Decay and half-life

The one equation

Radioactive decay is a first-order process: the number of transformations per second is proportional to the number of atoms present, and nothing else. Temperature, pressure, chemical form and dilution do not change it. That gives A(t) = A₀·e^−λt, where λ is the decay constant, ln 2 divided by the half-life. Every mode of this calculator is that equation solved for a different unknown.

Because the exponent depends only on the ratio of elapsed time to half-life, it is often easier to think in half-lives than in years. One half-life leaves 50%, two leave 25%, seven leave under 1%, and ten leave about one part in a thousand. The rule of ten half-lives is why waste storage periods are quoted the way they are.

Finding a half-life from two measurements

Two activity measurements separated by a known interval determine a half-life exactly: T½ = ln 2 · Δt / ln(A₀/A₁). The arithmetic is trivial and the interpretation is not. The calculation cannot distinguish decay from anything else that removed activity between the two readings — a source leaking, a tracer adsorbing onto the container wall, a detector drifting, or a geometry that changed when the sample was repositioned. All of those produce an apparent half-life shorter than the true one.

For that reason the result here is shown against the published half-life for the nuclide you picked. A difference of a few percent over a short interval usually means measurement uncertainty. A difference of tens of percent usually means something other than decay is happening, and the useful response is to look for it rather than to report a new half-life.

Choosing the interval

An interval much shorter than the half-life gives a ratio close to 1, and the logarithm of a number close to 1 amplifies counting uncertainty enormously. An interval much longer than the half-life drives the second measurement into background. The best precision comes from an interval of roughly one to three half-lives, where the ratio is between about 2 and 8.

Decay chains and ingrowth

When a daughter is itself radioactive, its activity starts at zero, rises as the parent feeds it, peaks, and then follows the parent down. The Bateman equations describe this, and the shape depends on which half-life is longer. When the parent is much longer-lived, the daughter reaches secular equilibrium and the two activities become equal — this is why a sealed Cs-137 source is really emitting the 662 keV photon of its daughter Ba-137m, and why a Sr-90 source is a Y-90 beta source in practice.

The chain calculation here assumes a simple series with no branching. Real chains in the uranium and thorium series branch, and a branching ratio has to be carried through each step.

What the answer assumes

A single nuclide, no ingrowth from a parent unless you build the chain explicitly, no losses other than decay, and a half-life taken from the evaluated nuclear data. Half-life values themselves carry uncertainty, and for a few nuclides different evaluations differ by enough to matter — Sr-90 is quoted between about 28.8 and 28.9 years depending on the compilation, which shifts a thirty-year projection by roughly half a percent.