The chain that runs at its slowest member's rate

Every rung of this ladder so far has treated one decay. A nucleus has a constant chance per unit time of decaying, and no clock of its own; the population falls exponentially; the exponential is a Fourier transform and fails at both ends; and what comes out has a spectrum that counts the bodies in the final state.

Almost no decay in nature is one decay. A uranium nucleus decaying produces thorium, which decays, producing protactinium, which decays, and so on through fourteen steps to lead. Every heavy nuclide that exists is at some position in such a chain, and the behaviour of a chain is not the behaviour of its members.

What the equations say #

For two members, the arithmetic is short enough to do here. The parent decays at $\lambda_1 N_1$; the daughter is created at that rate and destroyed at $\lambda_2 N_2$, so

$$\frac{\mathrm{d}N_2}{\mathrm{d}t} = \lambda_1 N_1 - \lambda_2 N_2.$$

Starting from pure parent, the solution is

$$N_2(t) = \frac{\lambda_1}{\lambda_2 - \lambda_1}\left(e^{-\lambda_1 t} - e^{-\lambda_2 t}\right),$$

and Bateman generalised it to any number of members in 1910. The whole of a chain’s behaviour is in that one expression and in which of its two exponentials dominates.

The interesting statement is what it settles into. If $\lambda_2 \gg \lambda_1$ the second exponential dies quickly, and after a few daughter half-lives

$$\lambda_2 N_2 \to \lambda_1 N_1.$$

The two activities are equal. Not the two amounts — the two products of amount and rate. That is secular equilibrium, and its content is a balance rather than a coincidence: the daughter is being made as fast as it is disappearing, so its population has nowhere to go.

The three regimes, from one ratio #

Whether that happens at all is decided by the ratio of the two decay constants and by nothing else.

The middle case is the one usually left out and it is the one that matters commercially.

Transient equilibrium, where the daughter is shorter-lived but not by much, gives an activity ratio of $\lambda_2/(\lambda_2 - \lambda_1)$ — above one, and computable. Molybdenum-99 and technetium-99m are the standard example: half-lives of 66 hours and 6 hours, so the ratio is about 1.1. A hospital’s technetium generator is a column of molybdenum from which the technetium is washed off with saline every day, and the exponential that decides everything is doing the deciding twice over, and the amount available is set entirely by that ratio and by how long since the last wash.

No equilibrium happens when the daughter outlives the parent, and then the chain simply pauses: the long-lived member accumulates the whole of the short-lived one’s output and then decays at its own pace. Every chain has such a bottleneck, and the bottleneck is what makes the chain observable at all.

How long “eventually” is #

Every statement above is about what happens after enough time, and the amount of time is worth making explicit because it is the practical constraint.

The daughter’s approach to equilibrium goes as $1 - e^{-\lambda_2 t}$: it is within a per cent after about seven daughter half-lives, and within a part in a thousand after ten. So the timescale for reaching equilibrium is set by the daughter’s half-life and not by the parent’s, which is why a chain with one very long-lived intermediate takes that long to equilibrate however short everything else is.

For the uranium series the slowest intermediate is uranium-234 at 245,500 years, so a fresh sample of pure ²³⁸U needs of order two million years to come into equilibrium — during which time the parent has decayed by less than a twentieth of a per cent. That separation of scales is what makes the whole picture consistent: the chain equilibrates in a time during which the parent has not noticeably changed, so the equilibrium is established against a background that is effectively constant.

The thorium series equilibrates in about fifty years, because its slowest intermediate is ²²⁸Ra at 5.75. That difference has a practical consequence: a thorium mineral separated chemically returns to equilibrium within a human lifetime and a uranium one does not, so the two series carry memories of quite different lengths.

What is equal and what is not #

The most persistent misreading of equilibrium is that it means equal amounts, and the numbers say otherwise by twelve orders of magnitude.

Equal activity is $\lambda N$ equal, so $N$ is inversely proportional to $\lambda$: the abundance ratio is the half-life ratio. Radium’s 1,600 years against uranium’s 4.468 billion gives one part in 2.8 million — a ratio of two exponentials with nothing else in it. Radon’s 3.8 days gives two parts in a million million.

Those numbers are why the early history of radioactivity reads as it does. Marie and Pierre Curie processed something over a tonne of pitchblende residues to obtain a decigram of radium chloride, and the yield was not a matter of technique — it is what a half-life ratio of $3.6 \times 10^{-7}$ permits. Nobody has ever seen a visible quantity of radon; a room’s worth at a hazardous concentration contains perhaps a hundred thousand atoms.

