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. For dosing, the note on decay correction for radiopharmaceuticals gives Tc-99m and F-18 at 15-minute steps.

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.

This calculator follows one nuclide; it does not build chains. The Bateman solution is applied along a single branch in the note on natural decay series. 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, 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.

Worked examples

A technetium-99m dose drawn later in the morning

A vial is calibrated at 800 MBq of Tc-99m at 07:00. The first patient is booked for 10:30 and another for 13:00. How much activity is left in the vial at each time?

Six hours is almost exactly one half-life of Tc-99m, so the vial holds half its calibrated activity by early afternoon. The same calculation run backwards gives the activity to order: a dose needed at 13:00 must be drawn as about twice that amount at 07:00. The vial is still measured in the dose calibrator before injection; the calculation tells you what to expect. Write the clock time of each draw on the syringe label as well as the activity. Without it the activity at injection cannot be reconstructed afterwards, which is the first thing asked when a scan looks wrong.

Identifying a vial from two measurements

An unlabelled vial is found in a hot lab. In the same counting position its net count rate is 1520 s⁻¹, and 24 hours later 1181 s⁻¹. Enter both rates and the interval in the “Find the half-life” mode.

The result matches Mo-99, but Au-198 is close by. Over one day the rate only falls by about a fifth, so a one-percent error in either count rate moves the half-life by about 4%, which is wider than the gap between the two candidates. A gamma spectrum settles the question: Mo-99 shows its 740 keV line and the 140 keV line of its Tc-99m daughter, Au-198 a single line at 412 keV. Counting the vial again a few days later helps as well: the longer the interval, the less a counting error moves the half-life, as the section on choosing the interval explains.

FAQ

How do I calculate the remaining activity after a given time?

Use A = A₀·exp(−λt) with λ = ln 2/T½, which is the same as A = A₀·2^(−t/T½). Time and half-life must be in the same unit. The formula is exact for a single nuclide with no parent feeding it.

How long does it take for activity to fall to one thousandth?

About ten half-lives, because 2¹⁰ = 1024. For I-131 that is 80.0 days. Decay-in-storage rules such as US 10 CFR 35.92, which covers nuclides with half-lives up to 120 days, do not fix a number of half-lives; they require that the waste cannot be distinguished from background in a survey before it is released.

What is the difference between half-life and mean life?

The half-life T½ is the time for half of the atoms to decay; the mean life τ = 1/λ = T½/ln 2 is the average time an atom survives, about 1.443 half-lives. After one mean life the activity has fallen to 1/e, about 37% of the start. Neither should be confused with the effective half-life used in nuclear medicine, which also counts biological removal: 1/T_eff = 1/T_phys + 1/T_bio.

Why does my half-life from two measurements come out wrong?

The two count rates must be net of background, taken in the same geometry and corrected for dead time, and the interval should be a large fraction of a half-life. Over a short interval the ratio is close to one and a small counting error moves the result a lot. A mixture of nuclides gives an apparent half-life that drifts as the shorter-lived component disappears.

Can temperature, pressure or chemistry change a half-life?

Not measurably for alpha and beta emitters. For nuclides that decay by electron capture, such as Be-7, the chemical form changes the half-life by a fraction of a percent at most, because the decay depends on electron density at the nucleus. The evaluated half-life is the right value for any practical calculation.