In-Situ Radiocarbon Calculator

There is a real mechanism that could do this.  Uranium and thorium in rock throw off alpha particles;  those knock neutrons out of light nuclei;  a neutron hitting a nitrogen-14 nucleus turns it into carbon-14.  It is the same reaction that makes radiocarbon in the atmosphere, running underground instead.

So the question is not whether this happens.  It does.  The question is how much, and this page answers it.  Set the nitrogen content, the uranium, and the rock the sample is sitting in, and read off the percent modern carbon, the radiocarbon age a laboratory would report, and the calibrated calendar age.

What the answer depends on

Five things, in roughly this order of importance.

Uranium and thorium in the surrounding rock. These are the whole neutron supply.  Double the uranium and you double the radiocarbon.  Average continental crust holds 2.7 ppm uranium and 10.5 ppm thorium;  a granite can hold twenty times that;  an ore body a thousand times more again.

Nitrogen in the sample. Nitrogen-14 is what actually catches a neutron and becomes carbon-14, and it is about 107,000 times better at it, gram for gram, than the carbon-13 that is the only alternative in a pure-carbon sample.  Bone collagen is around 16 % nitrogen, coal about 1.5 %, diamond a few hundred parts per million, and wood and graphite essentially none.

What else is in the sample, competing for the neutrons. This is the part people leave out, and it is often decisive.  Every neutron produced is eventually absorbed by something, and nitrogen only makes radiocarbon out of the ones it wins.  Hydrogen absorbs strongly and there is a lot of it in coal.  Calcium and phosphorus dominate bone.  A few parts per million of boron or gadolinium — routine trace elements — will out-absorb percent-level nitrogen.

Carbon in the sample. The answer is a ratio, ¹⁴C to total C.  A sample with little carbon shows a higher ratio for the same amount of ¹⁴C, which is exactly why uranium ores, which have almost no carbon, are the one place in-situ radiocarbon has actually been detected.

Not burial time. Production and decay reach a steady state, and after about 38,000 years the sample has forgotten how long it has been buried.  This is the structural point that decides the whole argument:  in-situ production predicts that a 70-million-year-old bone and a 300,000-year-old bone in the same rock should read the same, and that the reading should track nitrogen and uranium rather than age.

How it works

Five steps, no adjustable parameters at any of them.

1. Neutron production. Alpha particles from the ²³⁸U, ²³⁵U and ²³²Th decay chains strike light nuclei — aluminium, sodium, silicon, magnesium, oxygen — and knock out neutrons; ²³⁸U also fissions spontaneously. Yields come from Šrámek et al. 2017, Table 7, which tabulates the neutrons per kilogram per year per weight fraction of each target element per weight fraction of uranium or thorium.
2. Absorption. In a body large compared with the thermal neutron diffusion length, every neutron produced is absorbed somewhere, so each nuclide's share of the absorptions is its share of the total cross section. The flux follows as production ÷ absorption. Bulk density cancels out of both, which is why the calculator never asks for one.
3. The three reactions that make ¹⁴C. ¹⁴N(n,p)¹⁴C at 1.83 barns, ¹³C(n,γ)¹⁴C at 1.37 millibarns, and ¹⁷O(n,α)¹⁴C at 0.235 barns. The third is usually forgotten and should not be: rock is half oxygen by weight, so a nitrogen-free mineral still behaves like about 20 ppm of nitrogen.
4. Steady state. Production balances decay at N = P/λ, on a 5,700-year half-life.
5. Reporting. Divide by the sample's carbon, express as percent modern carbon against the 1.176 × 10⁻¹² modern ratio, and convert to a conventional radiocarbon age on the Libby 5,568-year half-life — because Stuiver & Polach 1977 require it of every published age, and using the true half-life would give a number comparable to nothing. Calibration is IntCal20, 95.4 % highest posterior density, flat prior.

Every default is chosen to favour in-situ production.  The largest measured neutron production rather than the typical one, nitrogen at the top of its range, no trace neutron poisons unless you add them, no leakage out of the sample, and the sample placed inside a host rock that outproduces it.  If the answer still comes out small, it is small.

Check it yourself

The Verify figures below are produced by sim/tests.ts, which grades this code against published numbers — including two from the young-earth side, on purpose.  If this tool can reproduce their own arithmetic, then any disagreement is about inputs, not physics.

• Šrámek et al. 2017 Table 6: average upper continental crust makes 10,680 neutrons kg⁻¹ yr⁻¹, and spontaneous fission is 11 % of them.
• Baumgardner, RATE volume II ch. 8 p. 616: 1,000 ppm nitrogen in diamond at 7.3 × 10⁷ n m⁻² yr⁻¹ gives 9 × 10⁻⁶ pMC.
• Giem, Origins 51:6–30 (2001): nitrogen-14 beats carbon-13 as a neutron target by about 110,000×. Computed from cross sections alone it comes out at 107,000. Giem is right.
• Barker & Torkelson, Geophysical Research Letters 12(10):737–740 (1985): (2.2 ± 0.5) × 10⁻¹⁵ ¹⁴C atoms per uranium atom, measured by AMS in real uranium minerals — the only place in-situ radiocarbon has ever actually been detected, and an empirical ceiling this calculator sits comfortably above.
• And two deliberately wrong inputs that must fail: no uranium must give exactly zero, and no flux must give exactly zero however much nitrogen is present.

The code

• sim/data.ts — every nuclear and compositional constant, with its source
• sim/insitu.ts — the calculation, and nothing else
• sim/calibrate.ts — IntCal20 calibration
• sim/tests.ts — the self-tests above
• sim/report.ts — every figure quoted in the radiocarbon article, regenerable with deno run
• app.ts — the interface, which contains no science

Where this is used

The radiocarbon dating article uses this calculator in its section on in-situ production, and applies it to Brian Thomas's 2018 University of Liverpool PhD catalogue of radiocarbon measurements on dinosaur bone.