The other direction is more useful. Because the abundances are fixed by half-lives, a mineral’s composition is a prediction, and a measured departure from it is information. Uranium ores are assayed by measuring the gamma rays of a short-lived member several steps down the chain, and converting to uranium content by a ratio that comes from a table of half-lives rather than from a calibration.

What breaks a chain #

The prediction holds only while nothing removes a member, and the ways a chain can be broken are as informative as the equilibrium.

Radon escapes. It is a noble gas, chemically inert, and it diffuses out of a mineral grain before decaying, so the members below it in the chain are depleted and the members above it are not. That is why radon is a domestic hazard at all — it is the only step in the sequence that can leave the rock — and it is why measurements of radium by its daughters have to be made on a sealed sample left for a month.

Chemistry separates. Uranium is soluble in oxidising groundwater and thorium is not, so water moving through rock removes uranium and leaves its daughters behind. The resulting disequilibrium then decays back toward equilibrium at a rate set by the slowest member involved, and the extent of the departure is a clock. Uranium-series dating of corals and cave deposits works exactly this way: the mineral forms with uranium and no thorium, and the thorium grows in.

And a decay can branch. Bismuth-212 decays two ways, to thallium or to polonium, and both routes rejoin at lead. Branching does not break the equilibrium — the activities still balance — but it means that a member’s abundance depends on the branching ratio as well as on the half-lives.

Each of those is a case where the departure from the simple result is the measurement. The equilibrium is the baseline against which anything interesting is seen, which is a common shape for a physical result and worth recognising: the useful thing is not that the prediction holds, but that it holds well enough for a departure to mean something — the same relationship an equilibrium has to the fluctuations about it.

The chain as a clock #

Two of the most important dating methods in existence are consequences of this rung, and they use opposite ends of the equilibrium.

Uranium–lead dating ignores the intermediate members entirely. Because every step below uranium is fast compared with uranium’s own half-life, the whole chain can be treated as a single decay from uranium to lead — the chain’s throughput is set by its slowest step, which is the first one. That is exactly the statement this rung is about, applied as a simplification rather than as a subject, and it is what makes the method robust: nothing about the fourteen intermediate half-lives enters the age at all.

Uranium-series dating uses the intermediates and nothing else. A coral takes up uranium from seawater and no thorium, so its ²³⁰Th grows in with the 75,000-year half-life of that member, and the ratio dates the coral over the last half-million years. The method’s range is set by the intermediate’s half-life, and different pairs cover different windows — ²³⁰Th/²³⁴U to half a million years, ²²⁶Ra/²³⁰Th to about eight thousand.

Both rest on the same fact. The chain has a memory of how long ago it was disturbed, and the length of that memory is the half-life of whichever member was disturbed.

The chain as a source of heat #

A consequence of equilibrium that has nothing to do with dating is worth stating, because it is where most of the energy released by radioactivity on Earth actually comes from.

Every member of a chain in equilibrium decays at the same rate, so a chain of fourteen steps releases fourteen decays’ worth of energy per uranium decay rather than one. For the uranium series that is about 51 MeV per ²³⁸U atom against the 4.27 MeV of the first step alone — a factor of twelve, and it arrives whether or not anybody is watching the intermediates.

That multiplication is why the Earth is as warm as it is. The radiogenic heat production of the planet is around twenty terawatts, roughly half its total heat flow, and essentially all of it is the summed output of three chains — uranium-238, uranium-235 and thorium-232 — plus the single decay of potassium-40. Every calculation of the mantle’s thermal history uses the chain totals rather than the head decays, and getting the factor of twelve wrong would change the answer by an order of magnitude.

The same arithmetic sets the output of a radioisotope thermoelectric generator, and it is one reason plutonium-238 is used instead: its decay is not the head of a long chain, so its heat output is predictable and its shielding requirement is modest. A source at the top of a chain is a source whose radiation gets harder with age as the daughters grow in, which for a spacecraft is a design problem and for a laboratory standard is a calibration problem.

Why the chains stop where they do #

Three long chains exist on Earth and a fourth does not, and the reason is this rung’s arithmetic applied to the age of the solar system.

