Why decay is exponential and what the half-life really is
Every nucleus of a given radioisotope has the same fixed probability of decaying in the next second, and that probability does not change with age. A nucleus that has survived a billion years is exactly as likely to decay in the next second as one created a moment ago. Nuclei do not wear out.
From that single fact the whole mathematics follows. If each nucleus has probability λ per unit time of decaying, then the number decaying in a short interval is −dN/dt = λN, proportional to how many are left. Integrating gives N = N₀e−λt. Set N = N₀/2 and you get the half-life: t½ = ln2 / λ = 0.693147/λ.
The consequence people find counter-intuitive is that the half-life is the same at every starting point. From 1,000 atoms you reach 500 in one half-life; from 500 you reach 250 in another; from 8 you reach 4. Nothing about the process knows how much you started with, which is why the decay curve is a straight line on a logarithmic plot.
Two other timescales appear in the literature and are frequently confused with the half-life. The mean lifetime τ = 1/λ = t½/ln2 is the average survival time of a nucleus, and is about 1.44 times the half-life — so carbon-14's 5,730-year half-life corresponds to a mean life of 8,267 years. Particle physicists usually quote τ; nuclear and geological work usually quotes t½. The biological half-life of a radiopharmaceutical is the time for the body to excrete half of it, a completely separate process; the combined effective half-life is 1/t_eff = 1/t_phys + 1/t_bio, and it is always shorter than either.
Activity, and how decay becomes a clock
Activity is the rate at which decays happen: A = λN, measured in becquerels, where one becquerel is one decay per second. The older unit, the curie, is 3.7 × 1010 Bq — originally the activity of one gram of radium-226. Because activity is proportional to N, it decays with exactly the same exponential and the same half-life, so a detector counting clicks is a direct proxy for how much nuclide is left.
Note the tension built into A = λN. A short half-life means a large λ and therefore intense activity per gram, but the source is exhausted quickly. A long half-life means a nearly constant but feeble source. Cobalt-60 at 5.27 years gives 41.8 TBq per gram and must be replaced every few years; uranium-238 at 4.47 billion years gives only about 12 kBq per gram, which is why you can hold it.
Dating inverts the equation: t = (t½/ln2)·ln(N₀/N). The arithmetic is trivial; the hard part is always knowing N0. Radiocarbon dating solves it by assuming living things exchange carbon with the atmosphere and so start at the atmospheric ratio, which stops at death. Because the atmospheric 14C level has varied — with solar activity, with the burning of fossil carbon, and dramatically with 1950s–60s weapons testing — raw radiocarbon dates must be calibrated against tree rings and other dated records. Convention also retains Libby's original 5,568-year half-life for reporting raw dates, with the more accurate 5,730-year value applied during calibration, so a published radiocarbon date is not a bare application of this formula.
Long-lived systems avoid the N0 problem by measuring the daughter instead. Potassium-argon, uranium-lead and rubidium-strontium dating all compare accumulated daughter with surviving parent, which needs no assumption about the starting amount as long as the daughter was absent or measurable at the start. Uranium-lead on zircons is how the age of the Earth is established.
Worked example: dating a wooden beam
A wooden beam from an excavation gives a 14C activity of 9.6 counts per minute per gram of carbon. Living wood gives 13.6 counts per minute per gram. Using a half-life of 5,730 years, how old is the beam?
- Decay constant. λ = ln2 ÷ 5,730 = 0.693147 ÷ 5,730 = 1.20968 × 10−4 per year.
- Activity ratio. N0/N = 13.6 ÷ 9.6 = 1.41667. Activity is proportional to the number of nuclei, so the ratio of counts is the ratio of nuclei.
- Natural logarithm. ln(1.41667) = 0.348307.
- Age. t = 0.348307 ÷ 1.20968×10−4 = 2,879 years.
- Cross-check in half-lives. 2,879 ÷ 5,730 = 0.5024 half-lives, and (½)0.5024 = 0.7059, which multiplied by 13.6 gives 9.60 counts per minute. The answer closes.
- Mean lifetime. τ = 5,730 ÷ 0.693147 = 8,266.6 years, so the beam is about 35% of one mean lifetime old.
Now the same isotope viewed as a limit. After ten half-lives — 57,300 years — the remaining fraction is (½)10 = 0.0977%, meaning fewer than one atom in a thousand survives. Combined with the difficulty of excluding modern carbon contamination, that is why radiocarbon dating is generally quoted as reaching about 50,000 years, and why older materials need potassium-argon or uranium-series methods with far longer half-lives.
Reading the result, and matching the isotope to the job
Convert to half-lives first — it is the quickest way to judge an answer. One half-life leaves 50%, two leave 25%, three 12.5%, seven leave under 1%, and ten leave under 0.1%. The rule of thumb for storage and clearance is that after about ten half-lives a source is effectively spent.
Match the half-life to the application. Diagnostic imaging wants a few hours: technetium-99m at 6.01 hours gives a strong signal during the scan and has essentially vanished within a couple of days, and fluorine-18 at 109.8 minutes is short enough that PET centres need a cyclotron nearby. Therapy and industrial radiography want years, which is why cobalt-60 at 5.27 years and iridium-192 at 73.8 days dominate. Geological dating wants half-lives comparable to the age being measured: potassium-40 at 1.25 billion years and uranium-238 at 4.47 billion years are what date rocks.
Two limits of this model are worth stating. First, it assumes a single decay path with a single half-life. Many nuclides sit in a decay chain, so the daughter is itself radioactive and its own ingrowth and decay must be modelled with the Bateman equations, not with a single exponential. Uranium-238 passes through fourteen steps before reaching stable lead-206. Second, decay is statistical: with a handful of atoms the actual number remaining fluctuates around the exponential prediction by roughly the square root of the count, which is why low-activity counting needs long integration times.
