Radioactive Decay and Half-Life Explained: Carbon Dating to Smoke Alarms

Radioactive decay is the process by which an unstable atomic nucleus releases energy by emitting an alpha particle, a beta particle or a gamma ray, or by capturing one of its own electrons, and turns into a different nucleus or a lower energy state. No one can say when a single atom will decay, but a large group of identical atoms loses a fixed fraction per unit time, so the amount left follows N(t) = N₀ e^(−λt). The half-life, t½ = ln 2 / λ, is the time for half of the atoms to decay: about 5,700 years for carbon-14, about 433 years for the americium-241 in an ionisation smoke alarm, and about 6 hours for the technetium-99m used in hospital scans.

This post sits next to our explainer on E = mc² and what Einstein's equation really means, which covers where the energy of a decay comes from. Here we go one level deeper into the arithmetic: three worked examples with real numbers, the decay chain that puts radon in basements, and where the reference half-lives actually come from.

A note on numbers. Half-lives and decay modes in this post come from the Evaluated Nuclear Structure Data File (ENSDF), retrieved on 24 October 2026 through the IAEA Nuclear Data Section's Live Chart of Nuclides data service. Unit definitions come from NIST Special Publication 811, the BIPM SI Brochure and the US EPA. Radon figures come from the US EPA. Radiocarbon reporting conventions come from the IntCal working group, the IntCal20 paper and a University of Waikato Radiocarbon Dating Laboratory sample report. Smoke alarm and medical isotope statements come from the US EPA and the US NRC. Where a textbook or agency page rounds differently from the evaluated data, we say so. Starting amounts that we chose ourselves, such as a sample's activity, are labelled as illustrative. Every calculation was rerun with the short Python script in the walkthrough.

Safety note. This post explains the physics; it is not handling guidance. Never open, dismantle or tamper with a smoke detector or any device that carries a radiation symbol, and never try to remove, collect or handle radioactive material. The US EPA says ionisation smoke detectors pose no health risk as long as they are not damaged and are used as directed, and it says never to tamper with one or try to remove the americium. For disposal, follow the manufacturer's instructions or your local rules. For radon, follow your national regulator's testing and fixing guidance. If you think you have found a damaged source or an unknown radioactive item, leave it alone and contact your state or national radiation control program.

Key takeaways

• Random for one, predictable for many: each nucleus of a given isotope has the same constant probability per second of decaying, λ. That makes the timing of a single decay unpredictable but the behaviour of trillions of atoms smooth and exponential.
• The two formulas: N(t) = N₀ e^(−λt) gives the number of atoms left, and t½ = ln 2 / λ ≈ 0.693 / λ links the decay constant to the half-life. Activity, the decay rate, follows the same curve: A = λN.
• Units: one becquerel (Bq) is one decay per second. One curie (Ci) is exactly 3.7 × 10¹⁰ Bq, so 1 microcurie is 37,000 Bq and 1 picocurie is 0.037 Bq.
• Carbon-14: the evaluated half-life is 5,700 ± 30 years. A sample with 40% of the modern carbon-14 level is about 7,500 years old by the decay law, but archaeologists report a conventional radiocarbon age and then calibrate it against a curve such as IntCal20, because atmospheric carbon-14 has not been constant.
• Americium-241: with a 432.6 year half-life, a smoke alarm source loses only about 1.6% of its activity over a ten year service life. Alarms are replaced because the alarm wears out, not because the source runs down.
• Technetium-99m: its 6.0 hour half-life is why it is so useful in diagnostic imaging. About 94% of it has decayed within a day, so it is delivered as a parent, molybdenum-99 with a 65.9 hour half-life, inside a generator that is eluted at the hospital.
• Radon: radon-222 (3.8 day half-life) is a link in the uranium-238 decay chain. The EPA estimates about 21,000 US lung cancer deaths a year from radon and recommends fixing homes at or above 4 pCi/L (150 Bq/m³).
• Reference data: half-lives are measured by labs worldwide and evaluated into ENSDF and the Decay Data Evaluation Project. NIST's own half-life page now points readers to the DDEP data, which superseded the values NIST once published there.

What is radioactive decay?

Every atom has a nucleus made of protons and neutrons. Some combinations are stable and stay as they are forever, as far as anyone can measure. Others carry too much energy, or the wrong balance of protons and neutrons, and at some point they rearrange themselves into something more stable, releasing energy as they do. That rearrangement is radioactive decay. The US Nuclear Regulatory Commission's radiation basics page puts the practical consequence simply: through radioactive decay, radioisotopes lose their radioactivity over time, and that loss is measured in half-lives.

The energy released comes from mass. The products of a decay weigh slightly less than the parent nucleus, and the difference appears as kinetic energy of the emitted particles and the recoiling daughter, plus any gamma ray. Our E = mc² explainer covers that accounting.

The four decay modes you will meet most

OpenStax Chemistry 2e, the free college textbook from Rice University, lists the common modes of decay. Here is what each does to the nucleus.

A few notes on the table:

• Alpha particles are heavy and slow by radiation standards. The EPA's RadTown page on smoke detectors says alpha particles are very heavy relative to other types of radiation and cannot travel very far. A sheet of paper or the outer layer of skin stops them, which is why alpha emitters are a concern mainly if they get inside the body. The EPA's americium-241 page says the same about americium: it poses a more significant risk if swallowed or inhaled.
• Beta decay changes the element. Carbon-14 becomes nitrogen-14 when one of its neutrons turns into a proton. OpenStax writes it as carbon-14 going to nitrogen-14 plus an electron.
• Electron capture and positron emission have the same effect on the nucleus. OpenStax notes that both reduce the atomic number by one without changing the mass number, and both happen in "proton rich" nuclei. Which one occurs is, in the textbook's words, difficult to predict. Potassium-40 is a good illustration: the ENSDF evaluation gives 89.28% beta minus decay to calcium-40 and 10.72% electron capture or positron emission to argon-40.
• Gamma emission rarely happens alone. It usually follows an alpha or beta decay that leaves the daughter in an excited state. When the excited state lives long enough to be measured, it is called a metastable state or isomer, written with an "m", and its decay to the ground state is an isomeric transition. Technetium-99m is the famous one.

