Spectral moments

A muon field/frequency spectrum carries more than a peak position. Its moments — the mean field, the RMS width, the skewness, the lineshape asymmetry — reduce the whole distribution \(p(B)\) to a handful of numbers that map directly onto physics, and that can be trended across temperature or field like any fitted parameter. The RMS width is the headline: for a type-II superconductor in the mixed state the second moment of the vortex-lattice field distribution sets the magnetic penetration depth, \(\langle\Delta B^2\rangle \propto 1/\lambda^4\), and hence the superfluid density — see Superconducting penetration depth models.

The Spectral moments control sits in the advanced stack of the Fourier panel and under the reconstruction in the MaxEnt panel. Pick a unit, drag a range over the line, and the readout updates live; Send to trend records the moments of every selected run as a trendable series.

The Spectral moments readout, computed on the phase-corrected real FFT of a YBCO vortex-lattice line. The spectrum is the asymmetric field distribution p(B) — a sharp low-field peak and a long high-field tail — with the moments range (shaded) and cutoff (dotted) drawn over it. The readout gives B_pk ≈ 1987 G, the mean B_ave sitting above the peak, the RMS width, and the positive skewness and asymmetry (β > 0) that are the vortex lattice’s expected signature.

What each moment tells you

For a window over the line, above an amplitude cutoff, Asymmetry reports:

• \(B_{\mathrm{pk}}\) — the peak field, refined by a parabola through the five points around the maximum.

• \(B_{\mathrm{ave}}\) — the amplitude-weighted mean field; its shift from the applied field is the diamagnetic (or Knight) shift.

• \(\langle B_{\mathrm{ave}}-B_{\mathrm{pk}}\rangle\) — how far the mean sits from the peak; a direct read on the line’s asymmetry.

• \(B_{\mathrm{rms}}\) about the mean and about the peak — the width. About the mean it is the standard deviation \(\sqrt{\langle\Delta B^2\rangle}\), the quantity that feeds \(\lambda\).

• Skewness — the third-moment asymmetry. Asymmetry reports both WiMDA’s cube-root form \(\alpha=\operatorname{sign}(m_3)\,\sqrt[3]{|m_3|}/\sqrt{m_2}\) and the standard standardised skewness \(\gamma_1=m_3/m_2^{3/2}\).

• Asymmetry \(\beta=(B_{\mathrm{ave}}-B_{\mathrm{pk}})/B_{\mathrm{rms,pk}}\) — positive when the mean lies above the peak.

The vortex-lattice field distribution is the canonical use case: a sharp low-field cutoff at the lattice’s saddle-point field and a long tail to high field near the cores give it a positive skew, so \(\beta>0\) and \(\gamma_1>0\) are the expected signature, and their magnitude tracks the lattice geometry and its disorder. The sign convention here matches WiMDA and the literature for that distribution [1] [2] [3].

The range and the cutoff

Moments are window-dependent, so the window is always drawn on the plot — a shaded range with draggable edges and a dotted cutoff line at the chosen fraction of the peak. Drag them, or type exact values into the control; the choice is recorded in the run’s provenance so a trend is reproducible.

• The range isolates the line of interest and excludes neighbouring features. (This is a range, not an exclusion: it selects what to include.)

• The cutoff (a percentage of the peak) trims the wings and the spectral floor before the integral, so far-off baseline noise does not inflate the width.

Tighten the range toward the main line and the skewness and \(\beta\) collapse toward zero as the tail is excluded; raise the cutoff and the width narrows as the wings drop out. There is no single correct window — report the one you used.

A caveat on apodised spectra

Apodisation broadens every line it smooths, so moments read from a filtered FFT carry the filter as a systematic: the widths and skewness include the filter’s broadening, not just the sample’s. When the active spectrum was computed with a Lorentzian or Gaussian filter, the moments readout shows an amber caveat —

Apodised spectrum (lorentzian, τ = 1.8 µs): widths and skewness include the filter’s broadening.

