Two decay chains, one internal test
Two Clocks in One Crystal
Move one synthetic zircon age and calculate its two radiogenic lead-to-uranium clocks independently. Break their agreement, then impose a simple lead-loss event across several grains and recover both event times from the discordia intersections. The page proves the arithmetic of that model, not the history of any real zircon.
A uranium-bearing crystal can carry two clocks. 238U ends at 206Pb*, while 235U ends at 207Pb*. The star matters: it means radiogenic lead after common-lead correction, not all lead measured in the crystal. The decay rates differ, so agreement is an internal check. It is not immunity from errors the two clocks share.
Layer one: keep or break the agreement
Put the crystal on concordia
Move the age. Both ratios are generated from the same elapsed time, then inverted separately. Touch either ratio control and the point leaves the curve. The two clocks have not been averaged; their disagreement remains visible.
206Pb*/238U ratio and age
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207Pb*/235U ratio and age
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Absolute clock mismatch
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The curve is concordia. Cyan is the current pair of measured ratios. Axes rescale with the chosen age so that a young point does not collapse into the corner.
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Layer two: make failure carry information
Break several grains the same way
The first instrument is a closed-system toy. Now impose a later, instantaneous event. Each synthetic aliquot loses a different fraction of the radiogenic lead it had accumulated, but the same fraction of both lead isotopes. Their present-day ratios lie on a chord. A line fit knows neither event time; numerical intersections with concordia recover them.
Lower intersection
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Upper intersection
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Discordia line residual
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vertical ratio units, synthetic exact data
Gold points are six synthetic aliquots, each with a different lead-loss fraction. Coral is their fitted discordia. The two outlined circles are numerical intersections, not the event settings copied onto the graph.
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This result is conditional. Real analyses carry correlated uncertainty in both axes, so an ordinary unweighted line like this one would claim false precision. Continuous loss, several disturbances, inherited lead, uranium mobility, mixtures, or a wrong common-lead correction can all defeat the two-event interpretation. A real discordia needs analytical covariance, uncertainty ellipses, and an errors-in-variables regression.
The check
This panel is computed again in the browser. It includes a separate inversion route, published half-life anchors, and a deliberately damaged input that the concordance test must reject.
Closed-form anchor
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Independent route
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Published anchor
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Control on the control
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Current discordia recovery
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Constants in use
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- Free choices: Steiger and Jäger conventional decay constants, synthetic aliquot fractions, equal instantaneous loss of both radiogenic lead isotopes, and an unweighted vertical line fit because the synthetic points have no analytical uncertainty.
- Named uncertainty: the decay constants are conventions grounded in measurement, not exact laws. The half-life comparison below uses the statistical uncertainties reported by Jaffey and colleagues. Possible shared systematic errors are not removed by concordance.
- Scope: the page establishes the equations and identifiability of this selected model. It does not establish that any real discordant dataset records a single lead-loss event.
Sources and what each one supports
Wetherill, 1956, “Discordant uranium-lead ages, I” gives the concordia construction and graphical treatment of discordant U-Pb ages.
Jaffey and colleagues, 1971 measured the uranium half-lives used as the published anchor in the check.
Steiger and Jäger, 1977 records the conventional decay constants used by this page.
The complete method and the executable verifier live in research/two-clocks-in-one-crystal/.