Finally, the half-life really is a constant. It does not change with temperature, pressure, magnetic field or chemical bonding to any degree that matters. The only measurable exceptions involve decay modes that depend on the electron cloud — electron capture and internal conversion — where extreme chemical or ionisation states shift rates by fractions of a percent. Nothing you can do in a laboratory speeds up or slows down ordinary alpha or beta decay.
Half-lives of commonly encountered radionuclides
| Nuclide | Half-life | Mean lifetime | Typical use or significance |
|---|---|---|---|
| Fluorine-18 | 109.77 min | 158.35 min | PET imaging tracer |
| Technetium-99m | 6.0067 h | 8.6658 h | Most-used diagnostic imaging isotope |
| Iodine-131 | 8.0252 d | 11.578 d | Thyroid therapy and imaging |
| Radon-222 | 3.8235 d | 5.5161 d | Indoor air hazard from uranium decay |
| Iridium-192 | 73.83 d | 106.51 d | Industrial radiography |
| Cobalt-60 | 5.2714 yr | 7.6050 yr | Sterilisation, radiotherapy |
| Tritium (H-3) | 12.32 yr | 17.774 yr | Self-luminous signs, fusion fuel |
| Strontium-90 | 28.79 yr | 41.535 yr | Fission product, bone-seeking |
| Caesium-137 | 30.08 yr | 43.396 yr | Fission product, gauging sources |
| Radium-226 | 1,600 yr | 2,308.3 yr | Historic luminous paint; defines the curie |
| Carbon-14 | 5,730 yr | 8,266.6 yr | Radiocarbon dating of organic material |
| Uranium-235 | 7.04 × 10⁸ yr | 1.0157 × 10⁹ yr | Fissile isotope; uranium-lead dating |
| Potassium-40 | 1.248 × 10⁹ yr | 1.8005 × 10⁹ yr | K-Ar dating; natural body radioactivity |
| Uranium-238 | 4.468 × 10⁹ yr | 6.4460 × 10⁹ yr | Dating the Earth; head of a decay chain |
| Thorium-232 | 1.405 × 10¹⁰ yr | 2.0270 × 10¹⁰ yr | Fertile nuclide; longest-lived common actinide |
Mean lifetimes are each half-life divided by ln 2 = 0.693147. Half-life values are the conventionally cited figures; evaluated nuclear data files carry the current uncertainties.
Mistakes that give a wrong decay answer
- Confusing half-life with mean lifetime. They differ by a factor of ln 2, so τ is about 1.44 t½. Particle physics quotes τ, nuclear tables quote t½, and swapping them is a 44% error.
- Using different units for half-life and elapsed time. Only their ratio matters, so both must be in the same unit before you divide.
- Treating decay as linear. Half in the first period does not mean all gone by the second. Ten half-lives still leave a tenth of a percent.
- Assuming a known starting amount in dating. The whole difficulty of radiometric dating is establishing N0, which is why daughter-product methods and calibration curves exist.
- Using a single exponential inside a decay chain. When the daughter is itself radioactive you need the Bateman equations; a single half-life will not describe the mixture.
- Mixing physical and biological half-life. For a radiopharmaceutical the effective half-life combines both and is always shorter than either.
- Reporting a raw radiocarbon age as a calendar date. Atmospheric 14C has varied over time, so raw ages must be calibrated against tree-ring and other dated records.
- Applying the smooth exponential to a handful of atoms. Decay is statistical; with small numbers the scatter is roughly the square root of the count.
Where radioactive decay sits in physics
The three classical decay modes each change the nucleus differently. Alpha decay ejects a helium-4 nucleus, dropping the mass number by four and the atomic number by two; it is the dominant mode for heavy nuclides and proceeds by quantum tunnelling through the Coulomb barrier, which is why its rate is so extraordinarily sensitive to decay energy — the Geiger–Nuttall relation spans more than twenty orders of magnitude in half-life over a few MeV. Beta decay converts a neutron into a proton or the reverse, changing the atomic number by one and emitting an electron or positron plus a neutrino. Gamma emission carries away excess energy with no change of composition at all.
Exponential decay is not unique to nuclei. Excited atomic states, unstable particles, capacitor discharge, drug clearance and any first-order chemical reaction all obey the same equation, for the same reason: a fixed probability per unit time acting on a population. The mathematics transfers exactly, so a half-life and a time constant are the same quantity in different clothes.
Decay energy also matters at the macroscopic scale. Radiogenic heat from potassium-40, uranium and thorium is a large part of the Earth's internal heat budget, driving mantle convection and plate tectonics. And decay heat in a shut-down reactor core — from short-lived fission products that continue decaying after the chain reaction stops — is why cooling must be maintained after shutdown. Turning that decay power into a temperature rise is a Q = mcΔT calculation, and the gas volumes involved in handling radioactive gases such as radon follow the ideal gas law. On the instrumentation side, the exponential response of a detector and the resonance of a counting circuit are the same kind of first-order and second-order behaviour treated by the oscillator formulas, and shielding calculations for a point source use the same inverse-square geometry as the intensity fall-off with distance.
This is a physics calculator, not a radiation-safety assessment
Dose depends on the type and energy of the radiation, the geometry, the shielding and the exposure route — not on activity alone. Handling of radioactive material is governed by national regulation and licensing. Use these numbers for understanding decay arithmetic, and a qualified radiation protection adviser for anything involving real sources.