Radioactive half-life

If a single atom never ages and cannot know when it was made, how can a jar of them decay on so dependable a schedule?

Watch one radium nucleus and you learn nothing usable. It may go in the next second or sit there through the rest of your life, and no measurement made beforehand will tell you which. It has no moving parts, no wear, no record of how long it has waited.

Yet a gram of the same substance keeps time well enough to date rocks and bones. How does a crowd of things that individually have no clock produce, collectively, one of the most reliable clocks we possess? And why is the constant a half-life — why halves?

Start with the smallest honest assumption about one nucleus: it has no memory. Its chance of decaying in the next second is the chance it had in the first second of its existence, and the chance it will have in a thousand years if it survives. Call that chance per unit time λ. That is the whole physical input; the rest is arithmetic.

Now put N of them in a jar. In a short interval dt, each has probability λ dt of going, so the expected number lost is λN dt:

dN/dt = -λN

Notice what this equation is saying. The rate of decay is proportional to how many are left — not because the survivors influence one another, but simply because each of them is rolling the same dice independently. Separate the variables and integrate: dN/N = -λ dt gives ln N = -λt + constant, so N(t) = N₀ e^{-λt}.

Now ask when half are gone. Set N/N₀ = 1/2, so e^{-λT} = 1/2, which means λT = ln 2, and T = 0.693/λ. Substitute λ = ln2/T back into the exponential and it rearranges to the form people actually use:

N(t) = N₀ · 2^(−t/T)

That answers the "why halves" question by itself. The exponential has no preferred amount: the time to fall from 1000 to 500 is the time to fall from 12 to 6, and the time for any starting quantity to reach any fixed fraction of itself. A half-life is not a special physical event but one convenient reading of a scale-free constant. We could as well quote the mean lifetime τ = 1/λ = T/0.693, about 1.44 half-lives, and physicists often do. Halves are chosen because a factor of two is the easiest fraction to see in a counting experiment.

So where does the dependability come from? Purely from the size of the crowd. If each nucleus decays independently with the same probability, the number decaying in a fixed interval is binomially distributed, and the relative spread of such a count is roughly 1/√N. A microgram of a heavy isotope holds about 10⁻⁶/226 of a mole, some 2.7 × 10¹⁵ nuclei; the square root of 10¹⁵ is about 3 × 10⁷, so the fractional wobble is around 3 × 10⁻⁸. The randomness has not gone anywhere — it has been averaged down to invisibility, and that is a small sample by laboratory standards.

One more question is where the law could most easily have failed: why does λ not depend on temperature, pressure, or chemical bonding? Compare energies. Thermal energy at room temperature is Boltzmann's constant, 8.617 × 10⁻⁵ eV per kelvin, times 300 kelvin — about 0.026 electron-volts. Chemical bonds run to a few electron-volts. The barriers governing nuclear decay run to millions. The environment offers a nudge some ten million times too small to matter, which is why heating, freezing, compressing and dissolving a sample leave the decay rate alone.

That argument also says where to expect exceptions, and there are some. Decay modes involving the electrons — electron capture, internal conversion — do sample the atomic environment, and their rates shift by fractions of a per cent with chemical state. Strip an atom of its electrons entirely and a beta decay can change dramatically, because the emitted electron has nowhere to go but a bound orbital. These are real measured effects; I recall them qualitatively and would not quote figures from memory. They do not touch alpha decay or ordinary beta decay in a neutral atom, which is what dating relies on.

The memorylessness is not a modelling convenience; it is what an unstable quantum state is. The nucleus occupies a state that is not an exact stationary state of the full interaction, and the amplitude for finding it undecayed falls at a rate set by the coupling and the available final states. Nothing in that description contains elapsed time, so the survival probability can only be exponential. The honest caveat: that is an excellent approximation rather than a limitless theorem, since quantum mechanics predicts tiny deviations at extremely short and extremely long times, and short-time deviation has been seen in an engineered system.

"Half-life" is also often misheard as a lifespan, as though nuclei were built to last T. Nothing fails on schedule; the survivors at time T are statistically identical to freshly made ones, and have the same half-life measured onward from there.

And this is why decay dating works at all. Because λ belongs to the nuclide and not to its history, a sample that was heated, buried, crushed and chemically altered still carries an unmolested count — exactly the property a chronometer needs and almost nothing else in geology has.

• How a decay chain behaves when a parent feeds a daughter with its own half-life, and what secular equilibrium means.
• Why the same exponential appears in drug clearance and capacitor discharge, and what those share.

Every term the collection defines is gathered in the glossary.