Negative-muon capture-lifetime analysis (experimental)

Warning

Experimental — negative-muon analysis is a work in progress.

This page documents a scriptable, API-only μ⁻ capture-lifetime analysis that is unvalidated against real μ⁻ data. It has been exercised only on synthetic histograms. The element lifetimes are literature-anchored (Suzuki, Measday & Roalsvig 1987), but the fitting and capture-ratio machinery have not been checked against an established μ⁻ tool on measured data, and the API may change. There is no GUI for this feature. Verify any physical interpretation against the primary literature and an established tool (WiMDA, Mantid) before relying on it.

What this measures

A negative muon implanted in matter forms a muonic atom at the lattice site, cascading to the 1s muonic orbit on a picosecond timescale. From there it either decays freely — just as a positive muon does, with lifetime τμ = 2.197 μs — or is captured by the nucleus at a rate Λcap(Z) that grows steeply with atomic number. The observable disappearance lifetime

\[\tau(Z) = \frac{1}{\Lambda_\mathrm{cap}(Z) + \Lambda_\mathrm{decay}}\]

is therefore element-characteristic: τ is 2.195 μs for hydrogen (capture negligible) and falls to ≈ 72–96 ns for the heavy elements from Ru to U.

In a sample containing several elements the decay-electron histogram summed over all detectors is a sum of exponentials with those element-characteristic lifetimes, and the amplitude of each term is proportional to the fraction of muon captures on that element. Fitting the histogram and forming amplitude ratios gives the relative capture probabilities, which are directly related to the stoichiometric composition weighted by each element’s capture rate. This is nuclear-charge-selective elemental analysis from the muon decay electrons.

The model

The raw detector count histogram is modelled as

\[N(t) = \sum_i N_i\, e^{-t/\tau_i} + N_\mathrm{bg}\, e^{-t/\tau_\mu} + b,\]

where the sum runs over the declared elemental components (each with a literature lifetime τi and a free amplitude Ni), the second term is a free-μ⁻ decay background with the fixed muon lifetime τμ and a free amplitude Nbg, and b is a flat background per bin. The model is fit directly to raw counts — not to the reduced asymmetry — using a Poisson (Cash) or Gaussian cost. The free amplitudes carry the composition information; the lifetimes are held fixed at their table values by default.

The lifetime table

Lifetimes are transcribed from Table C.1 of Blundell, De Renzi, Lancaster & Pratt, Muon Spectroscopy: An Introduction (Oxford University Press, Oxford, 2022), Appendix C, which combines measurements from Suzuki, Measday & Roalsvig (1987). Values for 79 elements from H (Z = 1) to Np (Z = 93) are included. A small number of entries marked source="WiMDA-provisional" carry no published uncertainty and are drawn from WiMDA’s older table where the primary compilation has no entry.

Access lifetimes programmatically via tau_us() and lifetime():

from asymmetry.core.negmu.lifetimes import tau_us, lifetime

tau_us("Fe")             # → 0.206 μs
lifetime("Fe").sigma_us  # → 0.001 μs

Lifetimes are fixed at their table values by default. Free a specific element’s lifetime by including its symbol in CaptureModelSpec.free_tau:

from asymmetry.core.negmu.fit import CaptureModelSpec

spec = CaptureModelSpec(
    elements=("Ca", "Fe"),
    include_decay_background=True,
    free_tau=frozenset({"Fe"}),   # fit the Fe lifetime as a free parameter
)

WiMDA’s older element table has several transcription errors — a Ti/Tl symbol swap and divergent values for Ne, Zn, Sr, and Ba — that were identified during the porting study (docs/porting/negative-muon-analysis/). The Table C.1 values from the primary Suzuki/Measday/Roalsvig compilation are adopted throughout.

When to use this. Reach for the negative-muon capture-lifetime API when:

  • you have a raw μ⁻ histogram from a mixed-element sample and want relative capture probabilities — for example, the stoichiometric composition of a compound from a single implantation measurement;

  • you can assert which elements are present and use their fixed table lifetimes to constrain the fit.

Do not use this for:

  • μ+SR spin-relaxation analysis — this module fits raw counts with element-characteristic lifetimes, not the reduced asymmetry with a depolarisation function; use FitEngine with an asymmetry model for that;

  • muonic X-ray (μ-XRF) elemental analysis, which identifies elements from the characteristic X-ray spectrum rather than the decay-electron histogram;

  • samples where the elements of interest all have lifetimes shorter than the time resolution (the sub-100 ns regime for Z ≳ 45) — the binned histogram must resolve the exponential decays.

Expect to declare the candidate elements explicitly rather than free-scanning the whole table; the fit has one amplitude per element and becomes underdetermined when many lifetimes are simultaneously free.

