Guides / Radiation safety math

Radioactive decay and half-life

The decay law in plain terms: how to work out the activity left after a given time, how long a source takes to fall to a given level, and the mistakes to avoid.

Practical guide

By Steven Michael Barber · Published October 10, 2026 · Updated October 25, 2026 · 5 min read

A single radionuclide on its own, with nothing added and no radioactive daughters growing in, gets weaker with time at a rate fixed by the isotope and nothing else. Heat, pressure and chemistry do not change it in any way that matters for practical work. That makes decay one of the most predictable calculations in health physics: if you know the half-life and the activity on one date, you can work out the activity on any other date. This guide shows the formula, the two questions people most often ask of it, and a few traps.

The decay law

The activity of one isolated parent nuclide falls exponentially:

A = A₀ × e−λt

• A₀ is the activity at the start (the reference date on a source certificate, for example).
• A is the activity after an elapsed time t.
• λ (lambda) is the decay constant, the probability per unit time that a given atom decays.

The decay constant and the half-life T½ are two ways of saying the same thing: λ = ln 2 ÷ T½, and ln 2 ≈ 0.693. Substituting gives a form that is easier to reason about:

A ÷ A₀ = (½)t/T½

After one half-life, half the activity remains. After two, a quarter. After ten half-lives, the fraction is 1 ÷ 1,024, a little under 0.1 percent. The ratio t ÷ T½, the number of half-lives elapsed, is the only thing that matters, so the units of time cancel as long as you use the same unit for both.

A related figure is the mean life, τ = 1 ÷ λ, which is about 1.443 times the half-life. It is the average time an atom survives before decaying, and it shows up in some textbook problems.

Half-lives used here

The calculator's half-lives are evaluated values as listed on NuDat from the National Nuclear Data Center (evaluation dates vary, so the last digit can differ from the newest listing). A few that come up often:

The calculator uses a year of 365.25 days (the Julian year). Using 365 days instead makes a negligible difference over a few years but is worth knowing if you compare results to many decimal places.

Worked example: how much activity is left?

A cesium-137 gauge source. A portable moisture-density gauge typically holds a cesium-137 source of 8 to 10 mCi, according to the NRC (the 40 to 50 mCi figure belongs to the gauge's separate americium-241:beryllium neutron source). Take one certified at 8 mCi. Ten years later, how much is left?

The number of half-lives elapsed is 10 ÷ 30.08 ≈ 0.3324. The fraction remaining is (½)^0.3324 ≈ 0.7942, so the activity is 8 × 0.7942 ≈ 6.35 mCi. After a decade the source has lost only about a fifth of its activity, which is why long-lived gauges keep working for decades and why disposal still has to be planned.

An iridium-192 radiography source. A 100 Ci source arrives on a known date. After 30 days, 30 ÷ 73.829 ≈ 0.406 half-lives have passed, and (½)^0.406 ≈ 0.7545, so about 75.5 Ci remains. Radiographers recalculate exposure times as the source decays, because the same film or detector needs about a third more exposure time at 75 Ci than at 100 Ci.

A technetium-99m dose. A 20 mCi vial is prepared in the morning. Six hours later, almost exactly one half-life (6 ÷ 6.0067 ≈ 0.9989), about 10.0 mCi remains. After 24 hours, about four half-lives, only 1.25 mCi is left, roughly 6 percent of the starting activity.

Worked example: how long until it falls to a given level?

Turn the formula around to solve for time:

t = ln(A₀ ÷ A) ÷ λ = T½ × log₂(A₀ ÷ A)

Suppose a facility replaces its iridium-192 radiography sources once they fall from 100 Ci to 30 Ci. How long is that?

1. The ratio A₀ ÷ A is 100 ÷ 30 ≈ 3.333.
2. ln(3.333) ≈ 1.204.
3. λ = 0.693 ÷ 73.829 days ≈ 0.009389 per day.
4. t ≈ 1.204 ÷ 0.009389 ≈ 128.2 days, or about 1.74 half-lives.

The same method answers questions like "when will this source be below the activity my license or procedure names?" or "how long must this waste be held before it can be surveyed for release?" The answer always depends on the starting activity being right, so start from the certificate value and its reference date, not from a remembered number. Elapsed time alone does not authorize a release: holding and releasing waste follow the license's procedure, with surveys, and any radioactive daughters count too.

Estimating a half-life from two measurements

If you measure the same sample twice, you can solve for the half-life instead: T½ = ln 2 × t ÷ ln(A₀ ÷ A). Say a sample's net count rate drops from 1,000 to 800 counts per minute over 2 days. Then T½ = 0.693 × 2 ÷ ln(1.25) = 1.386 ÷ 0.2231 ≈ 6.21 days.

Counts per minute work here because only the ratio matters, but the counts must be net of background and taken with the same detector and geometry. With only two points close together in time, counting statistics can move the answer a lot. More points over a longer span give a better estimate.

Common mistakes

• Mixing time units. A half-life in days with an elapsed time in hours gives a result off by a factor of 24 in the exponent. Convert first.
• Using the decay constant from the wrong base. λ = ln 2 ÷ T½ uses the natural log. If you work in powers of ½, use t ÷ T½ directly and skip λ.
• Forgetting daughters. The simple law covers one isotope. Some sources grow radioactive daughters, so the total activity or dose rate does not follow the parent's curve. Cesium-137 reaches equilibrium with its 2.55-minute daughter barium-137m within about half an hour, so its gamma output tracks the cesium half-life; radium-226 and its daughters are more complicated. The calculator does not model daughter in-growth.
• Expecting it to reach zero. Exponential decay never reaches exactly zero. Ten half-lives is a common rule of thumb for "decayed to under 0.1 percent," not a guarantee that a source is gone.

Connecting decay to dose rate

As long as the photon spectrum, any equilibrium with daughters, the geometry and the shielding stay the same, the gamma dose rate is proportional to the activity and falls on the same curve. A source with ingrowing daughters may not follow it. A cesium-137 source, whose barium-137m stays in equilibrium, that read 8.25 mR/h at 2 meters when new would read about 79 percent of that, roughly 6.6 mR/h, ten years later at the same distance. Our guide to gamma dose rate, distance and shielding covers that formula.