Alpha decay removes four nucleons, so a chain’s members all share the same mass number modulo four. There are therefore four possible series, headed in principle by thorium-232, neptunium-237, uranium-238 and uranium-235. Three of those heads have half-lives comparable with or longer than the age of the Earth — 14 billion, 4.5 billion and 0.7 billion years — and survive. Neptunium-237’s is 2.14 million years, which is nothing on that scale, and its entire series has decayed away.

So the absence of one of the four families is a statement about a single half-life, and the presence of the other three is what makes the whole subject of this rung observable. Had uranium-238’s half-life been ten times shorter, there would be no uranium ore, no radium, no radon, and considerably less heat in the mantle.

The neptunium series has not disappeared entirely, and the exception is instructive. Trace amounts of its members are made continuously by neutron capture on uranium in ores, at a level set by the local neutron flux — so the series exists in a steady state maintained from outside rather than in a decaying equilibrium, and the abundance arithmetic of this essay does not apply to it at all.

What a chain does to a dose #

One more consequence, because it is the one that decides how radioactive material is handled and it follows from nothing but the equilibrium.

The radiation a sample emits is the sum over every member of its chain, and the members further down are usually the energetic ones. Freshly separated uranium is almost harmless — it emits a 4.27 MeV alpha particle that a sheet of paper stops, and essentially no penetrating radiation at all. Left alone for a few months it grows in thorium-234 and protactinium-234m, whose beta and gamma emissions are penetrating, and its external hazard rises by orders of magnitude.

That growth is the two-member solution of this rung, with the daughter’s half-life of twenty-four days setting the timescale. A drum of uranium that was safe to handle when it was filled is not safe six months later, and nothing was added to it.

The same arithmetic runs the other way for a medical source. Technetium-99m is used precisely because it does not build anything in: it decays to a nuclide with a two-hundred-thousand-year half-life, which is to say to something that emits nothing measurable on any clinical timescale. A patient injected with it is radioactive for a day and then is not.

So the choice of isotope for any application is a choice of chain rather than of nuclide, and the questions are the same three every time — what does it decay into, how fast does that decay, and what does that one leave behind. The Bateman solution answers all three at once.

Where this stops being right #

The Bateman solution assumes the decay constants are distinct. Two members with equal half-lives give a degenerate case with a different functional form, drawn above, and members whose half-lives are close give a numerically ill-conditioned version of the general formula — the differences $\lambda_j - \lambda_i$ appear in denominators.

Every member has been assumed to stay put. Chemistry, diffusion and recoil all move atoms, and the recoil from an alpha decay is enough to eject a daughter from a small grain entirely. In fine-grained material that is a systematic error in every uranium-series age, and correcting for it is a substantial part of the practice.

The chain has been treated as closed. Cosmic-ray production, neutron capture and mixing all add members from outside, and in some settings that matters more than the decay does.

And equilibrium takes time. “Old enough” means several half-lives of the longest-lived intermediate, which for the uranium series is about a million years. A young mineral is not in equilibrium and none of the abundance arithmetic applies to it.

What the pictures cannot show #

The activity curves are drawn as smooth functions, and an activity is a count rate. For radon at equilibrium in a gram of uranium the count is about twelve decays a second, and for the last members of the chain in a small sample it is a few per hour — so the smooth curve is what an average over many samples would look like, not what any single measurement gives.

Nor can the abundance figure show that it is a picture of one instant. Every quantity on it is falling, at the parent’s rate, and the ratios are fixed only because they are all falling together. A logarithmic scale with twelve decades on it makes a slow universal decline invisible, which is convenient and is also the thing most easily forgotten.

Where the ladder stands #

Four rungs stand on decay. The first found the rate has no memory. The second found the law is a transform and is wrong at both ends. The third found the spectrum counts the bodies. This one puts decays in series and finds a balance that no member has on its own.

The habit worth carrying away is about series of processes with very different rates. When a sequence of first-order steps is left alone, it settles into a state where every step runs at the rate of the slowest, and the populations adjust to make that possible. That is secular equilibrium here; it is the rate-determining step in chemical kinetics; it is the bottleneck in a production line; and it is why the abundance of an intermediate in any such sequence is inversely proportional to how fast it is consumed. The populations carry the information and the rates carry none.

What is left on this ladder returns to the first rung and contradicts it. Every result here treats each decay constant as a fixed property of its nuclide — the assumption the whole ladder has rested on since the beginning. It is very nearly true and it is not exactly true, and the exceptions are measured in factors of a billion rather than in per cent.