Count-domain fitting (α calibration & single histograms)

Count-domain fitting works on raw detector counts rather than the reduced asymmetry. Two modes are available from the Fit target selector at the top of the Multi-Group Fit window, alongside the existing All groups grouped fit:

• Forward + Backward (free α) — fit a forward and a backward count histogram simultaneously with α as a free fit parameter. This is the statistically proper way to obtain α from a transverse-field calibration run, superseding the grid estimators in the Grouping dialog.

• Single group — fit one detector histogram to the count model N₀·exp(−t/τ_μ)·(1 + A·P(t)) + bg. This is the musrfit single-histogram (fittype 0) analogue, used for calibration and diagnostic work, and it is the natural mode for continuous-source data where a single detector already carries the full decay envelope.

When to use this. Reach for the free-α fit whenever you need a defensible α from a calibration measurement and want its uncertainty and — importantly — its correlation with the signal amplitude. In a transverse field the count balance can trade against the asymmetry amplitude, so α and A are strongly correlated; the joint fit reports that correlation, which a grid estimate cannot. Reach for the single-histogram fit when one detector is the object of study: a continuous-source run where you want the bare envelope and background, or a diagnostic check that one detector’s N₀ and background behave as expected. For ordinary physics extraction from balanced data, the regular F–B asymmetry fit remains the faster path — count-domain fitting is a calibration and diagnostic tool, not a replacement for asymmetry fitting.

The Multi-Group Fit window in count-domain mode. The Fit target is “F + B (free α)”; the “Count-fit options” section (the Cost selector, the skip window, the nuisance toggles and the double-pulse field) and the “Calibration” section (the Promote α / t₀ / background / DT₀ actions) are expanded. A Poisson forward/backward fit of a transverse-field calibration run recovers the detector balance α = 1.250(1) at χ²ᵣ ≈ 1.

What these modes fit

Both modes fit raw counts. The count model is

\[N(t) = N_0\, e^{-t/\tau_\mu}\left[1 + s\,A\,P(t)\right] + \mathrm{bg}\]

where \(P(t)\) is the chosen fit function’s polarisation, \(A\) its amplitude, \(\tau_\mu\) the fixed physical muon lifetime, and \(s\) a sign that is +1 for a forward histogram and -1 for a backward one. The muon lifetime is held fixed at the physical value.

For the free-α fit the forward and backward histograms share one normalisation split by the balance,

\[\begin{split}N_F(t) &= N_0\sqrt{\alpha}\,e^{-t/\tau_\mu}\left[1 + A\,P(t)\right] + \mathrm{bg}_F \\ N_B(t) &= \frac{N_0}{\sqrt{\alpha}}\,e^{-t/\tau_\mu}\left[1 - A\,P(t)\right] + \mathrm{bg}_B\end{split}\]

so that \(\alpha = N_{0,F}/N_{0,B}\). The shared amplitude \(A\), the physics parameters and α are fitted together; each side keeps its own background. Forward and backward here are the two detector groups designated in the Grouping dialog; sum multi-detector banks into those two groups upstream.

Choosing the cost

A Cost selector chooses how the counts are weighted:

• Poisson (default) — the Cash statistic, the correct treatment for the low-count bins at late time and on continuous-source data, where the count distribution is visibly skewed and a Gaussian σ underweights the constraint.

• Gaussian √N — ordinary least squares with σ = √N. Faster and adequate when every fitted bin has high counts; it matches the historical WiMDA weighting.

The two agree where counts are high and diverge where they are sparse; when in doubt, the Poisson cost is the safe default. The reduced statistic reported for a Poisson fit is the Cash value per degree of freedom, which behaves like a reduced χ² near the minimum.

Reading the result

The result panel reports the fitted parameters with uncertainties — for example a recovered balance of α = 1.250(1) — together with the per-degree statistic. For the free-α fit, the forward result carries the full covariance, so the α–amplitude correlation is available for inspection.

A successful fit also draws its model curve over the data in the Individual groups plot — the single-histogram fit over its one group, the free-α fit over both the forward and backward banks. The fit minimises a raw-count model, but the plot shows lifetime-corrected counts, so the overlay is scaled by exp(t/τ_μ) to land on the displayed data; a good fit traces the group exactly, and a poor one shows where it departs.

Window and nuisance flexibility

Three optional controls refine the fit window without splitting the run; each is off by default and, when off, leaves the fit numerically unchanged.

• Skip window (μs) — drop an interior window of bins from the fit. Set the two fields so the upper bound exceeds the lower; the bins inside are removed (endpoints inclusive). Use it to reject a laser or RF artefact, a spike, or any localised corruption — the fitted parameters then match what the clean data alone would give, where leaving the artefact in pulls the amplitude. The label distinguishes this drop from the MaxEnt de-weight window of the same units (see Exclusions: a glossary of six mechanisms).

• Fit t₀ offset — add a free time-zero offset that shifts the model time axis. Enable it when a small time-zero error is suspected; on clean data it recovers an offset consistent with zero and changes nothing else.

• Fit baseline drift — add a stretched-exponential damping exp(−(λ_b·t)^β_b) on the polarisation (β_b held at 1, simple exponential, by default), for a slowly relaxing non-precessing baseline.