Worked example

The snippet below synthesises a two-element capture run (Ca + Fe at amplitude ratio ≈ 5:3), fits the single-group histogram, and recovers the ratio. In a real measurement, replace template and run with a loaded Run — everything downstream is identical.

import numpy as np
from asymmetry.core.data.dataset import Histogram, MuonDataset, Run
from asymmetry.core.negmu.fit import CaptureModelSpec, fit_capture_group
from asymmetry.core.negmu.ratio import capture_ratio_report
from asymmetry.core.simulate import simulate_capture_run

# --- geometry template --------------------------------------------------
# In practice supply a loaded μ⁻ run; it provides the bin structure, t0,
# and grouping.  This minimal template is sufficient for a synthetic test.
n_bins, bw = 1000, 0.016          # bins; 16 ns bin width (μs)
template = Run(
    run_number=1,
    histograms=[
        Histogram(
            counts=np.zeros(n_bins),
            bin_width=bw,
            t0_bin=0,
            good_bin_start=0,
            good_bin_end=n_bins - 1,
        )
        for _ in range(2)
    ],
    metadata={},
    grouping={
        "groups": {1: [1, 2]},
        "first_good_bin": 0,
        "last_good_bin": n_bins - 1,
        "bunching_factor": 1,
        "deadtime_correction": False,
    },
    source_file="",
)

# --- model spec and simulated run ----------------------------------------
spec = CaptureModelSpec(elements=("Ca", "Fe"), include_decay_background=True)

# Weights allocate the event budget per component.  The fitted amplitude is
# proportional to weight × (1−exp(−bw/τ)), which simplifies to weight/τ when
# bw ≪ τ.  For Ca and Fe with bw = 0.016 μs the linear approximation holds to
# within ~4%, so w_Ca/w_Fe ≈ (5/3) × (τ_Ca/τ_Fe) targets amplitude ratio ≈ 5/3.
# For heavier elements (τ ≲ 5×bw) use the exact formula directly.
run = simulate_capture_run(
    template,
    components=spec.components(),
    weights={"Ca": 1.68, "Fe": 0.618},   # → Ca/Fe amplitude ratio ≈ 5/3
    total_events=150_000,
    seed=5,
)

# --- fit ----------------------------------------------------------------
# fit_capture_group wraps build_count_group, which expects a MuonDataset
# whose .run carries the histograms; time/asymmetry arrays are not used.
dataset = MuonDataset(
    time=np.array([]), asymmetry=np.array([]),
    error=np.array([]), metadata={}, run=run,
)
result = fit_capture_group(dataset, group_id=1, spec=spec)

# --- capture-ratio report -----------------------------------------------
report = capture_ratio_report(result, spec, reference="Fe")
for r in report.ratios:
    print(f"{r.numerator}/{r.denominator} = {r.ratio:.2f}({round(r.sigma * 100):d})")
# Ca/Fe = 1.67(5)

The ratio Ca/Fe = 1.67(5) recovers the input weight ratio 5/3.

Forward/backward fitting and α

For a μ⁻ run where forward and backward detector banks are read out separately, fit_capture_fb_alpha() fits both banks simultaneously with a shared set of elemental amplitudes and a free calibration constant α:

\[\begin{split}N_F(t) &= \sqrt{\alpha}\; \sum_i N_i\, e^{-t/\tau_i} + b_F \\ N_B(t) &= \frac{1}{\sqrt{\alpha}}\; \sum_i N_i\, e^{-t/\tau_i} + b_B\end{split}\]

The amplitudes Ni are shared between the two banks (isotropic capture populations), so the per-side capture ratios are identical by construction. This diverges from WiMDA, which fits independent per-side amplitudes. Use per-group calls to fit_capture_group() on each bank when a genuine forward/backward amplitude difference is needed.

fb_capture_ratio_report() returns a dictionary of per-side CaptureRatioReport objects ({"forward": ..., "backward": ...}); for the shared-amplitude fit the two sides report identical ratios and differ only in the side label.

References

  • T. Suzuki, D. F. Measday, and J. P. Roalsvig, Phys. Rev. C 35, 2212 (1987) — primary lifetime compilation used for the element table.

  • D. F. Measday, Phys. Rep. 354, 243 (2001) — review of the nuclear physics of μ⁻ capture; Λcap(Z) systematics.

  • S. J. Blundell, R. De Renzi, T. Lancaster, and F. L. Pratt, Muon Spectroscopy: An Introduction (Oxford University Press, Oxford, 2022), Ch. 22 and Appendix C — the lifetime table (Table C.1) and negative-muon physics overview.