Illustrated Physics
physics, drawn
Concept

Half-life — where it appears

The time in which half of a decaying population is gone, a constant of the substance and not of the sample. It exists because the decay probability per unit time does not change with age, so no nucleus is closer to decaying than any other however long it has waited.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

Decay, and the ensemble it is a property of. 400 nuclei followed for 4 half-lives. The smooth curve is the exponential; the stepped traces are 3 independent runs in which every nucleus was given its own decay time and told nothing about the others. The number surviving halves at each dashed line — 200, 100, 50, 25 — and it halves again over the next interval regardless of how long the sample has already been sitting there, which is the property no ordinary clock has. The traces wander further from the curve as the numbers get small: at the end only about 25 are left and the scatter is a visible fraction of that.

A nucleus with no clock

A half-life is a precise number and no individual nucleus has one. Each has the same chance of decaying in the next second as it had on the day it formed, and the exponential curve is a property of the population rather than of any member of it.

quantum · Decay Twenty-four decades of lifetime from a factor of two in energy. The half-lives of 7 alpha emitters against the reciprocal square root of the alpha's energy — the Geiger–Nuttall coordinates — with the measured values as points and a one-line tunnelling model as the open ones. The energies span a factor of 2.2 and the half-lives span 24 decades, which is what an exponent does. The model has no fitted parameter in it and reproduces every lifetime to within 0.5 decades — bad arithmetic by any ordinary standard, and a hundred-thousandth of the range it is predicting.

A wall that a factor of two makes impassable

Polonium-212 lives three tenths of a microsecond. Thorium-232 lives fourteen billion years. The alpha particles they emit differ in energy by a factor of two, and the lifetimes differ by twenty-four decades — because the quantity that decides is not the energy but an exponent built from it, and an exponent is where small differences go to become enormous.

quantum · Tunnelling Flat, then exponential, then a power law. Survival probability against time in lifetimes, both logarithmic, for a resonance 20 linewidths above the bottom of its band and 400 below the top. The straight dashed line is exp(−Γt), and the computed curve sits on it through the middle — a fitted rate of 0.9998 per lifetime between one and eight — and leaves it at both ends. Below 1.57e-2 lifetimes the curve is flat, falling as (t/τ_z)² with τ_z = 0.1253 lifetimes; beyond 22.6 lifetimes it is an inverse square, which any exponential eventually loses to. Both departures are forced: the head by the state being normalisable and the tail by the band having a bottom.

The exponential that is only true in the middle

A decay law is not an assumption about nuclei; it is the Fourier transform of an energy distribution. Do that transform honestly and the exponential fails at both ends — flat at the start, because the state is normalisable, and an inverse square at the end, because no system has states of arbitrarily negative energy. Neither departure is a correction that could be made small.

quantum · Decay Everything in the chain decaying at the parent's rate. The activity of each member of ²³⁸U → ²³⁴Th → ²³⁴Pa, relative to the parent's, against time in days. The parent's half-life is far the longest, so its activity is effectively constant over the range drawn while each daughter rises to meet it. Once they have, every member of the chain is decaying at exactly the same rate — secular equilibrium, checked here to one per cent off the integrated solution — because each is being made as fast as it is disappearing. The amounts are not equal at all: the abundance of each member sits in the ratio of its half-life to the parent's, which for radium in uranium is one part in three million and is why radium had to be extracted from tonnes of ore.

The chain that runs at its slowest member's rate

Put decays in series and something happens that no single decay does. The population settles where every member is being made exactly as fast as it disappears, so every activity in the chain is equal — while the amounts differ by twelve orders of magnitude, in the ratio of the half-lives. A gram of uranium contains a third of a microgram of radium and two hundred million million million atoms fewer of radon, and both numbers are read off a list of half-lives.

quantum · Decay The half-life that depends on the electrons. Half-lives of 3 nuclides as neutral atoms and as bare nuclei, measured at a heavy-ion storage ring where fully stripped ions can be kept circulating for months. The changes are not corrections. ¹⁸⁷Re goes from 41.6 billion years to 32.9 — a factor of 1.26 billion — because a beta decay whose available energy is only 2.5 keV cannot put an electron into the continuum but can put one into an empty K orbital, and stripping the atom opens that channel. ¹⁶³Dy is stable as an atom and decays in 47 days as a bare nucleus, for the same reason. The rate is a property of the nucleus and of what surrounds it, not of the nucleus alone.

