Radiocarbon calibration plateaus

Why can a more precise radiocarbon measurement still leave several possible calendar dates?

Normally a better instrument buys a better answer: halve the measurement error, halve the uncertainty. Radiocarbon dating breaks that expectation. For certain periods a laboratory can hand you a radiocarbon age good to a couple of decades and the calendar answer still spreads across three or four centuries — sometimes as several separate humps of probability with gaps between them.

That is not an apology for a sloppy method but a statement about the map between what is measured and what is wanted. So: what shape must that map have, for extra precision to buy nothing?

The failure is not in what the laboratory measures. It measures the ratio of ¹⁴C to stable carbon and converts it to a conventional radiocarbon age by running the exponential decay law backwards — and that law is not in question: a constant per-nucleus decay probability gives survival 2^(−t/T), and the ¹⁴C half-life is well determined.

There is a curiosity in which half-life is used, worth understanding because it looks like an error and is not. Libby's original determination gave 5568 ± 30 years; later remeasurement gave 5730 ± 40, the "Cambridge" value, and the better one. Conventional ages are still computed with Libby's 5568, so that ages published over seventy years stay comparable. The ratio 5730/5568 is 1.029, so a conventional age runs about three per cent short of a naive decay age — which costs nothing, since it is never used as a calendar date but looked up in an empirically measured curve, and any consistent convention washes out of that lookup.

Why is a lookup needed? Because the founding assumption is false in detail: it assumes the atmosphere held the same ¹⁴C proportion when the organism died as the reference standard does. ¹⁴C is made when cosmic rays produce neutrons that strike atmospheric nitrogen, and that flux is modulated by solar activity and the geomagnetic dipole. Production has varied, and so has distribution, since the ocean holds far more carbon than the air and exchanges at rates that have changed.

So we measure the map rather than assume it. Take wood dated exactly by counting tree rings back from the present, measure its radiocarbon content, and you have one point: calendar year in, radiocarbon age out. Do that for every year available and you have the calibration curve.

Now the mathematics. Write the curve as r(c), radiocarbon age against calendar year; the laboratory gives a ± σ. Inverting for calendar uncertainty carries the slope:

δc ≈ σ / |dr/dc|

Everything follows from that denominator. Where the curve rises at one radiocarbon year per calendar year, δc ≈ σ and precision transfers intact. Where the slope is a fifth of that, a ±20-year measurement becomes ±100 calendar years. Where it approaches zero — a plateau — δc diverges: every calendar year in the flat stretch yields the same radiocarbon age, so every one stays equally consistent however small σ becomes. You have not measured badly; you have measured a quantity that does not vary over the interval you care about.

The multiple dates come from a related feature. The curve is not merely flat in places, it wiggles — descending as well as rising, on decadal to centennial scales. Wherever it is non-monotonic, a horizontal line at your measured age crosses it at several distinct calendar years, and the calibrated distribution comes out with separated peaks.

The best-known example is the Hallstatt plateau, roughly the eighth to fifth centuries BC, where the curve hovers near a radiocarbon age around 2450 BP — both the interval and that figure recalled approximately, and treated as soft. It is a notorious obstacle in European Iron Age chronology.

Two boundaries need marking. The first is against the marine reservoir problem, treated elsewhere in this collection. There the sample's carbon came from deep-ventilated seawater already depleted relative to the air, so the reading is offset, systematically too old, and the correction is a subtraction. Here the sample exchanged faithfully with the atmosphere and the atmosphere itself varied, so the map from calendar year to radiocarbon age is flattened and folded. One shifts the answer; the other destroys its uniqueness. A marine sample in a plateau suffers both.

The second: this is no failure of the decay law, which is exact for practical purposes. Everything that goes wrong goes wrong in the calibration — in the initial condition, not the clock.

Now the test, since a plateau might be an artefact of the tree-ring material or of one laboratory's chemistry. A genuine atmospheric feature must appear at the same calendar times in archives dated by methods unrelated to tree rings — uranium-thorium dated corals and speleothems, independently counted lake varves — and in both hemispheres, allowing for the known interhemispheric offset. A plateau in one region's dendrochronological series but absent from an independently dated coral sequence over the same centuries would be an artefact. The sharper version is nearly a bench exercise: wood from two rings a century apart within a plateau, both counted and certain, must return indistinguishable radiocarbon ages. If they came back a century apart, there is no plateau.

One consequence redeems the wiggles. Given a sequence of known relative spacing rather than one sample, you can slide it along the curve and fit its shape: the wiggles become fingerprints, and wiggle-matching pins a floating sequence to within a decade or two. The structure is information; it reads as noise only for a single sample.

Every term the collection defines is gathered in the glossary.