The clock read by counting atoms

A nucleus with no clock found that a half-life belongs to a population and not to any nucleus in it: each one has a fixed chance of decaying in the next second, the same as on the day it formed, and the smooth exponential is the average of a great many memoryless events. It named radiocarbon dating as the method that turned the statistics of decay into a clock for the last fifty thousand years of human history, and it noted that a decay count improves only as the square root of the time spent counting. The chain that runs at its slowest member’s rate followed decays in series to a balance where every member decays as fast as it is made.

This essay takes the radiocarbon clock apart. It turns out to be read, these days, without counting a single decay, and its limit, usually explained by the half-life, turns out to have very little to do with it. Both facts follow from asking, for a measurement of finite length, how many of the clock’s atoms actually tick.

A clock wound by cosmic rays #

Carbon-14 is made in the upper atmosphere, where neutrons knocked out of nuclei by cosmic rays strike nitrogen-14 and convert it, ejecting a proton. The new atoms oxidise to carbon dioxide, mix through the atmosphere within a few years and enter every living thing through photosynthesis and the food chain. Production is balanced by decay, with a mean life of 8,267 years, so the atmosphere holds a steady inventory, the balance the chain that runs at its slowest member’s rate reaches when production matches decay: about 1.18 atoms of carbon-14 for every million million of ordinary carbon.

A living organism keeps that ratio by exchanging carbon with its surroundings. When it dies the exchange stops, and the carbon-14 it holds decays without being replaced. Measuring the ratio that remains gives the time since death. Willard Libby worked out the method in the late 1940s, tested it on wood from Egyptian tombs of known age, and received the Nobel Prize for it in 1960.

The trouble is visible in the figure. A gram of living carbon holds about sixty thousand million carbon-14 atoms, and because each lives on average eight thousand years, only thirteen and a half of them decay in a minute. A gram of charcoal fifty thousand years old gives one decay every half hour. Libby’s counters, and those that followed for thirty years, detected each decay as a pulse in a gas or a liquid scintillator, shielded under lead and iron and surrounded by a second counter to reject cosmic rays, and they needed grams of carbon and days of counting for each date.

Ticks, or atoms #

The reason is the one a nucleus with no clock gave for the precision of any decay measurement. A count of N events has an uncertainty of $\sqrt{N}$, so a date good to one per cent needs ten thousand decays, and with thirteen a minute from a gram that is half a day for modern carbon and correspondingly longer for older or smaller samples. From a milligram, a day yields about twenty decays: an uncertainty of nearly a quarter.

Decay counting waits for the clock to tick, and in any time short compared with the mean life, almost none of the atoms do. Over a day only one carbon-14 atom in three million decays. The other two million nine hundred and ninety-nine thousand are sitting in the sample, undecayed and, if they could be found, countable. Counting atoms directly gains a factor of the mean life over the counting time, three million for a day, less whatever fraction of the atoms the counting loses.

The difficulty is finding them. Carbon-14 is outnumbered nearly a million million to one by carbon-12, and an ordinary mass spectrometer, which separates ions by their mass through the curve a magnetic field gives their paths — the force that does no work bending each by its momentum — cannot pick out such a rare isotope, because nitrogen-14 has almost exactly the same mass and is everywhere, and molecules such as ¹³CH and ¹²CH₂ have mass fourteen too.

An accelerator as a sorting machine #

The solution, demonstrated in 1977, was to put the mass spectrometer around a particle accelerator. The sample’s carbon is turned into graphite, and a beam of caesium knocks negative carbon ions off it. Nitrogen does not form stable negative ions, so the most troublesome contaminant never enters the machine at all. The negative ions are accelerated towards a positive terminal at a few million volts, where they pass through a thin foil or a gas that strips several electrons off each, turning them into positive ions and breaking apart every molecule, since a molecule left with three or four positive charges flies apart. The now-positive ions are accelerated away again from the terminal, and magnets and electric fields select those with the momentum, energy and charge of carbon-14, which are finally counted one by one in a detector that measures each one’s energy loss to confirm what it is.

The tandem accelerator, two accelerations with a charge exchange between them, is what makes the separation clean, and it is a large machine to count something so small. It counts about one carbon-14 atom in a hundred that leave the sample, and from a milligram of modern carbon it records more than half a million in under an hour, where decay counting would record twenty in a day. That made dating possible from single seeds, from a few hairs, from flecks of pigment scraped from cave paintings and from the small samples cut in 1988 from the Shroud of Turin, which three laboratories independently dated to between 1260 and 1390.

