \(A_\mu = 514\) MHz. Because the axis is a coupling and not \(\gamma_\mu B\), the MHz / Gauss / Tesla field-unit selector is disabled for this view — converting a coupling to “Gauss” would be meaningless.
When to use this. Reach for the correlation spectrum when you have transverse- field data on a muoniated radical and want \(A_\mu\): to identify which radical formed, or to track its coupling versus temperature, solvent, or structure. It shines in liquids, at high field, with resolvable precession and good radical yield — the conditions of classic radical-μSR.
Pitfalls. It is a high-transverse-field construction: at low field the observable pair and the coupling relation differ, and the method does not apply. It needs a continuous muon source (PSI, TRIUMF) — the radical spectrum runs to hundreds of MHz, beyond a pulsed source’s time resolution — and a promptly formed radical, since a slowly forming one dephases in the transverse field before it can be labelled. The diamagnetic line is skipped, and a single strong line with no partner is suppressed rather than reported.
Using it in Asymmetry
Compute a transverse-field FFT as usual, then select Correlation (radical) in the Fourier panel’s display-mode list. Two controls appear:
Correlation field (G) — the transverse field used to compute the Breit–Rabi pairs. Leave it blank to use the run’s applied field from metadata; set it to nudge the matching field if the header value is missing or slightly off.
Correlation order — how aggressively unequal-amplitude (spurious) pairs are penalised. The default of 2 follows WiMDA; raise it to sharpen against noise, set 0 for a plain product.
With several detector groups selected the correlation is built from the averaged spectrum; select a single group to correlate that group alone. The peak position is the coupling — read it straight off the axis.
Worked example: the cyclohexadienyl radical
The textbook muoniated radical is cyclohexadienyl, formed when a muon adds to benzene. Its transverse-field spectrum at a few kilogauss shows the diamagnetic line and a pair straddling it; their sum gives a muon hyperfine coupling of \(A_\mu = 514.4(1)\) MHz. Feed such a spectrum to the correlation mode and a single peak appears at 514 MHz — the radical’s signature. A radical with two inequivalent muon environments would show two peaks, one per coupling.
TF correlation spectrum vs ALC — complementary routes to radical hyperfine couplings
The correlation spectrum is one of two ways to measure a radical’s hyperfine couplings, and most radical studies use both. They probe the coupling network from orthogonal directions.
Transverse-field (TF) correlation — the method on this page — applies a high field across the initial muon spin and Fourier-transforms the precession. It delivers the isotropic muon coupling \(A_\mu\) from the line pair. It is at its best in liquids, at high field, where the precession is sharp and the radical yield is good, and it needs a continuous source and a promptly formed radical.
Avoided level crossing (ALC) instead applies the field along the muon spin and sweeps it, recording the time-integral asymmetry; a resonance appears as a dip where two spin states cross and mix. ALC reads couplings that TF cannot:
A \(\Delta_1\) resonance (muon spin flip) sits near \(A_\mu / 2\gamma_\mu\) and gives the same \(A_\mu\) as the TF correlation — a useful cross-check.
A \(\Delta_0\) resonance (muon–nucleus flip-flop) appears once for each coupled nucleus, at a field set by both \(A_\mu\) and that nucleus’s hyperfine coupling — so ALC maps the other (nuclear) couplings the TF pair is blind to. In solids and oriented media the \(\Delta_0\) resonance also carries the dipolar (anisotropic) part of the coupling, making it the route to molecular orientation and dynamics.
ALC therefore shines exactly where TF struggles: in solids, liquid crystals, polymers and oriented or complex media, and wherever the precession is too broad to resolve.
The practical workflow is to use TF correlation first to identify the radical and pin \(A_\mu\), then ALC to map the rest of the coupling network and the anisotropy and dynamics. Asymmetry already provides the ALC route — see Integral scan mode (avoided-level-crossing field scans) for the integral-asymmetry field-scan workflow and its resonance fitting.
References
McKenzie, Annu. Rep. Prog. Chem. Sect. C 109, 65 (2013).
I. McKenzie, R. Scheuermann, S. P. Cottrell, J. S. Lord, and I. M. Tucker, J. Phys. Chem. B 117, 13614 (2013).
Pratt, Physica B 289–290, 710 (2000).
S. J. Blundell, R. De Renzi, T. Lancaster, and F. L. Pratt, Muon Spectroscopy: An Introduction (Oxford University Press, Oxford, 2022).
Hillier, S. J. Blundell, et al., Nat. Rev. Methods Primers 2, 4 (2022).