A nucleus with no clock

Uranium-238 has a half-life of 4.47 billion years, known to three figures. A particular uranium nucleus has no such property: it may decay in the next second or persist for a hundred billion years, and nothing about it distinguishes the two cases in advance.

The scatter is not noise obscuring the law. It is the law: the smooth curve is what a large enough sample of independent coin-flippers does, and the traces show the number at which “large enough” stops being satisfied.

Memoryless, and what that rules out #

The defining property is that the chance of decaying in the next interval does not depend on how long the nucleus has already existed. A uranium atom formed in a supernova five billion years ago and one formed in a reactor last week have identical prospects.

That is a strong statement and it excludes the obvious mechanisms.

It excludes ageing. There is no internal process running down, no accumulating damage, no wearing out. A population of light bulbs has a failure rate that rises with age; a population of nuclei does not.

It excludes an internal clock. If each nucleus carried a timer set at formation, the decay times would cluster around the timer’s setting, and the survival curve would be a step rather than an exponential.

And it excludes an external trigger, at least for ordinary decays. Temperature, pressure, chemical state and magnetic field leave the rate essentially unchanged — heating a sample to a thousand degrees or compressing it to gigapascals moves most half-lives by less than a part in a thousand, because the nucleus is bound a million times more strongly than anything chemistry can supply.

The exponential, from one assumption #

Let the probability of decay in a short interval dt be λ dt, the same for every surviving nucleus and independent of the past. Then the expected number decaying is Nλ dt, so

$$\frac{dN}{dt} = -\lambda N, \qquad N(t) = N_0 e^{-\lambda t}.$$

The half-life is where the exponential reaches a half: t½ = ln2/λ = 0.693/λ. It is a derived quantity, not a fundamental one, and the more natural constant is λ, the probability per unit time.

The memorylessness shows up on the drawing as a property nothing else has: the ratio of survivors over any interval of one half-life is a half, wherever that interval begins. The site’s figure gate checks exactly that, reading the drawn curve at five interval pairs including two whose start times nothing in the construction singled out, and requiring every ratio to be 0.500.

Where the constant comes from #

A constant probability per unit time is a strange thing for a physical system to have, and the mechanism behind it is the exponential of the previous rung.

An alpha particle inside a heavy nucleus is held by the strong force and repelled outside it by the Coulomb barrier. Its energy is far below the barrier’s top — 4.3 MeV against roughly 30 MeV for uranium-238 — so classically it can never leave. It leaves by tunnelling.

The picture Gamow gave in 1928 is of a particle rattling inside the nucleus, striking the barrier something like 10²¹ times a second, with a fixed small probability T of getting through on each attempt. The decay constant is then λ ≈ 10²¹ T per second, and the memorylessness is immediate: each attempt is independent, so the nucleus has no memory of the failed ones.

The numbers are extreme. Uranium-238’s half-life corresponds to T ≈ 10⁻³⁹ — one success in 10³⁹ attempts — and polonium-212, whose alpha carries 8.95 MeV instead of 4.27, has a half-life of 0.3 microseconds. A factor of two in energy, twenty-four orders of magnitude in lifetime.

The dependence that produces that span is exponential in the barrier’s width, and the width is not fixed: a higher-energy alpha crosses the Coulomb funnel further up, where it is narrower. So a small change in decay energy moves the transmission by an enormous factor. That is precisely the Geiger–Nuttall relation between decay energy and half-life — an empirical curiosity noticed in 1911 and left unexplained for seventeen years, which this argument accounts for in a line.

Why the scatter matters #

A decay rate is measured by counting, and counting a random process has a characteristic uncertainty: for N events the standard deviation is √N.

The consequence is that a measurement’s precision improves only as the square root of the counting time. Halving the uncertainty means counting four times as long, which is why low-activity measurements — radiocarbon dating of small samples, searches for rare decays — are limited by patience rather than by instruments.

It also means a small sample is genuinely unpredictable. Ten nuclei with a one-second half-life will not leave five after one second; they will leave anywhere from about two to about eight, and the spread is not experimental error. The figures on this page draw that directly: four independent runs of the same 400 nuclei, and the site’s gate requires that they differ from each other and that their scatter about the smooth curve is consistent with the binomial expectation √(Np(1−p)) at four different times. A “simulation” that tracked the curve exactly would fail that check, and would be wrong.

The rate a sample actually shows #

The quantity a laboratory measures is not the number of surviving nuclei but the number of decays per second, and the relation between them is worth stating because it changes what is easy.

Activity is λN, so it falls with exactly the same exponential as the population. A gram of a long-lived isotope has an enormous N and a tiny λ; a microgram of a short-lived one has a small N and a huge λ. The two can have the same activity, which is why the becquerel — one decay per second — says nothing on its own about how much material is present.

The arithmetic runs a long way. One gram of radium-226, with a 1,600-year half-life, gives 3.7 × 10¹⁰ decays per second; that number was the original definition of the curie. One gram of uranium-238, with a half-life 2.8 million times longer, gives 12,400. And one gram of polonium-212, if it could be assembled, would deliver its entire activity in under a millisecond.

