.. _alpha-calibration: Alpha calibration ================= The measured forward/backward asymmetry .. math:: A(t) = \frac{N_F(t) - \alpha N_B(t)}{N_F(t) + \alpha N_B(t)} depends on the calibration constant :math:`\alpha = N_F^0/N_B^0`, the ratio of the two detector groups' efficiency × solid-angle products. With the correct :math:`\alpha` the asymmetry equals :math:`a_0 P_z(t)`; with the wrong one, a spurious baseline and a distortion proportional to the muon decay envelope leak into every fit. Alpha is a property of the instrument configuration *and* the sample position, so it must be recalibrated per mounting — and, because detector solid angles change with sample environment, ideally per cooldown. The grouping dialog's **Estimate α** control determines :math:`\alpha` from the reference run and applies it to all selected datasets (the same reference-run convention as deadtime estimation). Three methods are offered; all report the estimate with a statistical uncertainty, e.g. 1.3702(45), obtained from a seeded Poisson bootstrap of the histogram counts. WiMDA reports a bare number for the same operations. Choosing a method ----------------- ============================ ==================================================== Data in hand Method ============================ ==================================================== Weak-TF calibration run **Diamagnetic** (preferred whenever TF data exist) Only relaxing LF/ZF runs **General** — needs *visible* relaxation Quick TF cross-check **Count ratio** (Mantid ``AlphaCalc`` convention) Most demanding work **Count-fit α** (free count-fit parameter) ============================ ==================================================== The standard workflow is the WiMDA/ISIS one: take a weak transverse-field calibration run (a "T20", around 20 G) on the *same sample and mounting*, estimate :math:`\alpha` there with the diamagnetic method, and hold it fixed for the longitudinal- and zero-field runs that follow. The General method exists for the case where no TF calibration was recorded. Corrections and the estimate ---------------------------- :math:`\alpha` is estimated on the *corrected* forward/backward counts — the deadtime correction and the background subtraction configured in the grouping dialog are applied to the histograms **before** the estimate reads them, so the balance is measured on exactly the spectra the reduction forms the asymmetry from. This matters most for the background: a flat background pedestal does not track the detectors' efficiency ratio, so estimating :math:`\alpha` from the raw (unsubtracted) counts balances the *totals* and leaves the reduced weak-TF asymmetry off-centre once the background is subtracted. Estimating on the background-subtracted counts centres it. The correction order — deadtime, then grouping, then background subtraction, then the estimate — follows every reference program (WiMDA, Mantid, musrfit). The calibration dialog shows which corrections the estimate reflects beneath the result, and warns when a *requested* correction could not be applied to the selected calibration run — for example a reference-run background whose reference could not be resolved — so a flattened "after" curve never misrepresents a calibration the reduction will not reproduce. Diamagnetic method ------------------ Minimises the weighted asymmetry power .. math:: S(\alpha) = \sum_i \left( \frac{A_i(\alpha)}{\sigma_i(\alpha)} \right)^2, where :math:`\sigma_i` is the Poisson error propagated to the asymmetry. In a transverse field the muons in a diamagnetic environment precess, so the true asymmetry oscillates symmetrically about zero — any imbalance shows up as a net offset that inflates :math:`S`. The minimum of :math:`S(\alpha)` is the balanced point. *When to use this.* The default for any weak-TF run on a sample in a diamagnetic (non-magnetic, non-relaxing) state — the standard calibration measurement. It is the most precise of the three methods: the oscillation provides a zero-mean reference that makes the imbalance directly visible. Avoid it on magnetically ordered samples (spontaneous fields shift the oscillation) and on data where the diamagnetic fraction is small. This is WiMDA's diamagnetic estimate with the grid search replaced by a bounded continuous minimisation, run on internally packed equal-statistics bins so the result does not depend on the display bunching factor. General method -------------- Uses the lifetime-corrected balanced count .. math:: N(\alpha, t) = \left( \frac{N_F(t)}{\sqrt{\alpha}} + \sqrt{\alpha}\, N_B(t) \right) e^{t/\tau_\mu}, which is constant in time *exactly* when :math:`\alpha` is correct: the polarisation term cancels between the two groups regardless of the shape of :math:`P_z(t)`. Asymmetry solves this flatness condition in closed form by equating the mean corrected count density between an early and a late equal-statistics window — everything is linear in the counts, so the estimate is unbiased and fails loudly (rather than returning a number) when the two windows show no polarisation contrast. *When to use this.