The half-life that chemistry can change

A nucleus has no clock, the usual account goes, and nothing outside it touches the rate. That is very nearly true and it is not exactly true. Electron capture takes an electron from the nucleus's own position, so its rate is proportional to how many electrons are there — which is chemistry, worth a per cent. And a nucleus stripped of every electron can gain a decay channel it did not have: rhenium-187 goes from forty-two billion years to thirty-three.

quantum · Decay Each unit of angular momentum costs a factor of tens of thousands. Weisskopf's estimates of the half-life of a gamma-ray transition in a nucleus of mass number 100, against the photon's energy, for electric multipoles of order 1 to 5 — the photon carrying away 1 to 5 units of angular momentum. At 1 MeV the estimates are 10⁻¹⁵ s for E1, 10⁻¹¹ s for E2, 2.0 µs for E3, 292.5 ms for E4, 17.3 h for E5. Each extra unit multiplies the half-life at 1 MeV by between 60,000 and 210,000: about a thousand of that is (λ/2πR)², the square of the photon's reduced wavelength over the nucleus's radius, which grows as the energy falls, and the rest is a numerical factor that grows with the order. A transition that must carry away five units at 100 keV has an estimated half-life longer than the age of the universe; the same energy carried off with one unit takes less than a nanosecond.

The spin a photon has to carry away

An excited nucleus sheds its energy as a gamma ray in a millionth of a millionth of a nanosecond — or in minutes, or centuries, or never within the age of the universe. The energy available hardly matters. What matters is how much angular momentum the photon must carry off, because a source much smaller than its wavelength is very bad at launching a wave that twists, and each extra unit of twist costs a factor of tens of thousands in time. Selection rules are usually stated as what cannot happen; they are really a price list, and it spans forty decades.

quantum · Decay The oldest age a trace of modern carbon allows. The age a radiocarbon measurement would report against the sample's true age, when a fraction of the carbon in it is modern contamination — from groundwater, roots, handling or the chemistry of preparation — of 5, 1 and 0.1 per cent; dashed, no contamination. Young samples are barely affected. Old ones are ruled by the contamination, because their own carbon-14 has dwindled to less than it: with 1 per cent, a sample of any age beyond about 60,000 years reads 38.1 thousand years, and 5 per cent caps it at 24.8 thousand. Even 0.1 per cent, a microgram of modern carbon in a milligram sample, caps it at 57 thousand. The limit of radiocarbon dating, usually quoted as about fifty thousand years, is a limit on cleanliness, not on the half-life.

The clock read by counting atoms

Radiocarbon dating reads a clock that ticks slowly: a gram of living carbon holds sixty thousand million carbon-14 atoms, and only thirteen and a half of them decay each minute. Counting those decays to date a milligram of bone would take years. Counting the atoms themselves, one by one in an accelerator, takes an hour, because in any measurement shorter than the clock's mean life almost every atom is still there, undecayed and countable. And the method's famous limit of about fifty thousand years turns out not to be set by the half-life at all. It is set by how much modern carbon a sample has picked up, because a trace of contamination outweighs whatever carbon-14 a very old sample has left.

quantum · Decay

Named alongside it

The objects these essays reach for when they reach for this one.

DecayRadioactive decayExponential decayMemorylessnessProbability densityTunnellingWavefunctionAbundanceAccelerator mass spectrometryActivityAlpha decayAngular momentum

All concepts