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 :math:`A_\mu` that produced it. It is the frequency-domain tool for **identifying a muoniated radical and pinning** :math:`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** :math:`A_\mu` (in frequency units, MHz). For organic radicals :math:`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 |muSR| 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 :math:`\nu_\mathrm{d} = (\gamma_\mu/2\pi)B` but a **pair** of lines, :math:`\nu_{12}` and :math:`\nu_{34}`, straddling it. The two frequencies are fixed by the field and the coupling, and their **sum is the coupling itself**: .. math:: A_\mu = \nu_{12} + \nu_{34}. Equivalently, the splitting between the two lines *is* :math:`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 :math:`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 :math:`A` it computes the exact Breit–Rabi pair :math:`(\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 :math:`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 :math:`A_\mu = 514` MHz. Because the axis is a coupling and not :math:`\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 :math:`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-|muSR|. *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 :math:`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. .. _radical-correlation-vs-alc: 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** :math:`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 :math:`\Delta_1` resonance (muon spin flip) sits near :math:`A_\mu / 2\gamma_\mu` and gives the **same** :math:`A_\mu` as the TF correlation — a useful cross-check. - A :math:`\Delta_0` resonance (muon–nucleus flip-flop) appears once for **each coupled nucleus**, at a field set by both :math:`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 :math:`\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 :math:`A_\mu`, then **ALC** to map the rest of the coupling network and the anisotropy and dynamics. Asymmetry already provides the ALC route — see :doc:`alc_mode` for the integral-asymmetry field-scan workflow and its resonance fitting. References ---------- - I. 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). - F. L. 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). - A. D. Hillier, S. J. Blundell, *et al.*, Nat. Rev. Methods Primers **2**, 4 (2022). .. |muSR| replace:: μSR