The practical consequence is that a sample’s usefulness and its hazard both peak at the same place. An isotope with a half-life comparable to the measurement time is ideal: long enough to survive preparation, short enough to give a strong signal. Technetium-99m’s six hours is the reason it is in three quarters of all nuclear-medicine procedures, and the reason it has to be generated on site.

What it has in common with the second law #

The exponential runs one way, and the reason is the same one that makes the diffusion equation irreversible while the mechanics underneath it is not.

Nothing in the microscopic physics of a decay picks a direction. The time-reverse of an alpha emission — an alpha particle arriving from infinity, tunnelling in and being captured — is a perfectly legitimate process, and it happens in a laboratory whenever a beam is aimed at a target. What makes decay one-way in practice is that the products fly apart into an enormous space of final states and never come back, which is a statement about counting rather than about dynamics.

That puts this page beside entropy rather than beside anything nuclear. The arrow comes from the number of ways to be dispersed exceeding the number of ways to be assembled, and the exponential is the shape that counting takes when the per-unit-time probability is constant.

The structure is not special to nuclei. Random walkers stepping with no direction preferred spread in a way that never reverses, and the same statement applies: neither the decay curve nor the spreading is a law about any individual. Both are what a large number of independent random events looks like from far enough away, and both acquire their apparent inevitability from the number rather than from the mechanism.

Made into a clock #

The reliability of the ensemble average is what turns decay into the most widely used dating method in science, and the logic is worth stating carefully because it is often garbled.

Nothing dates a single atom. What is dated is a ratio — of a parent isotope to its daughter, or of a decaying isotope to a stable one — measured over an enormous number of atoms, where the fluctuations are negligible.

Radiocarbon. Cosmic rays make carbon-14 in the upper atmosphere; living things exchange carbon with their surroundings and hold the atmospheric ratio; once dead they stop, and the ratio falls with a 5,730-year half-life. The method reaches back about ten half-lives before the remaining signal is lost in the background.

Uranium–lead. Two uranium isotopes decay to two lead isotopes with different half-lives, so a mineral holding both provides two independent clocks that must agree — a self-check no single clock offers. This is how the age of the Earth was established at 4.54 billion years.

Two assumptions are doing the work in every such method and both are testable. The initial ratio has to be known or inferable, and the system has to have stayed closed. Radiocarbon dates are calibrated against tree rings precisely because the atmospheric ratio has not been constant; the discrepancy reaches several hundred years, and correcting it is a whole discipline.

Half-lives that span forty orders of magnitude #

The range of measured half-lives is one of the widest of any physical quantity, and the two ends need entirely different instruments.

At the short end, decays faster than about a nanosecond cannot be timed at all. What is measured instead is the width of the state’s energy, and the lifetime is inferred from ħ/Γ — the same trade between duration and spread that gives a spectral line its natural width. A resonance 100 MeV wide has a lifetime of 7 × 10⁻²⁴ seconds, which is short enough that the particle never travels a nuclear diameter.

At the long end, nothing can be waited out. Tellurium-128’s half-life is 2 × 10²⁴ years, fourteen orders of magnitude longer than the age of the universe, and it is measured by counting the daughter product accumulated in ancient minerals — a decay observed not by watching it happen but by finding what it left.

Between them lies the entire practical range, and the spread is a direct consequence of the exponential in the barrier. A quantity that depends exponentially on an energy will span decades whenever the energy varies by a little, and nuclear decay energies vary by a factor of three.

Where the picture stops #

Four limits, and the last one is the interesting one.

Not all decays are alpha decays. Beta decay is mediated by the weak interaction rather than by tunnelling through a Coulomb barrier, and its rates are set by different physics entirely — though the exponential law is the same, because it follows from the constant per-unit-time probability and not from the mechanism.

Some rates are not constant. Electron capture depends on the electron density at the nucleus, so it responds slightly to chemical environment and to pressure — beryllium-7’s half-life shifts by about a per cent between chemical forms, which is small and measurable. A fully ionised nucleus with no electrons to capture cannot decay that way at all, which matters in stellar interiors.

The exponential is not exact. Quantum mechanics predicts deviations at very short times — the survival probability starts quadratically rather than exponentially, which is what makes the quantum Zeno effect possible — and at very long times, where a power-law tail should replace the exponential. The short-time deviation has been observed; the long-time one has not.

The nucleus does not have a definite lifetime. A state that decays with rate λ has an energy width ħλ, so it does not have a sharp energy — the same trade between duration and spread that broadens a spectral line. For a half-life of billions of years the width is unmeasurably small; for a short-lived resonance it is the width that is measured and the lifetime is inferred from it.

The isotope that is not decaying, and the one that is #

A final distinction, because the exponential invites a reading in which everything is on its way somewhere.

Most nuclei do not decay at all. Of the roughly 250 stable nuclides, none has ever been observed to change, and the stability is not a very long half-life — it is a conservation law, since there is no lower-energy configuration of the same nucleons to decay into. Iron-56 and nickel-62 sit at the bottom of the binding-energy curve and have nowhere to go.