* Relaxing LF or ZF data when no TF calibration run exists — the situation the plain count ratio cannot handle. The polarisation must visibly relax within the time window: on non-relaxing (or fully decoupled) data, *no* method can extract :math:`\alpha` from flatness, because every :math:`\alpha` gives a flat corrected count. Expect a larger uncertainty than the diamagnetic method at the same statistics; if the reported uncertainty is comparable to the value itself, treat the result as a consistency check rather than a calibration. Background and deadtime corrections should be applied first — a residual flat background grows as :math:`e^{t/\tau_\mu}` after lifetime correction and biases the late window. WiMDA's General estimate minimises a weighted relative-scatter functional of the same :math:`N(\alpha, t)`; that functional loses its interior minimum at realistic counting statistics, so Asymmetry replaces it with the closed-form flatness condition above (study record: divergence D14). Count ratio ----------- .. math:: \alpha = \frac{\sum_i N_F(t_i)}{\sum_i N_B(t_i)} over the good-bin window — the Mantid ``AlphaCalc`` convention, kept for cross-checks against Mantid workflows and older analyses. *When to use this.* Only on transverse-field data spanning many precession cycles, where the oscillating polarisation integrates to zero and the ratio of summed counts is the efficiency ratio. On relaxing LF/ZF data the positive polarisation does **not** integrate to zero and the ratio is biased upward by approximately :math:`a_0 \langle P_z \rangle` — typically several percent, which is an order of magnitude larger than the statistical uncertainty of a good calibration. The reported bootstrap uncertainty does not include this bias. Count-fit α ------------ The three estimators above are *reduction-time* methods on the grouping dialog. A fourth route lives in the **Multi-Group Fit** window: the forward/backward count fit with the balance free (*Forward + Backward (free α)* target) recovers :math:`\alpha` as a parameter of a full count-level fit. The model fits the raw forward and backward histograms simultaneously with :math:`\alpha` parameterised as :math:`\sqrt{\alpha}`, under the Poisson likelihood, and reports the full :math:`(\alpha, \text{amplitude})` covariance — so the strong α–amplitude correlation of a TF run is estimated *with* the physics rather than before it. This is the WiMDA manual's "most accurate way" to determine :math:`\alpha`. *When to use this.* The most demanding calibrations, and any case where α and the oscillation amplitude are strongly correlated (precessing TF data): the joint fit propagates that correlation into the reported uncertainty, which the reduction-time estimators cannot. It needs a physics model for the run, so it is heavier than the one-click dialog estimators — reach for it when the extra rigour matters, and use the diamagnetic estimator for routine T20 calibrations. Once the fit converges, **Promote α** writes the fitted balance into the grouping with ``alpha_method="count_fit"`` provenance and a reference-run record, the same keys the dialog estimators write — so the reduction picks it up on the next re-reduce. The promote is suggest-only and reports the before/after value; see :doc:`../count_domain_fitting` for the count-fit controls and the sibling t₀/background promotes. Uncertainties ------------- All three methods report a statistical uncertainty from a Poisson bootstrap: the histogram counts are resampled 200 times from their observed values, the estimator is re-run on each replica, and the spread of the replicas (a percentile-based standard error, robust against the heavy tails the General method develops near its identifiability limit) is quoted. The bootstrap is seeded, so repeating an estimate gives identical digits. The uncertainty is statistical only — it does not include the count-ratio method's polarisation bias, nor systematic effects such as a mis-set t0 or an uncorrected background. For the most demanding work, :math:`\alpha` can instead be treated as a free parameter of a count-level fit (the WiMDA manual's "most accurate way"). That route is the **Count-fit α** section above; its uncertainty comes from the fit covariance, including the α–amplitude correlation, rather than the bootstrap used by the reduction-time estimators here. **References** - S. J. Blundell, R. De Renzi, T. Lancaster, and F. L. Pratt, *Muon Spectroscopy: An Introduction* (Oxford University Press, Oxford, 2022). - F. L. Pratt, Physica B **289–290**, 710 (2000). - A. Yaouanc and P. Dalmas de Réotier, *Muon Spin Rotation, Relaxation, and Resonance: Applications to Condensed Matter* (Oxford University Press, Oxford, 2011).