Histogram binning
Counts arrive in bins of the TDC resolution (16 ns at ISIS, sub-ns at continuous sources), far finer than most physics requires. Because the muon ensemble decays as \(e^{-t/\tau_\mu}\), the Poisson error per raw bin grows as \(e^{t/2\tau_\mu}\) — the signal-to-noise halves roughly every 3 μs — so a single binning choice cannot serve both the densely-sampled early times and the count-starved tail. Three modes are offered in the grouping dialog’s Binning control:
Mode |
Output bin width |
|---|---|
Fixed |
bunching factor × raw width, everywhere |
Variable |
grows exponentially from Initial bin at t = 0 to Bin at 10 μs |
Constant error |
grows as \(e^{t/\tau_\mu}\) from Initial bin — equal counts, flat error per bin |
All three are display/fit-input transformations: the raw histograms are never modified, and changing mode (or any width) is always reversible. For the non-fixed modes the counts are summed onto the output bins and the asymmetry formed per output bin — at late times the raw bins hold zero counts, where an asymmetry ratio per raw bin is undefined while summed counts remain exactly Poisson.
Fixed
The default. Every output bin merges the same number of raw bins.
When to use this. Oscillating (TF) data, Fourier analysis, and MaxEnt — anything that needs uniform time sampling. Keep the bunching factor small enough that the highest frequency of interest stays below the Nyquist limit \(f_c = 1/(2\Delta t)\); rebinning is a low-pass filter, which can also be used deliberately to suppress an unwanted high-frequency component.
Variable
Width grows smoothly as
set by the width at t = 0 and the width at 10 μs (WiMDA’s two-knob convention; defaults 0.08 μs and 0.25 μs).
When to use this. Relaxation data where the early-time shape matters (a fast front, a Kubo–Toyabe dip) but the tail evolves slowly — fine bins where the physics is fast, coarse bins where only the level matters. The growth is gentler than constant-error mode, so it preserves more late-time structure at the cost of growing error bars.
Constant error
Width grows at exactly the muon decay rate,
so the expected counts per output bin — and hence the Poisson error per point — stay constant while the polarisation varies slowly. One knob: the initial width sets the statistics level of every bin.
When to use this. Weak, slow relaxation followed to long times (20–32 μs pulsed-source work): every plotted point carries equal statistical weight, which is also the friendliest input for eyeballing weak trends. Two caveats: late output bins become microseconds wide, so any structure faster than the local width is averaged away — check with fixed binning first that nothing oscillates; and the final bin, truncated by the good-data window, carries fewer counts than the rest.
Rebinning programmatically
Fixed bunching is also available directly on a reduced dataset.
MuonDataset.rebin
merges every factor consecutive bins and returns a new dataset, leaving
the original untouched — convenient for high-rate continuous-source data
(e.g. PSI GPS 1.25 ns bins) without dropping to manual array work:
coarse = dataset.rebin(8) # merge 8 raw bins → 10 ns effective width
It is a thin wrapper over the array-level
rebin() primitive: factor = 1 is a
no-op copy, a length that is not a multiple of factor drops the trailing
remainder bins, and the per-point error shrinks as
\(\sigma_\text{new} = \sqrt{\sum \sigma^2} / factor\) (i.e.
\(\propto 1/\sqrt{factor}\) on flat data).
Notes
Fourier and MaxEnt analyses require uniform sampling and always use fixed binning regardless of this setting. Fits of the displayed curve use the displayed binning — each point enters the χ² with its own error, so the non-uniform widths are handled correctly — but note that heavily binned input has less information about fast components than the raw data.
References
S. J. Blundell, R. De Renzi, T. Lancaster, and F. L. Pratt, Muon Spectroscopy: An Introduction (Oxford University Press, Oxford, 2022) — rebinning schemes and counting statistics.
Pratt, Physica B 289–290, 710 (2000).