What decays is a nucleus with an accessible lower state and a barrier in the way. The barrier decides the rate; the energy difference decides whether there is a rate at all. Both are needed, and confusing them produces the common error of supposing that a long half-life means a small energy release — polonium-212 releases twice uranium-238’s energy and lives 10²³ times less long, which is the relationship the wrong way round.

The two conditions describe a shape. A system sitting in a local minimum with a deeper one available elsewhere is metastable: it will get out eventually, and how long that takes is set by the barrier in the way rather than by how much deeper the other minimum is. A system already in the global minimum is stable and simply stays. Nuclear stability, chemical metastability and a ball in a valley are one diagram with three labels, and only the mechanism for getting across differs — thermal agitation in the chemical case, tunnelling in this one.

That figure also names what is different about the quantum case. A classical ball needs energy from outside to leave the valley, so a metastable classical system is stable at zero temperature. A nucleus does not need any, because it crosses the barrier rather than climbing it — which is why decay rates are so nearly independent of temperature and why a sample cooled to a millikelvin decays at the same rate as one at room temperature.

Two exponentials with different rates, and a reactor that ran itself #

Uranium comes as two isotopes with very different half-lives — 704 million years for the fissile 235 and 4.47 thousand million for the 238 — so the proportion of the two changes with time. Today the fissile isotope is 0.72 per cent of natural uranium, which is why a reactor needs enrichment.

Run the two exponentials backwards. Two thousand million years ago the shorter-lived isotope had gone through three of its own half-lives fewer than the other had of its, and the fraction was around three per cent — which is exactly what a modern power reactor is fuelled with.

So the arithmetic predicts that a rich enough uranium deposit, wet enough for water to moderate the neutrons, should have gone critical on its own at that epoch. In 1972 a French laboratory found uranium ore from Gabon with a deficit of the fissile isotope, and the reason turned out to be that it had already been burnt: sixteen natural reactors had operated in that deposit for a few hundred thousand years, at an average of about a hundred kilowatts.

They even regulated themselves. Water moderated the neutrons; the heat boiled the water away; the reaction stopped until it cooled and the water returned — a cycle of roughly half an hour on and a few hours off, inferred from the distribution of the products.

The site has since been used for something else entirely. The fission products’ proportions depend sensitively on a nuclear resonance whose position depends on the fine-structure constant, so measuring them bounds how much that constant can have changed in two thousand million years. The answer is a few parts in a hundred million — one of the tightest such limits there is, from a reactor nobody built.

Watching a single one #

The essay’s central claim is that no individual has a half-life, only a probability per unit time. It is possible to watch an individual, and what is seen is exactly that.

Trap a single ion, and drive it on a strong transition so that it scatters photons at an enormous rate: the ion is visibly bright. Now add a weak drive to a second, long-lived state. Most of the time nothing happens and the ion goes on shining; occasionally it jumps into the metastable state, where it cannot scatter on the strong transition, and the light goes out — completely, and for as long as it stays there.

What an observer sees is a lamp switching on and off at random. The dark intervals are the lifetimes of one atom’s excited state, measured one at a time, and their distribution is a decaying exponential whose mean is the state’s lifetime. A long dark period is followed by no greater chance of the light returning than a short one, which is memorylessness measured on a single object rather than inferred from a population.

The experiments were done in 1986 and settled an argument of long standing. Whether “quantum jumps” were real events or merely a way of speaking about ensembles had been disputed since the 1920s — Schrödinger wrote that if quantum jumping were really to stay, he was sorry to have had anything to do with the theory. The single-ion traces are as direct an answer as an experiment can give: the jumps are individual, abrupt, and random in precisely the way the exponential on this page describes.

What the picture cannot show #

The figure draws survivors against time, which is the only quantity the theory predicts, and it hides the individual entirely — deliberately, because there is nothing to draw. No property of a nucleus that decayed at t = 1.7 differs from one that decayed at t = 3.2.

It also cannot show what a detector sees. An experiment does not observe a smooth decline; it observes discrete clicks at irregular intervals, whose rate declines. The curve is the integral of a Poisson process, and the drawn steps are the nearest the figure gets to the granularity of the real signal.

And it cannot show the daughters. Every decay produces something, and in a real sample the daughters are often radioactive too — a decay chain, whose intermediate populations rise and then fall, and whose arithmetic is a set of coupled exponentials rather than one.

Where the ladder goes next #

The rungs from here: the Geiger–Nuttall relation derived from the Gamow factor; beta decay and the neutrino, which was proposed to save energy conservation in a spectrum that appeared to violate it; decay chains and secular equilibrium; the quantum Zeno effect, where frequent measurement of an undecayed state suppresses the decay by exploiting the non-exponential short-time behaviour; and the width–lifetime relation, which is how the lifetimes of particles too short-lived to travel are measured at all.

The claim to carry forward is the division between the individual and the ensemble. A half-life is a precise property of a population made of members that have no such property, and the precision improves with the size of the population rather than with the quality of the instrument. That is the same relationship a temperature has to a molecule and an entropy has to an arrangement — one of the site’s recurring shapes, appearing here in the one setting where the underlying randomness is not merely practical but fundamental.