— so a filtered reading is never silently mistaken for the unfiltered physics. For quantitative widths, recompute the FFT with apodisation None (or deconvolve the known filter contribution when reporting).

A caveat on \(B_{\mathrm{pk}}\)

\(B_{\mathrm{pk}}\) is the fragile member of the set. It is a parabola fitted to five points around the discrete maximum; on a noisy or near-flat spectrum the maximum hops between bins and the parabolic vertex can swing wide. Everything built on it — \(\langle B_{\mathrm{ave}}-B_{\mathrm{pk}}\rangle\) and, especially, \(\beta\) — inherits that fragility. The robust members are \(B_{\mathrm{ave}}\), \(B_{\mathrm{rms}}\) and (where the third moment converges) the skewness, which are amplitude-weighted integrals that average noise down. When the spectrum is noisy, trust \(B_{\mathrm{ave}}\) and \(B_{\mathrm{rms}}\); read \(B_{\mathrm{pk}}\) and \(\beta\) as indicative. The bootstrap error bars make this visible: a fragile \(B_{\mathrm{pk}}\) shows a large uncertainty next to a well-determined \(B_{\mathrm{ave}}\).

Uncertainties

WiMDA gives single-spectrum moments no error at all. Asymmetry does better: when the spectrum carries per-point errors (the averaged-FFT error, or the MaxEnt error estimate), each moment is given a bootstrap uncertainty — the spectrum is resampled within its noise many times and the moments recomputed, so the error propagates correctly through the nonlinear peak, skewness and \(\beta\). A value reads as \(B_{\mathrm{rms}} = 18.4(3)\). For a zero-padded FFT the samples are sinc-interpolated and correlated (only \(1/n\) of them are independent at pad factor \(n\)), so moment uncertainties are scaled by \(\sqrt{n}\) — the same effective-sample-size correction the frequency-domain fits apply. Run-to-run scatter across a temperature scan is then handled, as for any series, by the trend layer.

Which spectra qualify

Moments are only meaningful for a lineshape-faithful spectrum: the MaxEnt reconstruction or the phase-corrected real FFT. Power, magnitude, phase, Burg and correlation modes are squared or diagnostic lineshapes that bias the width and the skewness, so the control greys out for them with a note explaining why. Switch to the phase-corrected real mode or the MaxEnt reconstruction to take moments.

Trending the moments

Send to trend computes the moments of every selected run’s spectrum, with the current range/cutoff/unit, and records them as one computed series — one point per run, indexed by field and temperature — that fits like any parameter series. Re-sending the same selection replaces it rather than duplicating it. Fit \(B_{\mathrm{rms}}(T)\) with a superconductor model (Superconducting penetration depth models) to extract \(\lambda(T)\), or trend the skewness to follow a lattice transition.

When to fit the lineshape instead

Moments are model-free: they summarise whatever \(p(B)\) the spectrum shows, which is exactly right for a first look, for tracking a width across a scan, and for distributions with no clean analytic form. Fit the lineshape instead when you have a physical model of \(p(B)\) — a Brandt vortex-lattice distribution, a Gaussian-broadened London model, a sum of diamagnetic and background lines — and want its parameters with proper covariances, or when the tails are too noisy for a stable third moment. The two are complementary: moments give the quick, assumption-light trend; a lineshape fit gives the interpreted physics.

See also

• Fourier analysis for the FFT and MaxEnt spectra;

• Frequency-domain conditioning for the conditioning ladder that prepares them;

• Parameter trending and Superconducting penetration depth models for fitting the resulting \(B_{\mathrm{rms}}(T)\).

References

[1] E. H. Brandt, Phys. Rev. B 37, 2349 (1988); Phys. Rev. B 68, 054506 (2003).

[2] J. E. Sonier, J. H. Brewer, and R. F. Kiefl, Rev. Mod. Phys. 72, 769 (2000).

[3] S. J. Blundell, R. De Renzi, T. Lancaster, and F. L. Pratt, Muon Spectroscopy: An Introduction (Oxford University Press, Oxford, 2022).