Imperial Physics Year Two Radiation Experiment

Measures how a ⁹⁰Sr beta source's count rate falls off with distance, and fits a chain of physical corrections (geometry, dead time, air attenuation) needed to recover a clean inverse-square law from real detector data.

Raw inverse-square-scaled rate u(d) = rate(d)·d² (points) against three nested correction models — a bare constant (red) badly fails both near the source and far from it; adding geometry, dead time, and air attenuation (magenta) tracks the data across the full 0–90 cm range.

Why it matters

For an ideal point source and point detector, count rate should scale exactly as 1/d², so u(d) = rate(d)·d² should be flat. Real measurements never are: near the source the detector's finite size and its dead time distort the count rate, and far away beta particles lose energy in air before reaching the detector. Fitting a bare 1/d² law without these corrections silently biases the inferred source activity — the same failure mode that shows up in any inverse-square measurement (radioactive sources, light sources, RF power) taken with a real, non-ideal detector. This repo works through that correction chain end-to-end and quantifies how much each term actually matters, rather than assuming the ideal law and moving on.

Key technical features

• Nested nonlinear model family (radiation/models.py) — five models (M0–M4) building up from a bare constant to full geometry + dead-time + air-attenuation, fit by orthogonal distance regression (radiation/fitting.py) so both distance and rate uncertainties are honored, not just rate.
• Physical beta-spectrum transmission model (radiation/beta_spectrum.py) — converts an air path length to an energy cutoff via the empirical range-energy relation, then integrates an allowed-spectrum shape (two mixed endpoints for the Sr-90/Y-90 decay chain) to get a transmission fraction — a physical model, not a fitted empirical curve.
• Paralyzable dead-time inversion (radiation/deadtime.py) — inverts r_obs = r·e^(-rτ) in closed form via the Lambert W function, with analytic uncertainty propagation through the inversion.
• Model comparison, not just model fitting — AIC/BIC and nested Δχ² tests (radiation/fitting.py) to justify each added correction term rather than just reporting the fanciest model's numbers.
• Independent cross-check — scripts/air_transmission_explore.py re-derives the beta-mixing weight f_low by a totally separate method (grid search on far-field data only) as a sanity check on the main ODR fit.

Installation and quick start

pip install -r requirements.txt
# or: pip install -e .

# Fit models M0-M4 to rate-vs-distance data and report the fully
# corrected source constant K_est(d).
python scripts/fit_inverse_square.py --data data/filtered_data.csv

# Independent cross-check: grid-search the beta-spectrum mixing weight
# against far-field data only.
python scripts/air_transmission_explore.py --data data/filtered_data.csv

# Measure detector dead time from oscilloscope pulse traces.
python scripts/measure_deadtime.py --data-dir data/deadtime

Each script prints its results and (re)writes the corresponding figures/tables under results/.

Main results

The five nested models for u(d) = rate(d)·d², which should be flat if the inverse-square law and no systematics held exactly:

• M0 — constant (far-field only)
• M1 — + geometric distance offset d0
• M2 — + finite-detector solid angle
• M3 — + paralyzable dead-time correction
• M4 — + beta transmission through air

Dead time, from five oscilloscope pulse traces: τ = 3.12 ± 0.06 μs.

Fitting all five to 30 distance points (0–90 cm), the full model (M4) is overwhelmingly preferred over any model missing the air term:

Model	k	χ²	dof	χ²_red
M1 + offset	2	1065.7	28	38.1
M2 + solid angle	2	1035.8	28	37.0
M3 + dead time	2	990.1	28	35.4
M4 + air transmission	3	67.9	27	2.52

Adding the air-transmission term alone (M3 → M4) drops χ² by 922 for one extra parameter (p ≈ 1×10⁻²⁰²). The fit gives a geometric offset d0 = 1.34 ± 0.08 cm and an effective low-energy beta-spectrum weight f_low = 0.169 ± 0.016.

Backing out the fully corrected source constant point-by-point and taking its weighted mean gives K̄ = 151,580 ± 540 cps·cm², consistent with a constant across the full distance range (χ² = 26.4, dof = 29, p = 0.61) with no significant residual trend with distance (slope p = 0.25):

With a 1 cm² detector area, this implies Aη = 4πK̄ ≈ (1.905 ± 0.007)×10⁶ Bq, and — relative to the source's labeled 5 MBq activity — an overall detection efficiency η ≈ 0.381 ± 0.001.

Repository structure

radiation/        reusable analysis package (physics models, fitting, I/O)
scripts/          thin CLI entry points that call into radiation/
data/             raw measurements (rate vs. distance, oscilloscope traces)
results/          generated figures and tables, committed for visibility
docs/             lab script and final report PDFs
archive/          original notebooks and first-pass outputs, kept for reference

Tests and reproducibility

• The ODR fit is verified robust, not just lucky: refitting M4 from five different initial guesses for the mixing-weight parameter converges to the same optimum every time (K≈151,280–151,300, d0≈1.336–1.337, f_low≈0.169).
• scripts/air_transmission_explore.py cross-checks f_low against scripts/fit_inverse_square.py's ODR fit using an unrelated method (grid search on far-field-only data) rather than trusting a single fitting routine.
• results/ is committed with the exact output of the current code, so a stale figure or number shows up as a diff the next time the scripts are rerun.
• There is no automated (pytest-style) unit test suite yet — see Limitations below.

Limitations and future work

• Detector solid angle is modeled as an equivalent-area disk (radiation/geometry.py), not the exact rectangular-aperture formula for the 1cm×1cm detector derived in the lab script's appendix — a reasonable approximation at these distances, but not an exact match.
• Even the best model (M4) does not formally pass a χ² goodness-of-fit test (χ²_red = 2.52 gives p ≈ 2×10⁻⁵): some smaller, unmodeled systematic remains beyond geometry, dead time, and air attenuation.
• Air density and the Sr-90/Y-90 beta endpoint energies are fixed defaults in DetectorConfig, not fit to ambient conditions or re-derived from the source's decay scheme.
• No automated test suite: correctness currently rests on the cross-checks described above rather than pinned regression tests. Adding a small pytest suite (e.g. asserting the ODR fit reproduces known parameters within tolerance) would catch silent regressions from future refactors.

License

MIT — see LICENSE.