The NRC describes gamma rays as high energy waves that can travel great distances at the speed of light.

Why is decay random for one atom but predictable for many?

Here is the central idea. Every nucleus of a given isotope in a given state has the same, constant probability of decaying in the next second. That probability does not depend on how old the nucleus is, what happened to it before, the temperature, or what chemical compound it sits in (with tiny exceptions for electron capture, which depends slightly on the electrons around the nucleus). A carbon-14 atom that formed in the atmosphere last week has exactly the same chance of decaying tomorrow as one locked in a 10,000 year old piece of charcoal.

Call that probability per unit time λ, the decay constant. In a short interval dt, the chance that a particular nucleus decays is λ dt. For one atom, that is all you can say. It might decay in the next microsecond or survive for ten half-lives. There is no internal clock.

For a large number of atoms N, the expected number of decays in dt is λN dt. So the population shrinks at a rate proportional to its size:

dN/dt = −λN

That is a first order process, the same mathematics as a first order chemical reaction, and OpenStax makes exactly that point: radioactive decay follows first order kinetics. The solution is the exponential decay law.

How predictable is "predictable"? Counting statistics answer that. If decays are independent and random, the number you count in a fixed time follows a Poisson distribution, whose standard deviation is the square root of the mean. Count 100 decays and the relative uncertainty is about 1/√100 = 10%. Count 10,000 and it is about 1%. Count a million and it is 0.1%. Real samples contain trillions of atoms or more, so the fraction that decays in any interval is very well defined even though each single decay is random.

A useful number to keep in mind: the mean lifetime τ = 1/λ = t½ / ln 2 ≈ 1.4427 × t½. It is the average time a nucleus survives. For carbon-14, τ = 5,700 / 0.6931 ≈ 8,223 years.

What do N(t) = N₀ e^(−λt) and t½ = ln 2 / λ say?

Integrating dN/dt = −λN from time zero, when there are N₀ atoms, gives:

N(t) = N₀ e^(−λt)

The half-life is the time at which N(t) = N₀/2. Setting e^(−λt½) = 1/2 and taking natural logarithms:

t½ = ln 2 / λ ≈ 0.693 / λ

OpenStax gives the same pair of relationships. Because e^(−λt) = 2^(−t/t½), you can also write the law in a form that is often easier to use in your head:

N(t) = N₀ × (1/2)^(t / t½)

After one half-life, half remains. After two, a quarter. After three, an eighth. After ten half-lives, 1/1024, a little under 0.1%. That last figure is a handy rule of thumb: ten half-lives takes a source down by a factor of about a thousand, and twenty half-lives by a factor of about a million.

To go the other way, from a measured fraction remaining to an elapsed time, rearrange:

t = −(1/λ) ln(N/N₀) = (t½ / ln 2) × ln(N₀/N)

Every worked example below uses one of these three lines.

Activity, becquerels and curies

You rarely count atoms directly. You count decays. The activity of a sample is its decay rate:

A = λN

Because A is proportional to N, activity falls on exactly the same exponential curve: A(t) = A₀ e^(−λt). This is why a laboratory can measure a half-life by watching a counter's rate fall over time.

Two units are in common use:

• Becquerel (Bq). The SI unit of activity. The EPA defines one becquerel as one radioactive decay per second, and the BIPM SI Brochure lists the becquerel as a derived unit equal to one per second.
• Curie (Ci). The older unit, still used in the United States. NIST Special Publication 811 gives the conversion exactly: 1 Ci = 3.7 × 10¹⁰ Bq. The EPA explains its origin as approximately the number of decays per second in one gram of radium.

Because the curie is large, everyday values use prefixes. The EPA's radiation units page notes that results are usually shown in picocuries.

Activity is not dose. A becquerel tells you how many decays happen; it says nothing about how much energy reaches tissue, which depends on the kind of radiation, its energy, distance, shielding and whether the material is inside or outside the body. NIST SP 811 lists the separate units for absorbed dose (the gray, which replaced the rad, with 1 rad = 0.01 Gy) and dose equivalent (the sievert, which replaced the rem, with 1 rem = 0.01 Sv). This post works only with activity and half-life.

From activity to the number of atoms

Rearranging A = λN gives N = A / λ, which lets you answer questions like "how much americium is actually in that source?" The decay constant must be in the same time unit as the activity. For becquerels, which are per second, convert the half-life to seconds. We use that in worked example 2.

Worked example 1: carbon-14 dating, from a measured fraction to an age

How carbon-14 dating works

The Nobel Prize in Chemistry 1960 went to Willard F. Libby "for his method to use carbon-14 for age determination in archaeology, geology, geophysics, and other branches of science." The Nobel Prize summary sets out the idea. Carbon has a stable isotope, carbon-12, and a radioactive one, carbon-14. Carbon-14 forms in the atmosphere under cosmic radiation. Living things take in carbon from the atmosphere, directly or through the food chain, so while they are alive their carbon-14 to carbon-12 ratio tracks the atmosphere's. When an organism dies, that supply stops and its carbon-14 declines by radioactive decay at a fixed rate. According to the Nobel summary, Libby developed the method in 1949.