Other clocks read the same way #

Once atoms could be counted, every long-lived cosmogenic isotope became a clock too, and several of them run far longer than carbon-14. Beryllium-10, made by cosmic rays in the atmosphere and in the surface of exposed rock, has a half-life of 1.4 million years; aluminium-26, 0.7 million; chlorine-36, 0.3 million. Counted by accelerator, they date how long a boulder has lain exposed on a moraine, how fast a hillside is eroding, and when a layer of ice in Antarctica was laid down, with ages reaching millions of years. None of them could be measured by counting decays in the amounts available, for exactly the reason carbon-14 could not be measured from a milligram.

The noble gas krypton-81, with a half-life of 230,000 years, is so rare — about one atom in a million million of krypton — and so chemically inert that it is counted by a different machine: atom-trap trace analysis, which catches single atoms of the one isotope in a trap made of laser beams, using the friction made of light tuned so precisely to that isotope’s transition frequency that its neighbours pass through untouched, and counts them one at a time by their fluorescence. It dates groundwater in deep aquifers and ancient ice up to about a million years old. Each of these is the same move: give up waiting for the rare event, and find and count the rare thing.

The ceiling is a matter of cleanliness #

With atoms counted by the hundred thousand, the method should reach far further back than fifty thousand years: at ten half-lives, about fifty-seven thousand years, a sample still holds a thousandth of its original carbon-14, and an accelerator registers a thousandth of half a million atoms, five hundred of them, easily. In practice dates beyond about fifty thousand years are rarely trusted, and the reason is in the figure.

A sample whose carbon is partly modern — from rootlets that grew into buried charcoal, from carbonates dissolved in groundwater, from the glue of a museum repair, or from the chemistry of the laboratory’s own preparation — holds two populations of carbon-14 atoms: its own, dwindling as e^(−t/τ), and the contaminant’s, at the full modern level. The measured fraction is

$$F = (1-f)\,e^{-t/\tau} + f,$$

and once the sample’s own share has fallen below the contaminant’s, the contamination is all the measurement sees. A sample with one per cent of modern carbon cannot read older than τ ln 100, thirty-eight thousand years, however old it really is; with a tenth of a per cent the ceiling is fifty-seven thousand. A microgram of modern carbon in a milligram sample, an amount far too small to see, sets the oldest age the measurement can give.

So the oldest radiocarbon dates are achieved not by better counting but by better chemistry: aggressive cleaning of bone collagen, isolation of single amino acids, and the measurement of blanks — samples of carbon known to be millions of years old, such as coal or ancient graphite, put through the same preparation — to find out how much modern carbon the laboratory itself adds. The best laboratories reduce that background to well under a tenth of a per cent.

How the uncertainty grows #

The statistical uncertainty in an age follows from the counts. Because the age is the logarithm of the measured fraction, a fixed fractional uncertainty in the fraction is a fixed uncertainty in years, about eighty years per per cent. For a young sample the counts are large and the uncertainty a few decades. As the sample ages its own counts fall exponentially while the background, which must be subtracted, stays the same, and the uncertainty grows faster than exponentially, from a dozen years at two thousand years old to hundreds at fifty thousand. Beyond that point the sample’s own carbon-14 is a small difference between two comparable counts, and the age becomes a lower limit — “older than” — rather than a date.

The two half-lives, and the calendar #

The dates reported by radiocarbon laboratories contain a historical quirk worth knowing. Libby used a half-life of 5,568 years, the best value of his day; it was remeasured in the early 1960s as 5,730, about three per cent longer. To keep the published literature consistent, laboratories continued to compute “conventional radiocarbon ages” with Libby’s value, counted from the year 1950. A conventional age is therefore not a calendar age, and it was never intended to be one, because there was a larger correction to make anyway.

Libby assumed that the atmosphere’s carbon-14 level had always been what it was in his day. It has not. The production rate varies with the Sun’s activity and the strength of the Earth’s magnetic field, which shield the atmosphere from cosmic rays, and the carbon cycle has moved carbon between the air and the ocean at different rates. Tree rings, counted year by year back more than twelve thousand years, and their carbon measured ring by ring, give the atmosphere’s past ratio directly, and lake sediments laid down in annual layers, cave formations and corals dated by uranium extend the record further. Calibrating a conventional age against that curve turns it into a calendar date, and the correction can be thousands of years: forty thousand radiocarbon years corresponds to about forty-four thousand calendar years.