  Note

  WiMDA applies its baseline drift only to a dedicated constant-offset component; Asymmetry has no single privileged offset parameter, so the drift multiplies the whole polarisation. The term is off by default.

• Free muon lifetime — the core API exposes an optional tau parameter that frees the muon lifetime (musrfit-style), defaulting to the physical value τ_μ = 2.197 μs. Free it to fit the bare decay of a single histogram or as a detector-time diagnostic; with it fixed the fit is byte-identical to the standard fixed-lifetime fit. It is not combined with the double-pulse model.

Count loss and double pulse

• Fit deadtime DT₀ — add a non-paralysable count-loss term to the fit. The observed counts are the true counts damped by 1 − DT₀·r, with r the per-frame, per-detector count rate; DT₀ is the detector deadtime in microseconds, the same quantity the Grouping dialog’s deadtime correction applies. A high-rate run is needed for DT₀ to be well determined.

  The core API offers WiMDA’s full set of count-loss forms via the deadtime_model argument — simple (DT₀ only), linear ((DT₀ + DT₁·evfr)·r, using the group event fraction evfr), polynomial (adding C₂·10³·r², C₃·10⁶·r³, C₄·10⁹·r⁴), and power-law ((evfr·DT₀)^C₂·exp(−(C₄·λ_μ·t)^C₃)). The loaders carry no ISIS event-fraction block, so evfr defaults to 1 unless the grouping supplies it. The GUI fits the simple DT₀ term, which is the dominant and best-determined coefficient.

  Once a deadtime fit converges, Promote DT₀ → grouping writes the fitted value into the grouping’s per-detector deadtime (WiMDA’s Send-to-Group), reporting the before/after values; tick accumulate to add to the existing value rather than replace it. Re-reduce the run to apply the promoted correction. This closes the calibration loop: fit the deadtime, promote it, and the reduced data is corrected. A polynomial or power-law promotion also records the model name and higher-order coefficients with the run, while the reduction still applies the dominant DT₀.

• Double pulse (μs) — set the pulse separation for an ISIS double-pulse source; 0 leaves the single-pulse model. The two pulses each carry the polarisation, evaluated at t ± dpsep/2 and weighted by exp(∓dpsep/2τ_μ). This applies to both the single-histogram and the free-α (F+B) targets — for the latter the two pulses ride the same √α-tied forward/backward model, so α and the double-pulse structure are recovered together. The separation defaults to a fixed instrument value; tick fit to refine it.

  Note

  The separation enters the model through a non-smooth pulse-onset gate, so gradient (migrad) fitting of dpsep is unreliable. With fit ticked the separation is instead located by a coarse→fine grid scan bracketing the instrument value — at each grid point migrad refines the other parameters, and the best χ² wins. This recovers dpsep robustly without depending on a near-truth start. With the separation at its true value the model fits cleanly (χ²ᵣ ≈ 1); a wrong separation visibly degrades the fit.

Promoting α, t₀ and the background

The deadtime promote above is one of a family: the same suggest-only Send-to-Group pattern promotes the other fitted calibrations into the grouping, each reporting before/after and a Re-reduce the run to apply message. Like deadtime, α and t₀ are per-sample, per-setup calibrations — α “needs to be determined for each sample”, and the analysis time-zero “is the beginning of the spin dynamics” — so a value fitted from the run’s own counts is a legitimate calibration to persist.

• Promote α writes the free-α forward/backward balance into the grouping with alpha_method="count_fit" provenance. This is the statistically best of the four α routes — see Alpha calibration. Available after a Forward + Backward (free α) fit.

• Promote t₀ converts the fitted continuous time-zero offset (μs) to the nearest integer t0_bin via the bin width and discloses the sub-bin residual the integer index cannot represent. The fitted t₀ is per group but t0_bin is a single run-level index, so the promote applies the fitted group’s value run-wide and says so. Available after a fit with Fit t₀ offset enabled.

• Promote background writes the fitted flat count background into the grouping’s fixed background mode as a [forward, backward] pair. Because the count fit reads raw counts, this background term measures the full flat rate — if the grouping already corrects the background, the fit still sees it, so do not fix the fit’s background to zero (the panel notes this when a grouping background correction is active).

Worked example — α from a TF calibration run

1. Load the calibration run and open its Grouping dialog; confirm the forward and backward groups.

2. Open the Multi-Group Fit window and choose a transverse-field model (an oscillation with a free amplitude and frequency) in the model builder.

3. Set Fit target to Forward + Backward (free α) and leave Cost at Poisson.

4. Fit. The recovered α is the calibration balance; cross-check it against the Grouping dialog’s diamagnetic / general / ΣF/ΣB estimators — they should agree to a few percent, with differences explained by the estimators flattening an asymmetry window while the fit weights the whole count trace.

References

• S. J. Blundell, R. De Renzi, T. Lancaster, and F. L. Pratt, Muon Spectroscopy: An Introduction (Oxford University Press, Oxford, 2022) — the count equation N(t) = N₀·e^(−t/τ)(1 + A·P(t)) + B, the detector-balance parameter α, and the Poisson nature of detector counts.

• A. Suter and B. M. Wojek, Phys. Procedia 30, 69 (2012) — musrfit, the single-histogram (fittype 0) count fit and its cost options.