OpenStax walks through the same logic: carbon-14 decays by beta emission to nitrogen-14, so the ratio of carbon-14 to carbon-12 in a dead plant falls steadily after it dies.

Which half-life?

This is the first place where sources differ, so it is worth being precise.

The difference between 5,700 and 5,730 years is about half a percent, within the stated uncertainty of the evaluated value. For a physics calculation of how much carbon-14 is left after a given time, use the evaluated value. For archaeology, the reported radiocarbon age is a conventional quantity that is converted to a calendar age through a calibration curve, so the choice of half-life is absorbed in the calibration. More on that below.

Step 1: the decay constant

λ = ln 2 / t½ = 0.693147 / 5,700 y = 1.2160 × 10⁻⁴ per year

That means each carbon-14 atom has about a 0.012% chance of decaying in any given year. Its mean lifetime is 1/λ ≈ 8,223 years.

Step 2: a simple case, one quarter remaining

Suppose a piece of charcoal has a carbon-14 to carbon-12 ratio one quarter that of living material (an illustrative figure). One quarter is (1/2)², so two half-lives have passed:

t = 2 × 5,700 = 11,400 years

OpenStax uses the same reasoning with one half-life: a wooden object whose ratio is half that of a living tree is one half-life old.

Step 3: a realistic case, 40% remaining

Now suppose the measurement gives 40.0% of the modern level (illustrative). Use the rearranged law:

t = (t½ / ln 2) × ln(N₀/N) = 8,223 × ln(1 / 0.400) = 8,223 × 0.91629 ≈ 7,535 years

With the textbook 5,730 year half-life, the same fraction gives 8,267 × 0.91629 ≈ 7,575 years. The 30 year difference in half-life becomes a 40 year difference in age, which shows why half-life precision matters most for old samples.

Step 3b: the conventional radiocarbon age

A radiocarbon laboratory does not report the 7,535 year figure. A sample report published by the University of Waikato Radiocarbon Dating Laboratory states that its results follow Stuiver and Polach, 1977 (Radiocarbon 19, 355 to 363), based on the Libby half-life of 5,568 years with a correction for isotopic fractionation applied. The Libby mean life is 5,568 / ln 2 ≈ 8,033 years, so:

conventional age = 8,033 × ln(1 / F)

where F is the fraction of modern carbon after the fractionation correction. For our 40% sample, 8,033 × 0.91629 ≈ 7,360 radiocarbon years BP. The Waikato example report shows the same arithmetic on a real shell sample: a fraction modern of 87.4 ± 0.4% gives 1,080 ± 40 BP, and 8,033 × ln(1 / 0.874) ≈ 1,082. The report also notes that this fraction is also known as percent modern carbon (pMC).

The deliberately "wrong" half-life does no harm, because the conventional age is only an intermediate number. Calibration converts it to calendar years, and the calibration curve is built from known age samples processed the same way.

Step 4: how precise is the measurement?

Counting statistics limit the method. If a lab measures carbon-14 by counting decays, the uncertainty in the fraction falls as one over the square root of the number of counts, and the age uncertainty follows. Differentiating the age formula gives a neat result: an error δF in the fraction F produces an age error of about τ × δF/F. With τ ≈ 8,223 years, a 1% uncertainty in the fraction means about 82 years of uncertainty in the age, whatever the age. Old samples have little carbon-14 left, so getting to a 1% measurement takes more material or more counting time. Modern laboratories often count carbon-14 atoms directly with accelerator mass spectrometry rather than waiting for them to decay, which needs far smaller samples.

Step 5: the calibration caveat

The decay law assumes that the starting ratio, N₀, was the same in the past as it is today. It was not. The IntCal working group, which publishes the internationally agreed calibration curves, says on its site that radiocarbon dates require calibration against known age material, normally dated by other methods such as tree rings (dendrochronology) for wood or uranium series methods for corals and cave formations. Its first comprehensive curve dates from 1986.

The current Northern Hemisphere curve, IntCal20, was published in the journal Radiocarbon (volume 62, issue 4) by Paula Reimer and colleagues, published online on 12 August 2020. The paper extends the curve to 55,000 calendar years before present and states that all ages in the paper and database are reported relative to AD 1950, which is 0 BP. A calibrated date is therefore not a single number but a probability distribution over calendar years, and wiggles in the curve can make one radiocarbon age correspond to several possible calendar ranges.

Practical consequences:

• Report what you measured, then calibrate. Laboratories report the conventional radiocarbon age with its uncertainty, then a calibrated range with the curve named (for example "calibrated with IntCal20").
• Hemisphere and reservoir matter. There are separate curves for the Southern Hemisphere and for marine samples, because ocean carbon is older on average than atmospheric carbon.
• The method has a range. After about ten half-lives, roughly 57,000 years, only about 0.1% of the original carbon-14 remains, which is why IntCal20 stops at 55,000 calendar years.
• The modern era is special. Fossil fuel burning dilutes atmospheric carbon-14 with old carbon, and atmospheric nuclear tests in the mid twentieth century raised it sharply for a time. Those effects are why recent samples need special treatment.

Worked example 2: the americium-241 source in an ionisation smoke alarm

How the alarm uses decay

The EPA's RadTown page explains the mechanism. Ionisation smoke detectors use americium-241 as a source of alpha particles. The alpha particles ionise air molecules between two charged plates, which sets up a small, constant flow of ions, an electric current. When smoke enters the chamber, it disrupts that flow, and the alarm sounds. The EPA says the americium is encased in a layer of foil and ceramic that stops the alpha particles from leaving the detector, and that because of this shielding the detector poses no radiation health risk when properly handled. The NRC's radiation basics page also mentions smoke detectors as an example of the very small amounts in which alpha emitting materials are used.