A pulse that dates by the year #

In the 1950s and early 1960s atmospheric nuclear tests produced carbon-14 the same way cosmic rays do, from neutrons striking nitrogen, and nearly doubled the amount in the northern atmosphere by 1963. After the partial test-ban treaty of that year the excess has been absorbed by the oceans and the biosphere, falling by a few per cent a year at first and more slowly since. On a curve like that, carbon-14 is a clock with a resolution of a year rather than of decades, because the level changes much faster through exchange with the ocean than through decay, which removes only an eighth of a per cent a decade.

Carbon fixed by a plant takes up the level of the year it grew, and everything that eats the plant inherits it. Measuring the carbon-14 in a tissue that does not renew its carbon — tooth enamel, the lens of the eye, the DNA of cells that do not divide — dates its formation to within a year or two. The method has been used to date the vintages of wines and the age of elephants whose ivory was seized, to settle the ages of fish, and, since 2005, to date the birth of cells in the human body from the carbon-14 in their DNA: the neurons of the cerebral cortex turned out to be as old as the person, while some in the hippocampus carry levels from years after birth and so were made in adulthood. The same atoms that ended Libby’s assumption of a constant atmosphere became a precise clock of their own.

The opposite effect has been running since the industrial revolution. Fossil fuels are far older than carbon-14’s life and contain none, and burning them dilutes the atmosphere’s carbon-14, an effect Hans Suess identified in the 1950s. Fossil dilution is now overtaking the remnant of the bomb pulse, and by the middle of this century new organic material will carry less carbon-14 than material centuries old did when fresh, making some recent material look ancient.

Where the clock’s model stops #

The figures use one modern ratio and one mean life, a sample that died at a single moment and exchanged nothing since, and contamination that is a fixed fraction of modern carbon. Real samples break each assumption a little. Organisms that took their carbon from the ocean, or from water that has dissolved ancient limestone, start with less carbon-14 than the air, and so begin life with an apparent age of centuries — the reservoir effect, which must be corrected for marine shells and for people who ate a lot of fish. Isotopes fractionate in photosynthesis and other chemistry, so a plant takes up slightly less carbon-14 than its ratio in the air, which laboratories correct by measuring carbon-13 in the same sample. Contaminants may themselves be old rather than modern, pulling dates the other way. The precision figure counts only Poisson statistics and a single background value, and leaves out the systematic uncertainty in that background, which dominates the oldest dates.

The decay itself is about as reliable as anything in physics. The half-life that chemistry can change found that the rates of some decays, those that capture an electron from the atom, shift by parts in a thousand with the chemical surroundings; carbon-14 decays by emitting an electron and is unaffected at any level radiocarbon could detect. And the exponential that is only true in the middle found that decay departs from the exponential at extremely short and extremely long times, far outside the range any sample spans.

The domain of the method is carbon that took up the atmosphere’s ratio at a definite time and was sealed from exchange afterwards, between a few centuries and about fifty thousand years ago, and the further back it reaches the more it is a question of chemistry rather than physics.

Still open: how far the oldest dates can be trusted #

Dates near the limit, around forty to fifty thousand years, bear on some of the most debated questions in prehistory: when modern humans reached Europe and Australia, how long they overlapped with Neanderthals, when megafauna disappeared. At those ages a fraction of a per cent of modern contamination shifts a date by thousands of years, and successive improvements in cleaning have repeatedly made previously published dates older. How much of the remaining spread among dates for the same events reflects real history and how much reflects contamination not yet removed is argued site by site, and the calibration curve itself is least certain where the dates are oldest.

The habit worth carrying away is to ask, of any measurement of a slow process, whether it needs the process to happen. A gram of living carbon holds 5.9 × 10¹⁰ carbon-14 atoms and 13.6 of them decay a minute; counting atoms instead of decays gains a factor of the mean life over the counting time, about three million for a day, and leaves the method limited not by the half-life but by modern carbon in the sample, which caps the age at τ ln(1/f) — 38,000 years for one per cent. A clock that ticks too slowly to hear can still be read by counting its parts.