Validation of the decay calculator

Scope

This report covers the decay calculator: the three questions it answers — how much is left after a given time, when the activity reaches a target, and what half-life two measurements imply — together with the half-lives those answers rest on and the cases in which the calculator must refuse. It does not cover the ingrowth of daughters, the specific-activity conversion, or any other calculator on this site.

Why this report is built differently from the converter's

The unit converter rests entirely on definitions, so its report could be complete: a finite list of units, each derived from a published definition. A decay calculation rests on two things of very different character, and treating them as one would make the result meaningless.

The first is mathematics. The decay law and its inverse are exact. One half-life leaves exactly one half, not approximately; the statement can be checked by hand with no data at all, and any disagreement beyond the resolution of the arithmetic is a defect. That part is validated to the same standard as the converter.

The second is measured data. A half-life is not defined, it is measured, and evaluators differ because they weight the underlying experiments differently. Asking whether a half-life is "correct" to some percentage is the wrong question. The right one is whether it agrees with an independent evaluation within the uncertainty that evaluation publishes, and that is how the second section judges it.

Choosing something independent to compare against

This matters more than it first appears. The nuclear data on this site come from the IAEA's Livechart interface, which serves the evaluated nuclear structure data file, ENSDF. The obvious place to check a half-life is the National Nuclear Data Center at Brookhaven — and it would be the wrong place, because it serves the same file. Agreement would demonstrate nothing except that two copies of one evaluation match.

The comparison here is against the Decay Data Evaluation Project, published by the BIPM as Monographie-5, the Table of Radionuclides, and maintained in per-nuclide form by the LNE-LNHB. It is a separate evaluation, it publishes a standard uncertainty with every recommended value, and those uncertainties are what make a quantitative verdict possible rather than a vague one. That it is the international metrology community's own table is worth noting: it is the reference a calibration laboratory would reach for.

Each value below is quoted in the form it appears in the DDEP sheet, so that a reader can open the sheet and compare characters rather than trusting a transcription. That is not decoration: an automatic reading of those sheets initially took the half-life of U-238 to be 24.10 days, because each sheet also carries the daughter's half-life and 24.10 d is thorium-234. The values here were taken from the line naming the right isotope.

The implementation under test

The three modes are three expressions:

lambda      = (tHalfS) => LN2 / tHalfS
decayActivity        = a0 * Math.exp(-lambda(tHalfS) * elapsedS)
elapsedFromRatio     = -Math.log(ratio) / lambda(tHalfS)
halfLifeFromTwoPoints = (LN2 * elapsedS) / Math.log(a0 / a1)

There is no series expansion and no iteration, so there is no accumulation of error to bound — the only question is whether these are the right expressions and whether the library's exponential and logarithm are faithful. Both are settled by the identities below, which fail immediately if either is wrong.

Exact identities

These hold for every half-life, which is what makes them identities rather than test values. Each is therefore checked at 5 half-lives spanning seventeen orders of magnitude, from one second to 10¹⁷ seconds; the table shows the value at one day and the worst disagreement seen across all of them.

Half-lives against an independent evaluation

The column headed σ is the difference expressed in standard uncertainties of the DDEP value: a figure below one means the two evaluations agree within the published uncertainty, and the acceptance criterion is 3, the usual coverage factor. Judging by percentage instead would be meaningless here, because the uncertainties differ by two orders of magnitude between nuclides — Ni-63 carries ±2.4% and Ir-192 ±0.018%.

The median difference across the 24 nuclides is 0.0328%, and the largest is Cs-137 at 2.82σ. That one is worth a sentence, because it is not noise: the DDEP re-evaluation of caesium-137 recommends a shorter half-life than the ENSDF value this site carries, and the difference of about 0.2% is real and traceable to the evaluations rather than to any calculation. Over a thirty-year storage period it amounts to about a tenth of a percent in the remaining activity, which is far below the uncertainty of any measurement it would be compared with, but it is a difference and this report does not hide it.

Source: BIPM Monographie-5, per-nuclide data sheets.

The year, which is two different years

DDEP quotes long half-lives in years, the data file carries seconds, and the calculator offers years as a time unit. A year is not one length: the Julian year of 365.25 days and the Gregorian mean year of 365.2425 days differ by 0.002%. That is small, but it is exactly the kind of difference that sits undetected in a chain of conversions, and it can be measured rather than assumed.

Take the nuclides where the two evaluations recommend the same number of years, so that no evaluation difference remains and only the conversion is left.

The Gregorian conversion leaves a residual at the rounding of the quoted digits; the Julian one leaves twenty times more. The seconds in the data therefore rest on the Gregorian mean year, and the calculator's own year input uses the same 365.2425 days. The two agree, which is the only thing that matters here — a tool whose year differed from its data's year would be wrong by 0.002% in a way no test of the arithmetic would ever reveal.

Worked cases

One per mode, and then some. The half-life is taken as given in these — it was the subject of the previous section, and mixing the two would leave any disagreement unattributable. What is checked here is that the arithmetic the tool performs on that half-life is the arithmetic the decay law calls for, end to end, including the secondary figure the screen prints alongside the answer.

Required refusals

Some inputs have no answer. Not a large answer, not a small one — none. A calculator that produces a number there is worse than one that produces nothing, because the number will be written down.

The number on the screen

The checks above exercise the engine. A separate check drives the published calculator in a browser, picks the nuclide, enters the values, presses the button and reads the figure the screen prints, comparing it with the hand calculation and allowing only the declared display rounding. It also reads the secondary line, because the number of half-lives elapsed is the figure a reader uses to sanity-check the answer, and a report that only looked at the headline would not have looked at it.

Coverage and result

Coverage of the nuclide list is deliberately not claimed. The calculator offers 147 nuclides and this report compares 24 of them, chosen because DDEP has evaluated them and because they are the ones that appear in practice. A complete comparison is not possible: DDEP has not evaluated every nuclide, and claiming otherwise would be the kind of overreach this report exists to avoid. What the sample establishes is that the pipeline carrying half-lives from the evaluation into the calculation does not distort them.

What this does not establish

• That the half-life of a nuclide not listed above is right. The 24 compared here all agree; the rest are carried by the same pipeline, which is an argument, not a measurement.
• That decay is the only thing changing the activity. Leakage, adsorption on a container wall, and detector drift all look exactly like a shorter half-life in a two-point fit, and the calculator cannot tell them apart.
• That the daughter is negligible. Ingrowth is a different calculation, and for a nuclide such as Sr-90 or Cs-137 the daughter carries most of the dose.
• That the starting activity is what the certificate says. Everything here is conditional on the number entered.
• Fitness for a regulatory submission. This site does not replace an accredited calibration or measurement.

Reporting a disagreement

A disagreement with a source you hold is worth sending — including a half-life, since the sample above is not the whole list. Contact details are on the contact page, the converter's report is at validation of the unit converter, and the method behind the rest of RadCalc is on the methods page.