The EPA's safety advice is plain and worth repeating: never tamper with an ionisation smoke detector, and never attempt to tamper with or remove the americium. Most detectors, the EPA says, are certified for a useful life of ten years, so check the expiration date when you change the batteries. On disposal, the RadTown page says there are no special disposal instructions for ionisation smoke detectors and that they may go in household garbage or a community recycling program, while the EPA's americium page says to follow manufacturer instructions or check with your local fire department. Rules vary by country and state. For example, the New South Wales EPA in Australia says small numbers can go in household rubbish or to a community recycling centre, but that disposing of ten or more at once may carry regulatory requirements.

The half-life, and a disagreement worth noticing

The EPA's 432.2 years sits inside the ENSDF value's stated uncertainty of 0.6 years, and the NSW figure is a rounding. None of this matters for a smoke alarm; it matters a great deal if you are calibrating an instrument. We use the evaluated 432.6 years.

Step 1: the decay constant

λ = 0.693147 / 432.6 y = 1.6023 × 10⁻³ per year

In seconds, the ENSDF record gives the half-life as 1.36515 × 10¹⁰ s, so:

λ = 0.693147 / 1.36515 × 10¹⁰ s = 5.0774 × 10⁻¹¹ per second

Step 2: how much activity is lost over a ten year service life?

Take an illustrative source of 37 kBq, which is exactly 1 microcurie. (Detector labels state the activity of each model; this round number is for the arithmetic.)

A(10 y) = 37,000 × e^(−1.6023 × 10⁻³ × 10) = 37,000 × 0.98410 = 36,412 Bq

After ten years the source still delivers 98.4% of its starting activity. It has lost about 1.6%. After 50 years it would still be at e^(−0.0801) ≈ 92.3%.

So a ten year replacement interval has nothing to do with the source running down. Detectors age for other reasons, including electronics, dust and the sensing chamber itself, which is why the EPA points to a certified useful life and an expiration date.

Step 3: how long until the activity falls to 90%?

t = ln(1 / 0.90) / λ = 0.10536 / 1.6023 × 10⁻³ ≈ 65.8 years

Long after any detector would be replaced.

Step 4: how many atoms, and what mass, is 37 kBq?

Use N = A / λ with λ in per second:

N = 37,000 / 5.0774 × 10⁻¹¹ ≈ 7.29 × 10¹⁴ atoms

The atomic mass of americium-241 is about 241.06 g/mol, so the mass is:

m = N × 241.06 / 6.02214 × 10²³ ≈ 2.9 × 10⁻⁷ g

That is about 0.29 micrograms, a speck you could not see. The example shows why activity, not mass, is the useful measure for radioactive sources: a fraction of a microgram of a 433 year isotope gives tens of thousands of decays every second.

Step 5: what does it decay into?

Americium-241 decays by alpha emission to neptunium-237, which has a half-life of 2.144 million years (ENSDF). Because neptunium-237 lives about 5,000 times longer than its parent, its own activity builds up extremely slowly.

Worked example 3: technetium-99m and why short half-lives matter in medicine

What technetium-99m is

The EPA's technetium page describes technetium-99m as a short lived form of technetium used as a medical diagnostic tool, with a half-life of about six hours, which "does not remain in the body or the environment for long." The evaluated ENSDF value is 6.0072 hours. The metastable state sits at 142.68 keV above the ground state and decays 99.9963% by isomeric transition to technetium-99, with a tiny 0.0037% beta minus branch. The isomeric transition passes through a level at 140.51 keV, which is the source of the 140 keV gamma ray that gamma cameras image.

Its daughter, technetium-99, is a very different isotope: the ENSDF half-life is 211,100 years, and the EPA notes that, because of its long half-life, technetium-99 remains in the environment for an extended period. The two isotopes have the same mass number and the same element; they differ only in nuclear energy state. That is a neat demonstration that "half-life" belongs to a specific nuclear state, not to an element.

Why short is good

For diagnostic imaging, you want three things: enough gamma rays to make an image in the time the scan takes, a gamma energy that passes out of the body and is easy to detect, and as little lingering activity afterwards as possible. A six hour half-life is a good compromise between the first and the last. The pure isomeric transition helps too: most decays produce a photon rather than charged particles that would deposit energy in tissue without contributing to the image.

Step 1: the decay constant

λ = 0.693147 / 6.0072 h = 0.11539 per hour

Step 2: decay during the working day

Suppose a dose is prepared at 07:00 with an activity of 740 MBq (20 mCi), an illustrative figure chosen for round arithmetic, not clinical guidance.

Two conclusions follow. First, a radiopharmacy has to correct for decay between preparation and administration; three hours of delay costs 29% of the activity. Second, by the next day about 94% of the technetium-99m has decayed, and after two days less than half a percent remains. That is "does not remain in the body for long" in numbers. (Biological clearance also removes some of the material from the body, which shortens the effective half-life further; the physical half-life is an upper bound.)

Step 3: the delivery problem, and the generator solution

A six hour half-life is wonderful in the patient and terrible in a supply chain. Ship technetium-99m across a continent and much of it is gone before it arrives. The solution is to ship the parent instead. Molybdenum-99, with an ENSDF half-life of 65.924 hours, decays by beta minus to technetium, much of it via the 99m state. Hospitals receive a "generator" containing molybdenum-99 and draw off (elute) the technetium-99m as needed.

How long does a generator last? After one week, 168 hours:

fraction of Mo-99 left = 2^(−168 / 65.924) = 2^(−2.548) ≈ 0.171

About 17% of the molybdenum remains, so a generator's output falls steadily through the week after delivery, and each elution yields less than the one before at the same point in the regrowth cycle.

Step 4: transient equilibrium

When a parent lives longer than its daughter but not enormously longer, the two settle into transient equilibrium: after several daughter half-lives, the daughter's activity tracks the parent's, falling with the parent's half-life, at a fixed ratio. For an ideal parent and daughter with decay constants λp and λd, that ratio is:

A_daughter / A_parent = λd / (λd − λp)

With λd = 0.11539 per hour and λp = 0.693147 / 65.924 = 0.010514 per hour, the ratio is 0.11539 / 0.10487 ≈ 1.10. In a real molybdenum generator, that ideal figure is multiplied by the fraction of molybdenum decays that actually feed the 99m state, so the technetium-99m activity ends up close to, rather than above, the molybdenum activity. Either way, the daughter regrows quickly after each elution, which is why a generator can be "milked" again later the same day.

Other short lived medical isotopes

OpenStax's table of medical radioisotopes lists iodine-131 (ENSDF half-life 8.0252 days, beta minus) for thyroid scans and treatment. Fluorine-18, used in positron emission tomography, has an ENSDF half-life of 109.77 minutes, decaying by electron capture or positron emission. Its short half-life limits how far it can travel between production and use.

How are half-lives measured, and where do the reference values come from?

Two measurement strategies

How you measure a half-life depends on how long it is compared with a human lifetime.

For the middle range, the main enemies are instrument drift, background and impurities with different half-lives, so labs measure over several half-lives with very stable detectors.

For long half-lives, watching the activity fall is hopeless: americium-241 loses 0.16% a year. Instead labs determine N and A. The precision then depends on how well you know the mass, isotopic composition and counting efficiency, which is why national metrology institutes are involved.

The evaluated data chain

Raw measurements are not reference data. Different experiments disagree, some have underestimated uncertainties, and new measurements arrive every year. Evaluation is the process of collecting all published measurements, judging them, and producing a recommended value with an uncertainty.

• NIST. The NIST Radiation Physics Division's Radioactivity Group maintained a page of radionuclide half-life measurements, NIST Standard Reference Database 120. The page now states that the data that formerly appeared there has been superseded by the data on DDEP, the Decay Data Evaluation Project, and was last updated on 18 January 2022. NIST still measures; the recommended values live in the evaluated databases.
• ENSDF. The Evaluated Nuclear Structure Data File is the international evaluated database for nuclear structure and decay. Each record carries the evaluator and a literature cut-off date. The values in this post show how old some evaluations are: the carbon-14 record has a cut-off of 1 July 1990, americium-241 of 30 September 2015, and radon-222 of 25 September 2023.
• NNDC. The National Nuclear Data Center at Brookhaven National Laboratory hosts NuDat 3, which it describes as a tool to search and plot nuclear structure and decay data interactively, along with ENSDF, the Evaluated Nuclear Data File (ENDF) for reactions, and the EXFOR experimental reaction database.
• IAEA. The IAEA Nuclear Data Section's Live Chart of Nuclides presents the same evaluated data as an interactive chart, and its data service returns ground state and level properties as CSV, which is how the values in this post were retrieved.

When you need a half-life for anything that matters, go to one of these evaluated sources, note the evaluation date, and record the uncertainty.

What is a decay chain, and what is secular equilibrium?

Decay chains

Many daughters are themselves radioactive. A decay chain, or decay series, is the sequence of decays from a long lived parent down to a stable end product. OpenStax describes these as radioactive families: the naturally occurring radioactive isotopes of the heaviest elements fall into chains of successive decays. Uranium-238, for example, decays in a series of steps into lead-206, which OpenStax uses to date rocks.

The part of the uranium-238 chain that matters for indoor air runs from radium-226 through radon-222. All half-lives below are ENSDF values.

Secular equilibrium

When a parent lives vastly longer than every daughter in a stretch of the chain, the system reaches secular equilibrium: after enough time, every member of that stretch has the same activity as the parent. The reasoning is short. The daughter is produced at the parent's rate, λp Np, and destroyed at its own rate, λd Nd. Its population grows until destruction matches production, at which point λd Nd = λp Np, so the activities are equal.

The growth follows a simple law when the parent's activity is effectively constant:

A_daughter(t) = A_parent × (1 − e^(−λd t))

Worked example: seal a radium-226 source in a container so that no radon escapes. How long before radon-222 reaches equilibrium?

• After one radon half-life (3.82 days): 50% of the radium activity.
• After one week: 1 − 2^(−7 / 3.8222) = 1 − 0.281 = 71.9%.
• After 30 days: 1 − 2^(−30 / 3.8222) = 1 − 0.0043 = 99.6%.

Then the short lived progeny, with half-lives of minutes, catch up with the radon within a few hours. This is why sealed sources and samples are often held for a few weeks before measurement, and why radon measurements must account for whether the progeny are in equilibrium with the gas.

Secular equilibrium is different from the transient equilibrium in the technetium generator. In secular equilibrium the parent is effectively immortal on the timescale of interest and the activities become equal. In transient equilibrium the parent decays noticeably, and the daughter follows it down at a fixed ratio above one (before any branching correction).

Why does radon matter at home?

Radon-222 is the one member of the uranium chain that is a gas. Formed in soil and rock, it can move through cracks and gaps into buildings, where it and its short lived progeny are breathed in. The alpha emitting progeny then decay inside the lungs. The EPA's health risk page makes the scale clear:

• The EPA's best estimate is about 21,000 lung cancer deaths a year in the United States from radon, with an uncertainty range of 8,000 to 45,000, consistent with the estimates of the BEIR VI report.
• About 2,900 of those deaths are among people who have never smoked.
• The EPA calls radon the second leading cause of lung cancer, after smoking.
• The EPA gives the average indoor radon level as about 1.3 pCi/L (50 Bq/m³) and the average outdoor level as about 0.4 pCi/L (15 Bq/m³).
• It recommends fixing your home if the level is 4 pCi/L (150 Bq/m³) or more, and considering fixing between 2 and 4 pCi/L (75 to 150 Bq/m³), while noting that reducing levels below 2 pCi/L is difficult.

The EPA's radon hub page says testing is the only way to know your level of exposure and that all homes should be tested. If you live outside the US, use your national authority's reference level and guidance; the EPA page itself points to the World Health Organization's international radon project.

Worked example: the 3.8 day half-life and a radon test

The radon half-life explains a practical detail of short term tests. A passive test device collects radon over a few days and is then sealed and posted to a laboratory. The radon trapped in it keeps decaying in the mail. Suppose (illustratively) the device is analysed four days after it was sealed:

fraction remaining = 2^(−4 / 3.8222) = 0.484

More than half of the collected radon has decayed before the lab counts it. Labs correct for this using the recorded sealing time, which is why test instructions insist that you record start and end times accurately and send the device promptly. An unrecorded delay of a few days would bias the result badly.

Common isotopes compared: half-life, decay mode and everyday context

All half-lives and decay modes are ENSDF values retrieved through the IAEA data service on 24 October 2026. "Context" describes where you are likely to read about the isotope; it is not a statement of risk.

The range runs from microseconds to more than the age of the universe. The same equation covers all of it.

Choosing which half-life matters: a decision table

Walkthrough: reproduce every number in Python

You can check every number above with a short script. The script needs only the Python standard library. Treat it like any other software: keep it in version control and pin anything you add.

Step 1: create a clean environment

python3 -m venv .venv
source .venv/bin/activate
python -m pip install --upgrade pip
python -m pip freeze > requirements.txt

The standard library is enough for the arithmetic. If you later add plotting or fitting libraries, install them inside this environment and run python -m pip freeze > requirements.txt again so the next machine installs exactly the same versions with python -m pip install -r requirements.txt.

Step 2: fetch evaluated half-lives

The IAEA Live Chart data service returns ENSDF ground state data as CSV. This command fetches americium-241:

curl -s "https://nds.iaea.org/relnsd/v1/data?fields=ground_states&nuclides=241am" -o am241.csv
head -c 400 am241.csv

The half_life, unc_hl and unit_hl columns hold the value, its uncertainty in the last digits and the unit; half_life_sec gives the half-life in seconds; decay_1 and decay_1_% give the main decay mode and branching. For a metastable state such as technetium-99m, request fields=levels&nuclides=99tc and read the row whose energy is 142.68 keV. Record the Extraction_date column with your results so you know which evaluation you used.

Step 3: write the calculations

Save this as decay.py:

import math

LN2 = math.log(2)
NA = 6.02214076e23  # Avogadro constant, exact in the SI

def lam(t_half):
    return LN2 / t_half

def remaining(t, t_half):
    return 2 ** (-t / t_half)

def age_from_fraction(f, t_half):
    return (t_half / LN2) * math.log(1 / f)

# Worked example 1: carbon-14 (ENSDF 5700 y; textbook 5730 y)
for th in (5700, 5730):
    print(f"C-14 t1/2={th} y: lambda={lam(th):.4e}/y, "
          f"mean life={1/lam(th):.0f} y, "
          f"age at 40%={age_from_fraction(0.40, th):.0f} y, "
          f"age at 25%={age_from_fraction(0.25, th):.0f} y")

# Worked example 2: americium-241, illustrative 37 kBq (1 microcurie)
th_am_y = 432.6
th_am_s = 1.36515261766e10  # ENSDF half_life_sec
A0 = 37000.0
print(f"Am-241 lambda={lam(th_am_y):.4e}/y, {lam(th_am_s):.4e}/s")
for t in (10, 50):
    print(f"  after {t} y: {A0*remaining(t, th_am_y):.0f} Bq "
          f"({100*remaining(t, th_am_y):.2f}%)")
print(f"  time to 90%: {math.log(1/0.9)/lam(th_am_y):.1f} y")
N = A0 / lam(th_am_s)
print(f"  atoms: {N:.3e}, mass: {N*241.0568/NA*1e6:.3f} micrograms")

# Worked example 3: technetium-99m and molybdenum-99
th_tc, th_mo = 6.0072, 65.924  # hours
for t in (0, 1, 3, th_tc, 12, 24, 48):
    print(f"  Tc-99m at {t:6.3f} h: {740*remaining(t, th_tc):7.1f} MBq")
print(f"Mo-99 left after 168 h: {remaining(168, th_mo):.3f}")
ld, lp = lam(th_tc), lam(th_mo)
print(f"ideal transient equilibrium ratio: {ld/(ld-lp):.3f}")

# Radon: ingrowth from radium-226 and decay in transit
th_rn = 3.8222  # days
for d in (3.8222, 7, 30):
    print(f"Rn-222 ingrowth after {d} d: {100*(1-remaining(d, th_rn)):.1f}%")
print(f"Rn-222 left after 4 d in the post: {remaining(4, th_rn):.3f}")

# Unit check
print(f"1 pCi/L = {0.037*1000:.0f} Bq/m3; 4 pCi/L = {4*37:.0f} Bq/m3")

Step 4: run it and compare

python decay.py

The output should match the figures in this post: a carbon-14 mean life of 8,223 years and an age of 7,535 years at 40% (7,575 years with the 5,730 year value); 36,412 Bq of americium after ten years, 65.8 years to reach 90%, and about 0.29 micrograms for 37 kBq; 523 MBq of technetium-99m after three hours and 46 MBq after a day; 17.1% of the molybdenum left after a week; and 71.9% radon ingrowth after one week. The last line shows that 4 pCi/L is 148 Bq/m³, which the EPA rounds to 150.

Step 5: keep the script's dependencies healthy

A calculation script is software, and once it grows plotting and fitting libraries it has a dependency tree. If it lives in a repository next to a lab information system, an instrument integration service or a monitoring dashboard, scan it like the rest of your code. With the Safeguard CLI installed, safeguard scan --dir . scans the project directory, including requirements.txt, and writes JSON results under ./output.

Common mistakes

• Mixing time units. λ from a half-life in years gives decays per year; a becquerel is per second. Convert before you compute N = A/λ, or your atom count is off by a factor of about 31.6 million.
• Using log base 10 where the formula needs ln. t½ = ln 2 / λ uses the natural logarithm. With log₁₀ 2 = 0.301 instead of 0.693 you will be wrong by a factor of 2.3.
• Thinking a half-life means "gone after two half-lives." After two half-lives a quarter remains. Decay is exponential, not linear.
• Confusing activity with dose. Becquerels and curies count decays. Gray and sievert describe energy absorbed and its biological weighting. A high activity alpha source sealed in a smoke alarm and the same activity inhaled are very different situations.
• Quoting a radiocarbon age as a calendar date. A conventional radiocarbon age needs calibration with a named curve, such as IntCal20, and the result is a range.
• Using a textbook half-life as a calibration constant. Use an evaluated value from ENSDF, DDEP or NuDat, with its uncertainty and evaluation date. Agency summary pages can round or lag: compare EPA's 432.2 years with ENSDF's 432.6 for americium-241.
• Forgetting the daughters. A radium sample's activity includes its radon and progeny once they grow in. A technetium generator's output depends on the molybdenum. Treat a chain as a system.
• Blaming the source for a smoke alarm's end of life. The americium has lost under 2% after ten years. Replace alarms by the date on the label, because the device ages, not the source.

FAQ

What is radioactive decay in simple terms?

It is an unstable atomic nucleus changing into a more stable one by giving off energy, as an alpha particle, a beta particle, a gamma ray, or by capturing one of its own electrons. Each nucleus does this at a random moment, but a large group decays at a predictable, exponential rate.

What is a half-life?

The time it takes for half of the atoms of a radioactive isotope to decay. The NRC defines it as the time for one half of the atoms of a radioisotope to decay by emitting radiation. After each half-life, half of what was left remains.

What is the formula for radioactive decay?

N(t) = N₀ e^(−λt), where N₀ is the starting number of atoms, λ is the decay constant and t is time. The half-life is t½ = ln 2 / λ, and the same law can be written N(t) = N₀ × (1/2)^(t/t½). Activity follows the same curve, A = λN.

How does carbon dating work, and what half-life does it use?

Living things keep a carbon-14 level that matches the atmosphere; after death it declines by decay. Measuring the remaining fraction gives an age through the decay law. The evaluated half-life is 5,700 ± 30 years (OpenStax uses 5,730). Labs report a conventional radiocarbon age relative to AD 1950 and calibrate it with a curve such as IntCal20, which reaches back 55,000 calendar years.

Is the americium in a smoke detector dangerous?

The US EPA says there is no health risk from ionisation smoke detectors as long as they are not damaged and are used as directed. The americium is sealed in foil and ceramic that stops its alpha particles. Never tamper with a detector or try to remove the source, and follow manufacturer or local instructions for disposal.

Why do smoke alarms need replacing if americium lasts 432 years?

Because the alarm ages, not the source. Americium-241 loses only about 1.6% of its activity in ten years. The EPA says most detectors are certified for a useful life of ten years and advises checking the expiration date.

Why is technetium-99m used so much in medicine?

Its 6.0 hour half-life is long enough for imaging but short enough that about 94% has decayed within a day, and it decays almost entirely by isomeric transition, giving a 140 keV gamma ray that cameras detect well. It is supplied from a molybdenum-99 generator because it would decay too much in transport.

What is the difference between a becquerel and a curie?

One becquerel is one decay per second, the SI unit. One curie is exactly 3.7 × 10¹⁰ becquerels, roughly the activity of one gram of radium. US radon results use picocuries per litre: 1 pCi/L is 37 Bq/m³.

What level of radon is safe at home?

The EPA recommends fixing your home at 4 pCi/L (150 Bq/m³) or more and considering fixing between 2 and 4 pCi/L. It estimates about 21,000 US lung cancer deaths a year from radon, and says testing is the only way to know your level. Outside the US, follow your national authority's reference level.

Where software fits: an honest bridge

Radioactive decay is physics, and Safeguard does not do physics. Its models are single purpose security models for software; they do not calculate half-lives, date samples, design shielding, assess radiation risk or analyse instruments, and nothing in this post was computed by them. The connection is narrower and real. Radiocarbon laboratories, radiopharmacies, radon testing services and hospital imaging departments run software: instrument control code, data reduction pipelines, laboratory information systems, web portals that report results, and scripts like the one in the walkthrough. Much of that code is built from open source dependencies. Where Safeguard helps is there: scanning those repositories, container images and SBOM files for known vulnerable open source components, showing which projects carry a given component version, and, when a fixed version exists, opening an upgrade pull request for review.

Every time a lab rebuilds a data reduction pipeline or a results portal it pulls its dependencies again, and the instrument controllers and imaging workstations around these measurements ship open source components of their own. The fit is indirect for the nuclear physics itself, but for the software your team builds and ships around it, the same dependency hygiene applies, and the steps below show how to close the gap whether or not upstream has a fix yet.

Fix it without waiting for upstream

Whatever the vulnerability, most of the risk sits in two gaps. The first is the time between a fix being published and that fix actually running in your production builds. The second, and worse, is the time before any fix exists at all, while a maintainer works out a patch and a release schedule. Waiting through both gaps is a habit, not a requirement. Safeguard gives teams three ways to close them.

1. Autonomous remediation: pull requests, not tickets

When a fixed version exists, Safeguard's Griffin models open a tested, policy gated pull request that moves the affected dependency to the safe version. You can run it on one repository or in bulk across every project that carries the vulnerable version. Remediation covers eight categories, including dependencies, licenses, security posture and findings. Reachability analysis ranks the vulnerable code paths your application actually calls first, so the first pull requests you merge are the ones that matter most. Autonomous remediation ships switched off, with every category set to Manual, so your team decides how much runs on its own and what still needs a human approval.

2. Gold Open Source: the safe version in seconds, free

gold.safeguard.sh is free and needs no login. Search any CVE or package to see affected and fixed versions, CVSS, EPSS, CISA KEV status and a recommended safe version, or use the "Am I affected?" checker. The same data is available through a free read API, README status badges and RSS threat feeds. The safeguard-sh/gold-scan-action GitHub Action can fail a build on a known exploited vulnerability, on any known vulnerability, or never, so a version you have just removed cannot quietly come back in next week's merge.

3. Gold Registry: start from components that are already fixed

registry.safeguard.sh provides hardened, drop in replacements for upstream packages and container images. Every Gold artifact is held continuously to zero exploitable critical or high CVEs, zero malware and a completed license check, with SLSA L3 or higher provenance, a Sigstore signature and a public evidence page you can hand to an auditor. An artifact loses its Gold badge if it regresses. Teams can point their own package directory at the registry so hardened equivalents resolve without changing how developers install anything.

When upstream has no fix yet, or will not ship one, Safeguard first works to get the fix accepted upstream. If the issue is severe or upstream will not move, Safeguard builds a fork, fixes the vulnerability there and publishes a hardened Gold build, so you are not stuck on a maintainer's release schedule. You can also request a hardened, compatibility tested build of a specific package. This fork and patch flow is in early access today, so ask which packages it covers before you plan around it.

One honest limit: when the vulnerable code lives in a vendor appliance, a hosted service or an operating system runtime you do not build yourself, the fix still has to come from that vendor or distribution. The steps above apply to the open source dependencies and container images you build and ship, which is where most teams carry the largest and least tracked share of their exposure.

See how Safeguard would help

None of these demos involve decay calculations; they show the software side described in the bridge above.

The first demo is a real Safeguard CLI session: it installs the CLI, opens the interactive terminal and asks it to fix a lodash CVE in a repository. It reads the finding and advisory, asks before each command, edits package.json to lodash 4.17.21, runs npm install and npm test with approval, prints a summary and pushes nothing. The same flow applies to a radon results portal or a lab reporting web app your team maintains: scan the repository, then let the assistant propose the dependency upgrade under your approval.

Click through it at your own pace, or open it full screen.

The second demo sets up the Safeguard runner on your own machine, approves it with its device code and then scans from the terminal. That suits teams who keep instrument integration code, data reduction scripts or analysis notebooks on their own hardware and want the scan to run where the code lives.

Click through it at your own pace, or open it full screen.

The third demo connects Claude to the Safeguard MCP server with OAuth and lets you choose which tools Claude can use. For an engineer who already works with Claude while maintaining a laboratory information system, it brings dependency findings into the same conversation, limited to the tools you allow.

Click through it at your own pace, or open it full screen.

To try it on your own repositories, download Safeguard and follow the CLI documentation.

Quick check

Sources

• IAEA Nuclear Data Section, Live Chart of Nuclides
• IAEA Live Chart data service, carbon-14 ground state (ENSDF)
• IAEA Live Chart data service, americium-241 ground state (ENSDF)
• IAEA Live Chart data service, technetium-99 levels including 99m (ENSDF)
• IAEA Live Chart data service, molybdenum-99 ground state (ENSDF)
• IAEA Live Chart data service, radon-222 ground state (ENSDF)
• NIST, Radionuclide Half-Life Measurements (Standard Reference Database 120)
• NIST Special Publication 811, Appendix B.9: conversion factors (curie to becquerel)
• BIPM, The International System of Units (SI Brochure)
• National Nuclear Data Center, Brookhaven National Laboratory
• NNDC, NuDat 3
• OpenStax, Chemistry 2e, 21.3 Radioactive Decay
• US NRC, Radiation Basics
• US EPA, Radiation Terms and Units
• US EPA, Radionuclide Basics: Americium-241
• US EPA, Radionuclide Basics: Technetium-99
• US EPA RadTown, Americium in Ionization Smoke Detectors
• US EPA, Radon
• US EPA, Health Risk of Radon
• NSW Environment Protection Authority, Household smoke detectors
• The Nobel Prize, Willard F. Libby, Facts (Chemistry 1960)
• IntCal working group, IntCal home
• University of Waikato Radiocarbon Dating Laboratory, example age report (conventional age, Libby half-life)
• Reimer et al., The IntCal20 Northern Hemisphere Radiocarbon Age Calibration Curve (0 to 55 cal kBP), Radiocarbon 62(4), 2020

Last reviewed: 24 October 2026.